]>NUPHB14593S0550-3213(19)30067-710.1016/j.nuclphysb.2019.03.005Quantum Field Theory and Statistical SystemsFig. 1Y junction of spin chains with bulk exchange couplings J1 and J2 and boundary three-spin interaction Jχ. The arrows indicate the chirality favored by Jχ > 0.Fig. 1Fig. 2Schematic boundary phase diagram. The stable fixed points have emergent time-reversal symmetry and correspond to open (O) boundary conditions at weak coupling and a three-channel Kondo fixed point (K) at strong coupling. At the critical values Jχ=±Jχc, the Y junction is described by unstable chiral fixed points with either clockwise or counter-clockwise circulation of spin currents (C− or C+, respectively).Fig. 2Fig. 3Double Y junction on a finite system with three-spin coupling Jχ at x = 0 and its mirror image −Jχ at x=L. At the chiral fixed point Jχ=Jχc>0, the chiral currents obey the same boundary conditions at both ends: Jα,R(0)=Jα+1,L(0) and Jα,R(L)=Jα+1,L(L). Modes connected by the boundary conditions are identified by the same color.Fig. 3Fig. 4Illustration of the Y junction from the ordinary DMRG point of view. The green and blue regions represent the system and environmental DMRG blocks, respectively. The black solid (dashed) lines represent the nearest-neighbor (long-range) interactions and the arrows indicate the chiral three-spin interactions with respective ordering at the boundary sites. The filled (open) circles are the center (renormalized) sites. (For interpretation of the colors in the figure, the reader is referred to the web version of this article.)Fig. 4Fig. 5DMRG results for the SSCO expectation value, 〈Cˆj〉, vs Jχ for L=60 and two values of j. The data for j = 2 were multiplied by a factor −4.Fig. 5Fig. 6Logarithm of the RL correlations vs ln[Lπsin(πjL)] for Jχ = 3.4 (left panel) and Jχ = 8 (right panel). In order to show the accuracy of the correlations obtained by the DMRG, numerical data are shown for two numbers m of kept states.Fig. 6Fig. 7DMRG results for the spin conductance G12 vs Jχ for different system sizes.Fig. 7Fig. 8Logarithm of the RL correlations obtained by DMRG vs ln[Lπsin(πjL)], for Jχ = 0.4 and Jχ = −3.4. In order to demonstrate that correlations decay faster than [sin(πjL)]−2, we also show a line with slope −2.Fig. 8Fig. 9Exponent ν of the power-law decay of the three-spin correlations G3(j) vs Jχ. The shown results were obtained from fits to DMRG data in the interval jin ≤ j ≤ 32. The minimum is around the chiral fixed point, in agreement with the theoretical prediction (60).Fig. 9Fig. 10QMC data for Jχ = 1 (in units with D = v/a = 1). Left panel: Scalar spin chirality at the boundary, CB, vs T. Notice the logarithmic T scale. Error bars for QMC data points denote one standard deviation due to the stochastic sampling. The green dot-dashed line indicates the expected low-T behavior (106) near the O fixed point, which essentially predicts a constant-in-T behavior in this temperature regime. The blue dashed curve is a logarithmic fit corresponding to the chiral fixed point, see equation (107). Right: Local boundary susceptibility, χloc, as a function of T for Jχ = 1.Fig. 10Fig. 11QMC data for Jχ = 1.4. Note the logarithmic T scales. Left panel: Scalar spin chirality at the boundary, CB, vs T. The blue dashed line is a fit of the QMC data to a constant. Right panel: Local boundary susceptibility, χloc, vs T. Except at very high T, the logarithmic scaling (blue dashed line) expected near the chiral fixed point, see equation (100), is consistent with the QMC data in the accessible T window. Note that the power-law scaling near the Kondo fixed point (red solid curve), see equation (101), is not consistent with the data.Fig. 11Table 1Comparison of the (absolute value of the) dimensionless spin conductance G12/G0 for three values of Jχ corresponding to the O, C and K points, respectively. The DMRG estimates for Jχ = 0.4 and 3.4 were obtained for L=68. The value for Jχ = 10 has been extrapolated to infinite size, as explained in the main text.Table 1Fixed pointJχDMRGBCFTRel. error

O0.40.0040

C3.40.4980.50.4%

K100.2170.2303…5.7%

Table 2The exponents ν of the three-spin correlation G3(j) for representative values of Jχ corresponding to the three fixed points. The estimates were obtained from DMRG data for L=68, see main text. BCFT predictions and relative errors are also shown.Table 2Fixed pointJχνBCFTRel. error

O0.43.313.55.4%

C3.41.441.54%

K102.002.14.7%

Table 3Primary fields of Z3(5) and corresponding conformal dimension.Table 3Symbolsu(3) weights notationΔ

IΦ1,11,1=Φ3,41,1=Φ1,13,40

εΦ2,21,2=Φ1,22,2=Φ2,22,2110

ε′Φ1,21,2=Φ3,21,2=Φ1,23,212

Ψ⁎Φ2,11,1=Φ1,12,4=Φ2,42,135

ΨΦ1,12,1=Φ2,41,1=Φ2,12,435

ΩΦ2,12,1=Φ2,11,4=Φ1,42,185

ζΦ3,11,1=Φ1,11,4=Φ1,43,12

ζ⁎Φ1,13,1=Φ1,41,1=Φ3,11,42

ξΦ1,11,2=Φ3,31,1=Φ1,23,319

ξ⁎Φ1,21,1=Φ3,31,2=Φ1,13,319

ηΦ1,31,1=Φ1,13,2=Φ3,21,379

η⁎Φ1,11,3=Φ3,21,1=Φ1,33,279

ϕΦ1,12,2=Φ2,31,1=Φ2,22,3245

ϕ⁎Φ2,21,1=Φ1,12,3=Φ2,32,2245

μΦ2,11,2=Φ1,22,3=Φ2,32,11745

μ⁎Φ1,22,1=Φ2,31,2=Φ2,12,31745

ρΦ1,32,1=Φ2,21,3=Φ2,12,23245

ρ⁎Φ2,11,3=Φ1,32,2=Φ2,22,13245

νΦ1,21,3=Φ3,11,2=Φ1,33,1139

ν⁎Φ1,31,2=Φ1,23,1=Φ3,11,3139

Table 4Fusion rules for neutral fields.Table 4ε′ × Ψ = εε × Ψ⁎ = ε + ε′ε × Ψ = ε + ε′

ε′ × Ψ⁎ = εΩ × ε = ε + ε′ζ × ε′ = ε′

ζ × ε = εζ⁎ × ε′ = ε′ζ⁎ × ε = ε

ε′×ε′=I+2ε′ζ × Ψ = Ωζ⁎ × ζ⁎ = ζ

Ψ×Ψ⁎=I+ΩΩ × ε′ = εε×ε=I+2ε+2ε′+Ω+Ψ+Ψ⁎

ζ × Ψ⁎ = Ψζ × ζ = ζ⁎ε × ε′ = 2ε + Ω + Ψ + Ψ⁎

ζ⁎ × Ω = ΨΨ × Ψ = Ψ⁎ + ζζ × Ω = Ψ⁎

ζ×ζ⁎=IΨ × Ω = ΨΩ×Ω=I+Ω

Ψ⁎ × Ω = Ψ⁎ζ⁎ × Ψ = Ψ⁎ζ⁎ × Ψ⁎ = Ω

Table 5Fusion rules involving Z3-charged fields used in (47).Table 5ξ × ζ = νξ × ζ⁎ = ηξ × Ψ = ϕξ × Ψ⁎ = μ

ξ × Ω = ρξ⁎ × ν = ζ + ε′ξ⁎ × η = ζ + ε′ξ⁎ × ϕ = Ψ + ε

ξ⁎ × μ = Ψ⁎ + εξ⁎ × ρ = Ω + εξ⁎×ξ=I+ε′

Chiral Y junction of quantum spin chainsF.Buccheria⁎buccheri@hhu.deR.EggeraR.G. PereiraabcF.B. RamosbaInstitut für Theoretische Physik, Heinrich-Heine-Universität, D-40225 Düsseldorf, GermanyInstitut für Theoretische PhysikHeinrich-Heine-UniversitätDüsseldorfD-40225GermanybInternational Institute of Physics, Universidade Federal do Rio Grande do Norte, Campus Universitario, Lagoa Nova, Natal, RN 59078-970, BrazilInternational Institute of PhysicsUniversidade Federal do Rio Grande do NorteCampus UniversitarioLagoa NovaNatalRN59078-970BrazilcDepartamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, Natal, RN 59078-970, BrazilDepartamento de Física Teórica e ExperimentalUniversidade Federal do Rio Grande do NorteNatalRN59078-970Brazil⁎Corresponding author.Editor: Hubert SaleurAbstractWe study a Y junction of spin-1/2 Heisenberg chains with an interaction that breaks both time-reversal and chain exchange symmetries, but not their product nor SU(2) symmetry. The boundary phase diagram features a stable disconnected fixed point at weak coupling and a stable three-channel Kondo fixed point at strong coupling, separated by an unstable chiral fixed point at intermediate coupling. Using non-abelian bosonization and boundary conformal field theory, together with density matrix renormalization group and quantum Monte Carlo simulations, we characterize the signatures of these low-energy fixed points. In particular, we address the boundary entropy, the spin conductance, and the temperature dependence of the scalar spin chirality and the magnetic susceptibility at the boundary.1IntroductionThe field of spintronics has received continued attention in the last several years [1–3]. This is partly due to the discovery of the spin Seebeck effect and the spin Hall effect, which allow for the conversion between spin and charge currents [4–6]. Importantly, the generation and detection of magnonic currents via the spin Hall effect has been recently demonstrated using a YIG/Pt heterostructure [7], in which the magnons of the ferromagnetic insulator are responsible for spin transport between the terminals. Analogous setups have recently been realized with antiferromagnetic materials [8] and applications to magnon-based computation have been proposed in a multi-terminal platform [9]. The essential feature of these experiments is the presence of magnetization-carrying excitations with a relaxation length larger than the separation between the terminals. In this respect, the antiferromagnetic Heisenberg chain represents an idealized case of a one-dimensional (1D) system in which spinful excitations can propagate ballistically in the low-temperature limit [10,11]. Moreover, it was shown to describe accurately the behavior of effectively 1D crystals such as CsCoCl3 [12,13], CsCoBr3 [14,15], KCuF3 [16], and Sr2CuO3 [17,18]. The spectrum of the model has been known for a long time by exact methods [19,20], and its low-energy properties are essentially captured by field theory [21].A key element of any quantum or classical circuit is a Y junction, a topic of established interest in condensed matter and statistical physics [22–31]. In this work, we study a Y junction of spin-12 chains with a boundary three-spin interaction analogous to the chiral spin liquid order parameter proposed in [32,33]. This coupling preserves Z3 cyclic permutation of the chains as well as the spin SU(2) symmetry. On the other hand, it breaks time-reversal and leg-exchange (reflection) symmetries, while preserving the symmetry under the combined action of the two. The main question is how this chiral interaction affects the spin transport properties of the Y junction. More specifically, our goal is to identify an interaction regime in which a Y junction made of long chains can redirect spin currents in a clockwise or counter-clockwise manner, in analogy with quantum circulators that have been realized in photonic and superconducting circuits [34–36].The low-energy, long-wavelength physics of the Y junction is governed by conformally invariant boundary conditions [37,38] connecting the collective spin modes in different chains. In our earlier contribution [39], we showed that the Y junction model discussed below exhibits a remarkable boundary phase diagram as a function of the boundary interaction strength. It features two stable fixed points, corresponding to disconnected chains and to a three-channel-Kondo fixed point. The two fixed points are separated by an unstable chiral fixed point. The latter realizes an ideal quantum spin circulator but requires parameter fine tuning [39]. The experimental realization of such a spin circulator would be of practical interest in the field of antiferromagnetic spintronics, aiming at the realization of circuits using Mott insulators [40–42].Controlled and tunable setups for the simulation of quantum chains are also provided by cold-atom experiments, which have proven useful tools for studying Heisenberg chains and multi-spin exchange interactions [43,44]. An interesting perspective on the realization of a junction of quantum chains in these platforms is the use of ‘synthetic dimensions’ encoded by internal atomic states. This technique has been applied successfully for creating spin ladders with a synthetic magnetic flux [45,46].In this paper, we apply non-abelian bosonization and boundary conformal field theory (BCFT) to characterize physical observables of Y junctions of spin chains in the vicinity of the above fixed points. Going significantly beyond the results reported in our earlier work [39], here we present a detailed derivation of the boundary entropy and of the spin conductance matrix for each of the three fixed points. Moreover, we present numerical results for the spin conductance obtained by density matrix renormalization group (DMRG) methods, and for finite-temperature properties from quantum Monte Carlo (QMC) simulations. Put together, these numerical results strongly support our analytical predictions, in particular the existence of a chiral fixed point at intermediate coupling. Despite its instability, the chiral fixed point governs the physics of Y junctions at finite length scales and/or at finite temperatures over a wide parameter regime.The remainder of this article is organized as follows. After introducing our model for the Y junction in section 2, we discuss the bosonization approach and the boundary phase diagram in section 3. The BCFT formalism is presented in section 4, with particular attention to the operator content at the various fixed points. Observables characterizing the different fixed points are discussed in section 5. We then present our numerical DMRG and QMC results in section 6. Finally, we present our conclusions in section 7. Several technical details have been delegated to the Appendix.2ModelWe start with three decoupled isotropic spin chains, each labeled by a leg index α=1,2,3. The Hamiltonian for a single chain includes the usual antiferromagnetic nearest-neighbor exchange interaction J1>0 supplemented by a next-nearest-neighbor interaction J2,(1)Hα=J1∑j=1L−1Sj,α⋅Sj+1,α+J2∑j=1L−2Sj,α⋅Sj+2,α. Here Sj,α is a spin-12 operator that acts in the Hilbert space of site j in chain α, and L is the length of the chains with open boundary conditions. We shall be mainly interested in the limit of semi-infinite chains, L→∞, but see sections 4 and 6.1. In the presence of a Z3 cyclic chain permutation symmetry, which takes α↦α+1 (mod 3), the interaction parameters J1 and J2 are constrained to be the same for the three chains. In addition, Hα is invariant under time reversal, acting on spin operators as(2)T:S↦−S, and under the Z2 leg exchange (say, of chains 1 and 2)(3)P:α↦−α(mod 3).The ground-state phase diagram of the J1-J2 chain in (1) is well known [47,48]. There is a critical value J2c≈0.2412J1 [49–51] separating a critical phase for 0≤J2≤J2c from a gapped dimerized phase for J2>J2c. We focus on the critical phase and add to the Hamiltonian the most general boundary interactions, coupling only spins at position j=1 in each chain. These interactions are allowed to break T and P symmetries but are required to preserve SU(2), PT and Z3 symmetries. The full Hamiltonian takes the form(4)H=∑α=13Hα+HB, with the boundary interaction term(5)HB=J′∑α=13S1,α⋅S1,α+1+JχCˆ1. Here we define the scalar spin chirality operator (SSCO) as [32](6)Cˆj=Sj,1⋅(Sj,2×Sj,3). The first term in (5) is a simple exchange interaction between the end spins. This term preserves both P and T symmetries but by itself cannot lead to chiral spin transport through the junction. The second term involves the SSCO for the end spins. We note that expectation values of SSCOs have been proposed as order parameters [32,33] for characterizing chiral spin liquid phases [52,53]. The Jχ term in (5) can be derived starting from the Hubbard model in the limit of strong on-site repulsion and treating virtual hopping processes in the presence of a magnetic flux [54]. Alternatively, it appears in effective Floquet spin models for a Mott insulator driven by circularly polarized light [55,56]. Since we are mostly interested in studying the possibility of chiral spin transport, hereafter we set J′=0 and retain only the boundary parameter Jχ, see Fig. 1.3The boundary phase diagramIn the critical phase 0≤J2≤J2c, the low-energy effective field theory describing the continuum limit of Hamiltonian (1) is the SU(2)1 Wess-Zumino-Witten (WZW) model with an open boundary at the origin [21],(7)Hα=∫0∞dx{2πv3[Jα,L2(x)+Jα,R2(x)]−2πvγJα,L(x)⋅Jα,R(x)}. Here, v is the spin velocity and γ≥0 is the dimensionless coupling constant of the marginally irrelevant operator. The critical point J2=J2c corresponds to γ=0 in the effective Hamiltonian (7). At this point, the model becomes equivalent to free bosons (up to strictly irrelevant perturbations) and logarithmic corrections to correlations functions vanish [50]. In order to simplify our analysis and focus on the essential physics of the Y junction, we will henceforth consider the bulk of the spin chains to be tuned to this critical point. The chiral spin current operators Jα,La(z) and Jα,Ra(z¯) (with a=1,2,3) represent collective spin modes moving in the direction of decreasing or increasing x, respectively. They are functions of the complex coordinates z=vτ−ix or z¯=vτ+ix, where τ denotes imaginary time. These chiral spin currents are not be independent; instead, their relation is fixed by the boundary conditions, which have not yet been specified in (7). With our normalization choice, their operator product expansion (OPE) satisfies the SU(2)k Kac-Moody algebra [57],(8)Jα,La(z)Jα,Lb(w)∼18π2kδab(z−w)2+12πiεabcz−wJα,Lc(w), with k=1 and the Levi-Civita tensor ε. The boundary conditions will be determined as a function of Jχ in (5) according to a ‘delayed evaluation of boundary conditions’ [26]. The boundary term HB acting at x=0 will determine the boundary conditions, which in turn fix the dependence of the right from the left currents and the full set of OPEs.The effective field theory can be formulated in terms of just one (left- or right-moving) boson field for each chain α. These chiral boson fields, φα,μ(x) with μ=L/R=+/−, obey the commutation relations(9)[φα,μ(x),∂x′φα′,μ′(x′)]=iμδαα′δμμ′δ(x−x′). One can write the currents and the spin operators, as well as the boundary SSCO, in terms of these fields (see Appendix A). Without imposing boundary conditions yet, we write the boundary interaction as(10)HB≃Jχπ2∑α(12π(∂xφα,L−∂xφα,R)+Asin[π(φα,L−φα,R)])×sin[π(φα+1,L+φα+1,R−φα−1,L−φα−1,R)]×(πA+cos[π(φα+1,L−φα+1,R)])×(πA+cos[π(φα−1,L−φα−1,R)]), where the real positive number A is a nonuniversal constant of order unity.In [39], as Jχ is varied, we have argued for the phase diagram shown in Fig. 2. First, we note that the fixed point of disconnected chains with open boundary conditions (O fixed point). The open boundary condition(11)φα,R(x)=φα,L(−x)+C, where the constant C is constrained to C=0 or C=±π by virtue of SU(2) invariance, is stable for |Jχ| below a critical value Jχc. In fact, after imposing (11) we obtain the boundary SSCO in the form(12)Cˆ1∝∑α∂xφα,L(0)sin[4πφα+1,L(0)−4πφα−1,L(0)]. This is a dimension-three boundary operator that represents a highly irrelevant perturbation near the O point. In non-abelian bosonization, we can write the boundary SSCO as the triple product of the chiral spin currents,(13)Cˆ1∝∑αJα,L(0)⋅[Jα+1,L(0)×Jα−1,L(0)]. This SU(2)-symmetric form can be obtained directly from equation (6) by using that, in the continuum limit with open boundary conditions, Sα,j=1∝Jα,L(0) [50]. Beyond first order in Jχ, we must take into account all boundary operators allowed by symmetry, as they can be generated by the renormalization group (RG) flow. The leading boundary operators that perturb the O fixed point are given by(14)HB(O)=λ1(O)∑αJα,L(0)⋅Jα+1,L(0)+λ2(O)∑α[Jα,L(0)]2+λ3(O)∑αJα,L(0)⋅[Jα+1,L(0)×Jα−1,L(0)]+⋯, where ⋯ refers to more irrelevant operators. While λ3(O)∼Jχ, the leading irrelevant boundary operator that couples the chains is the time-reversal-invariant boundary exchange coupling λ1(O), a dimension-two operator generated by the RG to second order in Jχ (or first order in J′ in the more general model).It is important to underline that the argument for the stability of the O fixed point is only perturbative in the coupling to the boundary SSCO. For |Jχ|∼J1, we expect to find a different boundary condition whereby the chiral bosonic modes φα,R are connected with the modes φα±1,L in the other chains. Following [24,26,58], we consider conformally invariant boundary conditions of the form(15)φα,R(x)=∑β=13Mαβφβ,L(−x)+C, where M is a 3×3 matrix and C is a constant. At the O fixed point, M is simply the identity. More generally, the condition of preserving the algebra in equation (9) implies that M must be orthogonal. The orthogonality of M also implies that the scaling dimension of the chiral currents is not modified by the boundary condition. Imposing SU(2) symmetry for the Y junction leads to further constraints. First, we must have either C=0 or C=π. Second, the leading terms in the OPE of Jα,R+(x)Jα±1,L−(x′) must have the same scaling dimension as Jα,Rz(x)Jzα±1,L(x′), namely dimension two. One can conclude, using the bosonization formulas of the spin operators (A.7), that the only options that are compatible with the above conditions are as follows. (i) Mαβ=δα,β, which corresponds to open boundary conditions, (ii) Mαβ=δα,β−1, a cyclic permutation in one direction, or (iii) Mαβ=δα,β+1, a cyclic permutation in the other direction. The latter two options correspond to chiral boundary conditions,(16)C+:φα+1,R(x)=φα,L(−x)+C,(17)C−:φα−1,R(x)=φα,L(−x)+C. In terms of chiral currents, we have(18)C±:Jα±1,R(x)=Jα,L(−x). At the C± fixed points, the leading term in the OPE of the boundary SSCO is proportional to the identity. This is consistent with the fact that the boundary interaction in (10) is responsible for pinning the bosonic fields through chiral boundary conditions, similarly to other applications of the method of delayed evaluation of boundary conditions [26,59,60]. As a result, we find a nonzero expectation value,(19)C±:〈Cˆj〉=±(−1)jA32cos(πC), for sites j near the boundary, i.e., where we can approximate φα,L(x)≈φα,L(−x) for x→0 in the slowly varying fields. The choice of C=0 or C=π thus determines whether the scalar spin chirality is positive on even or odd sites near the boundary. For the boundary interaction acting at site j=1, we obtain the expectation value(20)C±:〈HB〉=∓JχA32cos(πC). Which chiral fixed point is then realized will depend on the sign of Jχ. For instance, C− with C=π corresponds to the case where the incoming spin current from chain α is channeled into chain α−1, and 〈Cˆ1〉<0. We expect this fixed point to be selected for a boundary interaction (5) with Jχ>0, since this choice lowers the ground state according to (20). The spin current then circulates clockwise in Fig. 1, as expected for a local negative-chirality state with ordering 3→2→1→3. This hypothesis can be checked by numerically calculating the spin conductance and the three-spin correlations, see section 6.We next analyze the stability of the chiral fixed points in the bosonization approach by considering other terms generated in the OPE of the boundary SSCO besides the identity. Imposing chiral boundary conditions in (10), we obtain(21)C±:HB≃∓Jχ2cos(πC)×{A3+A4π2∑αcos[πφα,L(0)−πφα+1,L(0)]}+⋯, where we omit irrelevant operators. The cosine term in (21) has scaling dimension 1/2 and thus corresponds to a relevant boundary operator. We can interpret it as being due to the backscattering of spin currents, in analogy to the relevant backscattering operator in a Y junction of quantum wires with Luttinger parameter K=1/2 (as appropriate for Heisenberg chains) [26]. In essence, the relevant boundary operator stems from the fact that for chiral boundary conditions, the boundary spin operator (A.4) does not reduce to the chiral currents Jα,L(0) but also contains a contribution proportional to the dimension-1/2 matrix field. Indeed, the most relevant perturbation to the chiral fixed point is given by(22)HB(C)=λ1(C)∑αTr[g˜α(0)]+⋯, where g˜α(x) are matrix fields obtained from the original gα(x) by replacing φα,R(x)↦φα±1,L(−x), cf. equation (A.9). If we choose the boundary conditions such that the prefactor in (21) is negative, we will have λ1(C)<0 with |λ1(C)|∼|Jχ| at weak coupling. As a consequence, the chiral fixed points cannot appear at weak coupling as they are destabilized by a relevant perturbation.However, the existence of only one relevant operator in (22) suggests that chiral fixed points can be realized as critical points at intermediate coupling, reached by tuning a single parameter in the lattice model. Let us suppose that there exists a critical value Jχc>0, where the relevant coupling vanishes and λ1(C)≈−b(|Jχ|−Jχc) for |Jχ|≈Jχc. We must have b>0 so that λ1(C)<0 for |Jχ|<Jχc as verified at weak coupling. What happens when λ1(C) changes sign across the putative critical point? Under the RG, the effective coupling λ1(C)<0 flows monotonically to strong coupling λ1(C)→−∞ at low energies. When this happens, the boson fields are pinned to the minima of the cosine potentials in (21),(23)φα,L(0)−φα+1,L(0)=2nαπ,nα∈Z. On the other hand, for λ1(C)>0 the effective coupling flows to λ1(C)→+∞ under the RG, and the cosine potentials pin the boson fields to(24)φα,L(0)−φα+1,L(0)=(2nα+1)π,nα∈Z. We can understand the difference between equations (23) and (24) by noting that when we impose chiral boundary conditions, the smooth part of the spin operator is represented by(25)C±:Sj,αz∼Jα,Lz(x)+Jα∓1,Lz(−x)=14π[∂xφα,L(x)+∂xφα∓1,L(−x)]. The magnetization measured out to large distances from the boundary is(26)∑j=1∞Sj,αz∼14π[∫0∞dx∂xφα,L(x)−∫0∞dx′∂x′φα∓1,L(x′)]=14π[φα∓1,L(0)−φα,L(0)]. The pinning conditions (23) and (24) thus correspond to(27)∑α∑j=1∞Sj,αz={n,λ1(C)→−∞n+12,λ1(C)→+∞, where n∈Z. This suggests that changing the sign of λ1(C) from negative to positive brings an effective spin 1/2 from infinity to the boundary. A similar effect occurs with the backscattering potential λcos[4πKϕ(0)] for the Kane-Fisher problem of a single impurity in a Luttinger liquid [61]. For repulsive interactions, corresponding to Luttinger parameter K<1, the backscattering operator is relevant for both signs of the coupling λ. The critical point λ=0 corresponds to a resonant tunneling condition, where the number of electrons on the scattering center fluctuates [61]. Changing the sign of the coupling switches between a predominantly repulsive and a predominantly attractive scattering potential. In the latter case, a bound state is formed for arbitrarily weak scattering amplitude, and exactly one extra unit of charge is bound near the origin. By analogy, we expect that in our SU(2)-symmetric spin chain problem the sign change of λ1(C) entails the formation of an effective spin-1/2 degree of freedom at the boundary for |Jχ|>Jχc.An analysis of our Y junction model in the strong coupling limit corroborates the above picture. For |Jχ|≫J1, one may, in a first approximation, neglect the coupling of the three end spins to the rest of the chain and diagonalize the reduced Hamiltonian in (5), where we now include J′≠0. The spectrum of this three-spin system can be labeled by the quantum numbers of total spin, s, magnetization, sz, and the scalar spin chirality, c. The spectrum is composed of a non-chiral (c=0) spin-3/2 quadruplet at energy 3J′/4 and two spin-1/2 doublets with opposite chiralities (c=±1) and energies(28)Ec(Jχ,J′)=−3J′4+cJχ34. As long as J′>−|Jχ|23, the ground state of HB is twofold degenerate. For Jχ>0, the ground state is the doublet with negative chirality, given by the states [32](29)|s=12,sz=12,c=−1〉=i3(|↓↑↑〉+ω|↑↑↓〉+ω2|↑↓↑〉),|s=12,sz=−12,c=−1〉=−i3(|↑↓↓〉+ω2|↓↑↓〉+ω|↓↓↑〉), where ω=ei2π/3 and ω2=ω⁎=ω−1. Both states are conjugated by PT with (PT)2=−1, and therefore their degeneracy is protected by PT symmetry. For Jχ<0, the ground state is given by the positive-chirality states obtained from (29) by applying either P or T.We can now define an effective spin-1/2 operator, S0, which acts in the Hilbert subspace spanned by the two states (29). For Jχ≫J1, we project the full Hilbert space of the boundary spins onto this low-energy subspace. The projection of the total Hamiltonian leads to(30)Heff=∑αH˜α+JKS0⋅∑α(S2,α+J2J1S3,α), where JK=J1/3 and(31)H˜α=∑j≥2(J1Sj,α⋅Sj+1,α+J2Sj,α⋅Sj+2,α) is the new Hamiltonian for each chain where the site j=1 has been removed to form the central spin S0. Remarkably, this projected Hamiltonian has an emergent time-reversal symmetry since it only contains two-spin (exchange) interactions. However, time-reversal symmetry breaking perturbations appear in the effective Hamiltonian to leading order in J1/Jχ once we take into account virtual transitions to the excited states of HB [39].We note that the coupling JK between the central spin S0 and the new boundary spins of the chains is suppressed by a factor 1/3 in comparison with the bulk exchange coupling. Thus, we can approach the Y junction in the strong coupling limit as the problem of an impurity spin weakly coupled to three chains with open boundary conditions at j=2. In contrast with the O fixed point, this new starting point is unstable because JK>0 is equivalent to an antiferromagnetic Kondo coupling which is marginally relevant. Taking the continuum limit of the projected Hamiltonian, we obtain (for J2=J2c)(32)Heff≈∑α∫−∞∞dx2πv3Jα,L2(x)−2πvλKS0⋅Jα,L(0), where λK∝JK/(2πv) is the dimensionless coupling constant. The perturbative RG equation takes the form [51,62](33)dλKdl=λK2−32λK3+⋯, where l=ln(Λ0/Λ) with Λ denoting the running high-energy cutoff. This equation coincides with the RG equation for the three-channel Kondo effect with itinerant electrons [21,63,64]. Even though our model lacks the charge degrees of freedom of the original Kondo problem, it is known that the single-channel and two-channel Kondo effects can be realized with spin chains [50,51,65]. Here we extend this analogy to the case of three spin chains and identify a low-energy three-channel Kondo (K) fixed point in the regime |Jχ|≫J1. It is also interesting to note that the scenario is connected to the results of [66], with the important difference that here the impurity is connected to the end of the spin chains. In order to provide a unified description of all fixed points of the Y junction, we now turn to the BCFT approach.4Non-abelian bosonization and boundary conformal field theoryThe key assumption in the BCFT approach is that the stable fixed points are described by conformally invariant boundary conditions, which can be generated using the fusion algebra of the primary fields of the theory [37,38,67]. This approach has been applied successfully to several quantum impurity problems, e.g., the multi-channel Kondo model [68–70], junctions of two spin chains [71], or electronic Y junctions [26,27]. In the following, we apply BCFT to describe the operator content at the fixed points of a Y junction of spin chains.4.1Conformal embedding and partition functionThe first step will be to identify a suitable conformal embedding [57,72] which can be used to describe the system at low temperatures. Here, we can proceed in analogy to the case of two spin chains [50,71]. The field theory for three decoupled spin chains at low energies is a CFT with total central charge c=3, arising from three copies of a spin SU(2)-symmetric model. The boundary interaction, HB with Jχ≠0, breaks the symmetry to SU(2)×Z3, as only the total spin of the three chains is conserved. The sum of three SU(2)1 chiral currents,(34)Ja(z)=∑α=13Jα,La(z), naturally defines the generator of an SU(2)3 Kac-Moody algebra, where Ja(z) obeys equation (8) at level k=3. The SU(2)3 WZW field theory has central charge c=9/5 [57,72].While the SU(2)3 WZW model describes the total spin degree of freedom, we also need to associate a CFT to the ‘flavor’ (or ‘channel’) degree of freedom carrying the remaining central charge. The Z3(5) CFT is a representative of a family of minimal models which contain parafermionic fields [73–75]. In addition to the conformal invariance, it possesses an additional infinite-dimensional symmetry, known as W3 algebra, generated by a local current. Its central charge is c=6/5, so the conformal embedding SU(2)×3Z3(5) has indeed ctot=3. This embedding was used to study a three-leg spin ladder in [76]. An alternative embedding, which also gives the correct central charge, employs the Ising and tricritical Ising CFTs [77]. However, the boundary conditions generated by fusion in these sectors would not be equivalent in general. We have been able to reproduce the properties of all three fixed points identified in section 3 only with the SU(2)×3Z3(5) embedding.The Z3(5) theory has 20 primary fields. The full list [78] can be found in Appendix B. The fusion algebra has been computed from the modular S matrix and the Verlinde formula in [79]. For convenience, the results are summarized in Appendix B. Importantly, as a consequence of the underlying symmetry algebra, not all the primary fields appear at the same time in the partition function, see below.The operator content of two-dimensional (2D) CFTs is organized into infinite-dimensional representations generated by a primary field. The characters χ of a representation yield a compact way of encoding the operator content of the theory [57,79,80]. In our case, the boundary conditions select a particular subspace of the tensor product of the spin, SU(2)3, and flavor, Z3(5), Hilbert spaces. This information is contained in the partition function, which has the general form(35)Z=∑s=03/2∑fnsfχsSU(2)3(q)χfZ3(q), where f runs over the set of primary fields of Z3(5), and the integers nsf determine the operator content for given boundary conditions [81]. The generally complex parameter q is specialized to q=e−πβL, with temperature T=1/β and length of a single chain L, in order to describe the partition function [71]. Importantly, for sufficiently strong interactions, the boundary degrees of freedom can reorganize the spectrum in a way encoded by the integers nsf. This is at the base of the fusion approach to Kondo problems and 2D quantum Brownian motion [77,82]. To describe the operator content for semi-infinite chains, a useful procedure is to map the upper half plane, (τ,x) with x≥0, onto an infinite strip of width L [70]. This is accomplished via the transformation(36)z↦w(z)=Lπlnz. In the Y junction, this conformal mapping relates the semi-infinite system to the finite system shown in Fig. 3, where the boundary at x=L is the mirror image of the boundary at x=0 [83,84]. This amounts to opposite signs of the boundary coupling, i.e., Jχ at x=0 and −Jχ at x=L.We start with the partition function for open boundary conditions, i.e., near the O fixed point. The partition function of decoupled spin chains can be formulated in terms of the characters of the primary fields of the SU(2)1 WZW CFT [71]. Let χsSU(2)1(q), with s=0,12, denote the character of the spin-s primary fields of SU(2)1. The latter have dimension 0 for s=0 and 12 for s=12. For three open chains with even (e) or odd (o) values of L (thus, with integer or half-integer values of the total spin), we have the partition functions(37)ZOOeee(q)=[χ0SU(2)1(q)]3,ZOOooo(q)=[χ1/2SU(2)1(q)]3. Using the conformal embedding, we can rewrite the partition function in terms of characters of the SU(2)3 WZW and Z3(5) theories. Details are provided in Appendix C, and the final expression is [76](38)ZOOeee(q)=χ0SU(2)3[χIZ3+χζZ3+χζ⁎Z3]+χ1SU(2)3[χΨZ3+χΨ⁎Z3+χΩZ3],ZOOooo(q)=χ3/2SU(2)3[χIZ3+χζZ3+χζ⁎Z3]+χ1/2SU(2)3[χΨZ3+χΨ⁎Z3+χΩZ3]. Here Ψ and Ψ⁎ denote the pair of conjugate fields of dimension 3/5 (see Appendix B) and Ω is the dimension-8/5 field generated from their fusion. The conjugate fields ζ and ζ⁎ have dimension 2. Explicit expression for the characters are given in [57,76,79].We can read off the finite-size spectrum for three decoupled spin chains from the partition functions in (38) [57]. In particular, the ground state for three even chains is a singlet, and all the conformal towers in ZOOeee are associated with operators of integer scaling dimension. By contrast, the ground state for three odd chains is eightfold degenerate, as expected from the direct product of three spin doublets, cf. equation (37). Furthermore, we may combine equation (38) with the physical intuition gained from the bosonization approach to analyze the symmetry properties of operators in the Z3(5) sector, as in [81] for the multi-channel Kondo model. Let us consider the partition function of three even chains, ZOOeee. There are nine fields with dimension 1 in both formulations (37) and (38). In equation (37), these can be identified with the chiral currents Jα,La, with α=1,2,3 and a=1,2,3. In order to identify the dimension-1 operators in the SU(2)×3Z3(5) embedding, it is convenient to introduce the helical combinations [85](39)Jha(z)=∑α=13ω−αhJα,La(z),h=−1,0,1. Clearly, J0a coincides with the SU(2)3 current Ja defined in equation (34). The combinations J±1a transform as J±1a↦ω±1J±1a under the cyclic permutation Jα,La↦Jα+1,La. Considering the Kac-Moody algebra (8), the helical currents obey the OPE(40)Jha(z)Jbh′(w)∼3δhh′δab8π2(z−w)2+iεabc2π(z−w)Jh+h′c(w), where the sum h+h′ is defined modulo 3 and restricted to the domain {−1,0,1}. Having identified J0a with the SU(2)3 current, we are left with the six components J±1a. The only other dimension-1 fields in the partition function (38) are the products ϕ1aΨ and ϕ1aΨ⁎. Here ϕ1 is the spin-1 primary field (the vector) of the SU(2)3 WZW model, which has scaling dimension 2/5, while Ψ and Ψ⁎ are conjugate fields of dimension 3/5. The helical currents J±1a must be linear combinations of ϕ1aΨ and ϕ1aΨ⁎ chosen such that the OPE (40) is compatible with the fusion algebra for su(2)3 and Z3(5). We then consider linear combinations with coefficients cα and dα,(41)ϕ1aΨ∼∑αcαJαa+⋯,ϕ1aΨ⁎∼∑αdαJαa+⋯, where ⋯ stands for operators with higher conformal dimension. Using the OPE for level-1 currents in (8), we find(42)∑α,βcαdβJαa(z)Jβb(w)∼(∑αcαdα)I(z−w)2+iεabc∑αcαdαJαc(w)z−w,∑α,βcαcβJαa(z)Jβb(w)∼(∑αcα2)I(z−w)2+iεabc∑αcα2Jαc(w)z−w, for the divergent part. On the other hand, the fusion rules calculated from the left hand side of (41) have to be satisfied, in particular Ψ×Ψ∼Ψ⁎ plus a regular part, see Table 4 and (A.11). This is sufficient to fix cα=ω−α=dα⁎, up to an overall phase ω or ω⁎ (corresponding to a Z3 permutation) and complex conjugation (corresponding to the exchange Ψ↔Ψ⁎ or 1↔2 for leg indices). Summing up, we conjecture that the helical combinations can be identified as(43)(J0aJ1aJ−1a)=(Jaϕ1aΨϕ1aΨ⁎)=(111ω⁎ω1ωω⁎1)(J1,LaJ2,LaJ3,La). A more careful determination of the OPE coefficients in this conformal embedding passes through the evaluation of four-point functions [81], but we do not tackle this problem here. The important point is that the transformation of the Ψ and Ψ⁎ fields in the Z3(5) sector are the same as for the helical currents J±1a under cyclic chain permutation. In particular, they transform under α↦α+1 as(44)Ψ↦ωΨ,Ψ⁎↦ω⁎Ψ⁎. Since both P and T exchange the helical currents J1a↔J−1a, these transformations must act in the Z3(5) sector by exchanging Ψ↔Ψ⁎. We can also deduce from the fusion rules that the dimension-8/5 operator Ω is invariant under cyclic chain permutation but odd under P and T (see [39] and section 5.1).We now apply the fusion procedure to generate other conformally invariant boundary conditions. Let us first consider the K fixed point, at which the three chains overscreen the effective spin-1/2 degree of freedom resulting from the strongly coupled end spins. Under the fusion hypothesis [82,86], the spectrum of the three-channel Kondo fixed point is obtained by fusing the partition function for open boundary conditions with the character of the spin-1/2 field in the SU(2)3 sector. The fusion rules are recalled in (A.11) in Appendix A. The partition function with the same boundary condition at both ends is generated by double fusion [87]. For even-length chains, we obtain(45)ZKKeee(q)=χ0SU(2)3[χIZ3+χζZ3+χζ⁎Z3+χΨZ3+χΨ⁎Z3+χΩZ3]+χ1SU(2)3[χIZ3+χζZ3+χζ⁎Z3+2χΨZ3+2χΨ⁎Z3+2χΩZ3]. Equation (45) allows us to identify the boundary operators in the effective Hamiltonian for the K fixed point. These operators must be SU(2) scalars and invariant under Z3 and PT symmetries. Thus, we can rule out the relevant (dimension 3/5) operators Ψ and Ψ⁎, which are not invariant under cyclic permutations of the chains. It turns out that all the boundary operators allowed by symmetry are irrelevant, implying that the K fixed point is stable and can thus describe the low-energy physics of the Y junction in the regime |Jχ|≫J1. This fusion hypothesis must be tested by comparing the analytical predictions against numerical results, as we shall do in section 6. As in the free-electron multi-channel Kondo model [68], the leading irrelevant operator is the Kac-Moody descendant of the spin-1 field, J−1⋅ϕ1. Here Jn denotes the modes of the SU(2)3 chiral current in the Laurent expansion, Ja(z)=∑n∈Zz−n−1Jna [57]. This boundary operator has dimension 7/5 and is time-reversal invariant. The leading operator which is odd under P and T is the dimension-8/5 field Ω acting in the Z3(5) sector. Note that this operator does not occur in the free-electron three-channel Kondo model, where the flavor sector is represented by an SU(3)2 WZW theory [68]. As a result, the leading perturbations to the Kondo fixed point are given by(46)HB(K)=λ1(K)J−1⋅ϕ1+λ2(K)Ω+⋯.Finally, we discuss how to obtain the chiral fixed points C± using fusion. Since these fixed points break the leg exchange symmetry P, we expect them to be generated from the open boundary partition function by fusion with judiciously chosen primary fields in the Z3(5) sector. The criteria to guide our choice of the fields are as follows. (i) The two chiral fixed points with opposite chirality should be associated with a pair of conjugate primary fields with the same dimension. (ii) The spectrum must reproduce the result for three chiral SU(2)1 models since the boundary conditions can, in the bosonization approach, be described by (18). (iii) The leading boundary operator allowed by symmetry must be a scalar with scaling dimension 1/2. (iv) The spin conductance matrix calculated from the corresponding boundary states (see section 5.2) must be asymmetric and reflect the transmission of spin currents according to the boundary conditions (18). These criteria lead us to postulate that the chiral fixed points are obtained by fusion with the dimension-1/9 fields denoted by ξ and ξ⁎ in Appendix B (see Table 3). Within double fusion, we must fuse first with one field (representing the first boundary in the strip geometry) and second with the conjugate field (representing the mirror image at the other boundary), according to the fusion rules in Tables 4 and 5. The order of the fusion, ξξ⁎ or ξ⁎ξ, selects the boundary state corresponding to either C+ or C−, but the partition function is the same in both cases,(47)ZCCeee(q)=χ0SU(2)3[χIZ3+3χε′Z3+χζZ3+χζ⁎Z3]+χ1SU(2)3[3χεZ3+χΨZ3+χΨ⁎Z3+χΩZ3]. Here ε and ε′ are the primary fields with dimension 1/10 and 1/2, respectively. Interestingly, the spectrum contains three towers of states associated with ε′, where the Z3 symmetric combination of such states corresponds to the backscattering operator ∑αTr[g˜α]. Note that there are, in addition, three towers associated with the spinful operator ϕ1ε, also with dimension 1/2. This operator can appear in the effective Hamiltonian for the chiral fixed point if combined with a free boundary spin, as in S0⋅ϕ1ε. In the bosonization approach, this corresponds to the boundary operator S0⋅∑αTr[g˜ασ]. In the strong coupling regime |Jχ|>Jχc, this operator destabilizes the chiral fixed points, signaling the RG flow towards the K fixed point where the boundary spin gets screened. Except for these two dimension-1/2 operators and their descendants, all other states in the partition function (47) come with integer scaling dimension as expected for chiral SU(2)1 WZW models.4.2Boundary entropyAs a consequence of nontrivial boundary conditions, 1D quantum systems can acquire a universal non-integer ground-state degeneracy, g, in the thermodynamic limit. The Affleck-Ludwig boundary entropy [82] appears as a non-extensive correction to the free energy for L≫vβ,(48)lnZ=πc6Lvβ+lng+⋯, where c is the central charge. For multi-channel Kondo fixed points, the boundary entropy can be computed from the modular S matrix of the SU(2)k WZW theory [57,82]. The components of the S matrix are labeled by the spin of the primary, j1,j2=0,12,1,32, and are given by(49)Sj1j2SU(2)3=25sin(π(2j1+1)(2j2+1)5). The ratio of the ground-state degeneracies at the K and O fixed points follows as(50)gKgO=S12,0SU(2)3S0,0SU(2)3=2cosπ5≈1.62. For the chiral fixed points, we use the S matrix for the Z3(5) theory [79,88], see Appendix B. The ratio between the ground-state degeneracy of the C± and O fixed points reads(51)gC+gO=gC−gO=Sξ,IZ3SI,IZ3=2. According to the g-theorem [82,89], the ground-state degeneracy can only decrease under the RG flow. Our boundary phase diagram in Fig. 2 is consistent with the g-theorem since the chiral fixed points C± have the highest g value, indicating that boundary perturbations trigger an RG flow towards either the O or the K fixed point.5Characterization of the fixed pointsWe now study physical observables that can be used in numerical or experimental tests of our proposed scenario.5.1Scalar spin chiralitySince it breaks time-reversal symmetry, a nonzero scalar spin chirality can in principle be probed by circular dichroism [55,90]. The expectation value of the SSCO also provides a numerical test of the boundary phase diagram, since the decay of the three-spin correlation function at large distances from the boundary is governed by the fixed point in each regime of Jχ [39]. To discuss three-spin correlations, let us first rewrite the SSCO (6) in the continuum limit using the conformal embedding. The term with the lowest bulk scaling dimension stems from the staggered magnetization for each chain,(52)Cˆ(τ,x)∼(−1)xεabcna1(τ,x)n2b(τ,x)n3c(τ,x)=(−1)xA3εabcTr[g1(τ,x)σa]Tr[g2(τ,x)σb]Tr[g3(τ,x)σc], where all fields are evolved in imaginary time. The operator in equation (52) is an SU(2) scalar with zero conformal spin and scaling dimension 3/2. Moreover, it is odd under P and T but invariant under Z3 cyclic chain permutation. Using these properties, we select its counterpart in the SU(2)×3Z3 formulation,(53)Cˆ(τ,x)∼(−1)xiTr[ϕ12(z)⊗ϕ12†(z¯)][Ψ(z)Ψ⁎(z¯)−Ψ⁎(z)Ψ(z¯)], where ϕ12 is the spin-12 primary field of the SU(2)3 WZW model and ϕ12(z)⊗ϕ12†(z¯) is the corresponding matrix field. Note the conformal dimensions Δ=Δ¯=320+35=34.For open boundary conditions, the boundary spins reduce to chiral currents, S1,α∝Jα,L(0), and the boundary SSCO becomes proportional to the triple product Cˆ(x=0)∝εabcJ1,La(0)J2,Lb(0)J3,Lc(0). The corresponding operator, which appears in the partition function (38) and is generated from the OPEs of the fields in equation (53) [72–75] in the boundary limit z¯→z→vτ, is given by(54)Cˆ(τ,x=0)∼J−1⋅ϕ1Ω(O fixed point). At the three-channel Kondo fixed point, the partition function (45) contains the boundary operator Ω, which has the same symmetries as the SSCO. This operator is obtained from equation (53) if the spin-12 fields are allowed to fuse to the identity at the boundary. Therefore, at the K fixed point the boundary SSCO is represented by the dimension 8/5 field(55)Cˆ(τ,x=0)∼Ω(K fixed point).Let us now discuss the large-distance decay of the three-spin correlation [39],(56)G3(x)=〈Cˆj=x〉. In the BCFT approach, this amounts to the calculation of a one-point function for given boundary conditions. At the chiral fixed point, the SSCO has a nonzero expectation value. From equation (53), we need to evaluate(57)〈ϕ12σ(z=ix)ϕ12†σ′(z¯=−ix)〉C∼δσσ′x−310, where σ,σ′=1,2 label the components of the fundamental spinor. We also need(58)i[〈Ψ(ix)Ψ⁎(−ix)〉C−〈Ψ(ix)Ψ⁎(−ix)〉C]=ImBΨξx6/5, where the modular S matrix in Appendix B is used to compute(59)BΨξ=Sξ,ΨZ3SI,IZ3SI,ΨZ3Sξ,IZ3=ω⁎. Combining equations (57) and (58), we obtain the decay(60)G3(C)(x)∼(−1)xx−32, which holds for both C+ and C− fixed points. We can obtain the same result from bosonization. Using the staggered part of the spin operators and the boundary conditions (11), the three-spin correlation at the chiral fixed point can be written as(61)G3(C)(x)∼(−1)x∏α=13〈eiπφα,L(x)e−iπφα,L(−x)〉∼(−1)xx−32.Turning next to the time-reversal symmetric O and K fixed points, we first note that the SSCO expectation values vanish identically right at these points. To obtain a nonzero result, we apply perturbation theory in the leading irrelevant boundary operators,(62)G3(O,K)(x)=〈Cˆ(x)e−∫−∞∞dτHB(O,K)(τ)〉O,K∼−∫−∞∞dτ〈Cˆ(x)HB(O,K)(τ)〉O,K. In the boundary Hamiltonian HB, we select the leading operator with the same symmetries as the SSCO.For the O fixed point, this is the operator ∼λ3(O) in equation (14). Using bosonization, we have(63)G3(O)(x)∼λ3(O)(−1)x∫−∞∞dτεabcεa′b′c′〈n1a(x)J1,La′(τ)〉×〈n2b(x)J2,Lb′(τ)〉〈n3c(x)J3,Lc′(τ)〉∝(−1)xx−7/2. Here we used that the chiral current Jα,L and the staggered magnetization nα can fuse to the dimerization operator, which has a nonzero expectation value in the open chain. On the other hand, using the conformal embedding, we can write the boundary SSCO as in equation (54). The correlation function then factorizes into SU(2)3 and Z3(5) sectors,(64)G3(O)(x)∼(−1)x∫−∞∞dτ〈J−1aϕ1a(τ)ϕ12σ(ix)ϕ12†σ(−ix)〉O×i[〈Ω(τ)Ψ(ix)Ψ⁎(−ix)〉O−〈Ω(τ)Ψ⁎(ix)Ψ(−ix)〉O], which can be evaluated using the fusion rules and again yields (63).Near the K fixed point, the leading contribution originates from the term ∼λ2(K) in (46). We obtain(65)G3(K)(x)∼(−1)x∫−∞∞dτ〈ϕ12σ(ix)ϕ12†σ(−ix)〉K×i[〈Ω(τ)Ψ(ix)Ψ⁎(−ix)〉O−〈Ω(τ)Ψ⁎(ix)Ψ(−ix)〉K]∝(−1)xx−21/10.In summary, the three-spin correlation function has the asymptotic (x→∞) power-law decay(66)G3(x)∼(−1)xx−ν, where the exponent is characteristic for the respective fixed point,(67)νO=72,νC=32,νK=2110. Note that the chiral fixed point has the smallest exponent corresponding to the slowest decay of G3(x).5.2Spin conductanceRecent experiments have shown that antiferromagnets can act as efficient conductors of spin currents, essentially without involving charge transport [8]. Here, we envision a setup in which a spin current is injected from a metal into a spin chain that forms one of the legs of a Y junction. The spin chain could be realized, for instance, by arranging spin-12 atoms on a surface using a scanning tunneling microscope (STM) [91,92]. The spin current could be generated by spin accumulation due to the spin Hall effect in the metal. The difference in chemical potential for spin-up and spin-down electrons in the terminals plays the role of a magnetic field Bαz=μα,↑−μα,↓ at the end of chain α=1,2,3. We note in passing that the spin chemical potential of magnons has recently been measured with high resolution [93]. Alternatively, Bαz could represent the external magnetic field of a magnetic STM tip. The field can be oriented in any direction by suitably modifying the setup. While the charge of the electrons cannot propagate into the antiferromagnetic insulator at low energies, the gradient of magnetic field at the metal-insulator interface drives a spin current into the spin chain. In this case, the elementary spin-carrying excitations are the spinons of the Heisenberg chain, as opposed to magnons in an ordered antiferromagnet. The spin current transmitted to the other legs of the Y junction could then be detected by converting it back to a charge current in the attached metallic terminal via the inverse spin Hall effect.One may exploit the similarities to charge transport in quantum wires to define a spin conductance for spin-1/2 chains [10]. Let Iαa denote the spin current component polarized along direction a flowing into the junction from chain α. Within linear response theory, spin transport through the Y junction is then characterized by a spin conductance tensor G,(68)Iαa=∑b,βGαβabBβb, where Bβb is the magnetic field or, more precisely, the spin chemical potential along direction b at the end of chain β. In the presence of SU(2) symmetry, the spin current is parallel to the field that drives it. The conductance tensor must therefore be diagonal in the spin indices,(69)Gαβab=δabGαβ. In analogy with charge conservation in quantum wires [26], total spin conservation in the junction implies, for arbitrary spin chemical potentials, the Kirchhoff node rule, ∑αIαa=0. This implies the constraint ∑αGαβ=0. Moreover, since spin currents only flow when there is a spin chemical potential difference between the terminals, we must have Iαa=0 if Bβ=B is identical for all chains. As a result, we also have ∑βGαβ=0. Finally imposing the Z3 symmetry of our Y junction, the general form of the linear spin conductance tensor compatible with all constraints is(70)Gαβab=12δab[GS(3δαβ−1)+GAεαβ], where εαβ is the Levi-Civita symbol with ε12=ε23=ε31=1 and εαβ=−εβα. The two parameters GS and GA characterize the symmetric and antisymmetric parts of the spin conductance tensor, respectively. Below we determine their values at the various fixed points of the Y junction model.5.2.1Hydrodynamic approach at the chiral fixed pointBefore tackling a more formal derivation, we provide an intuitive hydrodynamic approach to the problem of computing the spin conductance at the chiral fixed point. Following [94], we rewrite the Heisenberg equations of motion for the chiral currents Jα,L and Jα,R in each chain as coupled equations for the magnetization,(71)Mα(x)=Jα,R(x)+Jα,L(x), and for the spin current,(72)Jα(x)=v[Jα,R(x)−Jα,L(x)]. Here we have set γ=0 in equation (7) to neglect the marginally irrelevant bulk operator. We then obtain two equations by taking the sum and the difference of the Heisenberg equations. The first equation is simply the spin continuity equation,(73)∂tMα+∂xJα=0. The second equation is(74)∂tJα+v2∂xMα=0. Within the hydrodynamic approach, one replaces the operators Mα(x) and Jα(x) by their ‘classical’ expectation values [94]. In the steady state, ∂tMα→0, the continuity equation implies that the spin current is uniform in each chain, Jα(x)=Iα. The second equation implies that the magnetization Mα is also constant, varying only near the contacts. When different chains are coupled by very weak tunneling processes, i.e., we are near the O fixed point, the nonequilibrium distribution function in each chain can be described by a spin chemical potential vector Bα, where Mα=χ0Bα with the spin susceptibility χ0=1/(2πv). Note that Bα does not represent a magnetic field but characterizes only the distribution function due to attached spin reservoir [94]. However, near the C± fixed points, we should proceed in a different manner. Suppose that HB realizes ideal chiral boundary conditions, say C− with Jα,R(0)=Jα+1,L(0). We find(75)Jα+1+Jα=v(Mα+1−Mα). This in turn is consistent with the Kirchhoff node rule, ∑αJα=0, and we get(76)Jα=v(Mα+1−Mα−1). The spin chemical potentials here regulate only the incoming spin densities, Jα,L=χ0Bα. The spin currents are then given by(77)Jα=v(Jα,R−Jα,L)=v(Jα+1,L−Jα,L). Using the x-independence of the spin currents, Jα=Iα, we get(78)Iα=χ0v(Bα+1−Bα). Assuming that all spin chemical potentials and spin currents are taken along the z-axis, we have(79)Gαα′=−∂Jαz∂Bα′=12π(δα′,α−δα′,α+1). We can write this in the form of equation (70) with GS=GA=1/(2π). The associated spin conductance tensor is maximally asymmetric in the sense that Gα−1,α=−1/(2π) while Gα+1,α=0. Note that the magnitude of Gα−1,α equals the quantum of spin conductance,(80)G0=12π, in units where gμB=ħ=1 [10]. This means that the spin current injected into chain α is fully transmitted in a clockwise rotation to chain α−1, thus realizing an ideal spin circulator [39].5.2.2Kubo formulaThe linear spin conductance (taken at zero temperature) can alternatively be computed using the Kubo formula [10,26],(81)Gαβab=−limω→0+1ωL∫0Ldx∫−∞∞dτeiωτGα,βa,b(x,y;τ), with chain length L→∞ and the Matsubara Green's function for spin current operators,(82)Gαβab(x,y;τ)=〈TτJαa(τ,x)Jβb(0,y)〉, where Tτ is the imaginary-time ordering operator. Importantly, this Kubo formula neglects the resistance at the contacts between the spin chains and the corresponding spin reservoir. For a single ideal chain, the maximum value of the conductance predicted by this formula is G=14π, i.e., half of the conductance quantum in (80). This extra factor 2 can be traced back to the effective Luttinger parameter for the Heisenberg spin chain, which appears in the conductance for a Luttinger liquid only when one neglects the effects of noninteracting (or Fermi liquid) leads in the dc conductance [10,95]. We refer the reader to [26] for a discussion of how the contact to the leads affects the conductance tensor of the electronic Y junction.We now address the problem of computing the correlation function in (82) in the presence of a conformal boundary condition. Expanding the current operator in terms of the chiral currents, we obtain(83)Gαβab(x,y;τ)=v2[〈Jα,La(z1)Jβ,Lb(z2)〉+〈Jα,Ra(z¯1)Jβ,Rb(z¯2)〉−〈Jα,La(z1)Jβ,Rb(z¯2)〉−〈Jα,Ra(z1)Jβ,Lb(z¯2)〉], where z1=vτ+ix, z2=iy and we omit the time ordering operator on the right hand side. The absence of energy and momentum flow across the boundary implies that the two chiral sectors of the bulk CFT are not independent [38]. Correlation functions between spin currents of the same chirality (L/R) retain the bulk form,(84)〈Jα,La(z1)Jβ,Lb(z2)〉=18π2δabδαβ(z1−z2)2. Conversely, correlation functions for opposite chirality acquire a normalization which depends on the boundary condition B [96,97],(85)〈Jα,Ra(z¯1)Jβ,Lb(z2)〉=δab8π2AαβB(z¯1−z2)2. The coefficients AαβB are determined by the boundary state |B〉 associated with the boundary conditions [38,83]. Before proceeding with the calculation, we note that equations (84) and (85) imply that for two different chains, α≠β, the correlation function for the spin current operator reduces to(86)Gαβab(x,y;τ)=−δab8π2[AαβB(z¯1−z2)2+AβαB(z1−z¯2)2](α≠β).Inserting the above expression into the Kubo formula (81) and performing the integrals, we obtain(87)Gαβab=−δabAαβB4π(α≠β). Thus, the off-diagonal components of the conductance tensor are determined solely by the coefficient AαβB in the correlation function (85) [83,84].For the calculation of the conductance tensor, we first expand the chiral currents in terms of the operators in the embedding SU(2)3×Z3(5). Inverting (43), we obtain(88)(J1,LaJ2,LaJ3,La)=13(1ωω⁎1ω⁎ω111)(Jaϕ1aΨϕ1aΨ⁎). For the right-moving part, we write the same relation but treat the operators as the analytic continuation or mirror image of the left-moving part: OR(z¯)=OR(τ,x)↦OL(τ,−x)=OL(z¯). We can simplify the expression for the correlators by using the fusion rules and noting that the only terms that give nonzero contributions are those in which the operators can fuse to the identity. For currents in the same chiral sector, we get(89)〈Jα,La(z1)Jβ,Lb(z2)〉=19〈Ja(z1)Jb(z2)〉+19〈ϕ1a(z1)ϕ1b(z2)〉××[ωα−β〈Ψ(z1)Ψ⁎(z2)〉+ωβ−α〈Ψ⁎(z1)Ψ(z2)〉]. The normalization of the fields ϕ1, Ψ and Ψ⁎ is fixed so as to recover the correlations of chiral currents for the simplest case of open boundary conditions. Indeed, we then find the free form anticipated in equation (84) [21,57],(90)〈Jα,La(z1)Jβ,Lb(z2)〉=δab31+2cos[2π(α−β)3]8π2(z1−z2)2=δabδαβ8π2(z1−z2)2.The mixed left-right correlator depends on the boundary state |B〉. In our case, all the boundary states can be labeled as |B〉=|s,f〉, where s labels the primary in the SU(2)3 sector and f the primary in the Z3(5) sector generating the boundary state. In this notation, the O fixed point is identified with the identity, |O〉=|0,I〉, while the other fixed points are given by |K〉=|12,I〉, |C+〉=|0,ξ〉, and |C−〉=|0,ξ⁎〉. As for (89), the LR correlator can be written as(91)〈Jα,La(z1)Jβ,Rb(z¯2)〉=δab24π2(z1−z¯2)2[1+2Re(FBωα−β)], where FB is defined in analogy to (89) for the correlation functions of the spin-1 primary and the Ψ and Ψ⁎ fields. Given that the correlation functions of products of fields in different sectors factorize, it can be computed as [38,96](92)FB=Fs,f=XsYf, where Xs and Yf are given in terms of the modular S-matrices,(93)Xs=Ss,1SU(2)3S0,0SU(2)3S0,1SU(2)3Ss,0SU(2)3,Yf=Sf,ΨZ3SI,IZ3SI,ΨZ3Sf,IZ3. The application of the above considerations to the multichannel Kondo problem was given in [81].We can now calculate the spin conductance using the Kubo formula (81). We simplify the result using Re(ωα−β)=12(3δαβ−1) and Im(ωα−β)=−32εαβ. We obtain Gαβab=Gαβδab with(94)Gαβ=14π{[1−Re(FB)](δαβ−13)−Im(FB)3εαβ}, in agreement with the general expression (70). For open boundary conditions, FO=X0YI=1 and the conductance vanishes. For the three-channel Kondo fixed point, we have instead FK=X1/2=−[4cos2(π/5)]−1. Since FK∈R, the conductance tensor at the K fixed point is symmetric,(95)Gαβ=1πsin2(π5)(δαβ−13)(K fixed point). Finally, the chiral fixed points must have complex FC± in order for the conductance to have a nonzero antisymmetric part. Since Xs∈R ∀s, the factor Yf must be complex. This is only possible for fusion with Z3-charged fields in the Z3(5) sector (see Appendix B). In particular, the conjugate fields ξ and ξ⁎ have charges +1 and −1, respectively. Using the modular S-matrix for the Z3(5) theory, we find FC+=Yξ=ω⁎ and FC−=Yξ⁎=ω. More generally, all charged operators will generate an asymmetric conductance, which provides an intriguing physical intuition of this quantum number in the context of Y junctions. The conductance for the chiral fixed points thus becomes(96)Gαβ=14π[12(3δαβ−1)±12εαβ]=14π(δαβ−δα,β±1)(C±fixed point). This result differs from equation (79) by a factor 2 because it represents the Kubo conductance calculated without taking into account the contacts to spin reservoirs. The maximally asymmetric conductance tensor (96) is a direct consequence of the chiral boundary conditions (18). We note in passing that for Y junctions of electronic quantum wires [26] the maximally asymmetric charge conductance is obtained only asymptotically for Luttinger parameter K→1+, i.e., for infinitesimal attractive interactions at the edge of stability of the corresponding chiral fixed point.5.3Boundary susceptibility and scalar spin chirality at finite temperatureFor comparison with the numerical QMC results in section 6.2, in this section we present analytical predictions for the temperature dependence of boundary observables. We start with the local boundary susceptibility which describes the response to a boundary magnetic field,(97)H′=−hSBz=−h∑αSj,αz. The linear response has the form 〈SBz〉=χloch, where the local susceptibility χloc is determined by the correlation function(98)χloc(T)=∫0βdτ〈SBz(τ)SBz(0)〉. At the O fixed point, we take SBz=K∑αJα,Lz(0), where K is a nonuniversal prefactor. Using the finite-temperature correlation functions [21] at inverse temperature β=1/T, and introducing a short-time cutoff τ0 of order (J1)−1, we obtain(99)χloc(O)=K2∑α,α′∫τ0β−τ0dτ〈Jα,Lz(z=vτ)Jα′,Lz(0)〉=3K28π2v2∫τ0β−τ0dτ[π/βsin(πτ/β)]2≃3K24π2v2τ0(1−π2τ023T2+⋯). Therefore, at weak coupling |Jχ|≪Jχc, we expect the local susceptibility to approach a nonuniversal value at zero temperature and to decrease with a quadratic dependence upon increasing T. At the chiral fixed point, the dominant contribution comes from the staggered part of the spin operator, represented by SBz∼∑αTr[g˜α(0)σz]∼∑αsin[πφα,L(0)−πφα+1,L(0)]. The calculation of the boundary susceptibility in this case involves the two-point function for a dimension-1/2 boundary operator and gives(100)χloc(C)∼−ln(τ0T). Therefore, the susceptibility diverges logarithmically as T→0 at the chiral fixed points. Near the K fixed point, the leading operator representing the boundary spin is the spin-1 primary field of the SU(2)3 WZW model, SB∝ϕ1, with dimension 2/5 [81]. As a result, the boundary susceptibility diverges as a power law at low temperatures,(101)χloc(K)∼T−15. While this is a stronger divergence than at the chiral fixed point, we may anticipate difficulties in cleanly distinguishing (100) and (101) from stochastic QMC data.We now turn to the thermal average of the boundary SSCO,(102)CB(T)≡〈Cˆj=1〉=Z−1Tr(Cˆ1e−βH). Clearly, CB(T) vanishes at any temperature for Jχ=0 due to time-reversal symmetry. In the strong-coupling limit, Jχ→±∞, we have CB→∓34 as the chirality saturates at the eigenvalue of Cˆ1. In general, from the Hamiltonian (4), we see that CB can be obtained as(103)CB(T)=−1β∂∂JχlnZ. Jχ can thus be regarded as external parameter that couples to the boundary SSCO. In analogy with the response to an external field, we also define the chirality susceptibility,(104)−dCBdJχ=∫0βdτ〈Cˆ1(τ)Cˆ1(0)〉. Near the O fixed point, perturbation theory in the boundary Hamiltonian (14) yields the partition function(105)Z≃1+3(λ3(O))2(8π2v2)3β∫0βdτ[π/βsin(πτ/β)]6, where λ3(O)∝Jχ. Expanding lnZ for small Jχ and using (103), we find(106)CB(O)(T)≈CB(O)(0)(1+5π2τ023T2+…), where CB(O)(0)∝−Jχ is the nonuniversal T=0 value. Note that CB(O)(T) increases quadratically with increasing T. Similar contributions to the nonuniversal prefactor of the T2 term may come from the time-reversal-invariant irrelevant boundary operators in (14), assuming the corresponding coupling constants are even functions of Jχ. Next, we discuss the behavior near the chiral fixed point, where |Jχ|=Jχc∼J1. We then cannot calculate the chirality by perturbation theory in Jχ anymore. Nonetheless, we expect that the leading temperature dependence stems from the relevant perturbation in (22), with coupling constant λ1(C)≈−b(Jχ−Jχc) for Jχ≈Jχc. Computing the correction to the partition function to second order in λ1(C), we find(107)CB(C)(T)∼−6b2(Jχ−Jχc)ln(τ0T). Therefore, we predict a logarithmic temperature dependence with a sign change of the prefactor around the critical point. We note that the effective coupling constant gets renormalized at low temperatures, λ1,eff(C)(T)∼λ1(C)(τ0T)−1/2. Therefore, the perturbative result assuming chiral boundary conditions is only valid for T in the range(108)τ0T⁎≡(1−Jχ/Jχc)2<τ0T≪1. Here T⁎ sets the crossover temperature to the quantum critical regime of this boundary transition. Finally, near the K fixed point, we assume that the coupling constants of irrelevant boundary operators in (46) are smooth functions of Jχ. The dominant temperature dependence of CB(T) then stems from the leading irrelevant operator with dimension 7/5. Applying perturbation theory in λ1(K), we find(109)CB(K)(T)≈CB(K)(0)(1+c1T9/5+c2T2+⋯), where c1 and c2 are nonuniversal constants. The direct calculation from the correlation function gives c1>0 and c2<0. The precise temperature dependence of the boundary chirality in the strong coupling limit depends on the competition between these contributions.6Numerical resultsIn order to check our predictions, we employed both DMRG and QMC simulations. The first technique aims at ground-state properties and is therefore useful to identify the boundary fixed points and the associated scaling dimensions of operators through the large-distance decay of correlation functions. The second method, instead, computes equilibrium expectation values of local observables at finite temperature, using the effective field theory directly in the thermodynamic limit.6.1DMRGThe density matrix renormalization group is one of the most powerful techniques to investigate ground-state properties of (quasi-)1D quantum lattice systems. The power of this method lies on a systematic truncation of the Hilbert space, using the information provided by the reduced density matrix. Since its original development [98], several DMRG algorithms have been proposed to study systems with different geometries, such as Y junctions [99,100], finite-width strips [101,102], and two-dimensional systems [103].Some results for the chiral Y junction of spin chains obtained using the algorithm proposed in [99] have already been presented in [39]. The method of [99] works efficiently for a Y junction with boundary interaction among spins at the first site, j=1 and open boundary conditions at j=L. Here, we apply a different numerical scheme especially tailored for the calculation of the spin conductance, described in [83,84]. In this method, we implement the geometry illustrated in Fig. 3, featuring two Y junctions facing each other, where the system size (the length of each chain) is finite and one junction is a mirror image of the other. Implementing this geometry with the method used in [39] is equivalent to considering periodic boundary conditions and would require a much higher computational effort. Instead, we treat the double Y junction by mapping the system to a chain with long-range interactions and employing ordinary DMRG as shown schematically in Fig. 4. In our DMRG computations, we have kept up to m=3000 states per block. The largest truncation error of our results at the final sweep is of order 10−7. In order to check the accuracy of our correlations under truncation of the Hilbert space, for a fixed system size, we compared the numerical data obtained by keeping m=2400 versus m=3000 states. The errors in correlation functions are at least one order of magnitude smaller than the values acquired by DMRG. In addition, in our estimates, we did not find significant differences when fitting the correlations for these distinct numbers of kept states.In order to characterize the three different regimes of the Y junction using DMRG, we have calculated the expectation value of the SSCO at the boundary, the spin conductance from the Kubo formula, and the exponent in the power-law decay of the three-spin correlation. In all the DMRG results presented in this subsection, we have set J1=1 and J2=0. Thus, one should keep in mind that the results may be affected by logarithmic corrections due to the marginal operator in (7). Nonetheless, as we discuss below, we find remarkably good agreement with the analytical predictions that neglect logarithmic corrections as well as with our previous DMRG results [39] for three-spin correlations in the model with J2=J2c.In Fig. 5, we show the expectation value of the SSCO at positions j=1 and j=2 as a function of Jχ for a Y junction with length L=60. As expected, the chirality at the boundary site 〈Cˆ1〉 is negative for Jχ>0. Moreover, its absolute value increases monotonically and approaches the saturation value |〈Cˆ1〉|=3/4≈0.433 for Jχ→+∞, see equation (28). In contrast, the chirality at the second site, 〈Cˆ2〉, is positive and reaches a maximum value at intermediate coupling. The saturation of 〈Cˆ1〉 and the vanishing of 〈Cˆ2〉 support our picture that the boundary spins form a low-energy spin-1/2 doublet and time-reversal symmetry is effectively restored for the remaining spins in the limit Jχ→∞. The peak in 〈Cˆ2〉 also provides a rough estimate for the crossover scale separating the weak and strong coupling limits. Around this scale, one expects to find the chiral fixed point.To pinpoint the location of the chiral fixed point, we investigate the linear-response spin conductance of the Y junction. Here we follow the method developed by Rahmani et al. [83,84]. Although the Kubo formula (81) involves a dynamical correlation function, the dc conductance is uniquely determined by the prefactor of the correlator between L and R currents in (85). We can then rely on conformal invariance to extract this prefactor from the large-distance decay of static correlation functions, which are easy to access via time-independent DMRG. We then set z¯1=−z2=−ij, with j the distance from the boundary, and use the conformal map (36) to account for the finite system size. Using equation (87) to express the coefficient in terms of the conductance, we can write the RL correlation in the form(110)〈Jα,Rz(j)Jβ,Lz(j)〉=Gαβ/G0[4Lsin(πjL)]2(α≠β). From equation (110), we see that the problem of estimating the conductance resides in the computation of the correlation function of chiral currents in the ground state. In order to use DMRG, we need to write the chiral currents in terms of the spin operators in the lattice model. We can use the relation to the magnetization and the spin current in (71) and (72) and write(111)Jα,Rz(j)=12v[vMαz(j)+Jzα(j)],(112)Jα,Lz(j)=12v[vMαz(j)−Jzα(j)], where v=π/2 is the spin velocity for the Heisenberg chains. The magnetization and spin current operators are related to the spin operators by(113)Mαz(j)=12(Sα,jz+Sα,j+1z),(114)Jαz(j)=i2(Sα,j+S−α,j+1−Sα,j−Sα,j+1+). The spin operators obey the discrete-space version of the continuity equation in the bulk, ∂tSjz(t)+Jαz(j)−Jαz(j−1)=0. The linear combination in (113) is important to cancel out the staggered magnetization to leading order in the mode expansion (A.4).The spin conductance is estimated by fitting the correlations 〈Jα,Rz(x)Jβ,Lz(x)〉 using equation (110). Note that equation (110) is useful only when the spin conductance is finite. In the case of vanishing Gαβ, the current-current correlations do not scale linearly with [4Lsin(πjL)]−2. Instead, they are dominated by contributions of irrelevant operators which are responsible for a faster decay. It is worth mentioning that Gαβ(Jχ)=Gβα(−Jχ), but in general we have Gαβ(Jχ)≠Gβα(Jχ) due to the breaking of reflection and time-reversal symmetries. In particular, we expect the spin conductance to be maximally asymmetric at the chiral fixed point. For Jχ>0 (see Fig. 3), this means that G12 (and equivalent components obtained by cyclic permutations of the leg indices) must reach its maximum value at Jχ=Jχc, while it must vanish at Jχ=−Jχc.Indeed, in Fig. 6 we observe a linear behavior of the RL correlations with [4Lsin(πjL)]−2 for values of Jχ where we expect a finite spin conductance. In the same figure, we show DMRG results for two different numbers of states included in the calculation, showing the robustness of the estimate to truncation errors. Since the BCFT prediction (110) is only valid for distances far from the boundary, when extracting the spin conductance numerically, we made sure that the fitting interval covers only the region which exhibits such scaling behavior.In Fig. 7, we show our estimates of the spin conductance as a function of Jχ for different system sizes. The predictions for G12/G0 at the O, C− and K fixed points are 0, −12 and −23sin2(π/5), respectively, see equations (95) and (96). Note that if we were able to compute the correlation at asymptotically large distances in the limit L→∞, we would expect the conductance to be a discontinuous function of Jχ, taking the values G12=0 for Jχ<Jχc, G12/G0=−1/2 right at the critical point, and G12/G0=−23sin2(π/5) for Jχ>Jχc. In contrast, the result for G12(Jχ,L) for finite L is a smooth function of Jχ that must approach the fixed point values as we increase L.From Fig. 7, we identify a maximum in the conductance at Jχ≈3.4, where the peak conductance is close to the maximum value, |G12|/G0=1/2, predicted by the Kubo formula. On the other hand, the RL correlation at Jχ=−3.4 decays faster than 1/sin2(πj/L) at large distances, see Fig. 8. This is the same behavior as observed for Jχ=0.4, where we expect the conductance to vanish in the limit L→∞ because the regime of small Jχ is governed by the O fixed point. These results indicate that for L→∞, G12 will vanish for Jχ=−3.4, and hence G21 vanishes for Jχ=+3.4 as well. Altogether this provides strong evidence that the C− fixed point is located at Jχc≈3.4. This estimate for the nonuniversal critical coupling value differs only slightly from the one reported in our earlier work (Jχc≈3.1) [39], where the case J2=J2c has been studied instead of the model with J2=0 considered here.We now turn to the determination of the conductance at the K fixed point. For L→∞, our expectation is that the off-diagonal conductance should approach the plateau value corresponding to the K fixed point, for Jχ>Jχc. Conversely, for any finite size, the conductance is a continuous function of the coupling Jχ, approaching smoothly an asymptotic value. We have observed that the fit of the current-current correlation function by the form (110) is not reliable for Jχ>10, due to a combination of truncation errors, a residual staggered contribution and an evident dependence of the result on the fitting interval. On the other hand, as clear also from Fig. 5, the effective central spin is already fully developed at Jχ=10, as the expectation value of the SSCO at the boundary has reached over 98% of its asymptotic value for all system sizes; see discussion around (28). We therefore select this point as a representative of the strong coupling regime. Compared to the O and C points, finite-size effects are noticeably more important for large Jχ. The values of the conductance G12(Jχ,L) at Jχ=10 show a slow, but significant variation with the system size L. In order to extract the infinite-size limit, we considered the values of the conductance for different system sizes L=52,60,68 and performed an extrapolation in L, assuming the form G12(Jχ,L)=G12(Jχ,∞)+a1L−a2, with free fitting parameters a1 and a2. The asymptotic value G12(10,∞) is quoted in Table 1. Given the small fitting interval in L and the DMRG truncation errors, this result should be taken with some precaution. Nonetheless, the available evidence from DMRG is consistent with the three-channel Kondo fixed point scenario.Let us next discuss DMRG results for the three-spin correlation G3(j) described in section 5.1. This quantity oscillates with the distance from the boundary and exhibits power-law decay with an exponent governed by the low-energy fixed point, see equations (66) and (67). Using the conformal transformation (36), the three-spin correlation functions are brought into the form(115)G3(j)∼(−1)j[Lπsin(πjL)]ν. We have fitted our numerical results to this formula and thereby extracted the exponent ν. In Fig. 9, we show our estimates for ν as a function of Jχ. As discussed for the conductance, the function ν(Jχ,L) varies continuously with Jχ for finite L. In order to fit the data, we chose an interval jin<j<L/2 such that sites which are too close to the boundary or to the center of the double junction are not taken into account. We observe a robust minimum in ν(Jχ) at Jχ≈3.4, in remarkable agreement with the location of the maximum in the spin conductance. We have verified that this minimum is insensitive to changes of the values of jin and/or L. Moreover, the value of ν at the minimum is rather close to the prediction for the chiral fixed point, νC=1.5, see equation (67). Finally, for estimating the exponent νK near the K fixed point, we consider again Jχ=10 as a representative, as the limitations noticed for the conductance apply also to the three-spin correlation. Compared to the spin conductance, finite-size effects are here much smaller, well below truncation errors, and the largest system size under study (L=68) already provides an accurate answer. In Table 2, we collect our results for ν at the three fixed points, with the respective BCFT predictions and relative errors. Overall, we conclude that these numerical estimates are consistent with our analytical predictions.6.2QMC simulationsNext we turn to finite-temperature path-integral Monte Carlo simulations. Our QMC scheme employs a bosonized functional integral representation of the partition function, where the Gaussian bulk boson modes are integrated out analytically and the limit of infinite chain length can be taken from the outset. A similar method has previously been applied to study Kondo effects in Luttinger liquids [104], and we here describe a generalization of that approach for our Y junction problem. We will study the local spin susceptibility at the junction, χloc(T), and the boundary scalar spin chirality, CB(T)=〈Cˆ1〉, see section 5.3 for analytical predictions near the different fixed points. Importantly, our QMC approach is free from sign problems.6.2.1Simulation schemeWe start from the bosonized field theory and express the partition sum as imaginary-time functional integral,(116)Z=∫D[θα(x,τ)]e−S0[θ]−Jχ∫0βdτC1[θ]. The full action, S=S0+SB, contains a Gaussian bulk term (S0) and a boundary term SB due to HB. We here put the boundary at x=a, with a short-distance cutoff a, and impose hard-wall boundary conditions at x=0. After integration over the bulk (θα,ϕα) modes with x≠a, the action S0 is effectively replaced by a time-nonlocal dissipative action, S0→Seff, for the boson fields at the position x=a. We now define complex functions of the bosonic Matsubara frequency ω,(117)Fα(ω)=2∫0∞dk2πsin(ka)kθ˜α(k,ω),Gα(ω)=2∫0∞dk2πcos(ka)θ˜α(k,ω), where θ˜(k,ω) denotes the Fourier transform of the boson field θα(x,τ). The real-valued dual boson fields at x=a follow as(118)θα(a,τ)=T∑ωeiωτGα(ω)=Gα(τ),ϕα(a,τ)=T∑ω(−iω)eiωτFα(ω)=−∂τFα(τ). After some algebra, we obtain the effective action in the form(119)Seff=12∫0βdτ∫0βdτ′∑α(Gα(τ)Fα(τ))TK(τ−τ′)(Gα(τ′)Fα(τ′)). The real-valued kernel, K(τ)=T∑ωcos(ωτ)K˜(ω), is the Fourier transform of the non-negative matrix kernel(120)K˜(ω)=|ω|a|ω|coth(a|ω|)−1(2a|ω|1−e−2a|ω|−1−|ω|−|ω|ω2coth(a|ω|)). Importantly, no sign problem arises. Using a hard energy cutoff D=v/a, we keep all Matsubara components F(ω) and G(ω) with |ω|<D. Our QMC code uses the full action, S=Seff+SB, for the MC sampling. Each data point reported below has been obtained from ≈107 to 108 statistically independent samples.We now present our QMC results for CB(T) and χloc(T). We use units with D=v/a=1. Due to our regularization scheme in the low-energy theory behind the QMC approach, the values quoted for Jχ in this subsection differ from the corresponding values for the lattice model.6.2.2QMC resultsFirst, for very small Jχ≪1, we have established that the analytical results in equations (106) and (99), which are valid near the O fixed point, are accurately reproduced by our QMC data. In fact, for various different Jχ≪1, our data (not shown here) nicely fit the analytical expressions with the same choice for the product Dτ0≃3.5. This provides an important benchmark test for our scheme.For larger Jχ, we obtain substantial renormalizations of CB(T) and χloc(T) as compared to equations (106) and (99), respectively. In Fig. 10 we show the corresponding QMC results for Jχ=1. First, the left panel shows CB(T) with logarithmic scales of the T axis. At elevated temperatures, we observe a logarithmic scaling as predicted near the chiral fixed point in equation (107). This logarithmic scaling thus is interpreted as high-temperature signature of the unstable chiral fixed point. The positive slope indicates that Jχ=1 is below Jχc, see equation (107). At sufficiently low temperatures, the stable O point ultimately dominates and we find a crossover toward the essentially constant T-dependence of CB predicted by equation (106) near the O point. The right panel in Fig. 10 shows χloc(T) for the same value of Jχ (and also with a logarithmic T axis). For high T, we again observe a logarithmic scaling of χloc as expected near the chiral fixed point, see equation (100). In accordance with the analytical result, here the slope is negative. At lower T, the susceptibility saturates to a constant value, in agreement with equation (99) valid near the O point. (T2 corrections cannot be resolved within error bars.) To summarize, the data in Fig. 10 for Jχ=1 show that although the low-T behavior is dominated by the O point, the high-T behavior is already governed by the unstable chiral fixed point.Next we turn to the value Jχ=1.4, where the corresponding QMC data are shown in Fig. 11. This value appears to be quite close to Jχc (in the low-energy theory). Indeed, our data for both CB(T) and χloc(T) are consistent with the respective analytical expressions near the chiral fixed point. The left panel shows that CB(T) is basically constant, corresponding to a very small prefactor in front of the logarithm in equation (107). This is precisely as expected in the vicinity of the chiral point. The right panel shows the local boundary susceptibility, which exhibits a logarithmic T scaling over more than a decade. The power law (101), which is expected near the Kondo fixed point, is clearly inconsistent with the data. We conclude that Jχ=1.4 represents a boundary coupling in the near vicinity of the unstable fixed point. Unfortunately, probing even larger Jχ by means of QMC simulations turned out to be prohibitively costly. We therefore are not able to show results in the Kondo regime using this technique.7ConclusionsIn this paper, we have presented a detailed characterization of the boundary phase diagram of a Y junction of Heisenberg spin chains with a chiral boundary interaction Jχ. Using bosonization and boundary conformal field theory to construct a low-energy effective field theory, we have provided analytical predictions for the boundary entropy, the spin conductance, and for the low-temperature behavior of the boundary scalar spin chirality and of the local boundary susceptibility. The phase diagram exhibits two stable fixed points, namely a fixed point of disconnected chains at weak coupling (O) and a three-channel Kondo fixed point (K) at strong coupling. These stable points are separated at intermediate coupling by an unstable chiral fixed point (C). In BCFT language, the chiral fixed point is described by fusion with a Z3-charged operator in the Z3(5) theory associated with the ‘flavor’ degree of freedom of the junction.Using a DMRG scheme especially suitable for computing the spin conductance, we have tested the predicted phase diagram. In particular, the chiral fixed point is characterized by a maximally asymmetric spin conductance tensor, which has been unequivocally observed in our numerical calculations, even at small sizes of the system. In comparison, finite-size effects are more pronounced at strong coupling, which causes obstacles to the accurate numerical computation of the spin conductance in this limit. Nonetheless, we find good agreement between the BCFT and the DMRG predictions. In addition, by means of QMC calculations, we have probed the temperature dependence of the local spin susceptibility and the scalar spin chirality at the boundary. The reported results are in qualitative agreement with our analytical predictions.The Heisenberg spin chain is known to accurately describe a number of effectively 1D crystalline materials and its excitation spectrum has been probed by many experiments over the years. In view of the rapid developments in antiferromagnetic spintronics and given that spin currents can be readily generated and detected, it stands to reason that our setup can be experimentally realized and investigated in solid state and/or cold atom platforms. Once realized, one would have access to a circulator for spin currents. To the best of our knowledge, spin circulators have not yet been achieved, and hence this would represent a tremendous advancement in the control and manipulation of spin currents.AcknowledgementsWe thank C. Chamon, J. C. Xavier, F. Ravanini and E. Ercolessi for discussions and the High-Performance Computing Center (NPAD) at UFRN for providing computational resources. We acknowledge support by the Deutsche Forschungsgemeinschaft within the network CRC TR 183 (project C01), by the Alexander von Humboldt Foundation, and by the Brazilian ministries MEC and MCTIC.Appendix ABosonizationHere we recall some useful formulas commonly used when studying one-dimensional systems via bosonization [21]. Let us start with the case Jχ=0, where we have an open boundary and incoming chiral spin currents are fully reflected,(A.1)Jα,R(x=0)=Jα,L(x=0). One can see that the flow across the boundary of the full spin current(A.2)Jα(x)=v[Jα,R(x)−Jα,L(x)] is vanishing, with open boundary conditions. It is, in general, convenient to regard Jα,R as the analytic continuation of Jα,L to the negative-x axis,(A.3)Jα,R(x)=Jα,L(−x),x≥0. The effective Hamiltonian for a given chain in (7) (with γ=0) then becomes equivalent to a single chiral mode on the infinite line. The OPE of the right currents can be evaluated using (8), replacing z→z¯.For a description of the low-energy physics, spin operators are bosonized as [21](A.4)Sj,α∼Jα,L(x=j)+Jα,R(x=j)+(−1)jnα(x=j), where the staggered part,(A.5)nαa(x)=ATr[gα(x)σa], involves the standard Pauli matrices σa and the SU(2) matrix field gα(x) [57]. The SU(2) invariant trace of the matrix field appears in the dimerization operator,(A.6)Sj,α⋅Sα,j+1∼const.+(−1)jA′Tr[gα(x)], where A′ is another nonuniversal constant.The chiral spin currents, with Jα,L±=Jα,L1±iJα,L2, are given in terms of the chiral bosonic fields satisfying (9) by [50](A.7)Jα,L/R±(x)=12πe±2iπφα,L/R(x),Jα,L/Rz(x)=±12π∂xφα,L/R(x), while the staggered part takes the form(A.8)nα±(x)=Ae±iπ(φα,L+φα,R),nαz(x)=Asin[π(φα,L−φα,R)]. The dimerization operator involves(A.9)Tr[gα(x)]=cos[π(φα,L−φα,R)]. Here C=−π is equivalent to C=π because the boson fields are compactified. The two choices for C correspond to stronger bonds on either even or odd links whenever the dimerization field (A.9) acquires a nonzero expectation value.Within nonabelian bosonization, the currents (A.7) are instead level-1 descendents in the module of the identity [57]. Here, we only need to recall that the SU(2)k theory possesses k+1 WZW primary fields labeled by integrable representations of SU(2) [57,82], i.e., by the spin quantum number s=0,12,1,…,k2. The corresponding scaling dimensions are(A.10)Δs=s(s+1)k+2. The general form of the short-distance expansion of the primary fields is encoded into the fusion coefficients Ns1,s2s3 [72],(A.11)Ns1,s2s3={1,ifs1+s2+s3∈Nand|s1−s2|≤s3≤min(s1+s2,k2),0,otherwise.Appendix BZ3(5) toolboxHere we collect some notions about the Z3(p) CFT, focusing on one chiral sector in particular and p=5. When p is large, the models Z3(p) can be interpreted as critical solutions to the bosonic field theory defined by the action(B.1)S=∫d2r[∂μϕ†∂μϕ+V(ϕ,ϕ†)], where the polynomial V is invariant under the Z3 transformation,(B.2)ϕ→ωϕ,ϕ†→ω⁎ϕ†,ω=e2πi/3 has highest degree p−2 in (ϕ†ϕ), and its parameters have been tuned to a multi-critical point [74]. In addition to the energy-momentum tensor T(z), which generates the conformal transformations, the theory contains the additional local spin-3 currents W(z). In the same way as the modes Ln of the energy-momentum tensor T(z)=∑n∈Zz−n−2Ln generate the Virasoro algebra, the modes Wn of W(z)=∑n∈Zz−n−3Wn generate an additional symmetry algebra, denoted by W algebra [74]. The space A of local fields can be decomposed as(B.3)A=⊕i[Φi], where [Φi] denotes an irreducible representation of W. In particular, the representation [Φi] can be constructed starting from an ‘ancestor’ (or primary) field Φi, satisfying the properties(B.4)Ln>0Φi=Wn>0Φi=0,L0Φi=ΔiΦi,W0Φi=wiΦi for some real wi and non-negative Δi.The theory can be formulated in the Coulomb-gas formalism [57], in terms of a two-component free massless bosonic field φ=(φ1,φ2). A background charge makes the U(1) symmetry of the bosonic field theory anomalous and alters the central charge and the scaling dimension of the vertex operators. The energy-momentum tensor is written as(B.5)T=−14∑j=1,2(∂zφj)2+iα0∂z2φ1, for α0=1/30. Primary fields Φn′,m′n,m are labeled by two pairs of integers n,n′ and m,m′, such that n+n′≤4 and m+m′≤5. They can be written as free-field vertex operators(B.6)Vβ(z)=V(β1,β2)(z)=eiβ⋅φ, where β=βn′,m′n,m is given in terms of the su(3) weights,(B.7)ω1=12(1,13),ω2=12(1,−13), and the two real solutions α± of the equations(B.8)α+α−=−14,α++α−=α02. In particular [74], one has(B.9)βn′,m′n,m=2[(1−n)α++(1−m)α−]ω1+[(1−n′)α++(1−m′)α−]ω2. In this formalism, one identifies the fields(B.10)Φn′,m′n,m=Φn,m5−n−n′,6−m−m′=Φ5−n−n′,6−m−m′n′,m′ having conformal dimensionΔn′,m′n,m=3[6(n+n′)−5(m+m′)]2+[6(n−n′)−5(m−m′)]2−12360 The fusion algebra between the primary fields is invariant under the substitution(B.11)Φ→e2πi3qΦ, where the pertinent Z3-charge q is defined in [78] as(B.12)q=qn′,m′n,m=(m−m′)mod3. The three fields in the identification (B.10) have the same conformal dimension and Z3 charge. As a consequence, the OPE coefficients and the three-point functions vanish unless the total Z3 charge is zero, which severely constrains the possible fusion processes. Note that the dimension-3/5 fields Ψ,Ψ⁎ are the parafermions ψ1,ψ2 of [75], while the dimension 1/9 fields are the spin fields σ1,σ2. (See Table 3.)The modular S matrix describes the rearranging of the characters of the theory under modular transformations [57]. Moreover, it determines the fusion rules of conformal primary operators [105]. Its most general form can be found in [79,88]. Here we present, for the sake of clarity, only the case Z3(5), which we have tested against the fusion rules of [76,79] and used to generate the others necessary for this paper and to compute the spin conductance.We first recall a few basic notations from Lie algebras [57]. Denote the su(3) fundamental weights by ω1 and ω2 and the fundamental roots by α1 and α2. The Weyl vector ρ=ω1+ω2 is the sum of the fundamental weights. A generic weight is then expanded on this basis as ω=λ1ω1+λ2ω2. A Weyl reflection of the weight ω with respect to the root αj is denoted by sjω. It is possible to apply repeatedly a Weyl reflection and construct all the independent strings s of length len(s), which constitute the Weyl group W. For su(3), there are 6 elements:Element ssωSignature (−1)len(s)

Iλ1ω1 + λ2ω2+1

s1−λ1ω1+(λ1+λ2)ω2−1

s2(λ1+λ2)ω1−λ2ω2−1

s2s1λ2ω1−(λ1+λ2)ω2+1

s1s2−(λ1+λ2)ω1+λ1ω2+1

s1s2s1 = s2s1s2−λ2ω1 − λ1ω2−1

We now define the function(B.13)ϕa(b)=∑s∈W(−1)len(s)e−2πi(a+ρ)s(b+ρ), where the sum runs over the elements of the Weyl group. As a primary field Φn′,m′n,m is associated with the pair of weights λ=(n−1)ω1+(n′−1)ω2 and λ′=(m−1)ω1+(m′−1)ω2, such pairs of weights can be used to label the rows and the columns of the modular S matrix. In this notation,(B.14)S(λ,λ′),(μ,μ′)=e2πi[(λ+ρ)⋅(μ′+ρ)+(λ′+ρ)⋅(μ+ρ)]×ϕμ+ρ(6(λ+ρ)5)ϕμ′+ρ(5(λ′+ρ)6). The fusion coefficients are finally computed by using the Verlinde formula [105],(B.15)N(λ,λ′),(μ,μ′)(ν,ν′)=∑(η,η′)S(λ,λ′),(η,η′)S(μ,μ′),(η,η′)S(ν,ν′),(η,η′)SI,(η,η′), where the sum runs over the pairs of weight labeling the primary fields, taking into account the identification (B.10).Appendix CIdentity of partition functionsOne can show the equivalence of partition functions (37) and (38) by using a well-known formula [106] for the product of the su(2)k characters of the spin-l primary and su(2)1 character of the spin-l′ primary,(C.1)χl(k)(q)χl′(1)(q)=∑j=1k+2χj−12(k+1)(q)χ2l+1,jMk+2,k+3(q), where the sum runs over the values j=(2l+1+2l′)mod2. On the r.h.s. of this equation, we encounter products of a su(2)k character and a character of the Virasoro primary labeled by the integers (2l+1,j) in the Kac table of the minimal unitary model Mk,k+1. Applying twice this relation to the partition functions of three chains with open boundary conditions (37), one obtains(C.2)ZOOeee(q)=χISU(2)3(q)χ0TI(q)χ0I(q)+χ1SU(2)3(q)χ3/5TI(q)χ0I(q)+χISU(2)3(q)χ3/2TI(q)χ1/2I(q)+χ1SU(2)3(q)χ1/10TI(q)χ1/2I(q),ZOOooo=χ1/2SU(2)3(q)χ1/10TI(q)χ1/2I(q)+χ3/2SU(2)3(q)χ3/2TI(q)χ1/2I(q)+χ1/2SU(2)3(q)χ3/5TI(q)χ0I(q)+χ3/2SU(2)3(q)χ0TI(q)χ0I(q), expressed in terms of the characters of the Ising (I) and the Tricritical Ising (TI) models, here labeled by the dimension of the primary. The next step is to use the identities [77](C.3)χ0I(q)χ0TI(q)+χ1/2I(q)χ3/2TI(q)=χIZ3(q)+χζZ3(q)+χζ⁎Z3(q),χ0I(q)χ3/5TI(q)+χ1/2I(q)χ1/10TI(q)=χΨZ3(q)+χZ3Ψ⁎(q)+χΩZ3(q), relating the product of characters of the Ising and tricritical Ising models to sums of characters of the Z3(5) CFT. This brings (C.2) into the form (38) which is our starting point.References[1]G.A.PrinzMagnetoelectronicsScience2825394199816601663http://science.sciencemag.org/content/282/5394/1660[2]S.A.WolfD.D.AwschalomR.A.BuhrmanJ.M.DaughtonS.von MolnárM.L.RoukesA.Y.ChtchelkanovaD.M.TregerSpintronics: a spin-based electronics vision for the futureScience2945546200114881495http://science.sciencemag.org/content/294/5546/1488[3]D.D.AwschalomD.LossN.SamarthSemiconductor Spintronics and Quantum ComputationNanoScience and Technology2002Springerhttps://books.google.it/books?id=tlDSx_8_5v4C[4]K.UchidaS.TakahashiK.HariiJ.IedaW.KoshibaeK.AndoS.MaekawaE.SaitohObservation of the spin Seebeck effectNature455200877810.1038/nature07321[5]E.SaitohM.UedaH.MiyajimaG.TataraConversion of spin current into charge current at room temperature: inverse spin-hall effectAppl. Phys. Lett.8818200618250910.1063/1.2199473[6]S.MaekawaS.ValenzuelaE.SaitohT.KimuraSpin CurrentOxford Science Publications2017Oxford University Presshttps://books.google.de/books?id=iZ83DwAAQBAJ[7]L.J.CornelissenJ.LiuR.A.DuineJ.B.YoussefB.J.van WeesLong-distance transport of magnon spin information in a magnetic insulator at room temperatureNat. Phys.112015102210.1038/nphys3465[8]R.LebrunA.RossS.A.BenderA.QaiumzadehL.BaldratiJ.CramerA.BrataasR.A.DuineM.KläuiTunable long-distance spin transport in a crystalline antiferromagnetic iron oxideNature5617722201822222510.1038/s41586-018-0490-7[9]K.GanzhornS.KlinglerT.WimmerS.GeprägsR.GrossH.HueblS.T.B.GoennenweinMagnon-based logic in a multi-terminal YIG/Pt nanostructureAppl. Phys. Lett.1092201602240510.1063/1.4958893[10]F.MeierD.LossMagnetization transport and quantized spin conductancePhys. Rev. Lett.90200316720410.1103/PhysRevLett.90.167204https://link.aps.org/doi/10.1103/PhysRevLett.90.167204[11]F.LangeS.EjimaT.ShirakawaS.YunokiH.FehskeSpin transport through a spin-12 xxz chain contacted to fermionic leadsPhys. Rev. B97201824512410.1103/PhysRevB.97.245124https://link.aps.org/doi/10.1103/PhysRevB.97.245124[12]S.E.NaglerW.J.L.BuyersR.L.ArmstrongB.BriatPropagating domain walls in cscobr3Phys. Rev. Lett.49198259059210.1103/PhysRevLett.49.590https://link.aps.org/doi/10.1103/PhysRevLett.49.590[13]S.E.NaglerW.J.L.BuyersR.L.ArmstrongB.BriatSolitons in the one-dimensional antiferromagnet cscobr3Phys. Rev. B2819833873388510.1103/PhysRevB.28.3873https://link.aps.org/doi/10.1103/PhysRevB.28.3873[14]H.YoshizawaK.HirakawaS.K.SatijaG.ShiraneDynamical correlation functions in a one-dimensional Ising-like antiferromagnetic cscocl3: a neutron scattering studyPhys. Rev. B2319812298230710.1103/PhysRevB.23.2298https://link.aps.org/doi/10.1103/PhysRevB.23.2298[15]J.P.GoffD.A.TennantS.E.NaglerExchange mixing and soliton dynamics in the quantum spin chain cscocl3Phys. Rev. B521995159921600010.1103/PhysRevB.52.15992https://link.aps.org/doi/10.1103/PhysRevB.52.15992[16]D.A.TennantR.A.CowleyS.E.NaglerA.M.TsvelikMeasurement of the spin-excitation continuum in one-dimensional kcuf3 using neutron scatteringPhys. Rev. B521995133681338010.1103/PhysRevB.52.13368https://link.aps.org/doi/10.1103/PhysRevB.52.13368[17]N.MotoyamaH.EisakiS.UchidaMagnetic susceptibility of ideal spin 1 /2 Heisenberg antiferromagnetic chain systems, sr2cuo3 and srcuo2Phys. Rev. Lett.7619963212321510.1103/PhysRevLett.76.3212https://link.aps.org/doi/10.1103/PhysRevLett.76.3212[18]M.TakigawaN.MotoyamaH.EisakiS.UchidaDynamics in the S=1/2 one-dimensional antiferromagnet sr2cuo3 via Cu63 NMRPhys. Rev. Lett.7619964612461510.1103/PhysRevLett.76.4612https://link.aps.org/doi/10.1103/PhysRevLett.76.4612[19]H.BetheZur theorie der metalleZ. Phys.713193120522610.1007/BF01341708[20]M.TakahashiThermodynamics of One-Dimensional Solvable Models1st edition1999Cambridge University Press(in English)[21]A.GogolinA.NersesyanA.TsvelikBosonization and Strongly Correlated Systems2004Cambridge University Presshttps://books.google.it/books?id=BZDfFIpCoaAC[22]S.LalS.RaoD.SenJunction of several weakly interacting quantum wires: a renormalization group studyPhys. Rev. B662002165327https://link.aps.org/doi/10.1103/PhysRevB.66.165327[23]S.ChenB.TrauzettelR.EggerLandauer-type transport theory for interacting quantum wires: application to carbon nanotube y junctionsPhys. Rev. Lett.892002226404https://link.aps.org/doi/10.1103/PhysRevLett.89.226404[24]C.ChamonM.OshikawaI.AffleckJunctions of three quantum wires and the dissipative Hofstadter modelPhys. Rev. Lett.91200320640310.1103/PhysRevLett.91.206403https://link.aps.org/doi/10.1103/PhysRevLett.91.206403[25]X.Barnabé-ThériaultA.SedekiV.MedenK.SchönhammerJunction of three quantum wires: restoring time-reversal symmetry by interactionPhys. Rev. Lett.94200513640510.1103/PhysRevLett.94.136405https://link.aps.org/doi/10.1103/PhysRevLett.94.136405[26]M.OshikawaC.ChamonI.AffleckJunctions of three quantum wiresJ. Stat. Mech. Theory Exp.022006P02008http://stacks.iop.org/1742-5468/2006/i=02/a=P02008[27]C.-Y.HouC.ChamonJunctions of three quantum wires for spin-12 electronsPhys. Rev. B77200815542210.1103/PhysRevB.77.155422https://link.aps.org/doi/10.1103/PhysRevB.77.155422[28]A.AgarwalS.DasS.RaoD.SenEnhancement of tunneling density of states at a junction of three Luttinger liquid wiresPhys. Rev. Lett.1032009026401https://link.aps.org/doi/10.1103/PhysRevLett.103.026401[29]D.GiulianoP.SodanoY-junction of superconducting Josephson chainsNucl. Phys. B81132009395419http://www.sciencedirect.com/science/article/pii/S0550321308006457[30]D.N.AristovP.WölfleTransport properties of a y junction connecting Luttinger liquid wiresPhys. Rev. B842011155426https://link.aps.org/doi/10.1103/PhysRevB.84.155426[31]A.M.TsvelikMajorana fermion realization of a two-channel Kondo effect in a junction of three quantum Ising chainsPhys. Rev. Lett.1102013147202https://link.aps.org/doi/10.1103/PhysRevLett.110.147202[32]X.G.WenF.WilczekA.ZeeChiral spin states and superconductivityPhys. Rev. B391989114131142310.1103/PhysRevB.39.11413https://link.aps.org/doi/10.1103/PhysRevB.39.11413[33]G.BaskaranNovel local symmetries and chiral-symmetry-broken phases in s=(1/2 triangular-lattice Heisenberg modelPhys. Rev. Lett.6319892524252710.1103/PhysRevLett.63.2524https://link.aps.org/doi/10.1103/PhysRevLett.63.2524[34]M.ScheucherA.HilicoE.WillJ.VolzA.RauschenbeutelQuantum optical circulator controlled by a single chirally coupled atomScience35420161577http://science.sciencemag.org/content/early/2016/12/07/science.aaj2118[35]B.J.ChapmanE.I.RosenthalJ.KerckhoffB.A.MooresL.R.ValeJ.A.B.MatesG.C.HiltonK.LalumièreA.BlaisK.W.LehnertWidely tunable on-chip microwave circulator for superconducting quantum circuitsPhys. Rev. X72017041043https://link.aps.org/doi/10.1103/PhysRevX.7.041043[36]A.C.MahoneyJ.I.CollessS.J.PaukaJ.M.HornibrookJ.D.WatsonG.C.GardnerM.J.ManfraA.C.DohertyD.J.ReillyOn-chip microwave quantum hall circulatorPhys. Rev. X72017011007https://link.aps.org/doi/10.1103/PhysRevX.7.011007[37]J.L.CardyEffect of boundary conditions on the operator content of two-dimensional conformally invariant theoriesNucl. Phys. B27521986200218http://www.sciencedirect.com/science/article/pii/0550321386905961[38]J.L.CardyBoundary conditions, fusion rules and the Verlinde formulaNucl. Phys. B32431989581596http://www.sciencedirect.com/science/article/pii/055032138990521X[39]F.BuccheriR.EggerR.G.PereiraF.B.RamosQuantum spin circulator in y junctions of Heisenberg chainsPhys. Rev. B97201822040210.1103/PhysRevB.97.220402https://link.aps.org/doi/10.1103/PhysRevB.97.220402[40]P.WadleyB.HowellsJ.ŽeleznýC.AndrewsV.HillsR.P.CampionV.NovákK.OlejníkF.MaccherozziS.S.DhesiS.Y.MartinT.WagnerJ.WunderlichF.FreimuthY.MokrousovJ.KunešJ.S.ChauhanM.J.GrzybowskiA.W.RushforthK.W.EdmondsB.L.GallagherT.JungwirthElectrical switching of an antiferromagnetScience35162732016587590http://science.sciencemag.org/content/351/6273/587[41]T.JungwirthX.MartiP.WadleyJ.WunderlichAntiferromagnetic spintronicsNat. Nanotechnol.11201623110.1038/nnano.2016.18[42]V.BaltzA.ManchonM.TsoiT.MoriyamaT.OnoY.TserkovnyakAntiferromagnetic spintronicsRev. Mod. Phys.90201801500510.1103/RevModPhys.90.015005[43]M.BollT.A.HilkerG.SalomonA.OmranJ.NespoloL.PolletI.BlochC.GrossSpin- and density-resolved microscopy of antiferromagnetic correlations in Fermi-Hubbard chainsScience3536305201612571260http://science.sciencemag.org/content/353/6305/1257[44]H.-N.DaiB.YangA.ReingruberH.SunX.-F.XuY.-A.ChenZ.-S.YuanJ.-W.PanFour-body ring-exchange interactions and anyonic statistics within a minimal toric-code HamiltonianNat. Phys.132017119510.1038/nphys4243[45]A.CeliP.MassignanJ.RuseckasN.GoldmanI.B.SpielmanG.JuzeliūnasM.LewensteinSynthetic gauge fields in synthetic dimensionsPhys. Rev. Lett.112201404300110.1103/PhysRevLett.112.043001https://link.aps.org/doi/10.1103/PhysRevLett.112.043001[46]L.F.LiviG.CappelliniM.DiemL.FranchiC.ClivatiM.FrittelliF.LeviD.CalonicoJ.CataniM.InguscioL.FallaniSynthetic dimensions and spin-orbit coupling with an optical clock transitionPhys. Rev. Lett.117201622040110.1103/PhysRevLett.117.220401https://link.aps.org/doi/10.1103/PhysRevLett.117.220401[47]C.K.MajumdarD.K.GhoshOn next–nearest–neighbor interaction in linear chain. IJ. Math. Phys.10819691388139810.1063/1.1664978[48]F.D.M.HaldaneSpontaneous dimerization in the s=12 Heisenberg antiferromagnetic chain with competing interactionsPhys. Rev. B2519824925492810.1103/PhysRevB.25.4925https://link.aps.org/doi/10.1103/PhysRevB.25.4925[49]K.OkamotoK.NomuraFluid-dimer critical point in s = 12 antiferromagnetic Heisenberg chain with next nearest neighbor interactionsPhys. Lett. A1696199243343710.1016/0375-9601(92)90823-5http://www.sciencedirect.com/science/article/pii/0375960192908235[50]S.EggertI.AffleckMagnetic impurities in half-integer-spin Heisenberg antiferromagnetic chainsPhys. Rev. B4619921086610883https://link.aps.org/doi/10.1103/PhysRevB.46.10866[51]N.LaflorencieE.S.SørensenI.AffleckThe Kondo effect in spin chainsJ. Stat. Mech. Theory Exp.022008P02007http://stacks.iop.org/1742-5468/2008/i=02/a=P02007[52]V.KalmeyerR.B.LaughlinEquivalence of the resonating-valence-bond and fractional quantum hall statesPhys. Rev. Lett.59198720952098https://link.aps.org/doi/10.1103/PhysRevLett.59.2095[53]V.KalmeyerR.B.LaughlinTheory of the spin liquid state of the Heisenberg antiferromagnetPhys. Rev. B391989118791189910.1103/PhysRevB.39.11879https://link.aps.org/doi/10.1103/PhysRevB.39.11879[54]D.SenR.ChitraLarge-u limit of a Hubbard model in a magnetic field: chiral spin interactions and paramagnetismPhys. Rev. B5119951922192510.1103/PhysRevB.51.1922https://link.aps.org/doi/10.1103/PhysRevB.51.1922[55]S.KitamuraT.OkaH.AokiProbing and controlling spin chirality in Mott insulators by circularly polarized laserPhys. Rev. B96201701440610.1103/PhysRevB.96.014406https://link.aps.org/doi/10.1103/PhysRevB.96.014406[56]M.ClaassenH.-C.JiangB.MoritzT.P.DevereauxDynamical time-reversal symmetry breaking and photo-induced chiral spin liquids in frustrated Mott insulatorsNat. Commun.8120171192https://www.nature.com/articles/s41467-017-00876-y[57]P.Di FrancescoP.MathieuD.SénéchalConformal Field Theory1997Springer[58]B.BellazziniM.MintchevP.SorbaBosonization and scale invariance on quantum wiresJ. Phys. A, Math. Theor.401020072485http://stacks.iop.org/1751-8121/40/i=10/a=017[59]B.BellazziniM.MintchevP.SorbaQuantum wire junctions breaking time-reversal invariancePhys. Rev. B802009245441https://link.aps.org/doi/10.1103/PhysRevB.80.245441[60]S.MardanyaA.AgarwalEnhancement of tunneling density of states at a Y junction of spin-12 Tomonaga–Luttinger liquid wiresPhys. Rev. B92201504543210.1103/PhysRevB.92.045432https://link.aps.org/doi/10.1103/PhysRevB.92.045432[61]C.L.KaneM.P.A.FisherTransmission through barriers and resonant tunneling in an interacting one-dimensional electron gasPhys. Rev. B4619921523315262https://link.aps.org/doi/10.1103/PhysRevB.46.15233[62]V.BarzykinI.AffleckScreening cloud in the k-channel Kondo model: perturbative and large-k resultsPhys. Rev. B57199843244810.1103/PhysRevB.57.432arXiv:cond-mat/9708039[63]Ph.NozièresA.BlandinKondo effect in real metalsJ. Phys. France413198019321110.1051/jphys:01980004103019300[64]A.HewsonThe Kondo Problem to Heavy FermionsCambridge Studies in Magnetism1997Cambridge University Presshttps://books.google.it/books?id=fPzgHneNFDAC[65]B.AlkurtassA.BayatI.AffleckS.BoseH.JohannessonP.SodanoE.S.SørensenK.Le HurEntanglement structure of the two-channel Kondo modelPhys. Rev. B932016081106https://link.aps.org/doi/10.1103/PhysRevB.93.081106[66]A.M.TsvelikW.-G.YinPossible realization of a multichannel Kondo model in a system of magnetic chainsPhys. Rev. B8832013144401https://link.aps.org/doi/10.1103/PhysRevB.88.144401[67]J.CardyBoundary Conformal Field TheoryEncycl. Math. Phys.2006ElsevierarXiv:hep-th/0411189[68]I.AffleckA.W.W.LudwigThe Kondo effect, conformal field theory and fusion rulesNucl. Phys. B35231991849862http://www.sciencedirect.com/science/article/pii/055032139190109B[69]I.AffleckA.W.W.LudwigCritical theory of overscreened Kondo fixed pointsNucl. Phys. B36021991641696http://www.sciencedirect.com/science/article/pii/055032139190419X[70]I.AffleckA.W.W.LudwigExact conformal-field-theory results on the multichannel Kondo effect: single-fermion green's function, self-energy, and resistivityPhys. Rev. B48199372977321https://link.aps.org/doi/10.1103/PhysRevB.48.7297[71]I.AffleckLecture notes: conformal field theory approach to quantum impurity problemsarXiv:cond-mat/9311054[72]A.B.ZamolodchikovV.A.FateevOperator algebra and correlation functions in the two-dimensional Wess-Zumino SU(2) x SU(2) chiral modelSov. J. Nucl. Phys.431986657664Yad. Fiz.4319861031[73]A.ZamolodchikovV.FateevNonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in zn-symmetric statistical systemsJ. Exp. Theor. Phys.621985215http://www.jetp.ac.ru/cgi-bin/e/index/e/62/2/p215?a=list[74]V.FateevA.ZamolodchikovConformal quantum field theory models in two dimensions having z3 symmetryNucl. Phys. B280Supplement C1987644660http://www.sciencedirect.com/science/article/pii/0550321387901660[75]S.LukyanovV.FateevConformally-invariant models of a two-dimensional quantum field theory with zn symmetryZh. Ehksp. Teor. Fiz.943198823https://inis.iaea.org/search/search.aspx?orig_q=RN:20009480[76]K.TotsukaM.SuzukiPhase diagram of three-leg spin ladderJ. Phys. A, Math. Gen.291319963559http://stacks.iop.org/0305-4470/29/i=13/a=024[77]I.AffleckM.OshikawaH.SaleurQuantum brownian motion on a triangular lattice and c=2 boundary conformal field theoryNucl. Phys. B59432001535606http://www.sciencedirect.com/science/article/pii/S0550321300004995[78]V.FateevS.LykyanovThe models of two-dimensional conformal quantum field theory with zn symmetryInt. J. Mod. Phys. A0302198850752010.1142/S0217751X88000205arXiv:http://www.worldscientific.com/doi/pdf/10.1142/S0217751X88000205http://www.worldscientific.com/doi/abs/10.1142/S0217751X88000205[79]E.FrenkelV.KacM.WakimotoCharacters and fusion rules for w-algebras via quantized Drinfel'd-Sokolov reductionCommun. Math. Phys.14721992295328https://projecteuclid.org:443/euclid.cmp/1104250638[80]S.L.LukyanovV.A.FateevAdditional symmetries and exactly-soluble models in two-dimensional conformal field theorySov. Sci. Rev.1519881117[81]A.W.W.LudwigI.AffleckExact conformal-field-theory results on the multi-channel Kondo effect: asymptotic three-dimensional space- and time-dependent multi-point and many-particle green's functionsNucl. Phys. B4283199454561110.1016/0550-3213(94)90365-4http://www.sciencedirect.com/science/article/pii/0550321394903654[82]I.AffleckA.W.W.LudwigUniversal noninteger “ground-state degeneracy” in critical quantum systemsPhys. Rev. Lett.67199116116410.1103/PhysRevLett.67.161https://link.aps.org/doi/10.1103/PhysRevLett.67.161[83]A.RahmaniC.-Y.HouA.FeiguinC.ChamonI.AffleckHow to find conductance tensors of quantum multiwire junctions through static calculations: application to an interacting y junctionPhys. Rev. Lett.105201022680310.1103/PhysRevLett.105.226803https://link.aps.org/doi/10.1103/PhysRevLett.105.226803[84]A.RahmaniC.-Y.HouA.FeiguinM.OshikawaC.ChamonI.AffleckGeneral method for calculating the universal conductance of strongly correlated junctions of multiple quantum wiresPhys. Rev. B852012045120https://link.aps.org/doi/10.1103/PhysRevB.85.045120[85]K.IngersentA.W.W.LudwigI.AffleckKondo screening in a magnetically frustrated nanostructure: exact results on a stable non-Fermi-liquid phasePhys. Rev. Lett.95200525720410.1103/PhysRevLett.95.257204https://link.aps.org/doi/10.1103/PhysRevLett.95.257204[86]I.AffleckA current algebra approach to the Kondo effectNucl. Phys. B336199051710.1016/0550-3213(90)90440-O[87]I.AffleckConformal field theory approach to the Kondo effecteprintarXiv:cond-mat/9512099cond-mat/9512099[88]V.FateevS.LukyanovAdditional symmetries and exactly soluble models in two-dimensional conformal field theorySov. Sci. Rev. A1519901[89]D.FriedanA.KonechnyBoundary entropy of one-dimensional quantum systems at low temperaturePhys. Rev. Lett.932004030402https://link.aps.org/doi/10.1103/PhysRevLett.93.030402[90]L.N.BulaevskiiC.D.BatistaM.V.MostovoyD.I.KhomskiiElectronic orbital currents and polarization in Mott insulatorsPhys. Rev. B78200802440210.1103/PhysRevB.78.024402https://link.aps.org/doi/10.1103/PhysRevB.78.024402[91]R.ToskovicR.van den BergA.SpinelliI.S.EliensB.van den ToornB.BryantJ.S.CauxA.F.OtteAtomic spin-chain realization of a model for quantum criticalityNat. Phys.12201665666010.1038/nphys3722[92]K.YangY.BaeW.PaulF.D.NattererP.WillkeJ.L.LadoA.FerrónT.ChoiJ.Fernández-RossierA.J.HeinrichC.P.LutzEngineering the eigenstates of coupled spin-1/2 atoms on a surfacePhys. Rev. Lett.119201722720610.1103/PhysRevLett.119.227206https://link.aps.org/doi/10.1103/PhysRevLett.119.227206[93]C.DuT.van der SarT.X.ZhouP.UpadhyayaF.CasolaH.ZhangM.C.OnbasliC.A.RossR.L.WalsworthY.TserkovnyakA.YacobyControl and local measurement of the spin chemical potential in a magnetic insulatorScience3576347201719519810.1126/science.aak9611http://science.sciencemag.org/content/357/6347/195.full.pdfhttp://science.sciencemag.org/content/357/6347/195[94]L.BalentsR.EggerSpin-dependent transport in a Luttinger liquidPhys. Rev. B64200103531010.1103/PhysRevB.64.035310https://link.aps.org/doi/10.1103/PhysRevB.64.035310[95]D.L.MaslovM.StoneLandauer conductance of Luttinger liquids with leadsPhys. Rev. B521995R5539R554210.1103/PhysRevB.52.R5539https://link.aps.org/doi/10.1103/PhysRevB.52.R5539[96]J.L.CardyD.C.LewellenBulk and boundary operators in conformal field theoryPhys. Lett. B2593199127427810.1016/0370-2693(91)90828-Ehttp://www.sciencedirect.com/science/article/pii/037026939190828E[97]E.WongI.AffleckTunneling in quantum wires: a boundary conformal field theory approachNucl. Phys. B4173199440343810.1016/0550-3213(94)90479-0http://www.sciencedirect.com/science/article/pii/0550321394904790[98]S.R.WhiteDensity matrix formulation for quantum renormalization groupsPhys. Rev. Lett.6919922863286610.1103/PhysRevLett.69.2863https://link.aps.org/doi/10.1103/PhysRevLett.69.2863[99]H.GuoS.R.WhiteDensity matrix renormalization group algorithms for y-junctionsPhys. Rev. B742006060401https://link.aps.org/doi/10.1103/PhysRevB.74.060401[100]M.KumarA.ParvejS.ThomasS.RamaseshaZ.G.SoosEfficient density matrix renormalization group algorithm to study y junctions with integer and half-integer spinPhys. Rev. B93201607510710.1103/PhysRevB.93.075107https://link.aps.org/doi/10.1103/PhysRevB.93.075107[101]S.R.WhiteD.J.ScalapinoDensity matrix renormalization group study of the striped phase in the 2D t−J modelPhys. Rev. Lett.8019981272127510.1103/PhysRevLett.80.1272https://link.aps.org/doi/10.1103/PhysRevLett.80.1272[102]F.B.RamosJ.C.Xaviern-leg spin-s Heisenberg ladders: a density-matrix renormalization group studyPhys. Rev. B89201409442410.1103/PhysRevB.89.094424https://link.aps.org/doi/10.1103/PhysRevB.89.094424[103]S.R.WhiteA.L.ChernyshevNeél order in square and triangular lattice Heisenberg modelsPhys. Rev. Lett.99200712700410.1103/PhysRevLett.99.127004https://link.aps.org/doi/10.1103/PhysRevLett.99.127004[104]R.EggerA.KomnikScaling and criticality of the Kondo effect in a Luttinger liquidPhys. Rev. B571998106201062910.1103/PhysRevB.57.10620https://link.aps.org/doi/10.1103/PhysRevB.57.10620[105]E.VerlindeFusion rules and modular transformations in 2D conformal field theoryNucl. Phys. B300198836037610.1016/0550-3213(88)90603-7[106]P.GoddardA.KentD.OliveUnitary representations of the Virasoro and super-Virasoro algebrasCommun. Math. Phys.10311986105119http://projecteuclid.org/euclid.cmp/1104114626