]>NUPHB13569S05503213(15)00423X10.1016/j.nuclphysb.2015.12.004The AuthorsQuantum Field Theory and Statistical SystemsFig. 1(Color online.) The intrinsic sign structure may be measured by a sign average sgnR [defined in Eqs. (17) and (18)] for the ground state of the t↑–t↓–J model, which continuously interpolates the t–J and σ⋅t–J models as a function of the ratio t↓/t↑ (see text). The results here are obtained by the exact diagonalization with a chain length L=12. Here, R=t–J and R=σ⋅t–J represent two sets of basis [see Eq. (18)] used to measure sgnR of the ground state of t↑–t↓–J model, respectively. Clearly a critical point is indicated at t↓/t↑=0 where the sign structure shows a qualitative change in the ground state. A firstordertype of transition at this critical point is also to be confirmed by DMRG later.Fig. 2(Color online.) (a) The same onehole groundstate energy EG1hole vs, the chain length L for the t–J and σ⋅t–J models due to the absence of the interference of the phase string under an open boundary condition; (b) The charge and spin density distributions are also the same for both models. Note that the spincharge separation is clearly indicated in such a finitesize system.Fig. 3(Color online.) The ground state energy E0 of the t↑–t↓–J model as a function of t↓/t↑ [(a)], which exhibits a singularity at t↓/t↑=0 as indicated by the first order derivative of ground state energy [(b)]. It implies the breakdown of adiabatic continuity between the ground states of the t–J and σ⋅t–J models at t↓/t↑=1 and t↓/t↑=−1, respectively, and is consistent with the sign structure analysis shown in Fig. 1.Fig. 4(Color online.) The distinct characteristic momenta k0's as determined by the peaks of the singleparticle spectrum weight Zk. (a) The oneholedoped t–J chain; (b) The oneholedoped σ⋅t–J chain. Here the single hole state is realized by removing an upspin or downspin, with Stotz=−1/2 and Stotz=1/2, respectively; (c) Zk0 vanishes in the thermodynamic limit in a power law fashion L−α, with α≃0.49 for the t–J and α≃0.23 for the σ⋅t–J models, respectively.Fig. 5(Color online.) The momentum distributions, nk↑ and nk↓, for the t–J (a) and the σ⋅t–J (b) models. Stotz=±1/2 denotes the total spin of the onehole state. (c) The peak of the hole momentum distribution 1−n(k0)≡1−nk0↑−nk0↓ scales with L−α, with α≃0.47 for the t–J model while α≃0.22 (α≃0.24) for the peaks located at k0=0 (k0=π) in the σ⋅t–J model.Fig. 6(Color online.) (a) The ground state energies are no longer the same for the t–J and σ⋅t–J models under a periodic boundary condition. Their energy difference δEG1hole [cf. Eq. (24)] is shown to vanish ∼L−1 at large L; The ground state energy change under inserting a flux into the 1D ring [cf. Eq. (25)], ΔEG1hole, for the t–J model (b) and the σ⋅t–J model (c). ΔEG1hole oscillates and decays in a powerlaw fashion (∼L−3) for the t–J chain while it is nonoscillating and proportional to 1/L2 for the σ⋅t–J chain (Ref. [26]).Fig. 7(Color online.) The onehole groundstate energy for the variational wave function Eq. (28) (at t/J=3) calculated by VMC (red dots) is in excellent agreement with the DMRG (black crosses).Fig. 8(Color online.) Momentum distribution nkα calculated based on the variational wave function Eq. (28) for t–J model (a) and Eq. (29) for the σ⋅t–J model (b), by VMC with length L=101 at t/J=3. The sharp dip positions are the same as predicted in Table 1 as well as the DMRG results in Fig. 5; (c) The peak of the hole momentum distribution 1−n(k0)=1−nk0↑−nk0↓ vanishes in a power law fashion, L−α, in the thermodynamic limit.Table 1Characteristic momenta determined by Zk and nkα, which are dependent on the detailed sign structure in the ground states of σ⋅t–J and t–J models, respectively. Here the single hole is doped into the halffilling ground state by removing a spin of σ.ModelsσZknk↑nk↓
σ⋅t–J↑k=πk=πk=0
↓k=0

t–J↑k=±π/2
↓
Exact sign structure of the t–J chain and the single hole ground stateZhengZhua⁎zhuz10@mails.tsinghua.edu.cnQingRuiWangaD.N.ShengbZhengYuWengacaInstitute for Advanced Study, Tsinghua University, Beijing 100084, ChinaInstitute for Advanced StudyTsinghua UniversityBeijing100084ChinabDepartment of Physics and Astronomy, California State University, Northridge, CA 91330, USADepartment of Physics and AstronomyCalifornia State UniversityNorthridgeCA91330USAcCollaborative Innovation Center of Quantum Matter, Tsinghua University, Beijing 100084, ChinaCollaborative Innovation Center of Quantum MatterTsinghua UniversityBeijing100084China⁎Corresponding author.Editor: Hubert SaleurAbstractInjecting a single hole into a onedimensional Heisenberg spin chain is probably the simplest case of doping a Mott insulator. The motion of such a single hole will generally induce a manybody phase shift, which can be identified by an exact sign structure of the model known as the phase string. We show that the sign structure is nontrivial even in this simplest problem, which is responsible for the essential properties of Mott physics. We find that the characteristic momentum structure, the Luttinger liquid behavior, and the quantum phase interference of the hole under a periodic boundary condition can all be attributed to it. We use the density matrix renormalization group (DMRG) numerical simulation to make a comparative study of the t–J chain and a model in which the sign structure is switched off. We further show that the key DMRG results can be reproduced by a variational wave function with incorporating the correct sign structure. Physical implications of the sign structure for doped Mott insulators in general are also discussed.1IntroductionHow a doped hole propagates in a “vacuum” that is full of quantum spins is a central question in a doped Mott system [1,2]. On general grounds, one expects a “cloud of spin excitations” to be generated to accompany the motion of the hole. In a more conventional/weakly correlated system, a similar “cloud” forming around a testing particle is usually finite in size, remains featureless and rigid at low energy, which dresses only the particle's effective mass. The challenge arises, however, if the spin cloud becomes neither featureless nor rigid, or in other words, the motion of the hole becomes stronglycorrelated in nature. Generally speaking, a new mathematical description will be needed here. The issue of the single hole problem has attracted intense attention since the discovery of the highTc cuprate, which is considered to be a doped Mott insulator [3].In literature, the onedimensional (1D) doped Mott systems have been well studied. The exact Bethe ansatz or the Lieb–Wu solution [4] exists for the Hubbard model at an arbitrary doping concentration and ratio of the onsite Coulomb repulsion U and the nearestneighbor hopping integral t. In particular, the 1D t–J model can be solved exactly at t/J→∞ [5–7] and t/J=1/2 [8–11] (J is the superexchange coupling), both of which behave like a Luttinger liquid [12,13] at finite doping. Numerically, the phase diagram has been also given via exact diagonalization (ED) [14,15] and the density matrix renormalization group (DMRG) [16] methods. As a matter of fact, based on the 1D exact solution, Anderson proposed [17] the idea of the unrenormalizable manybody phase shift, which is argued [1,18] to be generally responsible for the Luttinger liquid behavior in the doped Mott insulator. The quantitative characterization of such phase shift was later analytically identified [19–21] in the 1D t–J model. The ground state properties of the doped Hubbard model at U≫t or the t–J model at J≪t can be also approximated by the socalled squeezed spin chain description [6,7,19,20,22–25].Nevertheless, a simple microscopic understanding is still much needed, even for the simplest oneholedoped 1D case. By answering the question raised at the beginning of this paper, one may gain a deeper insight into the strong correlation nature of the Mott physics, which goes beyond the specific 1D geometry. Utilizing exact analysis and numerical methods, one hopes to clearly illustrate the singlehole's motion in an antiferromagnetic spin background qualitatively and quantitatively, which are relatively easier to handle in 1D. It may then provide important insights for the problem in two dimensions (2D), which is more relevant to cuprate superconductors and other strongly correlated materials.In this paper, we investigate the ground state of the oneholedoped 1D Heisenberg chain using the exact analysis, DMRG, and wave function approach based on a variational Monte Carlo (VMC) method. The main results are obtained as follows. First of all, we explicitly show that a nontrivial sign structure or phase string emerges once a hole is doped into the Heisenberg spin chain, which otherwise is statisticalsign free. Since such sign structure is present for any dimension, the 1D limit provides the simplest example to show its novel consequences. The detailed analyses will be given in Sec. 2. Secondly, in contrast to a bare hole state created by annihilating an electron in the halffilled ground state, the true hole ground state differs by a fundamentally changed momentum distribution, as well as the vanishing singleparticle spectral weight, obeying a powerlaw scaling with the length of the chain. The phase string induced by one hole doping, as the singular manybody phase shift contributed by the spins in the vacuum, is responsible for the above momentum readjustment and the Luttinger liquid behavior. These will be confirmed by the DMRG simulations in Sec. 3. Furthermore, we show that a variational wave function constructed by incorporating the correct sign structure or the phase string can reproduce the DMRG results by using the VMC calculation in Sec. 4. Finally, in the summary section (Sec. 5), we will also discuss how the phase string sign structure plays a critical role in a general doped Mott insulator beyond the 1D case examined in the present work.2The model and exact analysis2.1The modelIn this paper, the main focus will be the ground state properties of a single hole injected into a 1D Mott insulator. It is described by the t–J Hamiltonian, Ht–J=Ht+HJ, generally defined in the Hilbert space constrained by the nodoubleoccupancy condition as follows(1)Ht=−t∑〈ij〉,σ(ciσ†cjσ+h.c.),HJ=J∑〈ij〉(Si⋅Sj−14ninj). Here, the operator ciσ† creates an electron at site i with spin σ, and Si is the spin operator, ni the number operator, respectively, with the summations running over all the nearestneighbors 〈ij〉 along the chain.For a comparative study, we will also consider the ground state of the socalled σ⋅t–J model [26], Hσ⋅t–J=Hσ⋅t+HJ, in which the superexchange term HJ remains the same as in the t–J model Eq. (1), but the hopping term is modified by(2)Hσ⋅t=−t∑〈ij〉,σσ(ciσ†cjσ+h.c.), in which an extra spindependent sign σ=± is inserted. The distinction between the ground states of the t–J and the σ⋅t–J models will reveal the critical role of the phasestring sign structure hidden in the t–J Hamiltonian, which however is precisely eliminated by the sign σ in Hσ⋅t in Eq. (2), as to be seen in the following [26].Finally, these two models may be connected by tuning the spindependent hopping integrals as follows(3)Ht↑–t↓=−t↑∑〈ij〉ci↑†cj↑−t↓∑〈ij〉ci↓†cj↓+h.c. Here, t↑ and t↓ are the hopping amplitudes for the upspin and downspin electrons, respectively, and again the constraint ni≤1 in the Hilbert space is always enforced. When t↑=t↓=t, the above model becomes the normal t–J model in Eq. (1). Similarly, when t↑=t and t↓=−t, this model becomes the σ⋅t–J model in Eq. (2). Then by fixing t↑=t and tuning the hopping integral t↓, one may continuously connect these two models to turn on or off the phase string sign structure. Note that the spin rotational symmetry is slightly broken here in the x–y plane by the hopping term involving a spin1/2 in the oneholedoping case, while the background spins governed by HJ still obey the spin rotational symmetry.2.2The exact sign structureFor oneholedoped t–J model Eq. (1) on a bipartite lattice, it has been previously demonstrated that the hopping of the hole will pick up a sequence of signs, i.e., (+1)×(−1)×(−1)×⋯, known as a phase string [24,27,28]. Here, the sign ± keeps track of the microscopic process of an ↑ or ↓spin exchanging with the hole at each step of hopping. The exact sign structure of the t–J model with one hole is precisely given by(4)τc=(−1)Nh↓[c] , with Nh↓[c] denoting the total number of exchanges between the hole and down spins along a path c which can be either open or closed.For example, τc appears in the singleparticle propagator of the hole from i to j as follows [24,27](5)G1h(j,i;E)∝∑cτcW[c] , where {c} includes all the spin and hole paths with the path weight W[c]≥0 at energy E<0. Among {c}, the path of the hole, which connects i and j, is an open one [24,27].On the other hand, τc also appears in the partition function [28](6)Zt–J=∑cτcZ[c] , where Z[c]≥0 for any closed path c, which is generally temperature as well as t and Jdependent.Furthermore, such phasestring sign structure can be artificially switched off by introducing the σ⋅t–J model Eq. (2) with inserting a spindependent sign σ in the hopping term. It is easy to show [26] that the phase string disappears in the σ⋅t–J model, with τc replaced by +1 in, say, Eq. (6)(7)Zσ⋅t–J=∑cZ[c] , where the nonnegative weight Z[c] for each path c remains unchanged. Similarly τc is also precisely eliminated in Eq. (5). Therefore, the t–J and the σ⋅t–J models are solely differentiated by the presence and absence of the phase string sign structure τc. In other words, the distinction between the two ground states will uniquely reveal the role of the sign structure.In order to examine the novel role of τc, let us consider in the most simplified case of the 1D chain, i.e., with an open boundary condition. Here only a selfretracing path will contribute to the partition function in Zt–J (in order for the whole spin–hole configurations to return to the same ones in carrying out the trace in Zt–J) such that Nh↓[c] in τc is generally an even number for any closed path in Eq. (6). Namely Zt–J=Zσ⋅t–J since τc=1. Correspondingly, the eigen energies are also the same for the two models.However, for such a 1D chain under open boundary condition, τc remains nontrivial for an open path, say, in the singleparticle propagator Eq. (5). In fact, without the interference effect due to τc=1 for a closed path, one may introduce a unitary transformation to “gauge away” the phasestring signs in the t–J model [24],(8)eiΘˆ≡e−i∑inihΩˆi , where Ωˆi≡∑lθi(l)nl↓ with the statistical angle θi(l) satisfying(9)θi(l)=Imln(i−l)={±π,if i<l,0,if i>l, such that(10)e−iΩˆi=e∓iπ∑l>inl↓≡(−1)∑l>inl↓ . Here, nih and nl↓ are the hole number and downspin number operator, respectively.Then, it is straightforward to show that the signfull t–J and signfree σ⋅t–J models can be related by such a unitary transformation(11)Ht–J=eiΘˆHσ⋅t–Je−iΘˆ. Consequently, any eigenstate Φ〉σ⋅t–J of the σ⋅t–J model, which has no “sign problem”, can be used to construct the corresponding eigenstate Ψ〉t–J of the t–J model of the same energy by(12)Ψ〉t–J=eiΘˆΦ〉σ⋅t–J. Although we shall focus on the single hole case below, the above construction is rigorous for an arbitrary hole concentration of the 1D chain under an open boundary condition [24].2.3More detailed sign structure in wave functionsOne has seen above that a doped hole in a Mott insulator will generally induce a phase (sign) shift, which is manybody (dependent on the background spins) and irreparable or unrenormalizable (as Z[c], W[c]≥0). The sign structure τc in Eq. (4) is obviously different from a conventional Fermi sign structure. It reflects a peculiar quantum “memory” effect of the hole moving on the quantum spin background. Namely it reflects a longrange entanglement between the charge and spin degrees of freedom.In order to further identify the sign structure in the wave function, we start from the undoped case. At halffilling, both the t–J model and σ⋅t–J model reduce to the same Heisenberg Hamiltonian HJ. According to Marshall [29], the groundstate wave function of the Heisenberg model for a bipartite lattice is real in the Ising basis and satisfies a Marshall sign rule. This sign rule requires that the flip of a pair of antiparallel spins at nearestneighbor sites will induce a sign change in the wave function, i.e., ↑↓ → (−1) ↓↑. If one introduces the socalled Marshall basis with the builtin Marshall sign by(13){s}〉=(−1)NA↓c1s1†c2s1†⋯cLsL†0〉, where L is the total site number or chain length and NA↓ denotes the total number of down spins belonging to the sublattice A, it is straightforward to verify that the offdiagonal matrix elements of HJ are nonpositive. Then the halffilling ground state of the Heisenberg model, denoted by ϕ0〉, has nonnegative coefficients in the Marshall basis according to the Perron–Frobenius theorem, namely,(14)ϕ0〉=∑{s}c({s}){s}〉, with c({s})≥0. Thus ϕ0〉 is indeed signfree in the Marshall basis {{s}〉}.The single hole doping can be then realized by removing a spinσ (σ=±1) from the half filling singlet state, leading to a total spin Stotz=−σ/2 singlehole state. Starting from the Marshall basis {{s}〉}, one may further construct a new signfree basis for the singlehole ground state of the σ⋅t–J mode (see Appendix A): {(−σ)iciσ{s}〉}, where the sign factor (−σ)i comes from the Marshall sign associated with site i. Namely the ground state of the σ⋅t–J Hamiltonian can be written as(15)Φ〉σ⋅t–J=∑i,{s}a(i,{s})(−σ)iciσ{s}〉 with a(i,{s})≥0. This is easy to understand as there is no nontrivial sign structure in the σ⋅t–J model, similar to the Heisenberg model at halffilling.Then, according to Eq. (12), the ground state of the t–J model can be precisely constructed in terms of Eq. (15) as follows (cf. Appendix A):(16)Ψ〉t–J=∑i,{s}a(i,{s})e−iΩˆi(−σ)iciσ{s}〉 , where the manybody phase shift (phase string) operator e−iΩˆi is defined by Eq. (10).2.4Breakdown of adiabatic continuityTwo ground states, Ψ〉t–J and Φ〉σ⋅t–J, differ by the phase string sign structure eiΘˆ according to Eq. (12). In the following, we explicitly show the breakdown of the adiabatic continuity between them by examining the t↑–t↓–J model introduced in Eq. (3), which interpolates the t–J and σ⋅t–J models under an open boundary condition.We define a sign average, motivated by Ref. [30], to directly “measure” the sign structure of a generic wave function ψ〉 in a given basis {nR〉}:(17)sgnR=∑nRsgn(〈nRψ〉)〈nRψ〉2∑nR〈nRψ〉2. The crucial feature in the above definition is that the signs are averaged according to the probability 〈nRψ〉2. If the coefficients 〈nRψ〉 of the wave function in this particular basis have the same sign, then we have sgnR=1. Otherwise the coefficients with different signs cancel each other at least partially, resulting in a relatively small sign average.For the models we focus on in this paper, the basis states nR〉 are chosen to be(18)nR〉={(−σ)iciσ{s}〉,R=σ⋅t–J,e−iΩˆi(−σ)iciσ{s}〉,R=t–J, which are the signfree bases for the σ⋅t–J model and the t–J model, respectively, as discussed above.Based on the sign structure analysis there, one can show that the sign average sgnσ⋅t–J defined above for the t↑–t↓–J model (t↑=t is fixed to be positive) is(19)sgnσ⋅t–J={1,t↓/t↑<0,〈ψ(−t↓/t↑)ψ(t↓/t↑)〉,t↓/t↑>0, where ψ(t↓/t↑)〉 is the ground state of the t↑–t↓–J model with parameter t↓/t↑. A similar but reversed result is expected for the sign average sgnt–J. The sign average sgnσ⋅t–J equals to one exactly for t↓/t↑<0, because the basis nσ⋅t–J〉 is the signfree basis in this parameter region. On the other hand, for t↓/t↑>0, the sign average measures the overlap between the two ground states for parameters ±t↓/t↑. Therefore, a diminished sign average for t↓/t↑=0+ (see numerical results below) indicates the orthogonality of the ground states with parameters t↓/t↑=0±, i.e., the discontinuity in connecting the σ⋅t–J model and the t–J model.We perform an exact diagonalization of the t↑–t↓–J model and calculate the sign average defined in Eqs. (17) and (18). The results on a chain with length L=12 are shown in Fig. 1. More abrupt changes of the sign averages are expected for larger system sizes at the transition point, which is clearly at t↓/t↑=0.2.5Distinct momentum structuresFor the 1D chain under an open boundary condition, the singlehole ground state energy does not depend on the sign structure, due to the absence of the nontrivial phase string τc for closed paths as to be verified by a DMRG calculation later. Nevertheless, as shown above, the two ground states cannot be smoothly connected, which means that the phase string, as captured by e−iΩˆi in Eq. (16), still leads to completely different physical properties of the ground states [24,31,32], even in the absence of the interference effect.In the following, we specifically examine the characteristic momenta in terms of the quasiparticle spectral weight Zk and the momentum distribution nkα for the two models. Other properties related to the sign structure, such as the total spin of the ground state and the ordering of energy levels for the t–J chain, can be found in Ref. [21].Firstly we start with the ground state wave function Eq. (15) of the σ⋅t–J chain. The quasiparticle spectral weight is defined as Zk=ak2, where ak is the overlap between the ground state Φ〉σ⋅t–J and the Blochlike state k〉=1L∑ie−ikiciσϕ0〉 constructed from the ground state of the Heisenberg model ϕ0〉 by removing an electron:(20)ak≡〈kΦ〉σ⋅t–J=1L∑i,{s}a(i,{s})eiki(−σ)i〈ϕ0ciσ†ciσ{s}〉. Because of the fact that a(i,{s})≥0 and 〈ϕ0ciσ†ciσ{s}〉≥0, it is straightforward to see that the quasiparticle weight Zk must be peaked at k=0 (k=π) if σ=↓ (σ=↑). Note that here σ denotes the spin removed from the halffilling in the onehole state, i.e., the ground state Φ〉σ⋅t–J has total spin Stotz=−σ/2.Similarly, the momentum distribution nkα (α=σ,σ¯) of the σ⋅t–J ground state wave function is given by(21)nkα=〈Φckα†ckαΦ〉σ⋅t–J=12−1Lδασ+1L∑i≠je−ik(j−i)〈Φciα†cjαΦ〉σ⋅t–J, where(22)〈Φciα†cjαΦ〉σ⋅t–J≡∑{s},{s′}a(j,{s′})a(i,{s})(−σ)i−j(−1)×{〈{s′}niσnjσ{s}〉,α=σ,〈{s′}Si∓Sj±{s}〉,α=σ¯. For α=σ (again σ denotes the spin removed in the single hole case), the momentum distribution nkσ will show a sharp dip (relative to the halffilling) at k=0 (k=π) if σ=↓ (σ=↑) because of a(i,{s})≥0. On the other hand, if α=σ¯, there is an another Marshall sign contribution (−1)i+j from 〈{s′}Si∓Sj±{s}〉. Hence the momentum distribution nkσ¯ should exhibit a sharp dip at k=π (k=0) at σ=↓ (σ=↑). These characteristic momenta manifested in Zk and nk for the σ⋅t–J chain are summarized in the Table 1.Then we similarly examine the case for the t–J model. The quasiparticle weight Zk=ak2 can be calculated similarly to Eq. (20), with 〈ϕ0ciσ†ciσ{s}〉 replaced by 〈ϕ0e−iΩˆiciσ†ciσ{s}〉. Since the ground state ϕ0〉 of the Heisenberg spin chain has an antiferromagnetic correlation, the phase string effect of e−iΩˆi will cause an “unrenormalizable phase shift” [17–20], whose leading order contribution is given by e±i(π/2)i, transforming the characteristic momentum at k=0 or π of the σ⋅t–J model to k=±π/2 in the t–J model. The fluctuation effect of e−iΩˆi around e±i(π/2)i will further contribute to the power of the vanishing quasiparticle weight at large L limit as to be discussed in the DMRG and VMC calculations later.For the momentum distribution, one finds an additional phase string contribution e−iΩˆieiΩˆj in Eq. (22), which in the leading order approximation, is given by e±i(π/2)(j−i). The phase string effect again shifts the momentum by π/2, leading to sharp dips of nkα at k=±π/2. All of the above theoretical predictions of the peak/dip positions of Zk and nkα, based on the sign structures of the σ⋅t–J model and the t–J model, are given in Table 1, which will be compared to the DMRG and VMC results in the following sections.3DMRG resultsThe present 1D problem can be accurately studied by the DMRG simulation [33]. Some numerical details are as follows. We shall make use of the conserved quantum numbers, i.e., the total particle number N=L−1 and total spin Sz, and fix t/J=3 throughout the paper, setting J as the unit of energy. The simulation ensures the truncation error of the order or less than 10−10 by keeping sufficient number of states (200∼6000) and performing at least 40 sweeps for different boundary conditions. By calculating the quasiparticle weight, we obtain the undoped ground state first, and then perform enough sweeps to get the one hole ground state by removing an electron in the center. The offdiagonal measurement Zj=〈Ψ0holecjσ†Ψ1hole〉 are made after obtaining the converged wavefunction.3.1Phase string effect under open boundary conditionSince the single hole doped t–J and σ⋅t–J chains can be connected by a unitary transformation [see Eq. (11)] under an open boundary condition, the eigen energies are the same for these two models. Indeed, the ground state energies calculated by the DMRG simulation are shown to coincide precisely in Fig. 2(a), where the onehole energy EG1hole is defined by(23)EG1hole=E01hole−E00hole, where E01hole and E00hole represent the groundstate energies of the oneholedoped and halffilled cases, respectively. Similarly, the physical quantities involving the local diagonal operators, like the hole density and spin distribution, nih=1−ni and Siz, are also the same on the ground states of t–J and σ⋅t–J chains as shown in Fig. 2(b).However, in Sec. 2.4, we have shown that these two ground states, Ψ〉t–J and Φ〉σ⋅t–J, differing by the phase string sign structure e−iΩˆi, cannot be smoothly connected by the t↑–t↓–J model Eq. (3), and there is an abrupt change of the sign average (cf. Fig. 1) at t↓/t↑=0.In Fig. 3(a), the ground state energy E0 of the t↑–t↓–J model computed by DMRG is presented as a function of the ratio t↓/t↑. E0 shows a singularity at t↓/t↑=0 by a sharp jump in the firstorder derivative curve [see Fig. 3(b)]. It indeed implies the breakdown of the adiabatic continuity between the two ground states by a firstordertype of transition, similar to Fig. 1 based on the sign structure.Even though there is no closed path for the interference effect of phase string to take place here, the nontrivial effect of the sign structure will still play a critical role in determining the quasiparticle weight Zk and the momentum distribution nk↑ and nk↓ as predicted in Sec. 2.5.Fig. 4 shows the quasiparticle weight Zk obtained by DMRG. Different peak position k0's of Zk here indicate that the momentum structures are totally different in the two models: i.e., k0=±π/2 in the t–J model [Fig. 4(a)] vs. k0=π and k0=0 for Stotz=−1/2 and Stotz=1/2, respectively, in the σ⋅t–J model [Fig. 4(b)]. These characteristic momentum peak positions in Zk are consistent with the exact analysis in Sec. 2.5 (cf. Table 1), where different sign structures are shown to be responsible for the distinction.Furthermore, a powerlaw decay of Zk at k0's in a fashion of Zk∼L−0.49 is found for the t–J model [cf. Fig. 4(c)]. By contrast, Zk0∼L−0.23 is identified for the σ⋅t–J model. The disappearance of Zk at L→∞ is usually called the Luttinger liquid behavior.For the σ⋅t–J model, the vanishing Zk0 may be purely attributed to the fact that the spin1/2 associated with the doped hole can move away to infinite along the chain, i.e., the socalled spincharge separation as the spin is gapless in the 1D chain. The charge and spin density distributions in Fig. 2(b) have clearly illustrated this. In fact, if one artificially turns on a gap in the spin chain, then a spincharge recombination may retake place in Fig. 2(b) and consequently a sharp peak with a finite Zk0, which corresponds to a conventional Bloch state, could be recovered.For the t–J model, the decay of Zk0 as a function of L is steeper, because the phase string e−iΩˆi will fluctuate around e±i(π/2)i to contribute to an additional powerlaw decay (cf. Sec. 2.5). Here the phase string sign structure plays the role of manybody phase shift. It determines not only the total momentum of the onehole ground state, but also the nonFermiliquid behavior through its fluctuation.Similarly the distinct momentum structure is also manifested in the momentum distribution nkα of the electrons as predicted in Table 1. As presented in Figs. 5(a) and (b), one sees that two sharp peaks (dips) of n↑ or n↓ at k0=±π/2 for the t–J model, whereas at k0=π and k0=0 for the σ⋅t–J model. Moreover, the height of the peak of the momentum distribution 1−n(k0) [n(k0)≡nk0↑+nk0↓] decays in a powerlaw fashion as shown in Fig. 5(c). It is again consistent with that the vanishing quasiparticle weight in the thermodynamic limit is much quicker for the t–J model because of the phase string factor.3.2Phase string interference under periodic boundary conditionThe phase string sign structure in the t–J model can be “gauged away” by performing the phasestring transformation Eq. (8) under an open boundary condition. Correspondingly the groundstate energies EG1hole are the same for the models with and without phase string signs, as shown in Fig. 2.However, if a periodic boundary condition is imposed, such a unitary transformation no longer exists. In this case, one expects the quantum interference effect to take place as the hole can circumvent the closed 1D ring of a finite size. Consequently the energy degeneracy of the t–J and σ⋅t–J models gets lifted due to the phase string sign structure:(24)δEG1hole≡EG1hole(t–J)−EG1hole(σ⋅t–J)=limβ→∞−1βln(Zt–JZσ⋅t–J)=limβ→∞−1βln(∑cτcZ[c]∑cZ[c])≡limβ→∞−1βln〈τc〉Z , which is nonvanishing as long as the phase interference takes place such that 〈τc〉Z<1 (instead of 〈τc〉Z=1 for the open boundary condition). The detailed discussion of the sign structures of the two models under periodic boundary condition can be found in Appendix B. The energy difference is shown in Fig. 6(a), where the scaling law is also altered as compared to Fig. 2(a).Since the phase string sign structure arises from the hole doping, one may detect it directly by measuring the behavior of the charge. For this purpose, one may insert a magnetic flux Φ threading through the 1D ring and then compute the ground state energy difference between Φ=π and 0, i.e.,(25)ΔEG1hole≡EG1hole(Φ=π)−EG1hole(Φ=0), which measures solely the charge sector that couples to the magnetic flux [26].Fig. 6(b) shows the DMRG result of ΔEG1hole for the single hole doped t–J model, which oscillates and decays in a powerlaw fashion L−3 [cf. the inset of Fig. 6(b)]. The period of the oscillation in ΔEG1hole is 4 lattice constant, which matches with the momentum k0=±π/2. Such an oscillation itself reflects the phase string signs, which disappears once the sign structure is turned off in the σ⋅t–J model as shown in Fig. 6(c), where ΔEG1hole is proportional to 1/L2 without any oscillation.4Variational wave functionSo far we have established, via the exact analysis and the DMRG simulation in Sections 2 and 3, that the phase string sign structure is essential in understanding the correct momentum structure and nonBlochwave behavior of a single hole injected into an AF Heisenberg spin chain.In the following, we shall further construct a variational ground state wave function based on the identified sign structure. By a VMC calculation, we show that such a trial wave function well reproduces the DMRG results, in which the role of the phase string is explicitly illustrated.4.1Wave function incorporating the correct sign structureThe general singlehole ground state of the t–J chain under the open boundary condition may be formally written as(26)Ψ〉t–J=∑iφh(i)[e−iΩˆiPˆi]ciσϕ0〉 , where an electron of spin σ is annihilated by ciσ from the singlet ground state ϕ0〉 at halffilling. Besides a singlehole wave function φh(i) which is presumably a smooth function, the nonBlochwave part of the wave function involves the nontrivial sign structure e−iΩˆi identified in Sec. 2.2, with Pˆi denoting the rest of nonsignrelated spin cloud (spinpolaron), in response to the bare hole state ciσϕ0〉.According to Eq. (12), the corresponding variational ground state for the σ⋅t–J model is given by(27)Φ〉σ⋅t–J=∑iφh(i)Pˆiciσϕ0〉 , where the phase string sign structure is gauged away, but Pˆi is still present.The wave function Eqs. (26) and (27) are similar to that previously proposed for the singlehole ground states of a twoleg ladder case [34]. The latter case is further simplified because the halffilling twoleg ladder is fully gapped such that the spinpolaron effect described by Pˆi is not essential for the longwavelength, lowenergy physics, and thus is neglected in Ref. [34]. Then Φ〉σ⋅t–J there becomes a Blochwave state with φh(i)∝eiki.However, in the present 1D chain case, the spin background is gapless with quasilongrange spin correlations in ϕ0〉. In the following, we shall see that Pˆi here will be related to the correction due to spincharge separation in the σ⋅t–J model: the spin1/2 associated with the doped hole can move away from the charge as a gapless spinon, which merely reflects the fact that the free gapless spinon already exists at halffilling. Pˆi will be approximately determined based on the socalled squeezed spin chain construction.4.2Squeezed spin chain constructionThe ground state of the doped Hubbard model at U≫t or the t–J model at J≪t can be well described by the socalled squeezed spin chain approximation [6,19,20,22,23,25], based on which the precise phase string sign structure has been first identified in Refs. [19,20,23,24].According to the squeezed spin chain approximation [6,19,20,22,23,25], one may construct the singleholedoped ground state as follows. Starting from the halffilling ground state ϕ0〉 and displacing the spin at site j (≥i) to site j+1 along an infinitelong chain, while leaving the spin at site j<i unchanged, a vacancy (hole) is then created at site i. The corresponding singlehole state reads(28)Ψ〉t–J=∑iφh(i)Tˆiϕ0〉, where Tˆi denotes an operation translating all the spins at sites j≥i by one lattice constant to j+1 along the 1D chain direction. A vacancy or hole is thus inserted at site i, and the hole wave function φh(i) can be regarded as the only variational parameter once the halffilling ground state ϕ0〉 is known. In Appendix C, we give an explicit check that the wave function Eq. (28) satisfies the sign structure requirement of the t–J model. The hole wave function φh(i) is also given by minimizing the hopping energy.In terms of Eqs. (13) and (14), one has Tˆiϕ0〉=∑{s}c({s})Tˆi{s}〉, with(29)Tˆi{s}〉→e−iΩˆii;{s}〉ss . Here in the newly defined Marshall basis i;{s}〉ss with a hole at site i, the Marshall sign in Eq. (13) is replaced by (−1)NA¯↓ on the displaced spin lattice excluding the site i, and Eq. (29) is obtained by noting(30)(−1)NA↓=(−1)NL(i)∩A¯↓+NR(i)∩B¯↓=(−1)NA¯↓+NR(i)↓≡(−1)NA¯↓e−iΩˆi, where L(i) [R(i)] denotes the sites on the left (right) hand side of the site i, and A¯ and B¯ are the two sublattices of the new lattice. With the hole inserted at site i, the sublattices A and B are switched for all the sites at j>i in the new (the socalled squeezed) spin chain [6,19,20,22,23,25].Comparing Eqs. (26) with Eq. (28), one finds the correspondence(31)Pˆiciσϕ0〉↔(−σ)i∑{s}c({s})i;{s}〉ss, where a sign factor (−σ)i arises from the annihilation of the spin σ by ciσ on the right hand side, which has already been seen in the exact expression Eq. (16). Therefore, the effect created by Pˆi, even though signfree, is still important in 1D due to the gapless spin excitations. In the next, we outline the VMC procedure based on such variational wave functions.4.3Variational Monte Carlo calculationThe variational wave functions Eqs. (28) and (29) can be fully constructed if we know the ground state ϕ0〉 of the halffilled Heisenberg spin model. The halffilled state ϕ0〉 can be approximated by the Liang–Doucot–Anderson type resonatingvalencebond (RVB) state as [35](32)ϕ0〉=∑v(∏(ij)∈vhij)v〉, where the valence bond state is given by(33)v〉=∑{σ}(∏(ij)∈vϵσi,σj)c1σ1†…cLσL†0〉. The LeviCivita symbol ϵσi,σj ensures the singlet paring between spins on sites i and j. hij's are nonnegative variational parameters depending on sites i and j belonging to opposite sublattices, respectively [35]. The most essential property of the RVB state Eq. (32) is that it satisfies the exact Marshall sign rule [29] for bipartite Heisenberg models.Based on the RVB approximation of the halffilled ground state ϕ0〉 in Eq. (32), we calculate the physical properties of the singleholedoped variational wave functions Eqs. (28) and (29) using the Monte Carlo method [35,36]. The details of the VMC simulations are present in Appendix D.The variational ground state energy of the wave function Eq. (28) is shown in Fig. 7, which is in excellent agreement with that obtained by DMRG. By the VMC, we also calculate the momentum distributions nkα of the variational ground states, Eqs. (28) and (29) for the t–J and σ⋅t–J models, respectively. The results are presented in Fig. 8. The overall shapes of the curves, in particular the dip positions, are in good agreements with the theoretical predictions in Table 1 and the DMRG results in Fig. 5. The peak values of 1−nk↑−nk↓ are also scaled in a power law fashion as ∼L−α for both models in Fig. 8(c), which are slightly different from the DMRG results possibly due to the reason that the squeezed spin chain approximation is only accurate at t≫J, but here we considered t/J=3. Furthermore, the variational wave functions constructed here are in the sector Stotz=0 based on the squeezed chain approximation, whereas the DMRG are calculated in the sector Stotz=±1/2. This may also explain why nk↑ and nk↓ are precisely the same in Fig. 8(a) in contrast to a small relative shift of the DMRG in Fig. 5(a).5DiscussionThe behavior of a single hole doped into a 1D Heisenberg spin chain has been examined by combined analytic, numerical, and variational approaches in the present paper. This is one of the simplest limits of doped Mott systems, involving only a single hole interacting with an antiferromagnetically correlated spin background. The basic message, not surprisingly, is that the doped hole does not propagate like a Blochwave due to strong correlation.In particular, we have established a general connection of the Mott strong correlation with a specific form of manybody phase shift. That is, the hole gains a “scattering” phase shift π by passing by (exchanged with) each down spin, and therefore accumulates a manybody phase string τc along an arbitrary path c [cf. Eq. (4)]. If the 1D chain is open on the two ends, we have further proved that τc can be explicitly incorporated into the ground state wave function by a nonlocal sign structure e−iΩˆi [cf. Eqs. (16) and (10)].In a conventional manybody fermion system, the ground state wave function satisfies the fermion sign structure, which dictates the momentum structure with a Fermi surface satisfying the Luttinger volume and a finite Zk at the Fermi surface. By contrast, the fermion sign structure gets completely altered [37] by strong onsite Coulomb repulsion and the new sign structure, i.e., τc and e−iΩˆi here, determines the momentum distribution of a nonFermiliquid or Luttinger liquid behavior with vanishing Zk in the large L limit.As it turns out, the chief role of τc is not a mass renormalization. As a matter of fact, the effective mass is even not affected by τc in the case of an open chain. However, the momentum structure is completely decided by it. The profile of the momentum distribution shown in Fig. 5 is categorically different from a residual Fermi distribution, even though the Luttinger volume seems unchanged as compared to a noninteracting electrons of the same density (which does not distinguish halffilling and oneholedoping in the thermodynamic limit). In fact, the momentum distribution is completely flat at halffilling due to the strong correlation. The emergent quasiparticle spectral weight Zk0 of the hole is found to vanish as ≈L−0.49 at large chain length L (at t/J=3) by DMRG. For the periodic boundary condition, the interference effect of τc has been also clearly identified.These results are in sharp contrast with those obtained by switching off the phase string sign structure. Indeed, in the σ⋅t–J model without τc, the characteristic momentum of the hole at k0=±π/2 is then shifted to π, and Zk0 follows a different scaling law, vanishing slower ≈L−0.23 at large L. In other words, the sign structure of the t–J model does play a critical role in shaping the motion of the hole on a quantum spin background and its singular effect must be treated with a great care. This has been further confirmed by the variational wave function approach based on the t–J and σ⋅t–J models via VMC in this work.Since the sign structure τc has been precisely identified in the t–J model for any dimensions [24,27,28], one expects that the novel properties of doped Mott insulators, due to the irreparable manybody phase shift, should persist beyond the 1D case as well as beyond the singleholedoping case [37].For example, recently a single hole doped into a twoleg Heisenberg spin ladder has been studied by both the DMRG [26,38–41] and a wave function approach [34] based on the VMC. In this system, the “vacuum” of the spin background is spin gapped at halffilling. So only a finitesize “cloud” of spin excitations can be created around the doped hole. In the strong anisotropic or strong rung limit, such a spin cloud or spin polaron effect is indeed found only to renormalize the effective mass without changing the Blochwave nature. But with reducing the anisotropy, a critical point can be reached [39,40], beyond which the momentum structure is fundamentally changed accompanied by the charge modulation and the divergence of the charge mass [26,39]. It has been demonstrated that the singular manybody phase shift or phase string τc becomes unscreened here and is responsible for these exotic properties [26,34,39,40].Such a novel phase of the twoladder system is shown to smoothly persist in the limit of strongchain/weakrung coupling. In the extreme limit of vanishing rung coupling, the twoleg ladder further reduces to two decoupled 1D t–J chains, with the hole localized within one of the chains. This corresponds to the singleholedoped Heisenberg spin chain studied in the present paper. As compared to the coupled twoleg ladder, there are two basic distinctions in this limit. One is that the spin background is now gapless in 1D, instead of spingapped in the coupled twoleg case at halffilling. In other words, the spinpolaron effect becomes more important in 1D, in a form of spincharge separation even for the σ⋅t–J model. The second distinction is that in the twoleg ladder there is an important quantum interference effect originated from the hole transversing from different paths [26], but it is obviously absent in the 1D case. Of course, as shown in this paper, the nontrivial quantum interference still takes place as a finitesize effect under the periodic boundary condition, where the hole may reach a lattice site either through a shorter path or by circumventing the closed ring.Therefore, the 1D nonFermiliquid behavior, the reconstruction of momentum structure in both 1D and the ladder systems, the charge modulation and possible selflocalization [26,39,40] in a ladder system with the leg number more than one, as well as the strong pairing mechanism in the evenleg ladder systems [38], have so far all been attributed to the exotic sign structure τc originated from the Mott physics. We expect the same sign structure to play a critical role in a 2D system as well, which might be directly relevant to the highTc cuprate.AcknowledgementsUseful discussion with J. Zaanen is acknowledged. This work is supported by the NBRC (973 Program, Nos. 2015CB921000), NSFC Grant No. 11534007 and US National Science Foundation Grant DMR1408560.Appendix AThe exact sign structures of the onehole ground states of the 1D σ⋅t–J and t–J models under open boundary conditionIn this appendix, we demonstrate that the onehole ground states of the 1D σ⋅t–J and t–J models in the sector Stotz=−σ/2 satisfy the sign structures given in Eqs. (15) and (16), respectively.First of all, the offdiagonal elements of the σ⋅t–J model in the basis {(−σ)iciσ{s}〉} are all nonpositive as shown below:(A.1)〈{s}cjσ†(−σ)j(−σtciσ†cjσ)(−σ)iciσ{s}〉=−(−σ)i+j+1t〈{s}niσnjσ{s}〉=−t〈{s}niσnjσ{s}〉≤0,(A.2)〈{s}cjσ†(−σ)j(−σ¯tciσ¯†cjσ¯)(−σ)iciσ{s′}〉=(−σ)i+j+1t〈{s}ciσ¯†ciσcjσ†cjσ¯{s′}〉=t〈{s}Si∓Sj±{s′}〉≤0,(A.3)〈{s}chσ†(−σ)h(J2Si+Sj−)(−σ)hchσ{s′}〉=J2〈{s}chσ†Si+Sj−chσ{s′}〉≤0, where i and j belong to the nearest neighbors in the σ⋅t–J model and the following property of the Marshall basis {{s}〉} is used:(A.4)〈{s}Si∓Sj±{s′}〉≤0.Similarly, the offdiagonal elements of the t–J model in the basis {e−iΩˆi(−σ)iciσ{s}〉} can be also shown to be nonpositive. Here, e−iΩˆi=(−1)NR(i)↓, where NR(i)↓=∑l>inl↓ denotes the number of downspins on the righthandside of site i. Then, a straightforward manipulation gives rise to(A.5)〈{s}cjσ†(−σ)j(−1)NR(j)↓(−tciσ†cjσ)(−1)NR(i)↓(−σ)iciσ{s}〉=σ(−σ)i+jt〈{s}niσnjσ{s}〉=−t〈{s}niσnjσ{s}〉≤0,(A.6)〈{s}cjσ†(−σ)j(−1)NR(j)↓(−tciσ¯†cjσ¯)(−1)NR(i)↓(−σ)iciσ{s′}〉=(−σ)(−σ)i+jt〈{s}ciσ¯†ciσcjσ†cjσ¯{s′}〉=t〈{s}Si∓Sj±{s′}〉≤0,(A.7)〈{s}chσ†(−σ)h(−1)NR(h)↓(J2Si+Sj−)(−1)NR(h)↓(−σ)hchσ{s′}〉=J2〈{s}chσ†Si+Sj−chσ{s′}〉≤0.According to the Perron–Frobenius theorem, if all the offdiagonal elements of a Hamiltonian in a given basis are nonpositive, then the ground state of the Hamiltonian must have a nonnegative coefficient in this basis. Therefore, similar to the ground state of the Heisenberg model at halffilling in the Marshall basis {{s}〉}, the onehole ground states of the 1D σ⋅t–J and the t–J models satisfy Eqs. (15) and (16), respectively, with the nonnegative coefficient a(i,{s}).Furthermore, the t↑–t↓–J model Eq. (3) (t↑ is fixed to be positive) has the same sign structure as the σ⋅t–J model (t–J model) if t↓/t↑<0 (t↓/t↑>0). Namely, its onehole ground state wave function always satisfies the same form as Eq. (15) or Eq. (16), depending on the sign of t↓/t↑. Of course, the nonnegative coefficient a(i,{s})≥0 will also depend on the ratio t↓/t↑.Appendix BSign structures of the σ⋅t–J model and the t–J model under periodic/antiperiodic boundary conditionsIn this appendix, we analyze the sign structures of the σ⋅t–J model and the t–J model under periodic/antiperiodic boundary conditions. We will show that the σ⋅t–J model under periodic boundary condition is still signfree. On the other hand, the t–J model under periodic/antiperiodic boundary condition has an interference effect due to the nontrivial sign structure.B.1The σ⋅t–J modelFirst let us consider the σ⋅t–J model on a lattice with even L:(B.1)Hσ⋅t–J=−t∑i=1L−1∑σσ(ciσ†ci+1σ+h.c.)+J∑i=1L−1(Si⋅Si+1−14nini+1)−ηt∑σσ(c1σ†cLσ+h.c.)+J(S1⋅SL−14n1nL), where η=±1 denotes the periodic or antiperiodic boundary conditions. Under a Marshall transformation UM=(−1)NA↓, which is diagonal in the Marshall basis {{s}〉}, the hopping term changes according to ciσ†ci+1σ→σciσ†ci+1σ, while the superexchange term changes according to Si+Si+1−→−Si+Si+1−. Therefore, the total Hamiltonian is transformed to(B.2)UMHσ⋅t–JUM†=−t∑i=1L−1∑σ(ciσ†ci+1σ+h.c.)+J∑i=1L−1(−12(Si+Si+1−+Si−Si+1+)+SizSi+1z−14nini+1)−ηt∑σ(c1σ†cLσ+h.c.)+J(−12(S1+SL−+S1−SL+)+S1zSLz−14n1nL). Under the periodic boundary condition (η=1), the offdiagonal terms of the Hamiltonian are always nonpositive (with the fermion sign in the hopping term considered for the onehole case). We therefore conclude that the oneholedoped σ⋅t–J model is signproblemfree not only under the open boundary condition, but also under the periodic boundary condition. In the partition function language, the partition function can be written as:(B.3)Zσ⋅t–JPBC=∑cZ[c], where Z[c] is nonnegative and c is the worldline path of the system.For the antiperiodic boundary condition (η=−1), the only nontrivial sign comes from the process that a hole hopping through the boundary. The partition function is(B.4)Zσ⋅t–JABC=∑c(−1)NbdyhZ[c], where Z[c] is nonnegative and Nbdyh is the number of times that the hole crosses the boundary. Therefore, in the squeezed spin chain approximation, the energy difference between the antiperiodic and periodic boundary condition ΔEG1hole is merely the energy difference for a Bloch particle under antiperiodic and periodic boundary conditions. The scaling of the latter is given by(B.5)ΔEG1hole∝1−cos(πL)∝1L2. This L−2 scaling of the energy difference is confirmed by the DMRG results in Fig. 6(c).B.2The t–J modelNow let us turn to the t–J model under the periodic boundary condition:(B.6)Ht–J=−t∑i=1L−1∑σ(ciσ†ci+1σ+h.c.)+J∑i=1L−1(Si⋅Si+1−14nini+1)−ηt∑σ(c1σ†cLσ+h.c.)+J(S1⋅SL−14n1nL). The Marshall transformation UM=(−1)NA↓ changes the above Hamiltonian to(B.7)UMHt–JUM†=−t∑i=1L−1∑σσ(ciσ†ci+1σ+h.c.)+J∑i=1L−1(−12(Si+Si+1−+Si−Si+1+)+SizSi+1z−14nini+1)−ηt∑σσ(c1σ†cLσ+h.c.)+J(−12(S1+SL−+S1−SL+)+S1zSLz−14n1nL). Now to absorb the sign in the front of the bulk hopping term, i.e. σciσ†ci+1σ→ciσ†ci+1σ, we perform the phase string transformation UPS≡eiΘˆ=∏i<l(−1)nihnl↓. The Hamiltonian is further transformed to(B.8)UPSUMHt–JUM†UPS†=−t∑i=1L−1∑σ(ciσ†ci+1σ+h.c.)+J∑i=1L−1(−12(Si+Si+1−+Si−Si+1+)+SizSi+1z−14nini+1)−ηt∑σσ(−1)N↓(c1σ†cLσ+h.c.)+J(12(S1+SL−+S1−SL+)+S1zSLz−14n1nL). Note that the boundary hopping term c1σ†cLσ (σ=±) and the boundary superexchange term S1+SL− acquire an additional sign (−1)N↓ and −1, respectively. Here (−1)N↓ denotes the total number of down spins.From the analysis above, under the periodic boundary condition (η=1), the nontrivial sign is tc1↓†cL↓ (tc1↑†cL↑) if N↓ is even (odd) and (J/2)(S1+SL−+S1−SL+). The partition function is given by(B.9)Zt–J=∑c(−1)Nbdy↑/↓Z[c], where Z[c] is nonnegative and Nbdy↑/↓ (for N↓ odd/even) is the number of times that an up/down spin crosses the boundary. On the other hand, under the antiperiodic boundary condition (η=−1), the partition function is still Eq. (B.9). But Nbdy↑ (Nbdy↓) is for N↓ even (odd).Since the number of down spins increases by one, if one increases the chain length L by two, Nbdy↑ and Nbdy↓ in the partition function are switched. Effectively, the periodic and antiperiodic boundary conditions are also switched (the combined quantity η(−1)N↓ determines the sign structure). Therefore, one expects the energy difference between the antiperiodic and periodic boundary condition ΔEG1hole to oscillate with increasing L. The DMRG result indeed shows the oscillation and L−3 scaling of the energy difference ΔEG1hole in Fig. 6(b).Appendix CSign structure of the variational wave functions in Eq. (28)In this appendix, we will show explicitly that the variational wave function Eq. (28) satisfies the sign structure requirement of the t–J model. The hole wave function φh(i) will be also determined by minimizing the hopping energy.Since the spin and charge are totally separated in the squeezed spin chain approximation, the state obtained from moving the hole from site i to j is exactly the one with hole at j initially. To be more precise, the matrix element at each step of hopping energy for the t–J model under Eq. (28), is given by(C.1)〈Ψ(−tciσ†cjσ)Ψ〉t–J=φh(j)φh(i)(〈ϕ0Tˆj†)(−tciσ†cjσ)(Tˆiϕ0〉)=φh(j)φh(i)(−t/2)≤0, where 1/2 comes from the two possibilities of the spin on sites i and j. As a result, to minimize the hopping energy, we choose φh(i) (i=1,2,⋯,L) to be the Bloch state for a free particle in the tight binding model:(C.2)φh(i)={1L,for periodic boundary condition,2L+1sin(iπL+1),for open boundary condition.As for the superexchange terms of the t–J model, the offdiagonal terms, such as Si+Sj−, always change the number of down spins on A sublattice by one, while leave the relative positions of the hole and spins unchanged. The Marshall sign mismatch in this process in the oneholedoped case is exactly the same as the halffilled spin chain wave function. Therefore, the offdiagonal terms of the superexchange terms are also nonpositive for the hole wave function basis in the variational wave function Ψ〉t–J, which inherits the Marshall sign from the halffilled ground state ϕ0〉.Thus, similar to the basis states used in Appendix A, the singlehole wave function basis {Tˆiϕ0〉} in the variational wave function Ψ〉t–J in Eq. (28) is indeed the signfree basis of the t–J model. Namely the variational wave function Ψ〉t–J with nonnegative hole wave function φh(i) satisfies the sign structure requirement of the model. The same argument can be apply to the σ⋅t–J model as its onehole ground state Φ〉σ⋅t–J is connected to Ψ〉t–J by a unitary transformation Eq. (12).Appendix DMonte Carlo for variational wave functionsIn this appendix, we present the Monte Carlo method in calculating nkα for the variational wave functions Ψ〉t–J and Φ〉σ⋅t–J. The procedures have some similarities to those used in Ref. [34].The singleholedoped basis is constructed from the singlet valence bond state v〉=∑{σ}(∏(ij)∈vϵσi,σj)c1σ1†⋯cLσL†0〉 specified by the dimer covering configuration v. By acting the operator Tˆh, the spins on site x≥h are translated by one lattice constant along xˆ direction, effectively creating a hole at site h. We denote this singleholedoped valence bond state by h,v〉, which is defined on a lattice with L sites (L is odd because of adding a hole site). The halffilled Liang–Doucot–Anderson type RVB state can be obtained by Monte Carlo as [35,36](D.1)ϕ0〉=∑vwvv〉, where wv=∏(ij)∈vhij, with nonnegative variational parameters hij's, is the nonnegative coefficient associated with the valence bond state v〉. The variational wave function Ψ〉t–J for the squeezed t–J chain is then given by(D.2)Ψ〉t–J=∑hφh(h)∑vwvh,v〉, where wv is the same as in Eq. (D.1), and the normalized hole wave function φh(h) is chosen according to Eq. (C.2). Therefore there are in fact no variational parameters to tune in Ψ〉t–J.The normalization of the basis states is given by(D.3)〈h′,v′h,v〉=δhh′〈v′v〉=2Nv,v′HF, where Nv,v′HF is the number of loops in the transition graph of the dimer covers v and v′ at halffilling. The normalization of the wave function Ψ〉t–J is then(D.4)〈ΨΨ〉t–J=∑hφh(h)2∑v,v′〈h′,v′h,v〉=(∑hφh(h)2)∑v,v′wv′wv2Nv,v′HF=∑v,v′wv′wv2Nv,v′HF, where we used the normalization of the hole wave function φh(h).From the above formula for the normalization of Ψ〉t–J, we can use MC method to calculate the momentum distribution of Ψ〉t–J:(D.5)〈nkα〉=〈ckα†ckα〉=1L∑i,jeik(i−j)〈ciα†cjα〉. We should first calculate(D.6)〈Ψciα†cjαΨ〉t–J〈ΨΨ〉t–J=∑h,h′φh(h′)φh(h)∑v,v′wv′wv〈h′,v′ciα†cjαh,v〉∑v,v′wv′wv2Nv,v′HF=∑v,v′wv′wv2Nv,v′HFTijα(v,v′)∑v,v′wv′wv2Nv,v′HF, where we have defined(D.7)Tijα(v,v′)≡∑h,h′φh(h′)φh(h)12Nv,v′HF〈h′,v′ciα†cjαh,v〉. The positive value wv′wv2Nv,v′HF can be used as the distribution weight in the MC procedure. The quantity Tijα(v,v′) is averaged over to obtain 〈ciα†cjα〉, and then the momentum distribution 〈nkα〉 according to Eq. (D.5).Now our task is to simplify Tijα(v,v′) in Eq. (D.7) to be calculated numerically. For i=j, we have(D.8)Tiiα(v,v′)=∑h,h′φh(h′)φh(h)12Nv,v′HFδhh′δi≠h2Nv,v′HF−1=∑h≠i12φh(h)2=12(1−φh(i)2). This is simply the probability 1−φh(i)2 of finding an electron at site i divided by two, because of two spin values (α=±). On the other hand, for i≠j, we have(D.9)〈h′,v′ciα†cjαh,v〉=−δihδjh′(〈j,v′cjα)(ciα†i,v〉)=−δihδjh′∑{s},si=sj=αδv,{s}δv′,{s}(−1)i+jη({s}i)η({s}j)=−δihδjh′∑{s},si=sj=αδv,{s}δv′,{s}(−1)i+j(−1)N(i,j)↓αi+j−1. Note that in the first line, ciα†i,v〉 is a halffilled spin state with fermion sign (−1)i−1 (coming from moving ciα† to the ith place in the sequence c1s1†⋯ci−1si−1†ci+1si+1†⋯cLsL†) and the Marshall sign η({s}i) on the original lattice with L−1 sites (removing the i site). δv,{s} specifies the constraint that the summation over the spin configuration {s} should be compatible with the dimer cover v, i.e., the spins on sites belonging to the same dimer should be opposite. From the second line to the third line, we simplified the product of the Marshall signs η({s}i)η({s}j) for the initial and final valence bond states v〉 and v′〉: The Marshall signs of the two states for spins on site x<i or x>j (suppose i<j) cancel each other; The Marshall signs for sites i and j contribute the factor αi+j−1; The product of Marshall signs for sites i<x<j is (−1)N(i,j)↓, where N(i,j)↓ is the number of down spins on sites i<x<j for the spin configuration {s}. As a result, Eq. (D.7) can be reduced to(D.10)Tijα(v,v′)=φh(j)φh(i)12Nv,v′HF∑{s},si=sj=αδv,{s}δv′,{s}(−1)i+j+1(−1)N(i,j)↓αi+j−1 for i≠j.For the variational wave function Φ〉σ⋅t–J of the σ⋅t–J model, the explicit form in the singleholedoped valence bond basis reads(D.11)Φ〉σ⋅t–J=∑hφh(h)∑v(−1)NR(h)↓wvh,v〉, where (−1)NR(h)↓ comes from the phase string transformation connecting the ground states of the t–J model and the σ⋅t–J model. Following the same analysis above, we can also use the positive value wv′wv2Nv,v′HF as the distribution weight in the MC simulations, and calculate the average value of the quantity Tijα(v,v′). For i=j, the formula Eq. (D.8) is the same. For i≠j, however, there is an additional phase string factor (−1)N(i,j)↓ from sites i<x<j and σ from the site j. 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