Timelike entanglement entropy is a complex measure of information that is holographically realized by an appropriate combination of spacelike and timelike extremal surfaces. This measure is highly sensitive to Lorentz invariance breaking. In this work, we study the timelike entanglement entropy in non-relativistic theories, focusing on theories with hyperscaling violation and Lifshitz-like spatial anisotropy. The properties of the extremal surfaces, as well as the timelike entanglement entropy itself, depend heavily on the symmetry-breaking parameters of the theory. Consequently, we show that timelike entanglement can encode, to a large extent, the stability and naturalness of the theory. Furthermore, we find that timelike entanglement entropy identifies Fermi surfaces either through the logarithmic behavior of its real part or, alternatively, via its constant imaginary part, with this constant value depending on the theory’s Lifshitz exponent. This provides a novel interpretation for the imaginary component of this pseudoentropy. Additionally, we examine temporal entanglement entropy, an extension of timelike entanglement entropy to Euclidean space, and provide a comprehensive discussion of its properties in these theories.

Taiwan

Mir Afrasiar

Jaydeep Basak

Dimitrios Giataganas

Holographic timelike entanglement entropy in non-relativistic theories

We propose a holographic formalism for a timelike entanglement entropy in non-conformal theories. This pseudoentropy is a complex-valued measure of information, which, in holographic non-conformal theories, receives contributions from a set of spacelike surfaces and a finite timelike bulk surface with mirror symmetry. We suggest a method of merging the surfaces so that the boundary length of the subregion is exclusively specified by holography. We show that in confining theories, the surfaces can be merged in the bulk at the infrared tip of the geometry and are homologous to the boundary region. The timelike entanglement entropy receives its imaginary and real contributions from the timelike and the spacelike surfaces, respectively. Additionally, we demonstrate that in confining theories, there exists a critical length within which a connected non-trivial surface can exist, and the imaginary part of the timelike entanglement entropy is non-zero. Therefore, the timelike entanglement entropy exhibits unique behavior in confining theories, making it a probe of confinement and phase transitions. Finally, we discuss the entanglement entropy in Euclidean spacetime in confining theories and the effect of a simple analytical continuation from a spacelike subsystem to a timelike one.

Timelike entanglement entropy and phase transitions in non-conformal theories

By using the notion of fractional derivatives, we introduce a class of massless Lifshitz scalar field theory in (1+1)-dimension with an arbitrary anisotropy index z. The Lifshitz scale invariant ground state of the theory is constructed explicitly and takes the form of Rokhsar-Kivelson (RK). We show that there is a continuous family of ground states with degeneracy parameterized by the choice of solution to the equation of motion of an auxiliary classical system. The quantum mechanical path integral establishes a 2d/1d correspondence with the equal time correlation functions of the Lifshitz scalar field theory. We study the entanglement properties of the Lifshitz theory for arbitrary z using the path integral representation. The entanglement measures are expressed in terms of certain cross ratio functions we specify, and satisfy the c-function monotonicity theorems. We also consider the holographic description of the Lifshitz theory. In order to match with the field theory result for the entanglement entropy, we propose a z-dependent radius scale for the Lifshitz background. This relation is consistent with the z-dependent scaling symmetry respected by the Lifshitz vacuum. Furthermore, the time-like entanglement entropy is determined using holography. Our result suggests that there should exist a fundamental definition of time-like entanglement other than employing analytic continuation as performed in relativistic field theory.

Adrita Chakraborty

Chong-Sun Chu

Himanshu Parihar

Massless Lifshitz field theory for arbitrary z

We analyze nonlocal rotating observables in holography corresponding to spinning bound states. To renormalize their energies and momenta we suggest and discuss different holographic renormalization schemes motivated by the static nonlocal observables. Namely the holographic renormalization and the rotating color singlet mass subtraction scheme. In the holographic renormalization we identify the infinite boundary terms and subtract them. In the mass subtraction scheme we evaluate the energy of a spinning trailing string corresponding to the color charged singlet which experiences dragging phenomena and we subtract it from the energy of the bound state to obtain the renormalized finite energy. Then we apply our generic framework to certain strongly coupled thermal theories with broken rotational symmetry. We find numerical solutions corresponding to spinning bound states with a fixed size while varying their angular frequency. By applying numerically the renormalization schemes, we find that there is a critical frequency where the bound state ceases to exist or dissociates. We also note that bound states require lower angular frequencies to dissociate when the theory has less symmetry.

Greece

Vangelis Giantsos

Holographic nonlocal rotating observables and their renormalization

We study holographically nonlocal observables in field theories at finite temperature and in the large <math><mi>d</mi></math> limit. These include the Wilson loop, the entanglement entropy, as well as an extension to various dual extremal surfaces of arbitrary codimension. The large <math><mi>d</mi></math> limit creates a localized potential in the near horizon regime resulting in a simplification of the analysis for the nonlocal observables, while at the same time retaining their qualitative physical properties. Moreover, we study the monotonicity of the coefficient <math><mi>α</mi></math> of the entanglement’s area term, the so-called area theorem. We find that the difference between the UV and IR of the <math><mi>α</mi></math> values, normalized with the thermal entropy, converges at large <math><mi>d</mi></math> to a constant value which is obtained analytically. Therefore, the large <math><mi>d</mi></math> limit may be used as a tool for the study and (in)validation of the renormalization group monotonicity theorems. All the expectation values of the observables under study show rapid convergence to certain values as <math><mi>d</mi></math> increases. The extrapolation of the large <math><mi>d</mi></math> limit to low and intermediate dimensions shows good quantitative agreement with the numerical analysis of the observables.

Nikolaos Pappas

Cyprus

Nicolaos Toumbas

Holographic observables at large <math><mi>d</mi></math>

We formulate an equivalence between the 2-dim <math><mi>σ</mi></math>-model spectrum expanded on a nontrivial massive vacuum and a classical particle Hamiltonian with variable mass and potential. By considering methods of analytic Galoisian nonintegrability on the appropriate geodesics of the Hamiltonian system we algebraically constrain the particle masses at fixed time, such that integrability is allowed. Through our equivalence, this explicitly constrains the masses of the excited spectrum of the dual 2-dim theory in such a way to imply the <math><mi>S</mi></math>-matrix factorization and no particle production. In particular, the integrability of the classical particle system implies the factorization of the <math><mi>S</mi></math>-matrix in the dual quantum 2-dim theory. Our proposal provides also nontrivial evidence on the connection between integrability and <math><mi>S</mi></math>-matrix factorization for large class of theories with interactions that break Lorentz invariance.

Analytic nonintegrability and <math><mi>S</mi></math>-matrix factorization

We propose the c-function as a new and accurate probe to detect the location of topological quantum critical points. As a direct application, we consider a holographic model which exhibits a topological quantum phase transition between a topologically trivial insulating phase and a gapless Weyl semimetal. The quantum critical point displays a strong Lifshitz-like anisotropy in the spatial directions, and the quantum phase transition does not follow the standard Landau paradigm. The c-function robustly shows a global feature at the quantum criticality and distinguishes, with great accuracy, the two separate zero temperature phases. Taking into account the relation of the c-function with the entanglement entropy, we conjecture that our proposal is a general feature of quantum phase transitions and that it is applicable beyond the holographic framework.

Spain

Matteo Baggioli

Detecting topological quantum phase transitions via the c-function

We propose a candidate <math><mi>c</mi></math>-function in arbitrary dimensional quantum field theories with broken Lorentz and rotational symmetry. For holographic theories we derive the necessary and sufficient conditions on the geometric background for these <math><mi>c</mi></math>-functions to satisfy the <math><mi>c</mi></math>-theorem. We obtain the null energy conditions for anisotropic background to show that they do not themselves assure the <math><mi>c</mi></math>-theorem. By employing them, we find that it is possible to impose conditions on the UV data that are enough to guarantee at least one monotonic <math><mi>c</mi></math>-function along the RG flow. These UV conditions can be used as building blocks for the construction of anisotropic monotonic RG flows. Finally, we apply our results to several known anisotropic theories and identify the region in the parameter space of the metric where the <math><mi>c</mi></math>-theorem holds for our proposed <math><mi>c</mi></math>-function.

<math><mrow><mi>c</mi></mrow></math>-theorem for anisotropic RG flows from holographic entanglement entropy

Baryonic states are sufficiently complex to reveal physics that is hidden in the mesonic bound states. Using gauge/gravity correspondence we study analytically and numerically baryons in theories with space-dependent <math><mrow><mi>θ</mi></mrow></math>-terms, or theories under strong magnetic fields. Such holographic studies on baryons are accommodated in a generic analytic framework we develop for anisotropic theories, where their qualitative features are common irrespective of the source that triggers the anisotropy. We find that the distribution of the quarks forming the state, depends on the angle between the baryon and the anisotropic direction. Its shape is increasingly elliptic with respect to the strength of the field sourcing the anisotropy, counterbalancing the broken rotational invariance on the gluonic degrees of freedom. Strikingly, the baryons dissociate in stages with a process that depends on the proximity of the quarks to the anisotropic direction, where certain quark pairs abandon the bound state first, followed by the closest pairs to them as the temperature increases. This observation may also serve as a way to identify the nature of certain exotic states. Finally, we investigate holographic baryons with a decreased number of quarks and explain why in theories under consideration the presence of anisotropy does not modify the universal stability condition in contrast to the usual trend.

Baryons under strong magnetic fields or in theories with space-dependent <math><mrow><mi>θ</mi></mrow></math>-term

We initiate a nonperturbative study of anisotropic, nonconformal, and confining gauge theories that are holographically realized in gravity by generic Einstein-axion-dilaton systems. In the vacuum, our solutions describe renormalization group flows from a conformal field theory in the UV to generic scaling solutions in the IR with generic hyperscaling violation and dynamical exponents <math><mrow><mi>θ</mi></mrow></math> and <math><mi>z</mi></math>. We formulate a generalization of the holographic <math><mi>c</mi></math> theorem to the anisotropic case. At finite temperature, we discover that the anisotropic deformation reduces the confinement-deconfinement phase transition temperature suggesting a possible alternative explanation of inverse magnetic catalysis solely based on anisotropy. We also study transport and diffusion properties in anisotropic theories and observe, in particular, that the butterfly velocity that characterizes both diffusion and growth of chaos transverse to the anisotropic direction saturates a constant value in the IR which can exceed the bound given by the conformal value.