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Physical review. D/Physical review. D.

Holographic dual of a quantum spin chain as a Chern-Simons-scalar theory

Naoto Yokoi, Yasuyuki Oikawa, Eiji Saitoh

Published Apr 25, 2024• 0 citations
Publisher

Abstract

We construct a holographic dual theory of one-dimensional anisotropic Heisenberg spin chain, which includes two Chern-Simons gauge fields and a charged scalar field. Thermodynamic quantities of the spin chain at low temperatures, which are exactly calculated from the integrability, are completely reproduced by the dual theory on three-dimensional black hole backgrounds and the exact matching of the parameters between the dual theory and the spin chain is obtained. The holographic dual theory provides a new theoretical framework to analyze the quantum spin chain and one-dimensional quantum many-body systems. Published by the American Physical Society 2024

DOI
10.1103/physrevd.109.086023

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Journal of the Physical Society of Japan•Apr 15, 1994•49 citations
We construct a holographic dual theory of one-dimensional anisotropic Heisenberg spin chain, which includes two Chern-Simons gauge fields and a charged scalar field. Thermodynamic quantities of the spin chain at low temperatures, which are exactly calculated from the integrability, are completely reproduced by the dual theory on three-dimensional black hole backgrounds and the exact matching of the parameters between the dual theory and the spin chain is obtained. The holographic dual theory provides a new theoretical framework to analyze the quantum spin chain and one-dimensional quantum many-body systems. Published by the American Physical Society 2024
We construct a holographic dual theory of one-dimensional anisotropic Heisenberg spin chain, which includes two Chern-Simons gauge fields and a charged scalar field. Thermodynamic quantities of the spin chain at low temperatures, which are exactly calculated from the integrability, are completely reproduced by the dual theory on three-dimensional black hole backgrounds and the exact matching of the parameters between the dual theory and the spin chain is obtained. The holographic dual theory provides a new theoretical framework to analyze the quantum spin chain and one-dimensional quantum many-body systems.
A holographic correspondence has been recently developed between higher spin theories in three-dimensional anti-de Sitter space (AdS3) and two-dimensional Conformal Field Theories (CFTs) with extended symmetries. A class of such dualities involves SL(N, R) × SL(N, R) Chern-Simons gauge theories in the (2 + 1)-dimensional bulk space-time, and CFTs with WN symmetry algebras on the (1 + 1)-dimensional boundary. The topological character of the bulk theory forces one to reconsider standard geometric notions such as black hole horizons and entropy, as well as the usual holographic dictionary. Motivated by this challenge, in this note we present a proposal to compute entanglement entropy in the WN CFTs via holographic methods. In particular, we introduce a functional constructed from Wilson lines in the bulk Chern-Simons theory that captures the entanglement entropy in the CFTs dual to standard AdS3 gravity, corresponding to SL(2, R) × SL(2, R) gauge group, and admits an immediate generalization to the higher spin case. We explicitly evaluate this functional for several known solutions of the bulk theory, including charged black holes dual to thermal CFT states carrying higher spin charge, and show that it reproduces expected features of entanglement entropy, study whether it obeys strong subadditivity, and moreover show that it reduces to the thermal entropy in the appropriate limit.
Abstract. We construct a one-dimensional (1D) spin chain Hamiltonian with generic interactions and prove that the thermal correlation functions of the model admit an explicit random matrix representation. As an application of the result, we show how the observables of U ( N ) Chern–Simons theory on S 3 can be reproduced with the thermal correlation functions of the 1D spin chain, which is of the XX type, with a suitable choice of exponentially decaying interactions between infinitely many neighbours. We show that for this model, the correlation functions of the spin chain at a finite temperature give the Chern–Simons partition function, quantum dimensions and the full topological S -matrix.
This thesis contains two parts. The _rst part consists of the _rst two chapters, studying the entanglement entropy. The second part consists of the third chapter, dealing with holographic superconductors. In chapter one we calculate the topological entanglement entropy of Chern-Simons theories in 2+1 dimensions. We employ a procedure of surgery to do the calculation, and _nd the general formula for the entanglement entropy for any state on any smooth surface. The key component of the result is the so-called S matrices of the WZW models associated with our Chern-Simons theories. We apply our results to the quantum Hall uids that are described by various kinds of Chern-Simons theories and calculate their S matrix elements. In chapter two we study the entanglement entropy for the nontrivial primary states of a two dimensional CFT, the free boson. We use several tricks to help us go through the calculation and get the exact results for several primary states. In chapter three we discuss a model of holographic superconductors. It is a (3+1)- dimensional Abelian-Higgs model of gravity with negative cosmological constant. We study the static, spherically symmetric solutions of this model numerically. These are hairy black holes.The nonzero hair corresponds to the nonzero expectation value of some composite operator in the dual _eld theory. We _nd the second order phase transition at certain critical temperature Tc and a mean _eld critical exponent of 0.5. The low temperature behavior of this model does not conform to that of a superconductor however. We also study the static, axially symmetric solutions (vortex solutions), especially their thermodynamical properties. A numerical algorithm to _nd such solutions is setup, with all the key steps solved.
The accessibility of the Hawking-Page transition in AdS 5 through a one-dimensional (1D) Heisenberg spin chain is demonstrated. We use the random matrix formulation of the Loschmidt echo for a set of spin chains, and randomize the ferromagnetic spin interaction. It is shown that the thermal Loschmidt echo, when averaged, detects the predicted increase in entropy across the Hawking-Page transition. This suggests that a 1D spin chain exhibits characteristics of black hole physics in 4 + 1 dimensions. We show that this approach is equally applicable to free fermion systems with a general dispersion relation. Published by the American Physical Society 2024
The Green's function formalism proposed by Kondo and Yamaji for low-dimensional spin systems with S =1/2 has been extended to the cases of S >1/2. We apply this formalism to the study of one- and two-dimensional Heisenberg ferromagnets over the whole temperature range. It is shown that our theory can reproduce the correct results obtained by the high-temperature expansion method. On the other hand, the results at low temperatures are similar to those of the modified spin-wave theory, which is considered to predict the low-temperature properties of such systems rather correctly. The gross behaviors of the calculated thermodynamic quantities of ferromagnetic Heisenberg chains agree with those obtained by the exact diagonalization method.