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XYZスピン1/2鎖における多数の局所保存量とその流れの研究

Yuji Nozawa

Published Mar 24, 2022• 0 citations
Publisher

Abstract

Integrable models are characterized by the existence of an extensive number of local conserved quantities.Their existence is closely related to various exact methods, such as the Bethe ansatz and quantum inverse scattering.They also play an important role in nonequilibrium dynamics in the integrable models.Their steady states and transport phenomena exhibit exotic properties, which are also related to this point.Local conserved quantities in a quantum integrable model in one dimension are generated from the transfer matrix of the corresponding classical two-dimensional model.However, it is still non-trivial to obtain their explicit expression.The meaning of "local" here is that each conserved quantity consists of the sum of observables acting on a finite range of sites called its support.We note that the local conserved quantities in this context are not O (1)-body observables in general.Their explicit expressions enable us to analyze theories of nonequilibrium dynamics using the local conserved quantities in detail, for example, the generalized Gibbs ensemble, the Mazur bound, and the generalized hydrodynamics.In this thesis, we have studied the local conserved quantities in the XYZ spin-1/2 chain, which is an integrable model corresponding to the classical eight-vertex model.We have obtained their explicit expression for arbitrary coupling constants of the XYZ chain.The model at the isotropic point called the Heisenberg or XXX chain is an exceptional case in which the expression had been known.We have checked that our results reproduce the known expression for the XXX chain as a special case.To obtain the explicit expression, we have employed the notation of doubling-product operators, which was introduced by Shiraishi to study the non-integrability of the XYZ spin-1/2 chain in a magnetic field.Using this notation, we have found recursion relations of the coefficients among different terms in the conserved quantities in the XYZ chain.Then, we have solved the recursion relations and obtained the explicit expression.We have shown that the local conserved quantity Q k for a fixed support k is essentially unique i ii ABSTRACT if 2 ≤ k ≤ L/2 with L being the number of sites.We have obtained the explicit expression for all {Q k } for 1 ≤ k ≤ L/2 and one Q k for each k with L/2 < k ≤ L.We have also studied the currents of these conserved quantities based on the respective continuity equations.We have obtained their explicit expression and numerically investigated their time evolution.In particular, we have studied the effects of an integrabilitybreaking perturbation on the time evolution of the currents and found that this term drastically changes their long-time behaviors as follows.As for the initial state, we have chosen several energy eigenstates of the XXX chain with a nonzero energy current, which is also known to be conserved.In particular, we have studied the time evolution with the Hamiltonian of the XXZ chain with and without a next-nearest-neighbor interaction and seen the effects of this integrability-breaking perturbation.We have found that, for these initial states, the long-time average of the total currents is nonzero for the integrable cases while zero for the non-integrable cases.We have also studied this behavior analytically and shown that the degeneracy of the energy eigenstates is essential.Our results for the XYZ spin-1/2 chain have revealed the structure of the conserved quantities and currents in detail.They provide materials for the development of algebraic studies in integrable models and will be useful for detailed analyses of nonequilibrium dynamics in these models.

DOI
10.15083/0002008962

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Integrable models are characterized by the existence of an extensive number of local conserved quantities.Their existence is closely related to various exact methods, such as the Bethe ansatz and quantum inverse scattering.They also play an important role in nonequilibrium dynamics in the integrable models.Their steady states and transport phenomena exhibit exotic properties, which are also related to this point.Local conserved quantities in a quantum integrable model in one dimension are generated from the transfer matrix of the corresponding classical two-dimensional model.However, it is still non-trivial to obtain their explicit expression.The meaning of "local" here is that each conserved quantity consists of the sum of observables acting on a finite range of sites called its support.We note that the local conserved quantities in this context are not O (1)-body observables in general.Their explicit expressions enable us to analyze theories of nonequilibrium dynamics using the local conserved quantities in detail, for example, the generalized Gibbs ensemble, the Mazur bound, and the generalized hydrodynamics.In this thesis, we have studied the local conserved quantities in the XYZ spin-1/2 chain, which is an integrable model corresponding to the classical eight-vertex model.We have obtained their explicit expression for arbitrary coupling constants of the XYZ chain.The model at the isotropic point called the Heisenberg or XXX chain is an exceptional case in which the expression had been known.We have checked that our results reproduce the known expression for the XXX chain as a special case.To obtain the explicit expression, we have employed the notation of doubling-product operators, which was introduced by Shiraishi to study the non-integrability of the XYZ spin-1/2 chain in a magnetic field.Using this notation, we have found recursion relations of the coefficients among different terms in the conserved quantities in the XYZ chain.Then, we have solved the recursion relations and obtained the explicit expression.We have shown that the local conserved quantity Q k for a fixed support k is essentially unique i ii ABSTRACT if 2 ≤ k ≤ L/2 with L being the number of sites.We have obtained the explicit expression for all {Q k } for 1 ≤ k ≤ L/2 and one Q k for each k with L/2 < k ≤ L.We have also studied the currents of these conserved quantities based on the respective continuity equations.We have obtained their explicit expression and numerically investigated their time evolution.In particular, we have studied the effects of an integrabilitybreaking perturbation on the time evolution of the currents and found that this term drastically changes their long-time behaviors as follows.As for the initial state, we have chosen several energy eigenstates of the XXX chain with a nonzero energy current, which is also known to be conserved.In particular, we have studied the time evolution with the Hamiltonian of the XXZ chain with and without a next-nearest-neighbor interaction and seen the effects of this integrability-breaking perturbation.We have found that, for these initial states, the long-time average of the total currents is nonzero for the integrable cases while zero for the non-integrable cases.We have also studied this behavior analytically and shown that the degeneracy of the energy eigenstates is essential.Our results for the XYZ spin-1/2 chain have revealed the structure of the conserved quantities and currents in detail.They provide materials for the development of algebraic studies in integrable models and will be useful for detailed analyses of nonequilibrium dynamics in these models.
The integrability of a quantum many-body system, which is characterized by the presence or absence of local conserved quantities, drastically impacts the dynamics of isolated systems, including thermalization. Nevertheless, a rigorous and comprehensive method for determining integrability or nonintegrability has remained elusive. In this paper, we address this challenge by introducing rigorously provable tests for integrability and nonintegrability of quantum spin systems with finite-range interactions. Our results significantly simplify existing proofs of nonintegrability, such as those for the S=1/2S=1/2S=1/2 Heisenberg chain with nearest-and next-nearest-neighbor interactions, the S=1S=1S=1 bilinear-biquadratic chain and the S=1/2S=1/2S=1/2 XYZ model in two or higher dimensions. Moreover, our results also yield the first proof of nonintegrability for models such as the S=1/2S=1/2S=1/2 Heisenberg chain with a non-uniform magnetic field, the S=1/2S=1/2S=1/2 XYZ model on the triangular lattice, and the general spin XYZ model. This work also offers a partial resolution to the long-standing conjecture that integrability is governed by the existence of local conserved quantities with small support. Our framework ensures that the nonintegrability of one-dimensional spin systems with translational symmetry can be verified algorithmically, independently of system size.
We present two directions of research in statistical mechanics of nonequilibrium onedimensional quantum systems. One is related to isolated integrable models, the other one to exactly solvable dissipatively driven spin chains. The device enabling their analysis is identified as the quantum group symmetry. In the framework of isolated systems we focus on the concept of integrable Floquet driven systems. In particular, we show how to build such systems out of the basic constituents of integrability structure. This allows substantiated and conclusive statements,for instance, about the transport phenomena. Despite their inherent relation, the presented integrable periodically driven models originate in two different ideas. On the one hand we have Trotterisations of integrable spin chains, concieved in an effort to better understand the dynamics of spin models. Then, there are systems originating in the attempts to solve quantum field theory in an algebraically closed form, for example, the quantum Hirota equation. The bulk of our consideration of isolated quantum systems consists of the construction of extensive conservation laws that either (i) constitute the generalised Gibbs ensemble and the hydrodynamic description of thermalisation after a quantum quench, or (ii) prevent the decay of current autocorrelations and thus characterise ideal transport in the system. The exact way in which these charges determine the dynamics stems from the symmetries of the quantum group representations. The second part of the exposition is dedicated to open quantum systems. Again we distinguish two formally related settings. The first one is that of a boundary driven quantum cellular automaton, our particular example being the integrable Trotterisation of the Heisenberg magnet. Starting from a repeated interaction protocol, in which the system under scrutiny is repeatedly coupled to the environment, we introduce the Kraus map as a general form of a dissipative time evolution of the density matrix. We then solve for its unique nonequilibrium steady state, using integrability structure of the model. The other setting is that of a dissipatively boundary driven spin chain in the continuous time. Here we present the recently developed formalism of inhomogeneous Lax structure. Using it we demonstrate the solvability of the XXZ and XYZ spin chains, acted upon by the Lindblad operators that polarise the boundary spins in arbitrary directions. The ansatz for the steady state is particularly interesting, since it exhibits a previously unknown integrability structure, differing from site to site in the spin chain. This structure can independently produce nontrivial conservation laws in an isolated spin chain with arbitrary boundary fields.
In this thesis, I study the local charges {Qk}\{Q_k\}{Qk​} in the one-dimensional Hubbard model, which is an integrable system used for theoretical studies of non-perturbative effects in strongly correlated electron systems. I obtained their explicit expressions in a closed form. An expression of the kkk-local charge QkQ_kQk​ for k>5k > 5k>5, where kkk denotes the support, had been unknown in previous studies. I find that QkQ_kQk​ is a linear combination of a particular kind of products of the hopping terms. I introduce a diagrammatic notation to represent these products efficiently, which enables the prediction of the general formula of the local charges. These linear combinations have non-trivial coefficients for k>5k > 5k>5, and some of the coefficients were proved to be identical to the generalized Catalan numbers, which also appear in the local charges of the Heisenberg chain. I derive the recursion equation for these coefficients. I also prove that the obtained local charges are exhaustive. Thus, any local charge is written as a linear combination of {Qk}\{Q_k\}{Qk​}. Although the completeness of the local charges is a natural conjecture in the transfer-matrix formulation, its proof has not been presented for almost all cases. Lastly, I emphasize that this is the first study demonstrating general explicit expressions of the local charges for an electron system that lacks the boost operator.
Generalized hydrodynamics is a recent theory that describes the large scale transport properties of one dimensional integrable models. At the heart of this theory lies an exact quantum-classical correspondence, which states that the flows of the conserved quantities are essentially quasiclassical even in the interacting quantum many body models. We provide the algebraic background to this observation, by embedding the current operators of the integrable spin chains into the canonical framework of Yang-Baxter integrability. Our construction can be applied in a large variety of models including the XXZ spin chains, the Hubbard model, and even in models lacking particle conservation such as the XYZ chain. Regarding the XXZ chain we present a simplified proof of the recent exact results for the current mean values, and explain how their quasiclassical nature emerges from the exact computations.
Introduction to Integrable Many-Body Systems II This is the second part of a three-volume introductory course about integrable systems of interacting bodies. The models of interest are quantum spin chains with nearest-neighbor interactions between spin operators, in particular Heisenberg spin-1/2 models. The Ising model in a transverse field, expressible as a quadratic fermion form by using the Jordan-Wigner transformation, is the subject of Sect. 12. The derivation of the coordinate Bethe ansatz for the XXZ Heisenberg chain and the determination of its absolute ground state in various regions of the anisotropy parameter are presented in Sect. 13. The magnetic properties of the ground state are explained in Sect. 14. Sect. 15 concerns excited states and the zero-temperature thermodynamics of the XXZ model. The thermodynamics of the XXZ Heisenberg chain is derived on the basis of the string hypothesis in Sect. 16; the thermodynamic Bethe ansatz equations are analyzed in high-temperature and low-temperature limits. An alternative derivation of the thermodynamics without using strings, leading to a non-linear integral equation determining the free energy, is the subject of Sect. 17. A nontrivial application of the Quantum Inverse Scattering method to the fully anisotropic XYZ Heisenberg chain is described in Sect. 18. Sect. 19 deals with integrable cases of isotropic spin chains with an arbitrary spin.