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arXiv:2003.02856 · Physical Review Letters

Explicit Construction of Local Conserved Quantities in the XYZ Spin- 1/2 Chain

Yuji Nozawa, Kohei Fukai

Published Aug 27, 2020• 32 citations
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Abstract

We present a rigorous explicit expression for an extensive number of local conserved quantities in the XYZ spin-1/2 chain with general coupling constants. All the coefficients of operators in each local conserved quantity are calculated. We also confirm that our result can be applied to the case of the XXZ chain with a magnetic field in the z-axis direction.

DOI
10.1103/physrevlett.125.090602

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We present a rigorous explicit expression for an extensive number of local conserved quantities in the XYZ spin-1/2 chain with general coupling constants. All the coefficients of operators in each local conserved quantity are calculated. We also confirm that our result can be applied to the case of the XXZ chain with a magnetic field in the z-axis direction.
Matrix product operator representations for the local conserved quantities of the spin-1/21/21/2 XYZ chain
We present explicit matrix product operator (MPO) representations for the local conserved quantities of the spin-1/21/21/2 XYZ chain. Through these MPO representations, we simplify the coefficients appearing in the local conserved quantities originally derived by one of the authors, and reveal their combinatorial meaning: the coefficients prove to be a polynomial generalization of the Catalan numbers, given by weighted sums over monotone lattice paths. We also prove that a single recurrence relation for the coefficients guarantees the conservation law. We further show that these MPO representations can also be recovered from a product of two eight-vertex transfer matrices built from Baxter's R-matrix, providing the first direct connection between the R-matrix formalism and closed-form expressions for all local conserved quantities. We also obtain a simple Lax operator for the XYZ chain that, unlike Baxter's R-matrix, involves no elliptic functions.
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.
We extend T. Prosen's construction of quasilocal conserved quantities for the XXZ model [Phys. Rev. Lett. 106, 217206 (2011)] to the case of periodic boundary conditions. These quasilocal operators stem from a two-parameter transfer matrix which employs a highest-weight representation of the quantum group algebra inherent in the algebraic structure of the Bethe ansatz. In contrast with the open chain, where the conservation law is weakly violated by boundary terms, the quasilocal operators in the periodic chain exactly commute with the Hamiltonian and other local conserved quantities.
We extend T. Prosen's construction of quasilocal conserved quantities for the XXZ model [Phys. Rev. Lett. 106, 217206 (2011)] to the case of periodic boundary conditions. These quasilocal operators stem from a two-parameter transfer matrix which employs a highest-weight representation of the quantum group algebra inherent in the Yang-Baxter algebra. In contrast with the open chain, where the conservation law is weakly violated by boundary terms, the quasilocal operators in the periodic chain exactly commute with the Hamiltonian and other local conserved quantities.