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arXiv:1807.03444 · Physical Review A

Analytical solutions for a boundary-driven XY chain

Chu Guo, Dario Poletti

Published Nov 21, 2018• 26 citations
arXivHTMLPDF

Abstract

We study noninteracting fermionic systems dissipatively driven at their boundaries, focusing in particular on the case of a non-number-conserving Hamiltonian, which for example describes an $XY$ spin chain. We show that despite the lack of number conservation, it is possible to convert the problem of calculating the normal modes of the master equations and their corresponding rapidities, into diagonalizing simply an $L\ifmmode\times\else\texttimes\fi{}L$ tridiagonal bordered 2-Toeplitz matrix, where $L$ is the size of the system. Such structure of matrix allows us to further reduce the problem into solving a scalar trigonometric nonlinear equation for which we also show, in the case of an Ising chain, exact analytical explicit, and system size independent, solutions.

DOI
10.1103/physreva.98.052126

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We study noninteracting fermionic systems dissipatively driven at their boundaries, focusing in particular on the case of a non-number-conserving Hamiltonian, which for example describes an XYXYXY spin chain. We show that despite the lack of number conservation, it is possible to convert the problem of calculating the normal modes of the master equations and their corresponding rapidities, into diagonalizing simply an L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL tridiagonal bordered 2-Toeplitz matrix, where LLL is the size of the system. Such structure of matrix allows us to further reduce the problem into solving a scalar trigonometric nonlinear equation for which we also show, in the case of an Ising chain, exact analytical explicit, and system size independent, solutions.
We show how to solve a large class of Lindblad master equations for noninteracting particles on LLL sites. In the first portion we concentrate on bosonic particles, while in the second we will address fermionic particles. In both cases we show how to reduce the problem to diagonalizing an $L\ifmmode\times\else\texttimes\fi{}L$ non-Hermitian matrix. In particular, for boundary dissipative driving of a uniform chain, the matrix is a tridiagonal bordered Toeplitz matrix which can be solved analytically for the normal master modes and their relaxation rates (rapidities). In the regimes in which an analytical solution cannot be found, our approach can still provide a speedup in the numerical evaluation. For bosonic particles, we use this numerical method to study the relaxation gap at nonequilibrium phase transitions in a boundary driven bosonic ladder with synthetic gauge fields. We conclude by showing how to construct the nonequilibrium steady state. The analysis for fermionic particles closely follows that of bosons, but with important differences due to the different commutation rules.
We review recent progress on constructing non-equilibrium steady state density operators of boundary driven locally interacting quantum chains, where driving is implemented via Markovian dissipation channels attached to the chain's ends. We discuss explicit solutions in three different classes of quantum chains, specifically, the paradigmatic (anisotropic) Heisenberg spin- chain, the Fermi–Hubbard chain, and the Lai–Sutherland spin-1 chain, and discuss universal concepts which characterize these solutions, such as matrix product ansatz and a more structured walking graph state ansatz. The central theme is the connection between the matrix product form of nonequilibrium states and the integrability structures of the bulk Hamiltonian, such as the Lax operators and the Yang–Baxter equation. However, there is a remarkable distinction with respect to the conventional quantum inverse scattering method, namely addressing nonequilibrium steady state density operators requires non-unitary irreducible representations of Yang–Baxter algebra which are typically of infinite dimensionality. Such constructions result in non-Hermitian, and often also non-diagonalisable families of commuting transfer operators which in turn result in novel conservation laws of the integrable bulk Hamiltonians. For example, in the case of the anisotropic Heisenberg model, quasi-local conserved operators which are odd under spin reversal (or spin flip) can be constructed, whereas the conserved operators stemming from orthodox Hermitian transfer operators (via logarithmic differentiation) are all even under spin reversal.
The Lindblad master equation for an arbitrary quadratic system of n fermions is solved explicitly in terms of diagonalization of a 4 n × 4 n matrix, provided that all Lindblad bath operators are linear in the fermionic variables. The method is applied to the explicit construction of non-equilibrium steady states (NESS) and the calculation of asymptotic relaxation rates in the far from equilibrium problem of heat and spin transport in a nearest neighbour Heisenberg XY spin-1/2 chain in a transverse magnetic field.
The XXX spin- \frac{1}{2} 1 2 Heisenberg chain with non-diagonal boundary fields represents a cornerstone model in the study of integrable systems with open boundaries. Despite its significance, solving this model exactly has remained a formidable challenge due to the breaking of U(1) U ( 1 ) symmetry. Building on the off-diagonal Bethe Ansatz (ODBA), we derive a set of nonlinear integral equations (NLIEs) that encapsulate the exact spectrum of the model. For U(1) U ( 1 ) symmetric spin- \frac{1}{2} 1 2 chains such NLIEs involve two functions a(x) a ( x ) and \bar{a}(x) a ‾ ( x ) coupled by an integration kernel with short-ranged elements. The solution functions show characteristic features for arguments at some length scale which grows logarithmically with system size N N . In the case considered here the U(1) U ( 1 ) symmetry is broken by the non-diagonal boundary fields and the equations involve a novel third function c(x) c ( x ) , which captures the inhomogeneous contributions to the T T - Q Q relation in the ODBA. The kernel elements coupling this function to the standard ones are long-ranged and lead for the ground-state to a winding phenomenon. In \log(1+a(x)) log ( 1 + a ( x ) ) and \log(1+\bar a(x)) log ( 1 + a ‾ ( x ) )
We obtain exact formulas for the time-dependence of a few physical observables for the open XX spin chain with Lindbladian dynamics. Our analysis is based on the fact that the Lindblad equation for an arbitrary open quadratic system of N N fermions is explicitly solved in terms of diagonalization of a 4N\times4N 4 N × 4 N matrix called structure matrix by following the scheme of the third quantization. We mainly focus on the time-dependence of magnetization and spin current. As a short-time behavior at a given site, we observe the plateau regime except near the center of the chain. Basic features of this are explained by the light-cone structure created by propagations of boundary effects from the initial time, but we can explain their more detailed properties analytically using our exact formulas. On the other hand, after the plateau regime, the magnetization and spin current exhibit a slow decay to the steady state values described by the Liouvillian gap. We analytically establish its O(N^{-3}) O ( N − 3 ) scaling and also determine its coefficient.
The nonequilibrium steady state (NESS) of integrable spin chains experiencing strong boundary dissipation is accounted by introducing quasiparticles with a renormalized -- dissipatively dressed -- dispersion relation. This allows us to evaluate the spectrum of the NESS in terms of the Bethe ansatz equations for a related coherent system which has the same set of eigenstates, the so-called dissipation-projected Hamiltonian. We find explicit analytic expressions for the dressed energies of the XXX and XXZ models with effective, i.e., induced by the dissipation, diagonal boundary fields, which are U(1) invariant, as well as the XXZ and XYZ models with effective non-diagonal boundary fields. In all cases, the dissipative dressing generates an extra singularity in the dispersion relation, substantially altering the NESS spectrum with respect to the spectrum of the corresponding coherent model.