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arXiv:2409.03208 · Physical review. B./Physical review. B

Boundary dissipative spin chains with partial solvability inherited from system Hamiltonians

Chihiro Matsui, Naoto Tsuji

Published Dec 26, 2024• 3 citations
arXivHTMLPDF

Abstract

Partial solvability plays an important role in the context of statistical mechanics since it has turned out to be closely related to the emergence of quantum many-body scar states, i.e., exceptional energy eigenstates which do not obey the strong version of the eigenstate thermalization hypothesis. We show that partial solvability of a bulk (closed) system Hamiltonian can lead to that of an open quantum many-body system coupled to boundary dissipators under certain conditions. We propose two mechanisms that support partially solvable structures in boundary dissipative systems: The first one is based on the restricted spectrum generating algebra, while the second one is based on the Hilbert space fragmentation. From these structures, we derive exact eigenmodes of the Gorini-Kossakowski-Sudarshan-Lindblad equation for a family of quantum spin chain models with boundary dissipators, where we find various intriguing phenomena arising from the partial solvability of the open quantum systems, including persistent oscillations (quantum synchronization) and the existence of the matrix product operator symmetry. We discuss how the presence of solvable eigenmodes affects long-time behaviors of observables in boundary dissipative spin chains based on numerical simulations using the quantum trajectory method.

DOI
10.1103/physrevb.110.224306

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Partial solvability plays an important role in the context of statistical mechanics since it has turned out to be closely related to the emergence of quantum many-body scar states, i.e., exceptional energy eigenstates which do not obey the strong version of the eigenstate thermalization hypothesis. We show that partial solvability of a bulk (closed) system Hamiltonian can lead to that of an open quantum many-body system coupled to boundary dissipators under certain conditions. We propose two mechanisms that support partially solvable structures in boundary dissipative systems: The first one is based on the restricted spectrum generating algebra, while the second one is based on the Hilbert space fragmentation. From these structures, we derive exact eigenmodes of the Gorini-Kossakowski-Sudarshan-Lindblad equation for a family of quantum spin chain models with boundary dissipators, where we find various intriguing phenomena arising from the partial solvability of the open quantum systems, including persistent oscillations (quantum synchronization) and the existence of the matrix product operator symmetry. We discuss how the presence of solvable eigenmodes affects long-time behaviors of observables in boundary dissipative spin chains based on numerical simulations using the quantum trajectory method.
The Lindblad equation for open quantum systems is central to our understanding of coherence and entanglement in the presence of Markovian dissipation. In closed quantum systems Hilbert-space fragmentation is an effective mechanism for slowing decoherence in the presence of constrained interactions. We develop a general mechanism for operator-space fragmentation of mixed states, undergoing Lindbladian evolution generated by frustration-free Hamiltonians and Pauli-string jump operators. The interplay of generator algebras of dissipative and unitary dynamics leads to a hierarchical partitioning of operator and real space into dynamically disconnected subspaces, which we elucidate using the bond and commutant algebras of superoperators. This fragmentation fundamentally constrains information spreading in open systems and provides new mechanisms to control highly entangled quantum states and dynamics. Our approach yields two key advances. Firstly, we introduce frustration graphs in operator space as a compact representation to construct effective non-Hermitian Hamiltonians in individual fragments and diagnose their free-fermion solvability. Secondly, using these methods we uncover a range of universal dynamical regimes in Pauli-Lindblad models, exhibiting symmetry enriched quantum chaos and integrability in operator-space fragments. Furthermore, we show dissipation-driven phase transitions corresponding to exceptional points in the Lindbladian spectrum whose signatures are captured by spectral statistics and operator dynamics. These results establish operator-space fragmentation as a fundamental principle for open quantum systems, with immediate implications for quantum error correction, where protected subspaces could emerge naturally from fragmentation. Our framework provides a systematic approach to discover and characterize novel non-equilibrium phases in open quantum many-body systems.
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We introduce families of one-dimensional Lindblad equations describing open many-particle quantum systems that are exactly solvable in the following sense: (i) The space of operators splits into exponentially many (in system size) subspaces that are left invariant under the dissipative evolution; (ii) the time evolution of the density matrix on each invariant subspace is described by an integrable Hamiltonian. The prototypical example is the quantum version of the asymmetric simple exclusion process (ASEP) which we analyze in some detail. We show that in each invariant subspace the dynamics is described in terms of an integrable spin-1/2 XXZ Heisenberg chain with either open or twisted boundary conditions. We further demonstrate that Lindbladians featuring integrable operator-space fragmentation can be found in spin chains with arbitrary local physical dimensions.
We propose two strategies to construct a family of nonintegrable spin chains with exactly solvable subspaces based on the idea of quasiparticle excitations from the matrix product vacuum state [Haegeman et al., Phys. Rev. B 88, 075133 (2013)]. The first one allows the boundary generalization, while the second one makes it possible to construct the solvable subspace with interacting quasiparticles. Each generalization is realized by removing the assumption made in the conventional method [Moudgalya et al., Phys. Rev. B 102, 085120 (2020)], which is the frustration-free condition or the local orthogonality, respectively. We found that the structure of the embedded equally spaced energy spectrum is not violated by the diagonal boundaries, as long as quasiparticles are identical and noninteracting in the invariant subspace. On the other hand, we show that there exists a one-parameter family of nonintegrable Hamiltonians, which shows the perfectly embedded energy spectrum of the integrable spin chain. Surprisingly, the embedded energy spectrum does not change by varying the free parameter of the Hamiltonian. The constructed models weakly break ergodicity, in which strong ETH is expected to be violated.
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.
We address dissipative dynamics of the one-dimensional nearest-neighbour XX spin-1/21/21/2 chain governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. In the absence of dissipation, the model is integrable. We identify a broad class of dissipative terms that generically destroy integrability but leave the operator space of the model fragmented into an extensive number of dynamically disjoint subspaces of varying dimensions. In sufficiently small subspaces, the GKSL equation in the Heisenberg representation can be easily solved, sometimes in a closed analytical form. We provide an example of such an exact solution for a specific choice of dissipative terms. It is found that observables experience the Wannier-Stark localization in the corresponding operator subspace. As a result, the expectation values of the observables are linear combinations of essentially a few discrete decay modes, the long time dynamics being governed by the slowest mode. We examine the complex Liouvillian eigenvalue corresponding to this latter mode as a function of the dissipation strength. We find an exceptional point at a critical dissipation strength that separates oscillating and nonoscillating decay. We also describe a different type of dissipation that leads to a single decay mode in the whole operator subspace. Finally, we point out that our exact solutions of the GKSL equation entail exact solutions of the Schr\"odinger equation describing the quench dynamics in closed spin ladders dual to the dissipative spin chains.