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arXiv:1902.03563 · Physical review. B./Physical review. B

Anomalous relaxation and multiple timescales in the quantum XY model with boundary dissipation

Shunyao Zhang, Ming Gong, Guang‐Can Guo, Zheng-Wei Zhou

Published Apr 30, 2020• 7 citations
arXivHTMLPDF

Abstract

The relaxation of a many-body system is still a challenging problem that has not been well understood. In this work we exactly calculate the dynamics of the quantum $XY$ model with boundary dissipation, in which the density matrix in terms of Majorana operators can be decoupled into independent subspaces ${\mathcal{K}}_{i}$ represented by different number of Majorana fermions. The relaxation is characterized by multiple timescales, and in the long-time limit it is determined by the single-particle relaxation process in a typical timescale ${T}^{*}$. For the bulk bands, we find ${T}^{*}\ensuremath{\propto}{N}^{3}/\ensuremath{\gamma}{n}^{2}$ in the weak dissipation limit and ${T}^{*}\ensuremath{\propto}\ensuremath{\gamma}{N}^{3}/{n}^{2}$ in the strong dissipation limit, where $N$ is the chain length, $\ensuremath{\gamma}$ is the dissipation rate, and $n$ is the band index. For the edge modes, ${T}^{*}\ensuremath{\propto}1/\ensuremath{\gamma}$ indicates the most vulnerable to dissipation in the long-chain limit. These results are counterintuitive because it means any weak dissipation can induce strong relaxation, while strong dissipation can induce weak relaxation. We find that these two limits correspond to two different physics, which are explained based on the first- and second-order perturbation theory in an equivalent non-Hermitian model. Finally, we show that even in the long-chain limit with weak dissipation, the relaxation may exhibit strong odd-even effect, which can be washed out by the strong dissipation. These results shed new insight into the dynamics of topological qubits in the environment, and the mechanism for many-body dissipation may also have universal significance in understanding the many-body dissipation in the other models.

DOI
10.1103/physrevb.101.155150

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The relaxation of a many-body system is still a challenging problem that has not been well understood. In this work we exactly calculate the dynamics of the quantum XYXYXY model with boundary dissipation, in which the density matrix in terms of Majorana operators can be decoupled into independent subspaces Ki{\mathcal{K}}_{i}Ki​ represented by different number of Majorana fermions. The relaxation is characterized by multiple timescales, and in the long-time limit it is determined by the single-particle relaxation process in a typical timescale T∗{T}^{*}T∗. For the bulk bands, we find T∗\ensuremath∝N3/\ensuremathγn2{T}^{*}\ensuremath{\propto}{N}^{3}/\ensuremath{\gamma}{n}^{2}T∗\ensuremath∝N3/\ensuremathγn2 in the weak dissipation limit and T∗\ensuremath∝\ensuremathγN3/n2{T}^{*}\ensuremath{\propto}\ensuremath{\gamma}{N}^{3}/{n}^{2}T∗\ensuremath∝\ensuremathγN3/n2 in the strong dissipation limit, where NNN is the chain length, \ensuremathγ\ensuremath{\gamma}\ensuremathγ is the dissipation rate, and nnn is the band index. For the edge modes, T∗\ensuremath∝1/\ensuremathγ{T}^{*}\ensuremath{\propto}1/\ensuremath{\gamma}T∗\ensuremath∝1/\ensuremathγ indicates the most vulnerable to dissipation in the long-chain limit. These results are counterintuitive because it means any weak dissipation can induce strong relaxation, while strong dissipation can induce weak relaxation. We find that these two limits correspond to two different physics, which are explained based on the first- and second-order perturbation theory in an equivalent non-Hermitian model. Finally, we show that even in the long-chain limit with weak dissipation, the relaxation may exhibit strong odd-even effect, which can be washed out by the strong dissipation. These results shed new insight into the dynamics of topological qubits in the environment, and the mechanism for many-body dissipation may also have universal significance in understanding the many-body dissipation in the other models.
We propose the Sachdev-Ye-Kitaev Lindbladian as a paradigmatic solvable model of dissipative many-body quantum chaos.It describes N strongly coupled Majorana fermions with random all-to-all interactions, with unitary evolution given by a quartic Hamiltonian and the coupling to the environment described by M quadratic jump operators, rendering the full Lindbladian quartic in the Majorana operators.Analytical progress is possible by developing a dynamical mean-field theory for the Liouvillian time evolution on the Keldysh contour.By disorder-averaging the interactions, we derive an (exact) effective action for two collective fields (Green's function and self-energy).In the large-N, large-M limit, we obtain the saddle-point equations satisfied by the collective fields, which determine the typical timescales of the dissipative evolution, particularly the spectral gap that rules the relaxation of the system to its steady state.We solve the saddle-point equations numerically and find that, for strong or intermediate dissipation, the system relaxes exponentially, with a spectral gap that can be computed analytically, while for weak dissipation, there are oscillatory corrections to the exponential relaxation.In this letter, we illustrate the feasibility of analytical calculations in strongly correlated dissipative quantum matter.
Understanding the dynamics of open quantum systems is intrinsically important to control and utilize quantum hardware in various applications. The generic behavior of purely dissipative open quantum many-body systems with kkk-local, normal dissipation processes can be investigated using random matrix theory, revealing different decay timescales of observables determined by their complexity, as shown by Wang et al. [K. Wang et al., Phys. Rev. Lett. 124, 100604 (2020)]. The time evolution of an open quantum system is entirely described by the eigenvalue spectrum of the Lindbladian, in which the hierarchy of decay timescales is reflected in the formation of well-separated eigenvalue clusters. Here, we extend the analysis of this spectrum to the presence of unitary dynamics using a random matrix model. In the case of strong dissipation, the unitary dynamics can be treated perturbatively and the kkk locality of the Hamiltonian determines how susceptible the spectrum is to such a perturbation. For the physically most relevant case of (dissipative) two-body interactions, the correction in the first order of the perturbation vanishes, leading to the relative robustness of the spectral features, which in turn implies the same robustness of the corresponding separation of decay timescales. We find the perturbative approach for the spectrum to yield an excellent agreement with exact-diagonalization results. For weak dissipation, the spectrum flows into clusters except for outlier eigenmodes, which we identify to be the local symmetries of the Hamiltonian. We support our analytic findings by numerical simulations and show that our generic model applies to a nonrandom Heisenberg chain.
Quantum many-body scars are special eigenstates that violate the eigenstate thermalization hypothesis while residing at finite energy density along with thermalizing eigenstates. The spin-1 XY model is known to host a family of such exceptional states originating from long-lived quasiparticle excitations that exhibit anomalously low entanglement entropy and long-time periodic revivals, resulting in weak ergodicity breaking. We study the stability of these scarred states against typical U(1) symmetry preserving perturbation in the XY chain. While perturbation theory can describe the deformed scar states at small system sizes, finite-size scaling of the perturbation matrix elements indicate that the scars ultimately thermalize in larger chains. Nonetheless, we demonstrate that the long-range order associated with the scars decays under the perturbation, and we estimate the relaxation timescale of oscillatory dynamics in certain local observables to be of order λ−2λ^{-2}λ−2, where λλλ is the perturbation strength.
We study symmetries and anomalies of dissipative Majorana chains described by Lindblad dynamics. Using the operator-state mapping, we represent the Lindbladian as a non-Hermitian operator on a doubled Hilbert space. In this formulation, the ket and bra contours admit strong fermion parities and strong translations, while their diagonal combinations define weak symmetries. For even and odd chain lengths, we diagnose anomalies by comparing symmetry algebras with and without weak fermion-parity defects. These algebras retain the lattice precursor of the modulo-888 anomaly of the closed Majorana chain and exhibit projective relations between strong fermion parity and weak translation, as well as between weak fermion parity and strong translation, which are intrinsic to open quantum systems. We establish the IR correspondence between the lattice symmetry algebras and those of the doubled Majorana CFT in four spin sectors, identifying chiral fermion parities emanated from the lattice translations. Finally, we develop the lattice bosonization of the dissipative chain. For even length with collective energy dephasing, combining the defect and no-defect sectors and imposing GSO projections gives rise to an Ising Lindbladian with non-invertible Kramers--Wannier symmetries descending from the anomalous strong translation and fermion parity of the Majorana chain. For odd length, constructing mutually commuting ket- and bra-sector spin operators instead yields an Ising Lindbladian with a Kramers--Wannier duality defect. Our results provide a controlled framework for extending lattice anomalies and non-invertible symmetries to open quantum systems.
We study the dissipative dynamics of a class of interacting “gamma-matrix” spin models coupled to a Markovian environment. For spins on an arbitrary graph, we construct a Lindbladian that maps to a non-Hermitian model of free Majorana fermions hopping on the graph with a background classical Z 2 gauge field. We show, analytically and numerically, that the steady states and relaxation dynamics are qualitatively independent of the choice of the underlying graph, in stark contrast to the Hamiltonian case. We also show that the exponentially many steady states provide a concrete example of mixed-state topological order, in the sense of strong-to-weak spontaneous symmetry breaking of a one-form symmetry. While encoding only classical information, the steady states still exhibit long-range quantum correlations. Afterward, we examine the relaxation processes toward the steady state by numerically computing decay rates, which we generically find to be finite, even in the dissipationless limit. However, we identify symmetry sectors where fermion-parity conservation is enhanced to fermion-number conservation, where we can analytically bound the decay rates and prove that they vanish in the limits of both infinitely weak and infinitely strong dissipation. Finally, we show that while the choice of coherent dynamics is very flexible, exact solvability strongly constrains the allowed form of dissipation. Our work establishes an analytically tractable framework to explore nonequilibrium quantum phases of matter and the relaxation mechanisms toward them.
The identification, description, and classification of topological features is an engine of discovery and innovation in several fields of physics. This research encompasses a broad variety of systems, from the integer and fractional Chern insulators in condensed matter, to protected states in complex photonic lattices in optics, and the structure of the QCD vacuum. Here, we introduce another playground for topology: the dissipative dynamics of pseudo-Hermitian many-body quantum systems. For that purpose, we study two different systems, the dissipative Sachdev-Ye-Kitaev (SYK) model, and a quantum chaotic dephasing spin chain. For the two different many-body models, we find the same topological features for a wide range of parameters suggesting that they are universal. In the SYK model, we identify four universality classes, related to pseudo-Hermiticity, characterized by a rectangular block representation of the vectorized Liouvillian that is directly related to the existence of an anomalous trace of the unitary operator implementing fermionic exchange. As a consequence of this rectangularization, we identify a topological index ν that only depends on symmetry. Another distinct consequence of the rectangularization is the observation, for any coupling to the bath, of purely real topological modes in the Liouvillian. The level statistics of these real modes agree with that of the corresponding random matrix ensemble and therefore can be employed to characterize the four topological symmetry classes. In the limit of weak coupling to the bath, topological modes govern the approach to equilibrium, which may enable a direct path for experimental confirmation of topology in dissipative many-body quantum chaotic systems.