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arXiv:2006.00746 · Progress of Theoretical and Experimental Physics

Quantum Ising chain with boundary dephasing

Naoyuki Shibata, Hosho Katsura

Published Aug 20, 2020• 23 citations
arXivHTMLPDF

Abstract

Abstract We study the quantum Ising chain with boundary dephasing. By doubling the Hilbert space, the model is mapped to the Su–Schrieffer–Heeger model with imaginary chemical potential at the edges. We show analytically and numerically that the Liouvillian gap, i.e. the inverse relaxation time of the model, scales with the system size $ N $ as $ N^{-3} $.

DOI
10.1093/ptep/ptaa131

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Abstract We study the quantum Ising chain with boundary dephasing. By doubling the Hilbert space, the model is mapped to the Su–Schrieffer–Heeger model with imaginary chemical potential at the edges. We show analytically and numerically that the Liouvillian gap, i.e. the inverse relaxation time of the model, scales with the system size NNN as N−3N^{-3}N−3.
We study a quantum Ising chain with tailored bulk dissipation, which can be mapped onto a non-Hermitian Ashkin-Teller model. By exploiting the Kohmoto-den Nijs-Kadanoff transformation, we further map it to a staggered XXZ spin chain with pure-imaginary anisotropy parameters. This allows us to study the eigenstates of the original Liouvillian in great detail. We show that the steady state in each parity sector is a completely mixed state. The uniqueness of each is proved rigorously. We then study the decay modes on the self-dual line corresponding to the uniform XXZ chain and obtain an exact formula for the Liouvillian gap ggg, the inverse relaxation time, in the thermodynamic limit. The gap ggg as a function of dissipation strength $\mathrm{\ensuremath{\Delta}}$ has a cusp, implying a kind of phase transition for the first decay mode.
Abstract Entanglement Hamiltonians provide the most comprehensive characterisation of entanglement in extended quantum systems. A key result in unitary quantum field theories is the Bisognano-Wichmann theorem, which establishes the locality of the entanglement Hamiltonian. In this work, our focus is on the non-Hermitian Su-Schrieffer-Heeger (SSH) chain. We study the entanglement Hamiltonian both in a gapped phase and at criticality. In the gapped phase we find that the lattice entanglement Hamiltonian is compatible with a lattice Bisognano-Wichmann result, with an entanglement temperature linear in the lattice index. At the critical point, we identify a new imaginary chemical potential term absent in unitary models. This operator is responsible for the negative entanglement entropy observed in the non-Hermitian SSH chain at criticality.
I study the universal finite-size scaling function for the lowest gap of the quantum Ising chain with a one-parameter family of ``defect'' boundary conditions, which includes periodic, open, and antiperiodic boundary conditions as special cases. The universal behavior can be described by the Majorana fermion field theory in 1+11+11+1 dimensions, with the mass proportional to the deviation from the critical point. Although the field theory appears to be symmetric with respect to the inversion of the mass (Kramers-Wannier duality), the actual gap is asymmetric, reflecting the spontaneous symmetry breaking in the ordered phase which leads to the two-fold ground-state degeneracy in the thermodynamic limit. The asymptotic ground-state degeneracy in the ordered phase is realized by (i) formation of a bound state at the defect (except for the periodic/antiperiodic boundary condition) and (ii) effective reversal of the fermion number parity in one of the sectors (except for the open boundary condition), resulting in a rather nontrivial crossover ``phase'' diagram in the space of the boundary condition (defect strength) and mass.
We study the boundary-dissipated transverse field Ising model described by a Lindblad master equation and exactly solve its Liouvillian spectrum in the whole parameter space. By mapping the Liouvillian into a Su-Schrieffer-Heeger model with imaginary boundary potentials under a parity constraint, we solve the rapidity spectrum analytically and thus construct the Liouvillian spectrum strictly with a parity constraint condition. Our results demonstrate that the Liouvillian spectrum displays four different structures, which are characterized by different numbers of segments. By analyzing the properties of rapidity spectrum, we can determine the phase boundaries between different spectrum structures analytically and prove the Liouvillian gap fulfilling a duality relation in the weak and strong dissipation region. Furthermore, we unveil the existence of a dynamical duality, i.e., the long-time relaxation dynamics exhibits almost the same dynamical behavior in the weak and strong dissipation region as long as the duality relation holds true.
We study the dynamics of entanglement in the quantum Ising chain with dephasing dissipation in a Lindblad master equation form. We consider two unravelings which preserve the Gaussian form of the state, allowing us to address large system sizes. The first unraveling gives rise to a quantum-state-diffusion dynamics, while the second one describes a specific form of quantum-jump evolution, suitably constructed to preserve Gaussianity. In the first case we find a crossover from area-law to logarithm-law entanglement scaling and draw the related phase diagram. In the second case we only find logarithm-law scaling, remarking on the different entanglement behavior for different unravelings of the same Lindblad equation. Finally, we compare these outcomes with the predictions of a non-Hermitian Hamiltonian evolution, finding conflicting results.
Abstract We investigate the thermodynamic properties of finite Su–Schrieffer–Heeger chains in thermal equilibrium at fixed temperature and chemical potential. Using the canonical and grand canonical ensembles, we calculate the energy density, particle number density, entropy, and heat capacity as functions of temperature, chemical potential, and hopping asymmetry. Our analysis reveals a double-peak anomaly in the heat capacity for non-dimerized configurations, in which two maxima are separated by a local minimum. We identify this feature as a Schottky-type thermal crossover governed by two distinct energy scales. The anomaly is most pronounced when the chemical potential is comparable to the band scale, and its two maxima separate further as the hopping asymmetry increases and the chain length grows, reflecting the widening gap between the two energy scales and the increasing density of states. We demonstrate that while the topological properties are determined by boundary states, the bulk thermodynamic behavior exhibits a rich crossover structure that can be tuned through the hopping parameter ratio. These findings provide insights into the interplay between topology, finite-size effects, and thermal fluctuations in one-dimensional topological systems, with potential implications for experimental realizations in cold atoms, photonic systems, and topoelectrical circuits.