Time glasses: Symmetry-broken chaotic phase with a finite gap

Topology in many-body physics usually emerges as a feature of equilibrium quantum states. We show that topological fingerprints can also appear in the relaxation rates of open quantum systems. To demonstrate this we consider one of the simplest models that has two topologically distinct phases in its ground state: the Kitaev chain model for the p-wave superconductor. After introducing dissipation to this model we estimate the Liouvillian gap in both strong and weak dissipative limits. Our results show that a nonzero superconducting pairing opens a Liouvillian gap that remains open in the limit of infinite system size. At strong dissipation this gap is essentially unaffected by the topology of the underlying Hamiltonian ground state. In contrast, when dissipation is weak, the topological phase of the Hamiltonian ground state plays a crucial role in determining the character of the Liouvillian gap. We find, for example, that in the topological phase this gap is completely immune to changes in the chemical potential. On the other hand, in the nontopological phase the Liouvillian gap is suppressed by a large chemical potential.

In this paper, we study the driven-dissipative p-spin models for $p\geq 2$. In thermodynamics limit, the equation of motion is derived by using a semiclassical approach. The long-time asymptotic states are obtained analytically, which exhibit multi-stability in some regions of the parameter space. The steady state is unique as the number of spins is finite. But the thermodynamic limit of the steady-state magnetization displays nonanalytic behavior somewhere inside the semiclassical multi-stable region. We find both the first-order and continuous dissipative phase transitions. As the number of spins increases, both the Liouvillian gap and magnetization variance vanish according to a power law at the continuous transition. At the first-order transition, the gap vanishes exponentially accompanied by a jump of magnetization in thermodynamic limit. The properties of transitions depend on the symmetry and semiclassical multistability, being qualitatively different among $p=2$, odd $p$ ($p\geq 3$) and even $p$ ($p\geq 4$).

It is highly nontrivial to what extent we can deduce the relaxation behavior of a quantum dissipative system from the spectral gap of the Liouvillian that governs the time evolution of the density matrix. We investigate the relaxation processes of a quantum dissipative system that exhibits the Liouvillian skin effect, which means that the eigenmodes of the Liouvillian are localized exponentially close to the boundary of the system, and find that the timescale for the system to reach a steady state depends not only on the Liouvillian gap Δ, but also on the localization length ξ of the eigenmodes. In particular, we show that the longest relaxation time τ that is maximized over initial states and local observables is given by τ∼Δ^{-1}(1+L/ξ) with L being the system size. This implies that the longest relaxation time can diverge for L→∞ without gap closing.

We study the classical analog of the quantum metric tensor and its scalar curvature for two well-known quantum physics models. First, we analyze the geometry of the parameter space for the Dicke model with the aid of the classical and quantum metrics and find that, in the thermodynamic limit, they have the same divergent behavior near the quantum phase transition, as opposed to their corresponding scalar curvatures which are not divergent there. On the contrary, under resonance conditions, both scalar curvatures exhibit a divergence at the critical point. Second, we present the classical and quantum metrics for the Lipkin-Meshkov-Glick model in the thermodynamic limit and find a perfect agreement between them. We also show that the scalar curvature is only defined on one of the system's phases and that it approaches a negative constant value. Finally, we carry out a numerical analysis for the system's finite sizes, which clearly shows the precursors of the quantum phase transition in the metric and its scalar curvature and allows their characterization as functions of the parameters and of the system's size.

We investigate the dynamics of the Su-Schrieffer-Heeger model with boundary dissipations described by Lindblad master equations and unravel distinct dynamical features in the topologically different phases of the underlying Hamiltonian. By examining the long-time damping dynamics, we uncover a dynamical duality phenomenon between the weak and strong dissipation region, which exists only in the topologically non-trivial phase, linked to the structure of the Liouvillian spectra,particularly the stripe closest to the steady state. When dissipation is confined to a single boundary, the dynamical duality phenomenon still exists. Under this condition, the Liouvillian gap fulfills an exponential size scaling relation in the topologically non-trivial phase and a power-law size scaling relation in the topologically trivial phase. Within the topologically non-trivial region, we identify the existence of boundary-localized dark states in the thermodynamical limit, which is responsible for the exponential size decay of Liouvillian gap.

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} $.

We study the stability of composite fermion fractional quantum Hall states in Harper-Hofstadter bands with Chern number $|C|>1$. We analyze the states of the composite fermion series for bosons with contact interactions and (spinless) fermions with nearest-neighbor interactions. We examine the scaling of the many-body gap as the bands are tuned to the effective continuum limit $n_\phi\to 1/|C|$. Near these points, the Hofstadter model realises large magnetic unit cells that yield bands with perfectly flat dispersion and Berry curvature. We exploit the known scaling of energies in the effective continuum limit in order to maintain a fixed square aspect ratio in finite-size calculations. Based on exact diagonalization calculations of the band-projected Hamiltonian, we show that almost all finite-size spectra yield the ground-state degeneracy predicted by composite fermion theory. We confirm that states at low ranks in the composite fermion hierarchy are the most robust and yield a clear gap in the thermodynamic limit. For bosons in $|C|=2$ and $|C|=3$ bands, our data for the composite fermion states are compatible with a finite gap in the thermodynamic limit. We also report new evidence for gapped incompressible states of fermions in $|C|>1$ bands, which have large entanglement gaps. For cases with a clear spectral gap, we confirm that the thermodynamic limit commutes with the effective continuum limit. We analyze the nature of the correlation functions for the Abelian composite fermion states and find that they feature $|C|^2$ smooth sheets. We examine two cases associated with a bosonic integer quantum Hall effect (BIQHE): For $\nu=2$ in $|C|=1$ bands, we find a strong competing state with a higher ground-state degeneracy, so no clear BIQHE is found in the band-projected Hofstadter model; for $\nu=1$ in $|C|=2$ bands, we present additional data confirming the existence of a BIQHE state.

Models of two-dimensional (2D) traps, with double-well structure in the third direction, for Bose-Einstein condensates are introduced with attractive or repulsive interactions between atoms. The models are based on systems of linearly coupled 2D Gross-Pitaevskii equations, where the coupling accounts for tunneling between the wells. Each well carries an optical lattice (OL) (stable 2D solitons cannot exist without OLs). The linear coupling splits each finite band gap in the spectrum of the single-component model into two subgaps. The main subject of the work is spontaneous symmetry breaking (SSB) in two-component 2D solitons and localized vortices (SSB was not considered before in 2D settings). Using variational approximation (VA) and numerical methods, we demonstrate that, in a system with attraction or repulsion, SSB occurs in families of symmetric or antisymmetric solitons (or vortices), respectively. The corresponding bifurcation destabilizes the original solution branch and gives rise to a stable branch of asymmetric solitons or vortices. The VA provides for an accurate description of the emerging branch of asymmetric solitons. In the model with attraction, all stable branches eventually terminate due to the onset of collapse. Stable asymmetric solitons in higher finite band gaps and vortices with a multiple topological charge are found too. The models also give rise to first examples of embedded solitons and embedded vortices (the states located inside Bloch bands) in two dimensions. In the linearly coupled system with opposite signs of the nonlinearity in the two cores, two distinct types of stable solitons and vortices are found, dominated by either the self-attractive component or the self-repulsive one. In the system with a mismatch between the two OLs, a pseudobifurcation is found: when the mismatch attains its largest value $(\ensuremath{\pi})$, the bifurcation does not happen, as branches of different solutions asymptotically approach each other, but fail to merge.

We investigate how particle pair creation and annihilation, within the quantum transverse XY model, affects the non-equilibrium steady state (NESS) and Liouvillian gap of the stochastic totally asymmetric exclusion process. By utilising operator quantization we formulate a perturbative description of the NESS. Furthermore, we estimate the Liouvillian gap by exploiting a Majorana canonical basis as the basis of super-operators. In this manner we show that the Liouvillian gap can remain finite in the thermodynamic limit provided the XY model anisotropy parameter remains non-zero. Additionally, we show that the character of the gap with respect to the anisotropy parameter differs depending on the phase of the XY model. The change of character corresponds to the quantum phase transition of the XY model.

In open quantum systems, the Liouvillian gap characterizes the relaxation time toward the steady state. However, accurately computing this quantity is notoriously difficult due to the exponential growth of the Hilbert space and the non-Hermitian nature of the Liouvillian superoperator. In this work, we propose a variational quantum algorithm for efficiently estimating the Liouvillian gap. By utilizing the Choi–Jamiołkowski isomorphism, we reformulate the problem as finding the first excitation energy of an effective non-Hermitian Hamiltonian. Our method employs variance minimization with an orthogonality constraint to locate the first excited state and adopts a two-stage optimization scheme to enhance convergence. Moreover, to address scenarios with degenerate steady states, we introduce an iterative energy-offset scanning technique. Numerical simulations on the dissipative XXZ model confirm the accuracy and robustness of our algorithm across a range of system sizes and dissipation strengths. These results demonstrate the promise of variational quantum algorithms for simulating open quantum many-body systems on near-term quantum hardware.

The gap of the Liouvillian spectrum gives the asymptotic decay rate of a quantum dissipative system, and therefore its inverse has been identified as the slowest relaxation time. Contrary to this common belief, we show that the relaxation time due to diffusive transports in a boundary dissipated many-body quantum system is determined not by the gap or low-lying eigenvalues of the Liouvillian but by superexponentially large expansion coefficients for Liouvillian eigenvectors with nonsmall eigenvalues at an initial state. This finding resolves an apparent discrepancy reported in the literature between the inverse of the Liouvillian gap and the relaxation time in dissipative many-body quantum systems.

We introduce the group-equivariant autoencoder (GE autoencoder), a deep neural network (DNN) method that locates phase boundaries by determining which symmetries of the Hamiltonian have spontaneously broken at each temperature. We use group theory to deduce which symmetries of the system remain intact in all phases, and then use this information to constrain the parameters of the GE autoencoder such that the encoder learns an order parameter invariant to these "never-broken" symmetries. This procedure produces a dramatic reduction in the number of free parameters such that the GE-autoencoder size is independent of the system size. We include symmetry regularization terms in the loss function of the GE autoencoder so that the learned order parameter is also equivariant to the remaining symmetries of the system. By examining the group representation by which the learned order parameter transforms, we are then able to extract information about the associated spontaneous symmetry breaking. We test the GE autoencoder on the 2D classical ferromagnetic and antiferromagnetic Ising models, finding that the GE autoencoder (1) accurately determines which symmetries have spontaneously broken at each temperature; (2) estimates the critical temperature in the thermodynamic limit with greater accuracy, robustness, and time efficiency than a symmetry-agnostic baseline autoencoder; and (3) detects the presence of an external symmetry-breaking magnetic field with greater sensitivity than the baseline method. Finally, we describe various key implementation details, including a quadratic-programming-based method for extracting the critical temperature estimate from trained autoencoders and calculations of the DNN initialization and learning rate settings required for fair model comparisons.

We study non-equilibrium stationary states of a cavity system consisting of many atoms interacting with a quantized cavity field mode, under a driving field in a dissipative environment. We derive a quantum master equation which is suitable for treating systems with a strong driving field and a strong atom–photon interaction. We do this by making use of the fact that the mean-field dynamics are exact in the thermodynamic limit thanks to a uniform coupling between atoms and photons. We find ordered states with symmetry-broken components of the photon field and atomic excitation driven by the external field. The mechanism by which these ordered states arise is discussed from the viewpoint of the quantum interference effect.

We study a multilane totally asymmetric simple exclusion process (TASEP) with narrow entrances under parallel update. The narrow entrances are modeled in this way: the entry of a particle is not allowed if the exit site of the previous lane is occupied. It is shown the results depend on the number of lanes, n. If n is an even number, the results are essentially the same as n=2: two symmetry-breaking phases--i.e., a high-density-low-density (HD-LD) phase and an asymmetric low-density-low-density phase--are identified. In contrast, if n is an odd number, a periodic structure is observed and the period is found to be proportional to the lane number n and system size L. It is also found when the injection rate alpha=1 that the seesaw phase observed in the case of n=2 disappears and the HD-LD phase or symmetric LD phase appears. Some mean-field calculations are also presented. We show that it is not possible to have high densities in two successive lanes and also not possible to have high density in one lane and low densities in all other lanes. For odd n, we have obtained the period T and it is in good agreement with simulations for a small removal rate beta, but deviates from simulation results for large beta because correlation is neglected in the mean-field approximation.

Recent experimental [Diao et al., J. Am. Chem. Soc.2011133, 3756.] and simulation results [Sear, J. Phys. Cond. Matt.201224, 052205.] are not consistent with a nucleation rate that is in the thermodynamic limit. This has consequences, if the rate is not in the thermodynamic limit, the time for nucleation will not necessarily scale as one over system size. Here, I show how to analyze data for nucleation times to test for the existence of a well-defined nucleation rate. I also show how to estimate the scaling of the nucleation time with the number of nucleation sites. The prediction is that the farther the system is from the thermodynamic limit, the more rapidly the nucleation time varies with system size. To make this prediction, I use extreme-value statistics. I also show how nucleation data can be analyzed to extract information on the heterogeneity in the surfaces on which nucleation is occurring.