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arXiv:1905.04128 · Physical review. B./Physical review. B

Mean-field thermodynamic quantum time-space crystal: Spontaneous breaking of time-translation symmetry in a macroscopic fermion system

K. B. Efetov

Published Dec 17, 2019• 8 citations
arXivHTMLPDF

Abstract

A model demonstrating the existence of a thermodynamically stable quantum time-space crystal has been proposed and studied. This state is characterized by an order parameter periodic in both real and imaginary times. The average of the order parameter over phases of the oscillations vanishes but correlation functions of two or more order parameters show nondecaying oscillations. An alternative interpretation of the results is based on the concept of an operator order parameter introduced for this purpose. The model studied here has been suggested previously, in particular, for describing the pseudogap state in superconducting cuprates. Although many properties of the time-space crystal considered here are close to those of a well known DDW state, static magnetic moments oscillating at $(\ensuremath{\pi},\ensuremath{\pi})$ do not exist. Instead, $\ensuremath{\delta}$ peaks at finite energies are predicted in the cross-section of inelastic spin-polarized neutron scattering.

DOI
10.1103/physrevb.100.245128

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A model demonstrating the existence of a thermodynamically stable quantum time-space crystal has been proposed and studied. This state is characterized by an order parameter periodic in both real and imaginary times. The average of the order parameter over phases of the oscillations vanishes but correlation functions of two or more order parameters show nondecaying oscillations. An alternative interpretation of the results is based on the concept of an operator order parameter introduced for this purpose. The model studied here has been suggested previously, in particular, for describing the pseudogap state in superconducting cuprates. Although many properties of the time-space crystal considered here are close to those of a well known DDW state, static magnetic moments oscillating at (\ensuremathπ,\ensuremathπ)(\ensuremath{\pi},\ensuremath{\pi})(\ensuremathπ,\ensuremathπ) do not exist. Instead, \ensuremathδ\ensuremath{\delta}\ensuremathδ peaks at finite energies are predicted in the cross-section of inelastic spin-polarized neutron scattering.
Investigation of states with a periodic time dependence of physical quantities attracts a considerable interest now. Although it has been proposed initially that such states (coined Quantum Time Crystals) might be macroscopic and thermodynamically stable, results of a more careful study of the problem seemed to indicate that quantum time crystals could be realized only in systems out of equilibrium. Here we show that, in contrast to the general belief, thermodynamically stable macroscopic quantum time crystals can exist. The order parameter of this new state of matter is periodic in both real and imaginary time but its average over the phase of the oscillations equals zero. At the same time, correlation functions characterizing physical quantities oscillate periodically in time without any decay. An alternative interpretation of the results is based on a concept of an operator order parameter. Calculations are performed for a rather general microscopic model that may in particular be suitable for describing the pseudogap state in superconducting cuprates.
Quantum field theories of many-body systems predict ``instantonic crystals'' connecting degenerate thermodynamic states with classically broken symmetries. The instantons break continuous translation symmetry of Euclidean space along the axis of imaginary time. The possibility for the emergence of ``thermodynamic quantum time crystals'' that extremize the Euclidean action of the system appeals to the imagination. However, the authors prove here that at least two general classes of ``instantonic crystals'', connecting states of Fermi systems with charge, spin, or superconducting orders can constitute neither thermodynamic equilibrium nor the ground state.
Temperature is a fundamental thermodynamic variable for matter. Physical observables are often found to either increase or decrease with it, or show a non-monotonic dependence with peaks signaling underlying phase transitions or anomalies. Statistical field theory has established connection between temperature and time: a quantum ensemble with inverse temperature β is formally equivalent to a dynamic system evolving along an imaginary time from 0 to iβ in the space one dimension higher. Here we report that a gas of hard-core bosons interacting with a thermal bath manifests an unexpected temperature-periodic oscillation of its macroscopic observables, arising from the microscopic origin of space-time locked translational symmetry breaking and crystalline ordering. Such a temperature crystal, supported by quantum Monte Carlo simulation, generalizes the concept of purely spatial density-wave order to the imaginary time axis for Euclidean action.
Discrete time crystals exhibit robust subharmonic oscillations, spontaneously breaking the discrete time-translation symmetry of a periodic drive. Since their emergence, the question of whether other types of temporal many-body order can exist has attracted considerable attention. Here we identify discrete time liquids -- a new form of spatiotemporal order characterized by short-range temporal coherence as a result of time-solitons. We study a two-dimensional spin system coupled to a finite-temperature cavity, whose fluctuations provide a noisy, approximately periodic drive with only finite temporal coherence. Remarkably, the spins nevertheless form a metastable state with robust period-doubled dynamics over an extended parameter regime. Unlike a time crystal, this state has only finite temporal coherence: individual trajectories stochastically switch between the two symmetry-related temporal configurations of the period-doubled response. We identify these switches as time-solitons: topological defects in temporal order that preserve local period doubling while destroying long-range temporal coherence, thereby producing the time liquid state. We propose a minimal model which reproduces the distribution of temporal correlations of the many-body system, suggesting a general description of the time liquid as a time crystal with a finite density of time-solitons. Because these solitons can occur at arbitrary times, ensemble averaging restores continuous time-translation symmetry, raising the intriguing possibility that discrete time liquids can exist in thermal equilibrium.
Dissipative time crystals can appear in spin systems, when the Z2{Z}_{2}Z2​ symmetry of the Hamiltonian is broken by the environment, and the square of total spin operator S2{S}^{2}S2 is conserved. In this paper, we relax the latter condition and show that time-translation-symmetry-breaking collective oscillations persist, in the thermodynamic limit, even in the absence of spin symmetry. We engineer an ad hoc Lindbladian using power-law-decaying spin operators and show that time-translation-symmetry breaking appears when the decay exponent obeys $0<\ensuremath{\eta}\ensuremath{\le}1$. This model shows a surprisingly rich phase diagram, including the time-crystal phase as well as first-order, second-order, and continuous transitions of the fixed points. We study the phase diagram and the magnetization dynamics in the mean-field approximation. We prove that this approximation is quantitatively accurate, when $0<\ensuremath{\eta}<1$ and the thermodynamic limit is taken, because the system does not develop sizable quantum fluctuations, if the Gaussian approximation is considered.
Abstract Time-translation symmetry breaking is a mechanism for the emergence of non-stationary many-body phases, so-called time-crystals, in Markovian open quantum systems. Dynamical aspects of time-crystals have been extensively explored over the recent years. However, much less is known about their thermodynamic properties, also due to the intrinsic nonequilibrium nature of these phases. Here, we consider the paradigmatic boundary time-crystal system, in a finite-temperature environment, and demonstrate the persistence of the time-crystalline phase at any temperature. Furthermore, we analyze thermodynamic aspects of the model investigating, in particular, heat currents, power exchange and irreversible entropy production. Our work sheds light on the thermodynamic cost of sustaining nonequilibrium time-crystalline phases and provides a framework for characterizing time-crystals as possible resources for, e.g. quantum sensing. Our results may be verified in experiments, for example with trapped ions or superconducting circuits, since we connect thermodynamic quantities with mean value and covariance of collective (magnetization) operators.