Instanton crystal is a fascinating phase which is encountered when the minimum of the free energy corresponds to a configuration with an imaginary-time-dependent order parameter in the form of a chain of alternating instantons and anti-instantons. We present the results of the investigation of the real-time correlation functions of the order parameter in the instanton crystal phase. In order to obtain the correlation functions in real time, we formulate an original method of analytic continuation from imaginary times, which is easily adapted into an efficient numerical scheme for the computations. The resulting correlation functions exhibit nontrivial slowly decaying oscillations in real time, which is reminiscent of prethermal time crystals.Instanton crystal is a fascinating phase which is encountered when the minimum of the free energy corresponds to a configuration with an imaginary-time-dependent order parameter in the form of a chain of alternating instantons and anti-instantons. We present the results of the investigation of the real-time correlation functions of the order parameter in the instanton crystal phase. In order to obtain the correlation functions in real time, we formulate an original method of analytic continuation from imaginary times, which is easily adapted into an efficient numerical scheme for the computations. The resulting correlation functions exhibit nontrivial slowly decaying oscillations in real time, which is reminiscent of prethermal time crystals.

Physical review. B./Physical review. BApr 26, 20220 citations

Thermodynamic Quantum Time Crystals

K. B. Efetov

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.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.

arXiv:1902.07520Feb 20, 20194 citations

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

K. B. Efetov

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.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.

arXiv:1905.04128Physical review. B./Physical review. BDec 17, 20198 citations

Critical properties of the prethermal Floquet time crystal

Muath Natsheh, Andrea Gambassi, Aditi Mitra

The critical properties characterizing the formation of the Floquet time crystal in the prethermal phase are investigated analytically in the periodically driven $O(N)$ model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and the absence of long-range spatial order from the nontrivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\ensuremath{\rightarrow}\ensuremath{\infty}$, we determine the exponent $\ensuremath{\nu}$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $\ensuremath{\beta}$ characterizing the growth of the amplitude of the order parameter, as well as the initial-slip exponent $\ensuremath{\theta}$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $\ensuremath{\nu},\ensuremath{\beta},\ensuremath{\theta}$ are found to be identical to those in the absence of the drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period doubled are also found to oscillate in space at the wave vector $\ensuremath{\omega}/(2v), v$ being the velocity of the quasiparticles, and $\ensuremath{\omega}$ being the drive frequency.The critical properties characterizing the formation of the Floquet time crystal in the prethermal phase are investigated analytically in the periodically driven $O(N)$ model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and the absence of long-range spatial order from the nontrivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\ensuremath{\rightarrow}\ensuremath{\infty}$, we determine the exponent $\ensuremath{\nu}$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $\ensuremath{\beta}$ characterizing the growth of the amplitude of the order parameter, as well as the initial-slip exponent $\ensuremath{\theta}$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $\ensuremath{\nu},\ensuremath{\beta},\ensuremath{\theta}$ are found to be identical to those in the absence of the drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period doubled are also found to oscillate in space at the wave vector $\ensuremath{\omega}/(2v), v$ being the velocity of the quasiparticles, and $\ensuremath{\omega}$ being the drive frequency.

arXiv:2103.10818Physical review. B./Physical review. BJun 28, 202118 citations

Real-time correlation functions from imaginary-time evolution

P. E. Kornilovitch

The problem of calculating real-time correlation functions is formulated in terms of an imaginary-time partial differential equation. The latter is solved analytically for the perturbed harmonic oscillator and compared with the known exact result. The first order approximation for the short-time propagator is derived and used for numerical solution of the equation by a Monte Carlo integration. In general, the method provides a reformulation of the dynamic sign problem, and is applicable to any two-time correlation function including single-particle, density-density, current-current, spin-spin, and others. The prospects of extending the technique onto multi-dimensional problems are discussed.The problem of calculating real-time correlation functions is formulated in terms of an imaginary-time partial differential equation. The latter is solved analytically for the perturbed harmonic oscillator and compared with the known exact result. The first order approximation for the short-time propagator is derived and used for numerical solution of the equation by a Monte Carlo integration. In general, the method provides a reformulation of the dynamic sign problem, and is applicable to any two-time correlation function including single-particle, density-density, current-current, spin-spin, and others. The prospects of extending the technique onto multi-dimensional problems are discussed.

arXiv:cond-mat/0010100Oct 6, 20001 citation

Critical properties of the Floquet Time Crystal

Muath Natsheh, Andrea Gambassi, Aditi Mitra

The critical properties characterizing the formation of the Floquet time crystal are investigated analytically in the periodically driven $O(N)$ model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and absence of long-range spatial order from the non-trivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\to\infty$ , we determine the exponent $\nu$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $\beta$ characterizing the growth of the amplitude of the order-parameter, as well as the initial-slip exponent $\theta$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $\nu, \beta, \theta$ are found to be identical to those in the absence of a drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period-doubled, are also found to oscillate in space at the wave-vector $\omega/(2 v)$ , $v$ being the velocity of the quasiparticles, and $\omega$ being the drive frequency.The critical properties characterizing the formation of the Floquet time crystal are investigated analytically in the periodically driven $O(N)$ model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and absence of long-range spatial order from the non-trivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\to\infty$, we determine the exponent $\nu$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $\beta$ characterizing the growth of the amplitude of the order-parameter, as well as the initial-slip exponent $\theta$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $\nu, \beta, \theta$ are found to be identical to those in the absence of a drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period-doubled, are also found to oscillate in space at the wave-vector $\omega/(2 v)$, $v$ being the velocity of the quasiparticles, and $\omega$ being the drive frequency.

Mar 19, 20210 citations

The critical properties characterizing the formation of the Floquet time crystal in the prethermal phase are investigated analytically in the periodically driven

O(N)

model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and the absence of long-range spatial order from the nontrivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\ensuremath{\rightarrow}\ensuremath{\infty}$, we determine the exponent $\ensuremath{\nu}$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $\ensuremath{\beta}$ characterizing the growth of the amplitude of the order parameter, as well as the initial-slip exponent $\ensuremath{\theta}$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $\ensuremath{\nu},\ensuremath{\beta},\ensuremath{\theta}$ are found to be identical to those in the absence of the drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period doubled are also found to oscillate in space at the wave vector $\ensuremath{\omega}/(2v), v$ being the velocity of the quasiparticles, and $\ensuremath{\omega}$ being the drive frequency.

The critical properties characterizing the formation of the Floquet time crystal are investigated analytically in the periodically driven

O(N)

model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and absence of long-range spatial order from the non-trivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for

N\to\infty

, we determine the exponent

\nu

that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent

\beta

characterizing the growth of the amplitude of the order-parameter, as well as the initial-slip exponent

\theta

of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents

\nu, \beta, \theta

are found to be identical to those in the absence of a drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period-doubled, are also found to oscillate in space at the wave-vector

\omega/(2 v)

v

being the velocity of the quasiparticles, and

\omega

being the drive frequency.