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The notorious fermion sign problem, arising from fermion statistics, constitutes one of the main obstacles of deciphering quantum many-body systems by numerical approach. The progress in overcoming sign problem will definitely lead to a great leap in various areas of modern physics. Here, by deviating from the conventional cognition that nonequilibrium studies should be more complicated than equilibrium cases, we propose an innovative framework based on nonequilibrium critical dynamics to preempt sign problem and investigate quantum critical point in fermionic model through numerically exact quantum Monte Carlo (QMC) simulation. By virtue of universal scaling theory of imaginary-time relaxation dynamics, we demonstrate that accurate critical point and critical exponents can be obtained in the short-time stage, in which the sign problem is not severe such that the QMC is accessible. After confirming the effectiveness of the method in two typical interacting fermionic models featuring Dirac quantum critical point (QCP), we for the first time reveal the quantum phase diagram in the Hubbard model hosting $\rm SU(3)$-symmetric Dirac fermions, and find that the QCP between Dirac semi-metal and a $\lambda_8$-antiferromagnetic phase belongs to a new universality class different from the previously known Gross-Neveu transitions., Comment: 7+13 pages

Various exotic phenomena emerge in non-equilibrium quantum many-body systems. The Mpemba effect, denoting the situation where a hot system freezes faster than the colder one, is a counterintuitive non-equilibrium phenomenon that has attracted enduring interest for more than half a century. In this Letter, we report a novel phenomenon of the Mpemba effect in the imaginary-time relaxation dynamics in quantum many-body systems, dubbed as imaginary-time Mpemba effect (ITME). Through numerically exact quantum Monte-Carlo (QMC) simulation, we unambiguously demonstrate that in different classes of interacting quantum models, the initial states with higher energy are relaxed faster than lower-energy initial states in the process of imaginary-time relaxation. The emergence of ITME is intimately associated with the low-energy excitations in quantum many-body systems. More crucially, since imaginary-time dynamics is broadly applied in numerical simulation on the quantum many-body ground states, the discovery of ITME potentially provides a new pathway to expedite the quantum many-body computation, particularly for QMC involving the sign problem., Comment: 4.5+8 pages, 4+6 figures

The nonequilibrium dynamics of quantum many-body systems have attracted growing attention due to various intriguing phenomena absent in equilibrium physics. One famous example is the quantum Mpemba effect, where the subsystem symmetry is restored faster under a symmetric quench from a more asymmetric initial state. The quantum Mpemba effect has been extensively studied in integrable and chaotic systems. In this Letter, we investigate symmetry restoration and quantum Mpemba effect in many-body localized systems with various initial states. We reveal that the symmetry can still be fully restored in many-body localization phases without approaching thermal equilibrium. Furthermore, we demonstrate that the presence of the quantum Mpemba effect is universal for any initial tilted product state, contrasting to the cases in the chaotic systems where the presence of the quantum Mpemba effect relies on the choice of initial states. We also provide a theoretical analysis of symmetry restoration and quantum Mpemba effects with the help of the effective model for many-body localization. This Letter not only sheds light on extending the quantum Mpemba effect to more non-equilibrium settings but also contributes to a deeper understanding of the many-body localization., Comment: 23 pages (including supplemental materials), 11 figures

Proposed as an elegant symmetry relating bosons and fermions, spacetime supersymmetry (SUSY) has been actively pursued in both particle physics and emergent phenomena in quantum critical points (QCP) of topological quantum materials. However, how SUSY casts the light on nonequilibrium dynamics remains open. In this letter, we investigate the Kibble-Zurek dynamics across a QCP with emergent $\mathcal{N}=2$ spacetime SUSY between the Dirac semimetal and a superconductor through large-scale quantum Monte Carlo simulation. The scaling behaviors in the whole driven process are uncovered to satisfy the full finite-time scaling (FTS) forms. More crucially, we demonstrate that the emergent SUSY manifests in the intimate relation between the FTS behaviors of fermionic and bosonic observables, namely the fermions and bosons acquire the identical anomalous dimensions. Our work not only brings a fundamental new ingredient into the critical theory with SUSY, but also provide the theoretical guidance to experimental detect of QCP with emergent SUSY from the perspectives of Kibble-Zurek mechanism and FTS., Comment: 7+2 pages, 3 figures

In conventional quantum critical point (QCP) characterized by order parameter fluctuations, the celebrated Kibble-Zurek mechanism (KZM) and finite-time scaling (FTS) theory provide universal descriptions of the driven critical dynamics. However, in strongly correlated fermionic systems where gapless fermions are usually present in vicinity of QCP, the driven dynamics has rarely been explored. In this Letter, we investigate the driven critical dynamics in two-dimensional Dirac systems, which harbor semimetal and Mott insulator phases separated by the QCP triggered by the interplay between fluctuations of gapless Dirac fermions and order-parameter bosons. By studying the evolution of physical quantities for different driving rates through large-scale quantum Monte Carlo simulation, we confirm that the driven dynamics is described by the FTS form. Accordingly, our results significantly generalize the KZM theory by relaxing its requirement for a gapped initial state to the system accommodating gapless Dirac fermionic excitation. Through successfully extending the KZM and FTS theory to Dirac QCP, our work not only brings new fundamental perspective into the nonequilibrium critical dynamics, but also provides a novel theoretical approach to fathom quantum critical properties in fermionic systems., Comment: 9+3 pages, 5+2 figures

We study the imaginary-time relaxation critical dynamics of the Neel-paramagnetic quantum phase transition in the two-dimensional (2D) dimerized S = 1/2 Heisenberg model. We focus on the scaling correction in the short-time region. A unified scaling form including both short-time and finite-size corrections is proposed. According to this full scaling form, improved short-imaginary-time scaling relations are obtained. We numerically verify the scaling form and the improved short-time scaling relations for different initial states using projector quantum Monte Carlo algorithm., Comment: 10 pages, 8 figures

Entanglement asymmetry, which serves as a diagnostic tool for symmetry breaking and a proxy for thermalization, has recently been proposed and studied in the context of symmetry restoration for quantum many-body systems undergoing a quench. In this Letter, we investigate symmetry restoration in various symmetric random quantum circuits, particularly focusing on the U(1) symmetry case. In contrast to non-symmetric random circuits where the U(1) symmetry of a small subsystem can always be restored at late times, we reveal that symmetry restoration can fail in U(1)-symmetric circuits for certain weak symmetry-broken initial states in finite-size systems. In the early-time dynamics, we observe an intriguing quantum Mpemba effect implying that symmetry is restored faster when the initial state is more asymmetric. Furthermore, we also investigate the entanglement asymmetry dynamics for SU(2) and $Z_{2}$ symmetric circuits and identify the presence and absence of the quantum Mpemba effect for the corresponding symmetries, respectively. A unified understanding of these results is provided through the lens of quantum thermalization with conserved charges., Comment: 4.5 pages, 5 figures, and Supplemental Material

Universal critical properties can manifest themselves not only in spatial but also in temporal directions. It has been found that critical point with emergent symmetry exhibits intriguing spatial critical properties characterized by two divergent length scales, attracting long-term investigations. However, how the temporal critical properties are affected by emergent symmetry is largely unknown. Here we study the nonequilibrium critical dynamics in the three-dimensional ($3$D) clock model, whose critical point has emergent $U(1)$ symmetry. We find that in contrast to the magnetization $M$, whose relaxation process is described by the usual dynamic exponent $z$ of the $3$D XY universality class, the angular order parameter $\phi_q$ shows a remarkable two-stage evolution characterized by different dynamic critical exponents. While in the short-time stage the relaxation dynamics is governed by $z$, in the long-time stage the dynamics is controlled by a new dynamic exponent $z'$. Further scaling analyses confirm that $z'$ is an indispensable dynamic critical exponent. Our results may be detected in the hexagonal RMnO$_3$ (R$=$rare earth) materials experimentally., Comment: 12 pages, 6 figures

Quantum criticality within Dirac fermions harbors a plethora of exotic phenomena, attracting sustained attention in the past decades. Nevertheless, the nonequilibrium dynamics therein has rarely been studied. To fill in the gap, we explore the imaginary-time relaxation dynamics in a typical Dirac quantum criticality belonging to chiral Heisenberg universality class. Performing large-scale quantum Monte Carlo simulation, we unveil rich nonequilibrium critical phenomena from different initial states. Particularly, a new dynamic exponent characterizing the non-stationary evolution in the short-time state is determined as $\theta=-0.84(4)$, in sharp contrast with the prevalent belief that $\theta$ is positive as demonstrated in classical cases. Furthermore, we propose a universal dynamic scaling theory governing the fruitful nonequilibrium properties in Dirac quantum criticality. Armed with the scaling theory, we develop a new framework to investigate fermionic quantum criticality based on short-time dynamics, paving a promising avenue to fathoming quantum criticality in diverse fermionic systems with high efficiency., Comment: 7+5 pages,5+4 figures

While the zeros of complex partition functions, such as Lee-Yang zeros and Fisher zeros, have been pivotal in characterizing temperature-driven phase transitions, extending this concept to zero temperature remains an open question. In this work, we propose a solution to this issue by calculating the imaginary-temperature zeros (ITZs), which are defined as the roots of the imaginary-temperature partition function. We illustrate the analytical properties of ITZs in the transverse-field Ising chain, showing that the ITZs' distribution can distinguish between various phases and signify the critical exponents. Universal singular behaviors manifest in such quantities as the edge density of ITZs and the magnetization, with the scaling exponents remarkably differing from those in Lee-Yang theory. We further illuminate the consistency between ITZs and the zeros of the spectral form factor, which offers a practical path for the experimental detection of ITZs., Comment: 10 pages, 10 figures

Quantum computers promise a highly efficient approach to investigate quantum phase transitions, which describe abrupt changes between different ground states of many-body systems. At quantum critical points, the divergent correlation length and entanglement entropy render the ground state preparation difficult. In this work, we explore the imaginary-time evolution for probing the universal critical behavior as the universal information of the ground state can be extracted in the early-time relaxation process. We propose a systematic and scalable scheme to probe the universal behaviors via imaginary-time critical dynamics on quantum computers and demonstrate the validness of our approach by both numerical simulation and quantum hardware experiments. With the full form of the universal scaling function in terms of imaginary time, system size, and circuit depth, we successfully probe the universality by scaling analysis of the critical dynamics at an early time and with shallower quantum circuit depth. Equipped with quantum error mitigation, we also confirm the expected scaling behavior from experimental results on a superconducting quantum processor which stands as the first experimental demonstration on universal imaginary-time quantum critical dynamics., Comment: 5 pages, 4 figures and supplemental materials

Local constraint in the lattice gauge theory provides an exotic mechanism that facilitates the disorder-free localization. However, the understanding of nonequilibrium dynamics in the non-Hermitian lattice gauge model remains limited. Here, we investigate the quench dynamics in a system of spinless fermions with nonreciprocal hopping in the $\mathbb{Z}_2$ gauge field. By employing a duality mapping, we systematically explore the non-Hermitian skin effect, localization-delocalization transition, and real-complex transition. Through the identification of diverse scaling behaviors of quantum mutual information for fermions and spins, we propose that the non-Hermitian quantum disentangled liquids exist both in the localized and delocalized phases, the former originates from the $\mathbb{Z}_2$ gauge field and the latter arises from the non-Hermitian skin effect. Furthermore, we demonstrate that the nonreciprocal dissipation causes the flow of quantum information. Our results provide valuable insights into the nonequilibrium dynamics in the gauge field, and may be experimentally validated using quantum simulators., Comment: 11+4pages, 6+4 figures; accepted by Communications Physics

Deconfined quantum criticality (DQC) arises from fractionalization of quasi-particles and leads to fascinating behaviors beyond the Landau-Ginzburg-Wilson (LGW) description of phase transitions. The unusual aspects of DQC in equilibrium also suggest exotic phenomena out of equilibrium, which are still largely unexplored. Here we study manifestations of DQC when driving (quantum annealing) a two-dimensional quantum magnet through a critical point separating antiferromagnetic and spontaneously dimerized ground states. Numerical simulations show that the celebrated Kibble-Zurek scaling (KZS) mechanism is inadequate for describing the annealing process. To explain our results, we introduce the concept of dual asymmetric KZS, where a deconfinement time enters in addition to the conventional relaxation time and the scaling also depends on the direction in which the system is driven through the critical point according to a duality principle connecting the topological defects in the two phases. These defects -- spinons in the dimerized phase and space-time hedgehogs in the antiferromagnetic phase -- require a much longer time scale for equilibration than the amplitude of the order parameter. Beyond the new insights into DQC, our scaling approach provides a new window into out-of-equilibrium criticality with multiple length and time scales., Comment: 15 pages, 9 figures

In this work, we explore the driven dynamics of the one-dimensional ($1$D) localization transitions. By linearly changing the strength of disorder potential, we calculate the evolution of the localization length $\xi$ and the inverse participation ratio (IPR) in a disordered Aubry-Andr\'{e} (AA) model, and investigate the dependence of these quantities on the driving rate. At first, we focus on the limit in the absence of the quasiperiodic potential. We find that the driven dynamics from both ground state and excited state can be described by the Kibble-Zurek scaling (KZS). Then, the driven dynamics near the critical point of the AA model is studied. Here, since both the disorder and the quasiperiodic potential are relevant directions, the KZS should include both scaling variables. Our present work not only extends our understanding of the localization transitions but also generalize the application of the KZS., Comment: 7 pages, 6 figures

In this paper, we explore quantum criticality in the disordered Aubry-Andr\'{e} (AA) model. For the pure AA model, it is well-known that it hosts a critical point separating an extended phase and a localized insulator phase by tuning the strength of the quasiperiodic potential. Here we unearth that the disorder strength $\Delta$ contributes an independent relevant direction near the critical point of the AA model. Our scaling analyses show that the localization length $\xi$ scales with $\Delta$ as $\xi\propto\Delta^{-\nu_\Delta}$ with $\nu_\Delta$ a new critical exponent, which is estimated to be $\nu_\Delta\approx0.46$. This value is remarkably different from the counterparts for both the pure AA model and the Anderson model. Moreover, rich critical phenomena are discovered in the critical region spanned by the quasiperiodic and the disordered potentials. In particular, in the extended phase side, we show that the scaling theory satisfy a hybrid scaling form as a result of the overlap between the critical regions of the AA model and the Anderson localization., Comment: 7 pages, 7 figures

We explore the imaginary-time relaxation dynamics near quantum critical points with semi-ordered initial states. Different from the case with homogeneous ordered initial states, in which the order parameter $M$ decays homogeneously as $M\propto \tau^{-\beta/\nu z}$, here $M$ depends on the location $x$, showing rich scaling behaviors. Similar to the classical relaxation dynamics with an initial domain wall in Model A, which describes the purely dissipative dynamics, here as the imaginary time evolves, the domain wall expands into an interfacial region with growing size. In the interfacial region, the local order parameter decays as $M\propto \tau^{-\beta_1/\nu z}$, with $\beta_1$ being an additional dynamic critical exponent. Far away from the interfacial region the local order parameter decays as $M\propto \tau^{-\beta/\nu z}$ in the short-time stage, then crosses over to the scaling behavior of $M\propto \tau^{-\beta_1/\nu z}$ when the location $x$ is absorbed in the interfacial region. A full scaling form characterizing these scaling properties is developed. The quantum Ising model in both one and two dimensions are taken as examples to verify the scaling theory. In addition, we find that for the quantum Ising model the scaling function is an analytical function and $\beta_1$ is not an independent exponent., Comment: 7 pages, 5 figures

Emergent symmetry is one of the characteristic phenomena in deconfined quantum critical point (DQCP). As its nonequilibrium generalization, the dual dynamic scaling was recently discovered in the nonequilibrium imaginary-time relaxation dynamics in the DQCP of the $J$-$Q_3$ model. In this work, we study the nonequilibrium imaginary-time relaxation dynamics in the $J$-$Q_2$ model, which also hosts a DQCP belonging to the same equilibrium universality class. We not only verify the universality of the dual dynamic scaling at the critical point, but also investigate the breakdown and the vestige of the dual dynamic scaling when the tuning parameter is away from the critical point. We also discuss its possible experimental realizations in devices of quantum computers., Comment: 18 pages, 11 figures

We propose scaling theories for Kosterlitz-Thouless (KT) phase transitions on the basis of the hallmark exponential growth of their correlation length. Finite-size scaling, finite-entanglement scaling, short-time critical dynamics, and finite-time scaling, as well as some of their combinations are studied. Relaxation times of both a usual power-law and an unusual power-law with a logarithmic factor are considered. Finite-size and finite-entanglement scaling forms somehow similar to a frequently employed ansatz are presented. The Kibble-Zurek scaling of topological defect density for a linear driving across the KT transition point is investigated in detail. An implicit equation for a rate exponent in the theory is derived and the exponent varies with the distance from the critical point and the driving rate consistent with relevant experiments. To verify the theories, we utilize the KT phase transition of a one-dimensional Bose-Hubbard model. The infinite time-evolving-block-decimation algorithm is employed to solve numerically the model for finite bond dimensions. Both a correlation length and an entanglement entropy in imaginary time and only the entanglement entropy in real-time driving are computed. Both the short-time critical dynamics in imaginary time and the finite-time scaling in real-time driving, both including the finite bond dimension, for the measured quantities are found to describe the numerical results quite well via surface collapses. The critical point is also estimated and confirmed to be $0.302(1)$ at the infinite bond dimension on the basis of the scaling form., Comment: 16 pages, 9 figures. A mistake in a finite-size scaling form corrected

As proposed to describe putative continuous phase transitions between two ordered phases, the deconfined quantum critical point (DQCP) goes beyond the prevalent Landau-Ginzburg-Wilson (LGW) paradigm since its critical theory is not expressed in terms of the order parameters characterizing either state, but involves fractionalized degrees of freedom and an emergent symmetry. So far, great efforts have been spent on its equilibrium properties, but the nonequilibrium properties therein are largely unknown. Here we study the nonequilibrium dynamics of the DQCP via the imaginary-time evolution in the two-dimensional (2D) J-Q$_3$ model. We discover fascinating nonequilibrium scaling behaviors hinging on the process of fractionization and the dynamics of emergent symmetry associated with two length scales. Our findings not only constitute a new realm of nonequilibrium criticality in DQCP, but also offer a controllable knob by which to investigate the dynamics in strongly correlated systems., Comment: 19 pages, 14 figures

Dynamical phase transition (DPT) characterizes the abrupt change of dynamical properties in nonequilibrium quantum many-body systems. It has been demonstrated that extra quantum fluctuating modes besides the conventional order parameter field can drastically change the properties of equilibrium phase transitions. However, the counterpart phenomena in DPTs have rarely been explored. Here, we study the DPT in the Dirac system after a sudden quench, and find that the fermion fluctuations can round a putative first-order DPT into a dynamical critical point, which is referred to as a fermion-induced dynamical critical point (FIDCP). It is also a nonthermal critical point, in which the universal short-time scaling behavior emerges despite the system goes through a first-order transition after thermalization. In the novel scenario of FIDCP, the quantum Yukawa coupling $g_q$ is indispensable for inducing the FIDCP albeit irrelevant in the infrared scale. We call these variables {\it indispensable irrelevant scaling variables}. Moreover, a dynamical tricritical point which separates the first-order DPT and the FIDCP is discovered by tuning this indispensable irrelevant scaling variable. We further mention possible experimental realizations., Comment: 6+4 pages, 4+4 figures

Symmetry breaking plays a pivotal role in modern physics. Although self-similarity is also a symmetry and appears ubiquitously in nature, a fundamental question is whether self-similarity breaking makes sense or not. Here, by identifying the most important kind of critical fluctuations dubbed as phases fluctuations and comparing the consequences of having self-similarity with those of lacking self-similarity in the phases fluctuations, we show that self-similarity can indeed be broken with significant consequences at least in nonequilibrium situations. We find that the breaking of self-similarity results in new critical exponents which give rise to violation of the well-known finite-size scaling or the less-known finite-time scaling and different leading exponents in the ordered and the disordered phases of the paradigmatic Ising model on two- or three-dimensional finite lattices when it subjects to the simplest nonequilibrium driving of linear heating or cooling through its critical point, in stark contrast to identical exponents and different amplitudes in usual critical phenomena. Our results demonstrate how surprising driven nonequilibrium critical phenomena can be. Application to other classical and quantum phase transitions is highly expected., Comment: 5 pages, 4 figures

The conventional Kibble-Zurek mechanism (KZM) describes the driven critical dynamics in the Landau-Ginzburg-Wilson (LGW) spontaneous symmetry-breaking phase transitions. However, whether the KZM is still applicable in the deconfined quantum criticality, which is beyond the LGW paradigm, has not been explored. In this paper, we study the driven critical dynamics near a one-dimensional incarnation of deconfined quantum critical point between a ferromagnetic (FM) phase and a valance-bond-solid (VBS) phase. By investigating the dependence of the density of the topological defects on the driving rate, we verify the KZM in this Landau-forbidden critical point. Moreover, we find that both the FM and the VBS order parameters satisfy the finite-time scaling in the whole driven process. The effects of the emergent symmetry in the nonequilibrium dynamics are also studied., Comment: 5+3 pages, 4+2 figures

Fluctuations can change the phase transition properties drastically. An example is the fermion-induced quantum critical point (FIQCP), in which fluctuations of the massless Dirac fermions turn a putative Landau-de Gennes first-order phase transition (FOPT) with a cubic boson interaction into a continuous one. However, for the Landau-Devonshire theory, which characterizes another very large class of FOPTs, its fate under the coupling with extra fluctuations has not been explored. Here, we discover a new type of FIQCP, in which the Dirac fermion fluctuations round the boson Landau-Devonshire FOPT into a continuous phase transition. By using the functional renormalization group analyses, we determine the condition for the appearance of this FIQCP. Moreover, we point out that the present FIQCP can be a supersymmetric critical point. We finally show that the low-temperature phase diagram can provide distinct experimental evidences to detect this FIQCP., Comment: 9 pages, 7 figures, Accepted in PRB

We study the driven critical dynamics of the quantum link model, whose Hamiltonian describes the one-dimensional $U(1)$ lattice gauge theory. We find that combined topological defects emerge after the quench and they consist of both gauge field and matter field excitations. Furthermore, the ratio of gauge field and matter field excitation is $1/2$ due to the constraint of the Gauss' law. We show that the scaling of these combined topological defects satisfies the usual Kibble-Zurek mechanism. We verify that both the electric flux and the entanglement entropy satisfy the finite-time scaling theory in the whole driven process. Possible experimental realizations are discussed., Comment: 8.1 pages, 7 figures

We study the behavior of the out-of-time-ordered correlator (OTOC) in a non-Hermitian quantum Ising system. We show that the OTOC can diagnose not only the ground state exceptional point, which hosts the Yang-Lee edge singularity, but also the \textit{dynamical} exceptional point at the excited state. We find that the evolution of the OTOC in the parity-time symmetric phase can be divided into two stages: in the short-time stage, the OTOC oscillates periodically, and when the parameter is near the ground state exceptional point, this oscillation behavior can be described by both the scaling theory of the $(0+1)$D Yang-Lee edge singularity and the scaling theory of the $(1+1)$D quantum Ising model; while in the long-time stage the OTOC increases exponentially, controlled by the dynamical exceptional point. Possible experimental realizations are then discussed., Comment: 5+2 pages, 6+4 figures

We study the prethermal dynamics of the Gross-Neveu-Yukawa quantum field theory, suddenly quenched in the vicinity of a quantum critical point. We find that the universal prethermal dynamics is controlled by two fixed points depending on the size of the quench. Besides the usual equilibrium chiral Ising fixed point for a shallow quench, a dynamical chiral Ising fixed point is identified for a deep quench. Intriguingly, the latter is a non-thermal fixed point without any equilibrium counterpart due to the participation of gapless fermionic fields. We also find that in the scaling regime controlled by the equilibrium fixed point, the initial slip exponent is rendered negative if there are enough flavors of fermions, thus providing a unique signature of fermionic prethermal dynamics. We then explore the temporal crossover between the universal scaling regimes governed by the two universality classes. Possible experimental realizations are also discussed., Comment: 4.2 pages + supplemental materials, 2 figures

In this paper we study the driven critical dynamics in the three-state quantum chiral clock model. This is motivated by a recent experiment, which verified the Kibble-Zurek mechanism and the finite-time scaling in a reconfigurable one-dimensional array of $^{87}$Rb atoms with programmable interactions. This experimental model shares the same universality class with the quantum chiral clock model and has been shown to possess a nontrivial non-integer dynamic exponent $z$. Besides the case of changing the transverse field as realized in the experiment, we also consider the driven dynamics under changing the longitudinal field. For both cases, we verify the finite-time scaling for a non-integer dynamic exponent $z$. Furthermore, we determine the critical exponents $\beta$ and $\delta$ numerically for the first time. We also investigate the dynamic scaling behavior including the thermal effects, which are inevitably involved in experiments. From a nonequilibrium dynamic point of view, our results strongly support that there is a direct continuous phase transition between the ordered phase and the disordered phase. Also, we show that the method based on the finite-time scaling theory provides a promising approach to determine the critical point and critical properties., Comment: 8 pages, 7 figures, 1 table

We study the equilibrium and dynamical properties of a ferromagnetic spinor atomic Bose-Einstein condensate. In the vicinity of the critical point for a continuous quantum phase transition, universal behaviors are observed both in the equilibrium state and in the dynamics when the quadratic Zeeman shift is swept linearly. Three distinct dynamical regions are identified for different sweeping time scales ($\tau$), when compared to the time scale $\tau_{\rm KZ}\sim N^{(1+\nu z)/\nu d}$ decided by external driving in a system with finite size $N$ ($\nu,z$ are critical exponents and $d$ the dimensionality). They are manifested by the excitation probability $\mathcal{P}$ and the excess heat density $\mathcal{Q}$. The adiabatic region of $\,\mathcal{P}\sim\mathcal{Q}\sim\tau^{-2}\,$ follows from the adiabatic perturbation theory when $\tau >\tau_{\rm KZ}$, while the non-adiabatic universal region of $\,\mathcal{P}\sim\mathcal{Q}\sim\tau^{-1}\,$ in the thermodynamic limit is described by the Kibble-Zurek mechanism when $\tau_{\rm KZ}>\tau >$ the time scale given by initial gap. The Kibble-Zurek scaling hypothesis is augmented with finite-size scaling in the latter region and several experimentally falsifiable features for the finite system we consider are predicted. The region of the fastest sweeping is found to be non-universal and far-from-equilibrium with $\mathcal{P}$ and $\mathcal{Q}$ essentially being constants independent of $\tau$., Comment: (7.5+1) pages, 5 figures, ask for feedbacks

The conventional Kibble-Zurek scaling describes the scaling behavior in the driven dynamics across a single critical region. In this paper, we study the driven dynamics across an overlapping critical region, in which a critical region (Region-A) is overlaid by another critical region (Region-B). We develop a hybridized Kibble-Zurek scaling (HKZS) to characterize the scaling behavior in the driven process. According to the HKZS, the driven dynamics in the overlapping region can be described by the critical theories for both Region-A and Region-B simultaneously. This results in a constraint on the scaling function in the overlapping critical region. We take the quantum Ising chain in an imaginary longitudinal-field as an example. In this model, the critical region of the Yang-Lee edge singularity and the critical region of the ferromagnetic-paramagnetic phase transition point overlap with each other. We numerically confirm the HKZS by simulating the driven dynamics in this overlapping region. The HKZSs in other models are also discussed., Comment: 6.1 pages, 6 figures, added more references

We study the driven dynamics across the critical points of the Yang-Lee edge singularities (YLESes) in a finite-size quantum Ising chain with an imaginary symmetry-breaking field. In contrast to the conventional classical or quantum phase transitions, these phase transitions are induced by tuning the strength of the dissipation in a non-Hermitian system and can occur even at finite size. For conventional phase transitions, universal behaviors in driven dynamics across critical points are usually described by the Kibble-Zurek mechanism, which states that the scaling in dynamics is dictated by the critical exponents associated with one critical point and topological defects will emerge after the quench. While the mechanism leading to topological defects breaks down in the YLES, we find that for small lattice size, the driven dynamics can still be described by the Kibble-Zurek scaling with the exponents determined by the $(0+1)$-dimensional YLES. For medium finite size, however, the driven dynamics can be described by the Kibble-Zurek scaling with two sets of critical exponents determined by both the $(0+1)$-dimensional and the $(1+1)$-dimensional YLESes., Comment: 4.5+3 pages, 4+8 figures; The Experimental feasibility and the discussion on the KZM in Parity-time symmetry-breaking phase transition have been added

We study the driven critical dynamics with an equilibrium initial state near a quantum critical point. In contrast to the original Kibble-Zurek mechanism, which describes the driven dynamics starting from an adiabatic stage that is far from the critical point, the initial adiabacity is broken in this scenario. As a result, the scaling behavior cannot be described by the original Kibble-Zurek scaling. In this work we propose a scaling theory, which includes the initial parameters as additional scaling variables, to characterize the scaling behavior. In particular, this scaling theory can be used to describe the driven scaling behavior starting from a finite-temperature equilibrium state near a quantum critical point. We numerically confirm the scaling theory by simulating the real-time dynamics of the one-dimensional quantum Ising model at both zero and finite temperatures., Comment: 6 pages, 5 figures

A system driven in the vicinity of its critical point by varying a relevant field in an arbitrary function of time is a generic system that possesses a long relaxation time compared with the driving time scale and thus represents a large class of nonequilibrium systems. For such a manifestly nonlinear nonequilibrium strongly fluctuating system, we show that there exists universal nonequilibrium critical behavior that is well described incredibly by its equilibrium critical properties. A dynamic renormalization-group theory is developed to account for the behavior. The weak driving may give rise to several time scales depending on its form and thus rich nonequilibrium phenomena of various regimes and their crossovers, negative susceptibilities, as well as violation of fluctuation-dissipation theorem. An initial condition that can be in either equilibrium or nonequilibrium but has longer correlations than the driving scales also results in a unique regime and complicates the situation. Implication of the results on measurement is also discussed. The theory may shed light on study of other nonequilibrium systems and even nonlinear science., Comment: 15 pages, 11 figures

We study the scaling behavior of the relaxation dynamics to thermal equilibrium when a quantum system is near the quantum critical point. In particular, we investigate systems whose relaxation dynamics is described by a Lindblad master equation. We find that the universal scaling behavior not only exhibits in the equilibrium stage at the long-time limit, but also manifests itself in the non-equilibrium relaxation process. While the critical behavior is dictated by the low-lying energy levels of the Hamiltonian, the dissipative part in the Lindblad equation also plays important roles in two aspects: First, the dissipative part makes the high energy levels decay fast after which the universal behavior controlled by the low-lying modes emerges. Second, the dissipation rate gives rise to a time scale that affects the scaling behavior. We confirm our theory by solving the Lindblad equation for the one-dimensional transverse-field Ising model., Comment: 5 pages, 4 figures

The Kibble-Zurek mechanism demands an initial adiabatic stage before an impulse stage to have a frozen correlation length that generates topological defects in a cooling phase transition. Here we study such a driven critical dynamics but with an initial condition that is near the critical point and that is far away from equilibrium. In this case, there is no initial adiabatic stage at all and thus adiabaticity is broken. However, we show that there again exists a finite length scale arising from the driving that divides the evolution into three stages. A relaxation--finite-time scaling--adiabatic scenario is then proposed in place of the adiabatic--impulse--adiabatic scenario of the original Kibble-Zurek mechanism. A unified scaling theory, which combines finite-time scaling with critical initial slip, is developed to describe the universal behavior and is confirmed with numerical simulations of a two-dimensional classical Ising model., Comment: 5+{\epsilon} pages, 5 figures

We present a scaling theory for the entanglement spectrum under an external driving. Based on the static scaling of the Schmidt gap and the theory of finite-time scaling, we show that the Schmidt gap can signal the critical point and be used to estimate the critical exponents no matter in the finite-size scaling region or in the finite-time scaling region. Crossover between the two regions is also demonstrated. We verify our theory using both the one-dimensional transverse-field Ising model and the one-dimensional quantum Potts model. Our results confirm that the Schmidt gap can be regarded as a supplement to the local order parameter., Comment: 7 pages, 7 figures

The Kibble-Zurek (KZ) mechanism has been applied to a variety of systems ranging from low temperature Bose-Einstein condensations to grand unification scales in particle physics and cosmology and from classical phase transitions to quantum phase transitions. Here we show that finite-time scaling (FTS) provides a detailed improved understanding of the mechanism. In particular, the finite time scale, which is introduced by the external driving (or quenching) and results in FTS, is the origin of the division of the adiabatic regimes from the impulse regime in the KZ mechanism. The origin of the KZ scaling for the defect density, generated during the driving through a critical point, is not that the correlation length ceases growing in the nonadiabatic impulse regime, but rather, is that it is taken over by the effective finite length scale corresponding to the finite time scale. We also show that FTS accounts well for and improves the scaling ansatz proposed recently by Liu, Polkovnikov, and Sandvik [Phys. Rev. B {\bf 89}, 054307 (2014)]. Further, we show that their universal power-law scaling form applies only to some observables in cooling but not to heating. Even in cooling, it is invalid either when an appropriate external field is present. However, this finite-time-finite-size scaling calls for caution in application of FTS. Detailed scaling behaviors of the FTS and finite-size scaling, along with their crossover, are explicitly demonstrated, with the dynamic critical exponent $z$ being estimated for two- and three-dimensional Ising models under the usual Metropolis dynamics. These values of $z$ are found to give rise to better data collapses than the extant values do in most cases but take on different values in heating and cooling in both two- and three-dimensional spaces., Comment: 14 pages, 14 figures

We study the imaginary-time relaxation critical dynamics of a quantum system with a vanishing initial correlation length and an arbitrary initial order parameter $M_0$. We find that in quantum critical dynamics, the behavior of $M_0$ under scale transformations deviates from a simple power-law, which was proposed for very small $M_0$ previously. A universal characteristic function is then suggested to describe the rescaled initial magnetization, similar to classical critical dynamics. This characteristic function is shown to be able to describe the quantum critical dynamics in both short- and long-time stages of the evolution. The one-dimensional transverse-field Ising model is employed to numerically determine the specific form of the characteristic function. We demonstrate that it is applicable as long as the system is in the vicinity of the quantum critical point. The universality of the characteristic function is confirmed by numerical simulations of models belonging to the same universality class., Comment: 12 pages, 14 figures, 2 tables

We propose a scaling theory for the universal imaginary-time quantum critical dynamics for both short times and long times. We discover that there exists a universal critical initial slip related to a small initial order parameter $M_0$. In this stage, the order parameter $M$ increases with the imaginary time $\tau$ as $M\propto M_0\tau^\theta$ with a universal initial slip exponent $\theta$. For the one-dimensional transverse-field Ising model, we estimate $\theta$ to be $0.373$, which is markedly distinct from its classical counterpart. Apart from the local order parameter, we also show that the entanglement entropy exhibits universal behavior in the short-time region. As the critical exponents in the early stage and in equilibrium are identical, we apply the short-time dynamics method to determine quantum critical properties. The method is generally applicable in both the Landau-Ginzburg-Wilson paradigm and topological phase transitions., Comment: 15 pages, 17 figures

We propose that nonequilibrium quantum criticality in open systems at both zero and finite temperatures can be described by a master equation of the Lindblad form. We derive this equation from a system coupling microscopic to a heat bath. It is suggested to be valid generally for studying dynamical quantum criticality and is thus designated as Model Q upon generalizing Hohenberg and Halperin's classification of classical critical dynamics. We find that the dissipation rate in the equation must be included in the scaling forms as an indispensable additional scaling variable in order to correctly describe the nonequilibrium quantum critical behavior. Yet, the equilibrium fixed point determines the nonequilibrium critical behavior in the weak coupling limit. Through numerically solving the Lindblad equation for the quantum Ising chain, we attest these propositions by finite-time scaling forms with the dissipation rate. Nonequilibrium dynamic critical behavior of spontaneous emissions in dissipative open systems at zero temperature near their quantum critical points is found and is also described well by the scaling forms. Model Q thus provides a general approach to study quantum critical behavior of a system itself through its weak coupling to the environment., Comment: 5 figures

Starting from the Liouville-von Neumann equation, under a weak coupling limit we derive the Lindblad master equation for the one-dimensional quantum Ising model in a Markov approximation and a rotating wave approximation. We also prove that the steady solution of the Lindblad equation is the canonical distribution independent of the dissipation rate.

We develop a theory of finite-time scaling for dynamic quantum criticality by considering the competition among an external time scale, an intrinsic reaction time scale and an imaginary time scale arising respectively from an external driving field, the fluctuations of the competing orders and thermal fluctuations. Through a successful application in determining the critical properties at zero temperature and the solution of real-time Lindblad master equation near a quantum critical point at nonzero temperatures, we show that finite-time scaling offers not only an amenable and systematic approach to detect the dynamic critical properties, but also a unified framework to understand and explore nonequilibrium dynamics of quantum criticality, which shows specificities for open systems., Comment: 5 pages, 4 figures