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.
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.
We propose that nonequilibrium quantum criticality in open systems under the Born-Markov approximation can be described by a master equation of the Lindblad form. This master equation is derived from a system coupling weakly to a heat bath microscopically and is suggested to provide an approach to study dynamic quantum critical behavior of the system at finite temperatures. We find that the dissipation rate in the equation representing the coupling 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 affirm 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 discovered and is also described well by the scaling forms.
Quantum Phase Transitions Quantum phase transitions, in contrast to temperature-driven critical phenomena, occur due to a competition between different parameters describing the basic interactions of the system. Their specific feature is the quantum character of the critical fluctuations. This implies, through the uncertainty principle, that energy fluctuations and time are coupled. Then at zero temperature time plays a crucial and fundamental role, the static properties being entangled with the dynamics (Continentino, 1994a; Sondhi et al ., 1997; Sachdev, 1999). In this book we are mainly interested in quantum phase transitions that occur in electronic many-body systems and how scaling concepts can be useful to understand their properties close to these transitions (Continentino, 1994a). Even though a similar approach has been used in the case of interacting bosons (Fisher et al ., 1989) the fermionic problem has its own idiosyncrasies and difficulties. For example, there is no natural order parameter associated with the localisation transition in the electronic case, while bosons at zero temperature are either localised or superfluid so that the superfluid order parameter can be used to distinguish between both phases. If the results of the study of quantum phase transitions were restricted to zero temperature, this would be an interesting but purely academic area of research. What is really exciting about this subject is the effect of quantum critical points (QCP) in the finite temperature phase diagram of actual physical systems, even far away from the QCP (Freitas, 2015). As we will show, there is a special line in this phase diagram, the quantum critical trajectory , where the temperature dependence of the thermodynamic and transport properties is governed by the quantum critical exponents, i.e. those associated with the QCP. This and the observed crossover effects induced by temperature in the non-critical side of the phase diagram of a material with a QCP are sufficient to make the study of quantum phase transitions an inevitable subject. We start this chapter by introducing the scaling theory of quantum critical phenomena, since scaling concepts are used throughout this book.
Originally, the term ‘dynamic critical phenomena’ was coined for time-dependent properties near second-order phase transitions in thermal equilibrium . The kinetics of phase transitions in magnets, at the gas–liquid transition, and at the normal- to superfluid phase transition in helium 4 were among the prominent examples investigated already in the 1960s. The dynamic scaling hypothesis , generalizing the scaling ansatz for the static correlation function and introducing an additional dynamic critical exponent, successfully described a variety of these experiments. Yet only the development of the systematic renormalization group (RG) approach for critical phenomena in the subsequent decade provided a solid conceptual foundation for phenomenological scaling theories. Supplemented with exact solutions for certain idealized model systems, and guided by invaluable input from computer simulations in addition to experimental data, the renormalization group now provides a general framework to explore not only the static and dynamic properties near a critical point, but also the large-scale and low-frequency response in stable thermodynamic phases. Scaling concepts and the renormalization group have also been successfully applied to phase transitions at zero temperature driven by quantum rather than thermal fluctuations. It is to be hoped that RG methods may help to classify the strikingly rich phenomena encountered in far-from-equilibrium systems as well. Recent advances in studies of simple reaction-diffusion systems, active to absorbing state phase transitions, driven lattice gases, and scaling properties of moving interfaces and growing surfaces, among others, appear promising in this respect.
Taking the quantum Kitaev chain as an example, we have studied the universal dynamical behaviors resulting from quantum criticality under the condition of environmental temperature quench. Our findings reveal that when the quantum parameter is at its critical value, both the excess excitation density at the end of linear quench and the free relaxation exhibit universal scaling behaviors. The scaling laws observed upon quenching to the zero-temperature quantum critical point and nonzero temperature points exhibit distinct scaling exponents, which are all intimately related to the dynamical critical exponents of the quantum phase transition. Additionally, for the case of linear quench to finite temperatures, we have also discovered an intrinsic universal dynamical behavior that is independent of quantum criticality. Our research offers profound insights into the relationship between quantum criticality and nonequilibrium dynamics from two perspectives: Kibble-Zurek-like scaling behavior and free relaxation dynamics. Notably, the Kibble-Zurek-like scaling behavior in this context differs from the standard Kibble-Zurek mechanism. These two aspects jointly open up a new avenue for us to understand quantum criticality through real-time dynamical behavior, even at finite temperatures.
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 appears in the equilibrium stage at the long-time limit but also manifests in the nonequilibrium 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.
We propose a method to study dynamical response of a quantum system by evolving it with an imaginary-time-dependent Hamiltonian. The leading nonadiabatic response of the system driven to a quantum-critical point is universal and characterized by the same exponents in real and imaginary time. For a linear quench protocol, the fidelity susceptibility and the geometric tensor naturally emerge in the response functions. Beyond linear response, we extend the finite-size scaling theory of quantum phase transitions to nonequilibrium setups. This allows, e.g., for studies of quantum phase transitions in systems of fixed finite size by monitoring expectation values as a function of the quench velocity. Nonequilibrium imaginary-time dynamics is also amenable to quantum Monte Carlo (QMC) simulations, with a scheme that we introduce here and apply to quenches of the transverse-field Ising model to quantum-critical points in one and two dimensions. The QMC method is generic and can be applied to a wide range of models and nonequilibrium setups.