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.
We propose a quantum Monte Carlo (QMC) algorithm for nonequilibrium dynamics in a system with a parameter varying as a function of imaginary time. The method is based on successive applications of an evolving Hamiltonian to an initial state and delivers results for a whole range of the tuning parameter in a single run, allowing for access to both static and dynamic properties of the system. This approach reduces to the standard Schr\"odinger dynamics in imaginary time for quasi-adiabatic evolutions, i.e., including the leading nonadiabatic correction to the adiabatic limit. We here demonstrate this quasi-adiabatic QMC (QAQMC) method for linear ramps of the transverse-field Ising model across its quantum-critical point in one and two dimensions. The critical behavior can be described by generalized dynamic scaling. For the two-dimensional square-lattice system, we use the method to obtain a high-precision estimate of the quantum-critical point
(h/J)c=3.04458(7), where
h is the transverse magnetic field and
J is the nearest-neighbor Ising coupling. The QAQMC method can also be used to extract the Berry curvature and the metric tensor.
We investigate the short-time quantum critical dynamics in the imaginary-time relaxation processes of finite-size systems. Universal scaling behaviors exist in the imaginary-time evolution. In particular, the system undergoes a critical initial slip stage characterized by an exponent $\ensuremath{\theta}$, in which an initial power-law increase emerges in the imaginary-time correlation function when the initial state has a zero order parameter and a vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo (QMC) simulations. In the one-dimensional case, we locate the quantum critical point at
(h/J)c=1.00003(8) in the thermodynamic limit, and we estimate the critical initial slip exponent $\ensuremath{\theta}=0.3734(2)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)$ by analyzing data on chains of length
L=32−−256 and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by
3.04451(7) in the thermodynamic limit, while the critical exponents are $\ensuremath{\theta}=0.209(4)$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)$ estimated by data on systems of size
L=24--64 and 32--64, respectively. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts due to the essential differences between classical and quantum dynamics. The short-time critical dynamics and the imaginary-time relaxation QMC approach can be readily adapted to various models.
Quantum phase transitions occur strictly at zero temperature and are accessed by varying a physical parameter such as a magnetic field or chemical potential. While these transitions themselves are experimentally inaccessible, the existence of a zero temperature quantum critical point has remarkable effects on the dynamical properties of interacting quantum many-body systems at finite temperatures. In this thesis, we study dynamics in four strongly interacting many-body systems near quantum critical points in the context of NMR relaxation, quantum quenches, many-body quantum chaos, and charge diffusion. In chapter 1, we begin by revisiting the issue of NMR relaxation in the quantum disordered phase of Ising spin chains. We find that the imaginary part of the spin susceptibility at low frequences is dominated by rare quantum processes which leads to an NMR relaxation rate that scales as
1/T1∼exp(−T2Δ) near the quantum critical point rather than the previously conjectured
1/T1∼exp(−TΔ). These results resolve certain discrepancies between the energy scales measured with different experimental probes in the quantum disordered paramagnetic phase of the Ising chain system
CoNb2O6. In chapter 2, we study the non-equilibrium dynamics of Sachdev-Ye-Kitaev (SYK) models with
q body interactions after a quantum quench, considering a quench protocol which drives the system from an initial state with long-lived quasiparticle excitations to a final state lacking any long-lived quasiparticle excitations. For
q=4, we find that the timescale for the system to locally thermalize
τeq∼kBTℏ and for
q→∞ we find that the system thermalizes instantaneously. In chapter 3, we study the real time dynamics of quantum electrodynamics in 2+1 dimensions with
Nf flavors of fermions. We focus on thermalization and the onset of many-body quantum chaos and compute the quantum Lyapunov exponent in a
Nf1 expansion. We briefly comment on chaos, locality, and gauge invariance. Finally, in Chapter 4 we consider whether the butterfly velocity - a speed at which quantum information propagates - may provide a fundamental bound on diffusion constants in dirty incoherent metals by computing the charge diffusion constant and the butterfly velocity in charge-neutral holographic matter with long wavelength ``hydrodynamic" disorder. In this limit, we find that the butterfly velocity does not set a sharp lower bound for the charge diffusion constant.
We give an overview of the scaling of density of quasi-particles and excess energy (heat) for nearly adiabatic dynamics near quantum critical points (QCPs). In particular we discuss both sudden quenches of small amplitude and slow sweeps across the QCP. We show close connection between universal scaling of these quantities with the scaling behavior of the fidelity susceptibility and its generalizations. In particular we argue that the Kibble-Zurek scaling can be easily understood using this concept. We discuss how these scalings can be derived within the adiabatic perturbation theory and how using this approach slow and fast quenches can be treated within the same framework. We also describe modifications of these scalings for finite temperature quenches and emphasize the important role of statistics of low-energy excitations. In the end we mention some connections between adiabatic dynamics near critical points with dynamics associated with space-time singularities in the metrics, which naturally emerges in such areas as cosmology and string theory.
We give an overview of the scaling of density of quasi-particles and excess energy (heat) for nearly adiabatic dynamics near quantum critical points (QCPs). In particular we discuss both sudden quenches of small amplitude and slow sweeps across the QCP. We show close connection between universal scaling of these quantities with the scaling behavior of the fidelity susceptibility and its generalizations. In particular we argue that the Kibble-Zurek scaling can be easily understood using this concept. We discuss how these scalings can be derived within the adiabatic perturbation theory and how using this approach slow and fast quenches can be treated within the same framework. We also describe modifications of these scalings for finite temperature quenches and emphasize the important role of statistics of low-energy excitations. In the end we mention some connections between adiabatic dynamics near critical points with dynamics associated with space-time singularities in the metrics, which naturally emerges in such areas as cosmology and string theory.