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Non-equilibrium real time dynamics of quantum spin systems and the quantum critical dynamics of 'dirty' bosons

Ray Jia Hong Ng

Published Sep 24, 2015• 0 citations
Publisher

Abstract

This thesis presents our work on two separate subfields and has for this reason been divided into two parts: (I) the development and implementation of a numerical technique for non-equilibrium dynamics and a (II) detailed investigation of the dirty-boson problem. In part (I), we optimize and develop the quantum phase space method known as the positive-P representation (PPR) specifically for the simulations of the real time quench dynamics of quantum spin systems. The main benefit of this approach is that the dynamics of the density operator is mapped onto $\sim N$ stochastic variables (where $N$ is the size of the system) that obey Langevin-type stochastic differential equations, thereby greatly reducing the complexity of the problem. The first publication presents our initial use of the PPR on spin systems by using a Schwinger mapping for spin operators on to Bosonic operators since the underlying basis comprises of Bosonic coherent states. We simulate the quench dynamics of the generalized transverse-field spin-1/2 XXZ model in 1d showing that simulations of up to 100 spins are possible, albeit for relatively short simulation times. In our second publication, we reformulate the PPR using SU(2) spin coherent states and further optimize simulation lifetimes by implementing an extrapolation scheme, to achieve several-fold improvements over the results of our first publication. We focus solely on the transverse Ising model and simulate its quench dynamics in 1d and 2d using up to $10^4$ spins, while significantly extending simulation lifetimes. The second part of this thesis is a numerical study of the universality class of the Superfluid-Bose Glass transition of the dirty-boson system. Recently the longstanding exact result $z=d$, where $z$ is the dynamic critical exponent and $d$ is the dimensionality of the system, has been challenged by a series of numerical studies, suggesting the alternative scenario that $z$ should be instead unconstrained. To address this controversy, we use large scale quantum Monte-Carlo simulations on two independent quantum models, and average over $5\times10^4-10^5$ disorder realizations to numerically determine the universality class of the dirty-boson transition, paying particular attention to the dynamic critical exponent, $z$ and without any biasing assumptions.

DOI
10.71548/9858

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Recently, the scaling result $z=d$ for the dynamic critical exponent at the Bose glass to superfluid quantum phase transition has been questioned both on theoretical and numerical grounds. This motivates a careful evaluation of the critical exponents in order to determine the actual value of $z$. We study a model of quantum bosons at $T=0$ with disorder in 2D using highly effective worm Monte Carlo simulations. Our data analysis is based on a finite-size scaling approach to determine the scaling of the quantum correlation time from simulation data for boson world lines. The resulting critical exponents are $z=1.8\ifmmode\pm\else\textpm\fi{}0.05$, $\ensuremath{\nu}=1.15\ifmmode\pm\else\textpm\fi{}0.03$, and $\ensuremath{\eta}=\ensuremath{-}0.3\ifmmode\pm\else\textpm\fi{}0.1$, hence suggesting that $z=2$ is not satisfied.

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This thesis presents our work on two separate subfields and has for this reason been divided into two parts: (I) the development and implementation of a numerical technique for non-equilibrium dynamics and a (II) detailed investigation of the dirty-boson problem. In part (I), we optimize and develop the quantum phase space method known as the positive-P representation (PPR) specifically for the simulations of the real time quench dynamics of quantum spin systems. The main benefit of this approach is that the dynamics of the density operator is mapped onto ∼N\sim N∼N stochastic variables (where NNN is the size of the system) that obey Langevin-type stochastic differential equations, thereby greatly reducing the complexity of the problem. The first publication presents our initial use of the PPR on spin systems by using a Schwinger mapping for spin operators on to Bosonic operators since the underlying basis comprises of Bosonic coherent states. We simulate the quench dynamics of the generalized transverse-field spin-1/2 XXZ model in 1d showing that simulations of up to 100 spins are possible, albeit for relatively short simulation times. In our second publication, we reformulate the PPR using SU(2) spin coherent states and further optimize simulation lifetimes by implementing an extrapolation scheme, to achieve several-fold improvements over the results of our first publication. We focus solely on the transverse Ising model and simulate its quench dynamics in 1d and 2d using up to 10410^4104 spins, while significantly extending simulation lifetimes. The second part of this thesis is a numerical study of the universality class of the Superfluid-Bose Glass transition of the dirty-boson system. Recently the longstanding exact result z=dz=dz=d, where zzz is the dynamic critical exponent and ddd is the dimensionality of the system, has been challenged by a series of numerical studies, suggesting the alternative scenario that zzz should be instead unconstrained. To address this controversy, we use large scale quantum Monte-Carlo simulations on two independent quantum models, and average over 5×104−1055\times10^4-10^55×104−105 disorder realizations to numerically determine the universality class of the dirty-boson transition, paying particular attention to the dynamic critical exponent, zzz and without any biasing assumptions.
Recently, the scaling result z=dz=dz=d for the dynamic critical exponent at the Bose glass to superfluid quantum phase transition has been questioned both on theoretical and numerical grounds. This motivates a careful evaluation of the critical exponents in order to determine the actual value of zzz. We study a model of quantum bosons at T=0T=0T=0 with disorder in 2D using highly effective worm Monte Carlo simulations. Our data analysis is based on a finite-size scaling approach to determine the scaling of the quantum correlation time from simulation data for boson world lines. The resulting critical exponents are $z=1.8\ifmmode\pm\else\textpm\fi{}0.05$, $\ensuremath{\nu}=1.15\ifmmode\pm\else\textpm\fi{}0.03$, and $\ensuremath{\eta}=\ensuremath{-}0.3\ifmmode\pm\else\textpm\fi{}0.1$, hence suggesting that z=2z=2z=2 is not satisfied.
We reformulate the full quantum dynamics of spin systems using a phase-space representation based on SU(2) coherent states which generates an exact mapping of the dynamics of any spin system onto a set of stochastic differential equations. This representation is superior in practice to an earlier phase-space approach based on Schwinger bosons, with the numerical effort scaling only linearly with system size. By also implementing extrapolation techniques from quasiclassical equations to the full quantum limit, we are able to extend useful simulation times severalfold. This approach is applicable in any dimension including cases where frustration is present in the spin system. The method is demonstrated by simulating quenches in the transverse-field Ising model in one and two dimensions.
The study of quantum dynamics is an increasingly important aspect of atomic, molecular and optical physics, as well as condensed-matter physics and other disciplines. It is essential in order to understand non-equilibrium phenomena which do not fit within the framework of equilibrium thermodynamics, such as many-body localization, dynamical quantum phase transitions, and non-equilibrium steady states in driven-dissipative systems. However, the exact calculation of dynamical observables is extremely difficult in general for several reasons, which makes it challenging for theoretical predictions and analyses to keep pace with rapidly advancing experiments. In this thesis I use and develop several computational methods to investigate the dynamics of quantum many-body systems, with the two goals of better understanding the evolution of local observables and correlation functions in spin and Fermion systems as well as creating and testing more effective numerical methods for solving quantum dynamics problems. I explore quantum dynamics by using Hamiltonian parameter quenches in several commonly-used models in AMO physics and elsewhere: the Fermi-Hubbard model, the XXZ model, and the transverse-Ising model. The first represents a system of short-range interacting Fermions, while the latter two represent spin systems with different spin-spin couplings and magnetic field terms. All of these models can be realized in current experiments. First, I derive an exact solution for an extreme interaction quench in the Fermi-Hubbard model, from a strongly-interacting initial state to a noninteracting final Hamiltonian. From this result I show that there are nontrivial transient connected correlations, despite the fact that the system has very high temperature and a noninteracting Hamiltonian. Those features would suppress such correlations in equilibrium, but the dynamical correlations are still significant. Then, I develop and examine two numerical methods for solving dynamics of the spin expectation values and correlation functions in the transverse-Ising and XXZ models. Both methods rely on approximating a thermodynamically large system with a much smaller and more computationally tractable cluster of lattice sites, or a series of clusters. The first method is a dynamical numerical linked cluster expansion (NLCE). NLCEs have already been used successfully on systems in thermodynamic equilibrium, and I extended the technique to directly compute the dynamics of an observable after a quantum quench. I review the general NLCE method, describe my dynamical variant of it, and compute the quench dynamics of spin and correlation observables in the transverse-Ising and XXZ models. The second method, which I term the adaptive-boundary cluster method, relies on recently refined bounds on information propagation in quantum systems to choose a cluster of lattice sites that minimizes the finite size error bound of a dynamical computation. I briefly review these bounds, and describe a new algorithm which uses them to construct a cluster whose boundary is chosen according to a related error estimate. Such an adaptive-boundary cluster has the property that there exists a computable upper bound on the finite size error of observables measured at the cluster center, and this error bound is nearly minimal when compared to that of other clusters of the same size. I then compute the quench dynamics of spin and correlation observables in the transverse-Ising model using the adaptive boundary clusters. I apply this method to both uniform and strongly disordered systems. To evaluate the effectiveness of both dynamical NLCE and adaptive-boundary clusters, I compare the results obtained using them to results from a standard and general purpose numerical method: exact solvers applied a rectangular cluster with periodic boundary conditions. Both of the new methods usually converge faster and are more accurate than the standard method for comparable cluster sizes. This thesis illustrates that choosing numerical methods which take advantage of the geometric and entanglement structure of the initial state and Hamiltonian can substantially enhance our ability to study quantum dynamics, by reducing the computational cost of numerically exact solutions. I conclude with a discussion of various avenues for future research to build on this work.
The work presented in this thesis is divided into two main parts: strong zero modes in quantum spin chains with open boundaries and large deviations in classical spin systems. The results concerning the strong zero modes are presented in chapters 2 and 3, while chapter 4 contains the results about large deviations in systems of many, but independent, classical degrees of freedom. These two distinct classes of problems in spin chains are both about aspects of slow relaxation to equilibrium in many-body systems. Nowadays slow dynamics and non-ergodicity in many-body quantum systems are receiving much attention due to their notable technological potential for storing and processing, for very long times, the quantum information encoded in boundary degrees of freedom. In the first part of the work presented in this thesis we mainly focus on quantum spin chains with open boundaries, which can display coherence times for edge spins, that diverge with the system size, as a consequence of the emergence, in the thermodynamic limit, of conserved local quantities, called strong zero modes. First, we discuss the fate of these coherence times when dynamics becomes dissipative. Understanding the behavior of these long correlation times in open quantum systems, undergoing Markovian (memoryless) dynamics, would allow for the possibility of exploiting the protected information encoded in the boundary spins for applications in quantum devices in realistic non-equilibrium settings. Second, we study one possible simple generalization of the transverse field Ising model to a class of ladder spin systems with plaquette interactions in the presence of a transverse field. Also for these systems one can explicitly construct the strong zero mode operators and understand the behavior of the consequential long coherence times for the edge degrees of freedom under isolated (unitary) dynamics. This is possible because the models we consider, while appearing complex superficially, can be brought to simple forms by exploiting an extensive number of conserved quantities they possess. The results presented in chapters 2 and 3 are closely related, even if they focus on different models and investigate different aspects of their dynamics. Nevertheless, the underlying concept they are based on is the same one, that is, an eigenstate phase transition for the strong zero modes that has interesting consequences for the full energy spectrum. In the second part of the thesis we focus on a different type of phase transitions. Here we address the issue of dynamical large deviation phase transitions from a disordered phase to an ordered one, in classical systems which consist in a collection of independent Ising spins evolving stochastically. Surprisingly, by reweighing trajectories accordingly to a certain type of time-integrated observable, even a system of independently evolving spins can display some critical behavior, at the fluctuation level, in trajectory space in the large size and long time limit. By making use of the tilted generator techniques we study the statistics of two particular time-integrated observables. Their associated tilted generators have an exact Kramers-Wannier duality and they look like, up to a sign and an additive constant, some specific well known quantum Hamiltonians. In this sense, the dynamical large deviation phase transitions we observe in these classical stochastic systems translate in ground state quantum phase transitions taking place at the self dual point of the tilted generators we study.
Phase transitions and their associated critical phenomena are of fundamental importance and play a crucial role in the development of statistical physics for both classical and quantum systems. Phase transitions embody diverse aspects of physics and also have numerous applications outside physics, e.g., in chemistry, biology, and combinatorial optimization problems in computer science. Many problems can be reduced to a system consisting of a large number of interacting agents, which under some circumstances (e.g., changes of external parameters) exhibit collective behavior; this type of scenario also underlies phase transitions. The theoretical understanding of equilibrium phase transitions was put on a solid footing with the establishment of the renormalization group. In contrast, non-equilibrium phase transition are relatively less understood and currently a very active research topic. One important milestone here is the Kibble-Zurek (KZ) mechanism, which provides a useful framework for describing a system with a transition point approached through a non-equilibrium quench process. I developed two efficient Monte Carlo techniques for studying phase transitions, one is for classical phase transition and the other is for quantum phase transitions, both are under the framework of KZ scaling. For classical phase transition, I develop a non-equilibrium quench (NEQ) simulation that can completely avoid the critical slowing down problem. For quantum phase transitions, I develop a new algorithm, named quasi-adiabatic quantum Monte Carlo (QAQMC) algorithm for studying quantum quenches. I demonstrate the utility of QAQMC quantum Ising model and obtain high-precision results at the transition point, in particular showing generalized dynamic scaling in the quantum system. To further extend the methods, I study more complex systems such as spin-glasses and random graphs. The techniques allow us to investigate the problems efficiently. From the classical perspective, using the NEQ approach I verify the universality class of the 3D Ising spin-glasses. I also investigate the random 3-regular graphs in terms of both classical and quantum phase transitions. I demonstrate that under this simulation scheme, one can extract information associated with the classical and quantum spin-glass transitions without any knowledge prior to the simulation.
This thesis investigates the physics of strongly interacting driven–dissipative bosonic systems in one dimension, combining field-theoretical approaches with various numerical simulation techniques. While weakly interacting condensates are accurately described by a generalized Gross–Pitaevskii equation, the strongly correlated regime of the same microscopic models remains far less understood, particularly out of equilibrium. This thesis aims to bridge this gap by developing controlled quantum-many-body simulations for open systems governed by Lindblad dynamics, as well as investigating the corresponding action under the functional renormalization group.In the first part, we formulate a field-theoretical description of driven–dissipative one-dimen-sional bosons using the Schwinger–Keldysh formalism and the functional renormalization group (fRG).Two models are analyzed: (i) a driven-dissipative condensate whose phase dynamics is expected to exhibit emergent Kardar–Parisi–Zhang (KPZ) behavior, for which we search the emergent KPZ fixed point starting from the bosonic field theory; and (ii) a two-photon–driven system with two-body losses realizing dissipative cat qubits. For the latter, we determine the phase diagram and identify a continuous phase transition between symmetry-broken and symmetric phases, which motivates the construction of an fRG Ansatz capturing this critical behavior.The second part of the thesis focuses on the open system dynamics using exact diagonalization and tensor-network methods. For the Bose–Hubbard model with incoherent pump and loss, we characterize the steady state, and study spectral functions across the full parameter space. We show that key ground-state features are absent under dissipation, while the high-temperature unitary spectrum captures aspects of the open-system dynamics. In an extended model with additional two-body and non-local losses, we establish the semi-classical emergence of KPZ scaling and extract the KPZ dynamical exponent from the spectral function. A finite-size analysis exposes the current limitations of the tensor-network methods for reaching the high occupancy regime in the full quantum evolution.Finally, a last project develops a two-loop perturbative RG analysis of a spin-1 chain at its continuous symmetry breaking quantum critical point, revealing critical exponents distinct from the Ising universality class.Together, these results advance the understanding of strongly interacting open quantum systems, elucidating their symmetries, effective field theories, and dynamical properties, and outlining key directions for future theoretical and numerical work.