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Computational studies of thermal and quantum phase transitions approached through non-equilibrium quenching

Chengwei Liu

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

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.

Related papers

Quantum phase transitions in transverse field spin models: from statistical physics to quantum information

Amit Dutta, G. Aeppli, Bikas K. Chakrabarti +3 more

We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.

arXiv:1012.0653•Dec 3, 2010•171 citations

Concluding Comments

Amit Dutta, G. Aeppli, Bikas K. Chakrabarti +3 more

In this book, we have discussed quantum phase transitions in transverse field Ising and XY models in one and higher dimensions, particularly in the context of quantum information processing and non-equilibrium dynamics. We have illustrated how these models relate to actual quantum annealing protocols in physical systems and their connection to efficient quantum computation. Quantum phase transitions for pure models as well as for models with random interactions or random fields have also been considered. These models were introduced in the early 1960's in the context of order-disorder ferroelectric systems. This book focuses on the salient issues for which these models continue to be important and interesting even fifty years after their first appearance. Our starting point involved generic theories of behavior of information theoretic measures, which were then illustrated using the transverse field models. Related models, e.g. those which are exactly soluble using Jordan-Wigner transformations and those exhibiting topological quantum phase transitions, were presented in the book at some length. These models are relevant to low-dimensional condensed matter physics as well as to quantum information and quantum dynamical studies. On the theoretical side, the integrability of the transverse field Ising/ XY models in one dimension has provided an ideal testing ground for field theoretical and information theoretical studies. Numerous predictions of theoretical studies have been verified experimentally in recent years. [180]. The one-dimensional version of these models have been extremely useful in studying information theoretic measures like concurrence, entanglement entropy, fidelity and fidelity susceptibility, and also the scaling of the defect density generated by quantum critical and multicritical quenches, namely, the Kibble–Zurek scaling. These models have played a crucial role in the development of quantum annealing techniques and adiabatic quantum algorithms. The equilibration or thermalization following quantum quenches have been studied using variants of transverse field models. An advantageous feature of transverse field Ising models is the quantum-classical mapping which renders these models ideally suited for quantum Monte Carlo studies for d > 1.

Cambridge University Press eBooks•Jan 28, 2015•0 citations

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The Kibble-Zurek mechanism (KZM) was originally proposed to describe the evolution and "freezing" of defects in the early universe, but later it was generalized to study other quantum and classical systems driven by a varying parameter. The basic idea behind the KZM is that, as long as the changing rate (velocity) of the parameter is below a certain critical velocity, 𝑣_crit, the system will remain adiabatic (for isolated quantum systems) or quasi-static (for classical systems with a heat bath). \nThe nonequilibrium finite-size scaling (FSS) method based on KZM has been exploited systematically. Through applying the scaling hypothesis, we can extract the critical exponents and study the dynamic properties of the system. \n \nIn the first few chapters of this dissertation, we discuss the applications of KZM in several classical systems: first, we study the dynamics of 2D and 3D Ising model under a varying temperature as well as a varying magnetic field. Secondly, we examine the classical ℤ₂ gauge model, in which we show that KZM also works for topological phase transitions. Moreover, we also investigate the dynamics of other models with topological ordering only at T=0, where KZM cannot be applied. Lastly, we explore the 2D Ising spin glass with bimodal and gaussian couplings. With bimodal couplings, we find dual time scales associated with the order parameter and the energy correspondingly, while in the gaussian case one unique time scale is involved. \n \nThe systems mentioned above are all classical and the dynamics are approached through simulated annealing (SA), in which thermal fluctuations drives systems to explore the energy landscape in finding the ground state. In the last chapter, we explore the efficiency of Quantum Annealing (QA) on a fully-connected spin glass (or Sherington-Kirkpatrick model) with a transverse field. QA is the counterpart of SA, where quantum fluctuations drive the system toward the ground state when the quantum terms are reduced. QA is currently widely explored as a paradigm for quantum computing to solve optimization problems. Here we compare the scaling of the dynamics (with system size) of the fully-connected spin glass through QA versus SA.

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Crossover from Classical to Quantum Kibble-Zurek Scaling

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The Kibble-Zurek (KZ) hypothesis identifies the relevant time scales in out-of-equilibrium dynamics of critical systems employing concepts valid at equilibrium: It predicts the scaling of the defect formation immediately after quenches across classical and quantum phase transitions as a function of the quench speed. Here, we study the crossover between the scaling dictated by a slow quench, which is ruled by the critical properties of the quantum phase transition, and the excitations due to a faster quench, where the dynamics is often well described by the classical model. We estimate the value of the quench rate that separates the two regimes and support our argument using numerical simulations of the out-of-equilibrium many-body dynamics. For the specific case of a ϕ^{4} model we demonstrate that the two regimes exhibit two different power-law scalings, which are in agreement with the KZ theory when applied to the quantum and classical cases. This result contributes to extending the prediction power of the Kibble-Zurek mechanism and to providing insight into recent experimental observations in systems of cold atoms and ions.

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Quantum Monte Carlo studies of phase transitions

Pranay Patil

Phase transitions have been an active area of research in statistical mechanics for almost a century and have recently been integrated into quantum mechanics. Many phenomena such as superconductivity and unconventional magnetism are understood to arise from exotic quantum phases and at points describing quantum phase transitions. A detailed understanding of these phase transitions requires numerical simulations of models which benchmark realistic models against theoretical frameworks. The topic of this thesis is the implementation of Quantum Monte Carlo simulation, which is a powerful technique to understand quantum condensed matter, in interesting models to illustrate novel phenomena in magnetic systems. The novel features of condensed matter systems described in this thesis consist of emergent symmetries at critical points, interesting dynamical features of such systems and the drastic effects of defects in spin systems used in the field of adiabatic quantum computing. Emergent symmetries are shown by condensed matter systems especially at critical points and are features which cannot be shown by individual or a small number of spins. Examples of this in one and two dimensions are presented in an early chapter of this thesis. In addition to this, spin systems can show excitations which have an interesting spatial structure as a consequence of restricted dynamics which only allow the excitations to spread in a particular region. This is presented in the context of a simple model in the following chapter along with numerical support. The following chapter contains a description of adiabatic quantum computing along with a particular model which we study. The phase transition and the effects on the performance of adiabatic quantum computing are studied in this context.

•Jan 1, 2020•0 citations
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.
We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.
In this book, we have discussed quantum phase transitions in transverse field Ising and XY models in one and higher dimensions, particularly in the context of quantum information processing and non-equilibrium dynamics. We have illustrated how these models relate to actual quantum annealing protocols in physical systems and their connection to efficient quantum computation. Quantum phase transitions for pure models as well as for models with random interactions or random fields have also been considered. These models were introduced in the early 1960's in the context of order-disorder ferroelectric systems. This book focuses on the salient issues for which these models continue to be important and interesting even fifty years after their first appearance. Our starting point involved generic theories of behavior of information theoretic measures, which were then illustrated using the transverse field models. Related models, e.g. those which are exactly soluble using Jordan-Wigner transformations and those exhibiting topological quantum phase transitions, were presented in the book at some length. These models are relevant to low-dimensional condensed matter physics as well as to quantum information and quantum dynamical studies. On the theoretical side, the integrability of the transverse field Ising/ XY models in one dimension has provided an ideal testing ground for field theoretical and information theoretical studies. Numerous predictions of theoretical studies have been verified experimentally in recent years. [180]. The one-dimensional version of these models have been extremely useful in studying information theoretic measures like concurrence, entanglement entropy, fidelity and fidelity susceptibility, and also the scaling of the defect density generated by quantum critical and multicritical quenches, namely, the Kibble–Zurek scaling. These models have played a crucial role in the development of quantum annealing techniques and adiabatic quantum algorithms. The equilibration or thermalization following quantum quenches have been studied using variants of transverse field models. An advantageous feature of transverse field Ising models is the quantum-classical mapping which renders these models ideally suited for quantum Monte Carlo studies for d > 1.
The Kibble-Zurek mechanism (KZM) was originally proposed to describe the evolution and "freezing" of defects in the early universe, but later it was generalized to study other quantum and classical systems driven by a varying parameter. The basic idea behind the KZM is that, as long as the changing rate (velocity) of the parameter is below a certain critical velocity, 𝑣_crit, the system will remain adiabatic (for isolated quantum systems) or quasi-static (for classical systems with a heat bath). \nThe nonequilibrium finite-size scaling (FSS) method based on KZM has been exploited systematically. Through applying the scaling hypothesis, we can extract the critical exponents and study the dynamic properties of the system. \n \nIn the first few chapters of this dissertation, we discuss the applications of KZM in several classical systems: first, we study the dynamics of 2D and 3D Ising model under a varying temperature as well as a varying magnetic field. Secondly, we examine the classical ℤ₂ gauge model, in which we show that KZM also works for topological phase transitions. Moreover, we also investigate the dynamics of other models with topological ordering only at T=0, where KZM cannot be applied. Lastly, we explore the 2D Ising spin glass with bimodal and gaussian couplings. With bimodal couplings, we find dual time scales associated with the order parameter and the energy correspondingly, while in the gaussian case one unique time scale is involved. \n \nThe systems mentioned above are all classical and the dynamics are approached through simulated annealing (SA), in which thermal fluctuations drives systems to explore the energy landscape in finding the ground state. In the last chapter, we explore the efficiency of Quantum Annealing (QA) on a fully-connected spin glass (or Sherington-Kirkpatrick model) with a transverse field. QA is the counterpart of SA, where quantum fluctuations drive the system toward the ground state when the quantum terms are reduced. QA is currently widely explored as a paradigm for quantum computing to solve optimization problems. Here we compare the scaling of the dynamics (with system size) of the fully-connected spin glass through QA versus SA.
Quantum phase transitions (QPTs) involve transformations between different states of matter that are driven by quantum fluctuations. These fluctuations play a dominant role in the quantum critical region surrounding the transition point, where the dynamics are governed by the universal properties associated with the QPT. The resulting quantum criticality has been explored by probing linear response for systems near thermal equilibrium. While time dependent phenomena associated with classical phase transitions have been studied in various scientific fields, understanding critical real-time dynamics in isolated, non-equilibrium quantum systems is of fundamental importance both for exploring novel approaches to quantum information processing and realizing exotic new phases of matter. Here, we use a Rydberg atom quantum simulator with programmable interactions to study the quantum critical dynamics associated with several distinct QPTs. By studying the growth of spatial correlations while crossing the QPT at variable speeds, we experimentally verify the quantum Kibble-Zurek mechanism (QKZM) for an Ising-type QPT, explore scaling universality, and observe corrections beyond simple QKZM predictions. This approach is subsequently used to investigate novel QPTs associated with chiral clock model providing new insights into exotic systems, and opening the door for precision studies of critical phenomena and applications to quantum optimization.
The Kibble-Zurek (KZ) hypothesis identifies the relevant time scales in out-of-equilibrium dynamics of critical systems employing concepts valid at equilibrium: It predicts the scaling of the defect formation immediately after quenches across classical and quantum phase transitions as a function of the quench speed. Here, we study the crossover between the scaling dictated by a slow quench, which is ruled by the critical properties of the quantum phase transition, and the excitations due to a faster quench, where the dynamics is often well described by the classical model. We estimate the value of the quench rate that separates the two regimes and support our argument using numerical simulations of the out-of-equilibrium many-body dynamics. For the specific case of a ϕ^{4} model we demonstrate that the two regimes exhibit two different power-law scalings, which are in agreement with the KZ theory when applied to the quantum and classical cases. This result contributes to extending the prediction power of the Kibble-Zurek mechanism and to providing insight into recent experimental observations in systems of cold atoms and ions.
Phase transitions have been an active area of research in statistical mechanics for almost a century and have recently been integrated into quantum mechanics. Many phenomena such as superconductivity and unconventional magnetism are understood to arise from exotic quantum phases and at points describing quantum phase transitions. A detailed understanding of these phase transitions requires numerical simulations of models which benchmark realistic models against theoretical frameworks. The topic of this thesis is the implementation of Quantum Monte Carlo simulation, which is a powerful technique to understand quantum condensed matter, in interesting models to illustrate novel phenomena in magnetic systems. The novel features of condensed matter systems described in this thesis consist of emergent symmetries at critical points, interesting dynamical features of such systems and the drastic effects of defects in spin systems used in the field of adiabatic quantum computing. Emergent symmetries are shown by condensed matter systems especially at critical points and are features which cannot be shown by individual or a small number of spins. Examples of this in one and two dimensions are presented in an early chapter of this thesis. In addition to this, spin systems can show excitations which have an interesting spatial structure as a consequence of restricted dynamics which only allow the excitations to spread in a particular region. This is presented in the context of a simple model in the following chapter along with numerical support. The following chapter contains a description of adiabatic quantum computing along with a particular model which we study. The phase transition and the effects on the performance of adiabatic quantum computing are studied in this context.