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arXiv:1806.02347 · Physical review. B./Physical review. B

Nonequilibrium quantum spin dynamics from two-particle irreducible functional integral techniques in the Schwinger boson representation

Alexander Schuckert, Asier Piñeiro Orioli, Jürgen Berges

Published Dec 10, 2018• 21 citations
arXivHTMLPDF

Abstract

We present a nonequilibrium quantum field theory approach to the initial-state dynamics of spin models based on two-particle irreducible (2PI) functional integral techniques. It employs a mapping of spins to Schwinger bosons for arbitrary spin interactions and spin lengths. At next-to-leading order (NLO) in an expansion in the number of field components, a wide range of nonperturbative dynamical phenomena are shown to be captured, including relaxation of magnetization in a 3D long-range interacting system with quenched disorder, different relaxation behavior on both sides of a quantum phase transition, and the crossover from relaxation to arrest of dynamics in a disordered spin chain previously shown to exhibit many-body localization. Where applicable, we employ alternative state-of-the-art techniques and find rather good agreement with our 2PI NLO results. As our method can handle large system sizes and converges relatively quickly to its thermodynamic limit, it opens the possibility to study these phenomena in higher dimensions in regimes in which no other efficient methods exist. Furthermore, the approach to classical dynamics can be investigated as the spin length is increased.

DOI
10.1103/physrevb.98.224304

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We present a nonequilibrium quantum field theory approach to the initial-state dynamics of spin models based on two-particle irreducible (2PI) functional integral techniques. It employs a mapping of spins to Schwinger bosons for arbitrary spin interactions and spin lengths. At next-to-leading order (NLO) in an expansion in the number of field components, a wide range of nonperturbative dynamical phenomena are shown to be captured, including relaxation of magnetization in a 3D long-range interacting system with quenched disorder, different relaxation behavior on both sides of a quantum phase transition, and the crossover from relaxation to arrest of dynamics in a disordered spin chain previously shown to exhibit many-body localization. Where applicable, we employ alternative state-of-the-art techniques and find rather good agreement with our 2PI NLO results. As our method can handle large system sizes and converges relatively quickly to its thermodynamic limit, it opens the possibility to study these phenomena in higher dimensions in regimes in which no other efficient methods exist. Furthermore, the approach to classical dynamics can be investigated as the spin length is increased.
Abstract We develop a unified framework for open quantum systems composed of many mutually interacting quantum spins, or any isomorphic systems like qubits and qudits, surrounded by one or more independent bosonic baths. Our framework, based on Schwinger–Keldysh field theory (SKFT), can handle arbitrary spin value S , dimensionality of space, and geometry, while being applicable to a large parameter space for system and bath. It can probe regimes in which non-Markovian dynamics and nonperturbative effects pose formidable challenges for other state-of-the-art theoretical methods. This is achieved by working with the two-particle irreducible (2PI) effective action, which resums classes of Feynman diagrams of SKFT to an infinite order. Furthermore, such diagrams are generated via an expansion in 1 / N , where N is the number of Schwinger bosons we employ to map spin operators onto canonically commuting ones, rather than via conventional expansion in system-bath coupling constant. We carefully benchmark our SKFT+2PI-computed results vs. numerically (quasi)exact ones from tensor network calculations applied to the archetypical spin-boson model where both methodologies are applicable. Additionally, we demonstrate the capability of SKFT+2PI to handle a much more complex spin-chain-boson model with multiple baths interacting with each spin where no benchmark from other methods is available at present. The favorable numerical cost of solving integro-differential equations produced by the SKFT+2PI framework with an increasing number of spins and time steps makes it a promising route for simulating driven-dissipative systems in quantum computing, quantum magnonics, and quantum spintronics.
In this thesis, far-from-equilibrium dynamics of fermionic quantum gases is discussed utilising functional quantum field theoretical methods. Employing the Schwinger-Keldysh path integral, real-time Schwinger-Dyson/Kadanoff-Baym dynamic equations for the two-point correlation functions are derived from the two-particle irreducible (2PI) effective action. For two specific models, these dynamic equations are investigated further. (a) For an N-fold spin-degenerate ultra-cold Fermi gas, non-perturbative approximation schemes based on either a loop or a 1/N expansion of the 2PI effective action are presented. Adopting these approximations, the long-time evolution of a homogeneous Fermi gas with N=2 after an initial preparation far from thermal equilibrium is thoroughly studied in one spatial dimension. Depending on the total energy, the gas is found to evolve into thermal as well as non-thermal states, the latter becoming manifest in violating the fluctuation-dissipation relation. (b) A similar 1/N expansion is derived for the SU(N) symmetric Kondo lattice model. At leading order, the mean-field dynamic equations of the U=0 Anderson model are recovered. At next-to-leading order (NLO), both spin-flip and direct interactions between localised atoms and conduction band atoms are taken into account non-perturbatively into the dynamic equations. This allows future studies of possibly existing novel phases in coupling regimes where the Kondo screening and RKKY-type interactions are competing.
We calculate the far-from-equilibrium dynamics and thermalization both for the quantum and the classical O(N)–model. The early and late-time behavior can be described from the 2PI–loop expansion for weak couplings or the nonperturbative 2PI–1/N expansion of the effective action beyond leading order. A comparison with exact simulations in 1+1 dimensions in the classical limit shows that the 2PI–1/N expansion at next-to-leading order gives quantitatively precise results already for moderate values of N. We derive a criterion for the validity of the classical approximation and verify it by comparing far-fromequilibrium quantum and classical dynamics. At late times one observes the expected deviations due to the difference between classical and quantum thermal equilibrium. In recent years we have witnessed an enormous increase of interest in the dynamics of quantum fields out of equilibrium. Strong motivation comes from a wide range of applications including current and upcoming relativistic heavy-ion collision experiments, phase transitions in the early universe or the dynamics of Bose-Einstein condensation. Directly simulating quantum fields in real time, such as solving the Schrödinger equation for the wave functional is prohibitively difficult and one has to find suitable approximations. The
A central challenge in strongly interacting many-body systems is understanding the far-from-equilibrium dynamics. Here we study the many-body magnetic dynamics of the two-component Bose-Hubbard model by developing a two-component extension of nonequilibrium bosonic dynamical mean-field theory. Using this numerical method, we uncover rich quantum spin dynamics via interspecies interaction quenches. A sudden ramp up of interactions induces slow thermalization, leading to a long-lived metastable state, whereas quenching to weak interactions results in rapid thermal equilibrium, featuring a two-step relaxation behavior through distinct exponential decays. Furthermore, under periodic modulation of the interspecies interactions, emergent Floquet dynamics drives a transition from a magnetic to an unordered phase.
We investigate the conditions under which quantum fluctuations are relevant for the quantitative interpretation of experiments with ultracold Bose gases. This requires to go beyond the description in terms of the Gross-Pitaevskii and Hartree-Fock-Bogoliubov mean-field theories, which can be obtained as classical (statistical) field-theory approximations of the quantum many-body problem. We employ functional-integral techniques based on the two-particle irreducible (2PI) effective action. The role of quantum fluctuations is studied within the nonperturbative 2PI 1∕N1∕\mathcal{N}1∕N expansion to next-to-leading order. At this accuracy level memory integrals enter the dynamic equations, which differ for quantum and classical statistical descriptions. This can be used to obtain a classicality condition for the many-body dynamics. We exemplify this condition by studying the nonequilibrium evolution of a one-dimensional Bose gas of sodium atoms, and discuss some distinctive properties of quantum versus classical statistical dynamics.
There has been substantial progress in recent years in the quantitative understanding of the nonequilibrium time evolution of quantum fields. Important topical applications, in particular in high energy particle physics and cosmology, involve dynamics of quantum fields far away from the ground state or thermal equilibrium. In these cases, standard approaches based on small deviations from equilibrium, or on a sufficient homogeneity in time underlying kinetic descriptions, are not applicable. A particular challenge is to connect the far-from-equilibrium dynamics at early times with the approach to thermal equilibrium at late times. Understanding the ``link'' between the early- and the late-time behavior of quantum fields is crucial for a wide range of phenomena. For the first time questions such as the explosive particle production at the end of the inflationary universe, including the subsequent process of thermalization, can be addressed in quantum field theory from first principles. The progress in this field is based on efficient functional integral techniques, so-called n-particle irreducible effective actions, for which powerful nonperturbative approximation schemes are available. Here we give an introduction to these techniques and show how they can be applied in practice. Though we focus on particle physics and cosmology applications, we emphasize that these techniques can be equally applied to other nonequilibrium phenomena in complex many body systems.