Session L18: Quantum Manybody Systems: Theory and Computation I

Entanglement Entropy from Nonequilibrium Work

The Rényi entanglement entropy of quantum many-body systems can be viewed as the difference in free energy of partition functions with different trace topologies. We introduce an external field λ that controls the partition function topology, allowing us to define a notion of nonequilibrium work as λ is varied smoothly. Nonequilibrium fluctuation theorems of the work provide us with statistically exact estimates of the Rényi entanglement entropy. We put these ideas to use in the context of quantum Monte Carlo simulations of SU(N) symmetric spin models in one and two dimensions. In both cases we detect logarithmic violations to the area law with high precision, allowing us to extract the central charge for the critical 1D models and the number of Goldstone modes for the magnetically ordered 2D models.   

Entanglement Entropy and Negativity in Inhomogeneous (1+1)D Systems: The Rainbow Chain, the SSD and their Holographic Dual

Starting with a system described by a conformal field theory (e.g. a critical spin chain or free fermions), one can find interesting violations to the typical logarithmic behavior of the bipartite entanglement entropy by introducing an inhomogeneous kinetic term in the Hamiltonian. Two examples of recent interest are the rainbow chain and the sine-squared deformed (SSD) model. Such systems can be equivalently described by placing the original CFT on a curved background manifold. Using the AdS/CFT correspondence, we develop a holographic dual description of inhomogeneous (1+1) dimensional systems by foliating the bulk spacetime with curved surfaces. Extending these foliations to the BTZ spacetime allows us to describe inhomogneous systems at finite temperatures. Using field-theoretic, holographic, and numerical techniques, we are able to compute the entanglement entropy and negativity, for various configurations of intervals, both at zero at finite temperatures, for the rainbow chain and the SSD.   

Crossover Behavior of Entanglement Entropy for Energy Eigenstates of 1d and 2d Fermionic Systems

The entanglement entropy in ground states of typical condensed matter systems obeys the area law or a log-area law for critical systems. Subsystem entropies in random and thermal states obey a volume law. Here, we discuss the distribution of entanglement entropy in energy eigenstates of quasi-free fermionic systems as a function of energy and subsystem size. Numerical results are obtained with a Monte Carlo approach. We characterize the crossover behavior from the area or log-area law in the vicinity of the ground state and for small subsystem size to the volume law at higher energy and larger subsystem size. The coefficients of the volume law scaling can be matched to entropy densities in equilibrium thermal ensembles. For critical 1d systems at low energies, the universal crossover function matches the prediction from 1+1d conformal field theory for systems at nonzero temperatures. For 2d systems, we find a similar crossover behavior.   

Relating different localization lengths via non-perturbative construction of local integrals of motion in the many-body localized phase

Many-body localization (MBL) is characterized by the absence of thermalization and the violation of conventional thermodynamics. The phenomenological model, which describes the system using a complete set of local integrals of motion (LIOMs), provides a powerful tool to understand MBL. We explicitly compute a complete set of LIOMs non-perturbatively by maximizing the overlap between LIOMs and physical spin operators. This method enables a direct mapping from real space Hamiltonian to the phenomenological model. We demonstrate the exponential decay of weight of LIOMs in real-space and interaction strength of LIOMs in range. We further compare the localization lengths extracted from LIOMs, their interactions and dynamics. Our scheme is immune to accidental resonances and can be applied even at the phase transition point, providing a novel tool to study the microscopic features of the phenomenological model of MBL.   

A numerical procedure for non-integrable many-body quantum systems

Numerical computations of properties of quantum many-body systems generally get drastically simplified by the presence of conserved quantities. Non-integrable quantum systems are those that lack, or have very few, conserved quantities and hence, are invariably intractable numerically. However, the eigenstate thermalization hypothesis postulates certain properties that non-integrable quantum systems must have and so far, computations on small systems have validated these properties. In this talk, a numerical procedure will be described that exploits the eigenstate thermalization hypothesis to discard, at the outset, vast amounts of useless quantum information and extract useful information about a non-integrable system more efficiently. The utility of the algorithm will be demonstrated via comparisons with exact diagonalization studies on prototypical non-integrable models.   

Product Spectrum Approximation

Calculating the physical properties of quantum thermal states is a difficult problem for classical computers, rendering it intractable for most quantum many-body systems. To address this problem, we propose a variational scheme to prepare approximate thermal states on a quantum computer by applying a series of two-qubit gates to a product state. We apply our method to a non-integrable region of the mixed field Ising chain and the Sachdev-Ye-Kitaev model. We demonstrate how our method can be easily extended to large systems governed by local Hamiltonians and the preparation of thermofield double states. By comparing our results with exact solutions, we find that our construction enables the efficient preparation of approximate thermal states on quantum devices. Our results imply that the details of the many-body energy spectrum are not needed to capture simple thermal observables.   

Non-Markovian Quantum Dynamics via General Quantum Master Equation and Tensor Networks

Computing real-time dynamics of a quantum many-body system is challenging due to the exponential growth of entanglement. While many different approaches have been investigated, how to compute the real-time dynamics of generic condensed matter systems, or how to best classify for which systems the dynamics cannot feasibly be computed on a classical computer, remain open questions. In this work, we treat the Nakajima-Zwanzig general quantum master equation with tensor network methods to identify the limits of classical time evolution algorithms. Though the memory kernel is difficult to compute even numerically, tensor networks provide a systematic means of obtaining the kernel for diverse complex systems, such as the spin-boson model with anharmonic bath sites. In this work we focus on the spin-boson model and analyze how the types and strengths of interactions affect the lifetime of the memory kernel. Our work provides general insight into the classical simulatability of quantum dynamics.
  

Collective Excitations in a Landau-Majorana Liquid

Landau-Fermi liquid theory is one of the foundational theories of interacting many-body fermions. Its simplicity has led to its continued application to real materials, and the rare instances where it fails (e.g., the case of a non-Fermi liquid) have become a major concern in the condensed matter community. In this study, we apply the formalism of Landau-Fermi liquid theory to describe a quantum liquid of interacting Majorana fermions. As opposed to Majorana zero modes, which characterize topologically non-trivial materials and obey anyonic statistics, Majorana fermions are spin-1/2 particles that experience mutual pairwise annihilation. Drawing on a previous work (arXiv:1709.04483), we describe the quasiparticles in such a system and construct the Landau-Silin kinetic equation. The presence of a robust Fermi surface in the screened Majorana system leads to a Lifshitz transition in the limit of large driving frequency. In addition, a calculation of the zero sound leads to the presence of an enhanced stability of the Landau-Majorana liquid against Pomeranchuk instabilities. These results lead to important differences in the fundamental physics of interacting Majorana fermion ensembles that has potential applications in various condensed matter systems and astrophysical phenomena.   

Divergences of the irreducible vertex functions in correlated metallic systems: Insights from the Anderson Impurity Model

We analyze the occurrence of divergences in the irreducible vertex functions of the Anderson impurity model (AIM) [1]. These divergences — a surprising hallmark of the breakdown of many-electron perturbation theory [2] — have been recently observed [3,4] in several contexts, including the dynamical mean-field solution of the Hubbard model. Hitherto, however, a clarification of their origin could be obtained only in the limit of high temperatures and/or large interactions, where the
underlying physics is greatly simplified by the absence of low-energy quasiparticle excitations. In this respect, our numerical calculations for the AIM, as well as their comparison with the corresponding results for the Hubbard model, allow us to clarify several open questions about the origin and the properties of vertex divergences occurring in a more interesting context, the correlated metallic regime at low-temperatures.

[1] P. Chalupa, et.al., Phys. Rev. B 97, 245136 (2018).
[2] O. Gunnarsson, et.al., Phys. Rev. Lett. 119, 056402 (2017).
[3] T. Schäfer, et.al., Phys. Rev. Lett. 110, 246405 (2013).
[4] T. Schäfer, et.al., Phys. Rev. B 94, 235108 (2016).   

Kondo impurities at the edges of a quantum wire

Quantum impurity systems, where a localized impurity is coupled to a large bath of particles appear naturally in many fields from mesoscopic physics and quantum dot systems to the Kondo lattice of Heavy fermion materials.
In this talk we study a miniature Kondo lattice consisting of a one dimensional quantum wire coupled to Kondo impurities at either end. We show that it is exactly solvable by means of Bethe Ansatz for arbitrary values of bulk spin exchange interaction and of the Kondo coupling. The model exhibits both gapped and gapless phases depending on the sign of the bulk interaction and a competition between dynamical mass generation and the Kondo effect. We construct the ground state in all regimes and study the excitations and their finite size corrections as well as the impurity magnetization as function of an applied field.   

Dynamics of Densities and Currents in Spin Ladders

The impact of integrability or nonintegrability on the dynamics of isolated quantum systems is a longstanding issue. For integrable models, a macroscopic set of (quasi)local conservation laws can lead to partially conserved currents and ballistic transport. In generic situations, however, integrability is lifted due to various perturbations and currents are expected to decay. Still, since the dynamics of interacting quantum many-body systems poses a formidable challenge to theory and numerics, it remains open whether nonintegrability as such already implies the emergence of diffusion.
In this context, we study the dynamics of spin and energy in the two-leg spin-1/2 ladder with up to 40 lattice sites, using an efficient pure-state approach based on the concept of typicality. We discuss correlation functions in real and momentum space, and in the time and frequency domain, providing a comprehensive picture of high-temperature dynamics in this archetypal nonintegrable quantum model. Particularly, we unveil the occurence of diffusion for both spin and energy.

[1] J. Richter, F. Jin, H. De Raedt, K. Michielsen, J. Gemmer, R. Steinigeweg, PRB 97, 174430 (2018).
[2] J. Richter, F. Jin, L. Knipschild, J. Herbrych, H. De Raedt, K. Michielsen, J. Gemmer, R. Steinigeweg, in preparation.

  

Semiclassical echo dynamics in the Sachdev-Ye-Kitaev model

The existence of a quantum butterfly effect in the form of exponential sensitivity to small perturbations has been under debate for a long time. Lately, this question gained increased interest due to the proposal to probe chaotic dynamics and scrambling using out-of-time-order correlators. In this work we study echo dynamics in the Sachdev-Ye-Kitaev model under effective time reversal in a semiclassical approach. We demonstrate that small imperfections introduced in the time-reversal procedure result in an exponential divergence from the perfect echo, which allows to identify a Lyapunov exponent λ. In particular, we find that λ is twice the Lyapunov exponent of the semiclassical equations of motion. This behavior is attributed to the growth of an out-of-time-order double commutator that resembles an out-of-time-order correlator.
This talk is based on arXiv:1802.06796.   

Generalized Hydrodynamics Revisited

A number of attempts to formulate a continuum description of complex states have been proposed to circumvent more cumbersome many-body and simulation methods. Typically these have been quantum systems and the resulting phenomenologies frequently called "quantum hydrodynamics". This objective is placed in a formally controlled context using the exact macroscopic conservation laws for number density, energy density, and momentum density, together with standard tools of non-equilibrium statistical mechanics. These continuum equations entail perfect fluid fluxes that are functionals of the chosen fields, plus unknown irreversible energy and momentum fluxes. Typically, the latter are obtained from a solution to the Liouville-von Neumann equation for small space and time variations. Instead, here we avoid that restriction by requiring that the unknown irreversible fluxes deliver the exact linear response functions for the fields. In this way, a non-linear generalized hydrodynamic description is obtained, valid across a broad range of length and time scales. The example of electrons in a given ion configuration is described to make contact with current phenomenological "quantum hydrodynamics".   

Quantum Fluid Dynamics (QFD ) or Bohmian Representation of Schrödinger Equation with Navier – Stokes Type Dissipation Attila Askar Koc University, Sariyer, Istanbul 34450, Turkey

The Quantum Fluid Dynamics (QFD) representation has its foundations in the works of Madelung, De Broglie and Bohm. It is an interpretation of quantum mechanics with the goal to find classically identifiable dynamical variables at the sub-particle level. The approach is partly motivated by Einstein’s questioning of the completeness of the quantum theory. Einstein expected the complete theory to have nonlinearity and admit solutions with “particle” nature, similar to solitons in contemporary terminology.
The QFD approach leads to two conservation laws, for ”mass” and ”momentum”, similar to those in fluid-dynamics for a compressible fluid as a set of nonlinear partial differential equations. The QFD formalism is utilized advantageously for solving the time dependent Schrödinger equation for scattering problems.
This paper extends the QFD formalism to include dissipation in the form of Navier – Stokes term in classical fluid dynamics. The introduction of dissipation in the Navier – Stokes sense transform the differential equations from hyperbolic to parabolic type, with nonlinear and dispersive terms. These offer a natural framework for fundamentally new phenomena, in particular possibility of dissipation in quantum mechanics, solitons, vortices and chaos.