Session L20: Quantum Many-Body Systems and Methods II

Entanglement dynamics in quantum many-body systems

The dynamics of quantum entanglement $S(t)$ has proven useful to distinguishing different quantum many-body phases. In particular, the growth of entanglement following a quantum quench can be used to distinguish between many-body localized($S(t) \sim \log t$) and ergodic($S(t) \sim t$) phases. Here, we provide a theoretical description of the growth of entanglement in a quantum many-body system, and propose a method to experimentally measure it. We show that entanglement growth is related to the spreading of local operators. In ergodic systems, the linear spreading of operators results in a universal, linear in time growth of entanglement. Furthermore, we show that entanglement growth is directly related to the decay of the Loschmidt echo in a composite system comprised of many copies of the original system, subject to a perturbation that reconnects different parts of the system. Using this picture, we propose an experimental set-up to measure entanglement growth by using a quantum switch (two-level system) which controls connections in the composite system. Our work provides a way to directly probe dynamical properties of many-body systems, in particular, allowing for a direct observation of many-body localization.   

Measuring Entanglement Spectrum via Density Matrix Exponentiation

Entanglement spectrum (ES), the eigenvalues of the reduced density matrix of a subsystem, serves as a powerful theoretical tool to study many-body systems. For example, the gap and degeneracies of the entanglement spectrum have been used to identify various topological phases. However, the usefulness of such a concept in real experiments has been debated, since it is believed that obtaining the ES requires full state tomography, at a cost which exponentially grows with the systems size. Inspired by a recent density matrix exponentiation technique, we propose a scheme to measure ES by evolving the system with a Hamiltonian that is the subsystem's own reduced density matrix. Such a time evolution can be induced by an ancilla photon that is coupled to multiple qubits at the same time. The phase associated with the time evolution can be detected and converted into ES through either a digital or an analogue scheme. The digital scheme involves a modified quantum phase estimation algorithm based on random time evolution, while the analogue scheme is in the spirit of Ramsey interferometry. Both schemes are not limited by the size of the system, and are especially sensitive to the gap and degeneracies. We also discuss the implementation in cavity/circuit-QED and ion trap systems.   

Flow equation holography

The Ryu-Takayanagi conjecture [1] about the holographic derivation of the entanglement entropy provides a remarkable geometric picture by relating minimal surfaces to the entanglement entropy. Underlying this conjecture is the AdS/CFT correspondence, which limits the applicability of this geometric picture in its original formulation to a very specific set of theories. In this talk I will show how the flow equation method [2,3] can be used to construct an emergent geometric picture for generic quantum many-body systems. Results for the emergent Riemannian geometry of certain low-dimensional quantum systems are presented based on analytical and numerical solution of the flow equations. Minimal surfaces on these Riemannian manifolds show behavior in agreement with the entanglement entropy of the corresponding quantum theory, both for gapped and critical systems. \newline [1] S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006) \newline [2] F. Wegner, Ann. Phys. (Leipzig) 3, 77 (1994) \newline [3] S. Kehrein, The Flow Equation Approach to Many-Particle Systems (Springer, 2006)   

Irreversibility in Quantum Many-Body Systems

The question of thermalization in closed quantum many-body systems has received a lot of attention in the past few years. An intimately related question is whether a closed quantum system shows irreversible dynamics. However, irreversibility and what we actually mean by this in a quantum many-body system with unitary dynamics has been explored very little. In our work we investigate the irreversibility of dynamics in quantum many-body systems by studying echo dynamics. In order to quantify the (ir)reversibility we study the time evolution involving an imperfect effective time reversal. Our measure for the recovery of the initial state are the echo peaks occurring in the time evolution of observables. Specifically, we investigate non-interacting and interacting one-dimensional spin chains. We study the characteristics of the echo peak decay and especially focus on whether this depends on the (non-)integrability of the model.   

Quantum Quench Dynamics in the Transverse Field Ising Model at Non-zero Temperatures

The recently discovered \emph{Dynamical Phase Transition} denotes non-analytic behavior in the real time evolution of quantum systems in the thermodynamic limit and has been shown to occur in different systems at zero temperature [Heyl \emph{et al.}, Phys. Rev. Lett. \textbf{110}, 135704 (2013)]. In this talk we present the extension of the analysis to non-zero temperature by studying a generalized form of the Loschmidt echo, the work distribution function, of a quantum quench in the transverse field Ising model. Although the quantitative behavior at non-zero temperatures still displays features derived from the zero temperature non-analyticities, it is shown that in this model dynamical phase transitions do not exist if $T>0$. This is a consequence of the system being initialized in a thermal state. Moreover, we elucidate how the Tasaki-Crooks-Jarzynski relation can be exploited as a symmetry relation for a global quench or to obtain the change of the equilibrium free energy density.   

Nonequilibrium Kondo physics in the Anderson impurity model: Auxiliary master equation approach

An accurate investigation of the evolution of the Kondo peak as a function of bias voltage is presented for the single impurity Anderson model (SIAM). We greatly enhance the capability of the recently introduced auxiliary master equation approach (AMEA) [1,2] by making use of matrix product states [3]. This allows us to obtain highly accurate spectral functions and observables for the SIAM at large values of the interaction and low temperatures $T$, well below the Kondo scale $T_K$. For $T\approx T_K/4$ and $T\approx T_K/10$ we find a clear splitting of the Kondo resonance into a two-peak structure at bias voltages just above $T_K$. A benchmark in the equilibrium case for $T\approx T_K/4$ reveals a remarkably close agreement to the numerical renormalization group. This, together with the high flexibility and the applicability to various problems such as dynamical mean field theory [1,4,5], demonstrates the great potential of AMEA for correlated systems, both in nonequilibrium as well as in equilibrium situations. \\ {[1] E. Arrigoni et al., PRL 110, 086403 (2013)} \\ {[2] A. Dorda et al., PRB 89, 165105 (2014)} \\ {[3] A. Dorda et al., PRB 92, 125145 (2015)} \\ {[4] I. Titvinidze et al., arXiv:1508.02953} \\ {[5] A. Dorda et al., arXiv:1509.09255} \\   

State of the Polaron: Criticality in the Spin Boson Model

In strongly coupled open quantum systems it can become necessary to take into account the behaviour of environmental degrees of freedom more rigorously than is usual with standard weak coupling techniques. We investigate the use of a variational ansatz, making use of a multitude of coherent states, to obtain more precise information about the interaction between an open system and its environment. We provide a thorough study of the ground state of the ubiquitous spin boson model; presenting analytic and numerical results concerning the nature of its much debated quantum phase transition.   

Quantum Phase Transitions detected by a local probe using Time Correlations and Violations of Leggett-Garg Inequalities

We introduce a new way of identifying quantum phase transitions of many-body systems by means of local time correlations and Leggett-Garg inequalities. This procedure allows to experimentally determine the quantum critical points not only of finite-order transitions but also those of infinite-order as the Kosterlitz-Thouless transition that is not always easy to detect with current methods. By means of an analytical calculation on a general spin-$1/2$ Hamiltonian, and matrix product simulations of one-dimensional $XXZ$ and anisotropic $XY$ models, we argue that finite-order quantum phase transitions can be determined by singularities of the time correlations or their derivatives at criticality. The same features are exhibited by corresponding Leggett-Garg functions, which remarkably indicate violation of the Leggett-Garg inequalities for early times and all the Hamiltonian parameters considered. In addition, we find that the infinite-order transition of the $XXZ$ model at the isotropic point can be revealed by the maximal violation of the Leggett-Garg inequalities.   

Scaling behaviors at discontinuous quantum transitions

First-order (or discontinuous) quantum phase transitions (FOQTs) are characterized by a vanishing energy gap and jumps in the values of some observables across the critical point in the thermodynamic limit. Unlike what happens at continuous transitions, the correlation lengths remain finite at FOQTs. Nevertheless, finite systems at FOQTs exhibit finite-size effects, in the form of a rounding and smoothing of the discontinuities. We show that a scaling theory, similar to the usual finite-size scaling, can be formulated at FOQTs, and that the relevant scaling variable is extremely sensitive to the choice of boundary conditions. We further consider the scaling effects due to the presence of spatial inhomogeneities, in analogy with trap-size scaling at continuous transitions. Our results are supported by numerical simulations on the ferromagnetic quantum Ising chain and on the $q$-state quantum Potts chain with $q>4$. We provide FSS predictions for the energy gap and the magnetization of finite quantum chains, which can be relevant for quantum computation applications.   

Order $O(1)$ algorithm for first-principles transient current through open quantum systems

First principles transient current through molecular devices is known to be extremely time consuming with typical computational complexity $T^3 N^3$ where $N$ and T are the dimension of the scattering system and the number of time steps respectively. Various algorithms have been developed which eventually brings the complexity down to $c T N^3$, a linear scaling in $T$, where $c$ is a large coefficient comparable to $N$. Here we provide an order $O(1)$ algorithm that reduces it further to $c_1 N^3+c_2 T N^2$ where $c_1$ and $c_2$ are $\sim$50 and 0.1 respectively. Hence for $TN$. Benchmark calculation has been done on graphene nanoribbons using Tight-binding (TB) Hamiltonian with a huge speed up factor of $100T$, confirmed the $O(1)$ scaling.   

Generalized non-equilibrium vertex correction method in coherent medium theory for quantum transport simulation of disordered nanoelectronics

In realistic nanoelectronics, disordered impurities/defects are inevitable and play important roles in electron transport. However, due to the lack of effective quantum transport method, the important effects of disorders remain poorly understood. Here, we report a generalized non-equilibrium vertex correction (NVC) method with coherent potential approximation to treat the disorder effects in quantum transport simulation. With this generalized NVC method, any averaged product of two single-particle Green’s functions can be obtained by solving a set of simple linear equations. As a result, the averaged non-equilibrium density matrix and various important transport properties, including averaged current, disordered induced current fluctuation and the averaged shot noise, can all be efficiently computed in a unified scheme. Moreover, a generalized form of conditionally averaged non-equilibrium Green's function is derived to incorporate with density functional theory to enable first-principles simulation. We prove the non-equilibrium coherent potential equals the non-equilibrium vertex correction. Our approach provides a unified, efficient and self-consistent method for simulating non-equilibrium quantum transport through disorder nanoelectronics.   

Efficient heat-bath sampling in Fock space

We introduce an algorithm for sampling many-body quantum states in Fock space. The algorithm efficiently samples states with probability approximately proportional to an arbitrary function of the second-quantized Hamiltonian matrix elements connected to the current state. We apply the new sampling algorithm to the recently-developed Semistochastic Full Configuration Interaction Quantum Monte Carlo method (S-FCIQMC), a semistochastic implementation of the power method for projecting out the ground state energy in a basis of Slater determinants. The heat-bath sampling requires modest additional computational time and memory compared to uniform sampling but results in newly-spawned weights that are approximately of the same magnitude, thereby greatly improving the efficiency of projection. A comparison in efficiency between uniform and approximate heat-bath sampling is performed on the all-electron nitrogen dimer at equilibrium in Dunning's cc-pVXZ basis sets with $X\in\left\{ D,T,Q,5\right\}$, demonstrating a large gain in efficiency that increases with basis set size.