Imaginary-Time Formulation of Steady-State Nonequilibrium: Application to Strongly Correlated Transport

Abstract

We extend the imaginary-time formulation of the equilibrium quantum many-body theory to steady-state nonequilibrium with an application to strongly correlated transport. By introducing Matsubara voltage, we keep the finite chemical potential shifts in the Fermi-Dirac function, in agreement with the Keldysh formulation. The formulation is applied to strongly correlated transport in the Kondo regime using the quantum Monte Carlo method.PACS numbers: 73.63. Kv, 72.10.Bg, 72.10.Di A coherent formulation of equilibrium and nonequilibrium is one of the ultimate goals of statistical physics. In the last two decades, this has become a particularly pressing issue with the advances in nanoelectronics. Although it has long been considered such Gibbsian description may exist in the steady-state nonequilibrium [1], implementation of time-independent nonequilibrium quantum statistics has produced limited success [2] without widely applicable algorithms.In nanoelectronics, the strong interplay between manybody interactions and nonequilibrium demands nonperturbative treatments of the quantum many-body effects. Perturbative Green function techniques [3,4] have been successful, but are often plagued by complicated diagrammatic rules and are limited to simple models. In the last few years, important advances have been made in this field to complement the diagrammatic theory. Timedependent renormalization group [5,6] and densitymatrix renormalization group method [7] were applied to calculate the real-time convergence toward the steadystate. Real-time methods [5,6,7] calculate the process toward the steady-state and therefore have clear physical interpretations. Unfortunately they often suffer from long-time behaviors associated with low energy strongly correlated states and finite size effects. Direct construction of nonequilibrium ensembles through the scattering state formalism [2,8,9, 10] and field theoretic approach [11] have provided new perspectives to the problem.The main goal of this work is to provide a critical step toward the time-independent description of equilibrium and steady-state nonequilibrium quantum statistics. In addition to the resolution of this fundamental problem, we provide a strong application. The steadystate nonequilibrium can be solved within the same formal structure as equilibrium, and therefore the powerful equilibrium many-body tools, such as the quantum Monte Carlo (QMC) method, can be easily applied to complex transport systems with many competing interactions. We demonstrate this point by applying this formalism to strongly correlated transport in the Kondo regime by using QMC. In contrast to the real-time methods, this approach starts from the steady-state and simulates the effect of many-body interaction. However, numerical analytic continuation and low temperature calculation, especially with the QMC application, are technical difficulties.In the following, we first construct a time-independent statistical ensemble of steady-state nonequilibrium [2] in the non-interacting limit with the i...

Nonequilibrium dynamical mean-field theory and its applications

“…The Hamiltonian approach has been used in combination with the numerical renormalization group (Joura, Freericks, and Pruschke, 2008), and very recently with the exact-diagonalization-based method (Arrigoni, Knap, and von der Linden, 2013;Gramsch et al, 2013). In the case of a steady state, the effective Matsubara method (Han and Heary, 2007) based on Hershfield's expression (Hershfield, 1993) for a steady-state density matrix has been tested as an impurity solver (Aron, Weber, and Kotliar, 2013). In the following, we focus on the diagrammatic approaches, which have been used in various types of applications.…”

“…To sample the configurations, one usually adopts the Metropolis-Hastings algorithm (Metropolis et al, 1953;Hastings, 1970). One proposes to insert an nth interaction vertex at t n ∈  with probability p prop ðn − 1 → nÞ ¼ jdt n j=ð2t max þ βÞ, or to remove the nth vertex with probability p prop ðn → n − 1Þ ¼ 1=n.…”

Impurity model for non-equilibrium steady states

“…Among several promising steady-state techniques [3], a remarkable leap forward has been recently achieved by Han and Heary who developed a non-equilibrium steadystate impurity solver based on a Matsubara-like formalism and a Hirsch-Fye algorithm (NESS-HF) [4,5]. Progress has also been made in describing the nonequilibrium steady-state dynamics of correlated electrons on finite dimensional lattices [6,7].…”

“…To demonstrate the practical relevance of the impurity model we introduced, we solve it and compute non-perturbatively the interaction contribution to the retarded self-energies Σ R U by generalizing to multiple leads the steady-state impurity solver that was recently developed by Han and Heary in the context of a two-lead environment [4,13]. Unlike the IPT solver, this solver provides conserving solutions even away from particle-hole symmetry.…”

Nonequilibrium quantum transport through a dissipative resonant level

“…Though much is known for quantum impurity systems in equilibrium, understanding their properties in non-equilibrium steady-state is still limited. Nevertheless, significant progress has been made by different approaches, such as (1) analytical approximations: perturbative renormalization group method (RG) 27,28 , Hamiltonian flow equations 29 , functional RG 30,31 , strong-coupling expansions 32 , master equations 33 ; (2) exact analytical solutions: field theory techniques 34 , the scattering Bethe Ansatz 35 , mapping of a steady-state non-equilibrium problem onto an effective equilibrium system [36][37][38][39] , non-linear response theory approach to current fluctuations 40 ; (3) numerical methods: time-dependent density matrix renormalization group (RG) 41 , time-dependent numerical RG 42 , diagrammatic Monte Carlo 43 , and imaginary-time nonequilibrium quantum Monte Carlo 44 .…”