Nonequilibrium functional renormalization group for interacting quantum systems

Coarse graining in time with the functional renormalization group: Relaxation in Brownian motion

We apply the functional renormalization group (fRG) to study relaxation in a stochastic process governed by an overdamped Langevin equation with one degree of freedom, exploiting the connection with supersymmetric quantum mechanics in imaginary time. After reviewing the functional integral formulation of the system and its underlying symmetries, including the resulting Ward-Takahashi identities for arbitrary initial conditions, we compute the effective action Γ from the fRG, approximated in terms of the leading and subleading terms in the gradient expansion: the local potential approximation and wave-function renormalization, respectively. This is achieved by coarse graining the thermal fluctuations in time resulting in, e.g., an effective potential incorporating fluctuations at all timescales. We then use the resulting effective equations of motion to describe the decay of the covariance and the relaxation of the average position and variance toward their equilibrium values at different temperatures. We use as examples a simple polynomial potential, an unequal Lennard-Jones type potential, and a more complex potential with multiple trapping wells and barriers. We find that these are all handled well, with the accuracy of the approximations improving as the relaxation's spectral representation shifts to lower eigenvalues, in line with expectations about the validity of the gradient expansion. The spectral representation's range also correlates with temperature, leading to the conclusion that the gradient expansion works better for higher temperatures than lower ones. This paper demonstrates the ability of the fRG to expedite the computation of statistical objects in otherwise long-timescale simulations, acting as a first step to more complicated systems.

Renormalizing Open Quantum Field Theories

The functional renormalization group flow of a scalar field theory with quartic couplings and a sharp spatial momentum cutoff is presented in four-dimensional Minkowski space-time for the bare action by retaining the entanglement of the IR and the UV particle modes. It is argued that the open interaction channels have to be taken into account in quantum field theory defined by the help of a cutoff, and a non-perturbative UV-IR entanglement is found in closed or almost closed models.

Nonequilibrium Properties of Berezinskii-Kosterlitz-Thouless Phase Transitions

We employ a novel, unbiased renormalization-group approach to investigate nonequilibrium phase transitions in infinite lattice models. This allows us to address the delicate interplay of fluctuations and ordering tendencies in low dimensions out of equilibrium. We study a prototypical model for the metal to insulator transition of spinless interacting fermions coupled to electronic baths and driven out of equilibrium by a longitudinal static electric field. The closed system features a Berezinskii-Kosterlitz-Thouless transition between a metallic and a charge-ordered phase in the equilibrium limit. We compute the nonequilibrium phase diagram and illustrate a highly nonmonotonic dependence of the phase boundary on the strength of the electric field: for small fields, the induced currents destroy the charge order, while at higher electric fields it reemerges due to many-body Wannier-Stark localization physics. Finally, we show that the current in such an interacting nonequilibrium system can counter-intuitively flow opposite to the direction of the electric field. This nonequilibrium steady state is reminiscent of an equilibrium distribution function with an effective negative temperature.

Auxiliary master equation approach within stochastic wave functions: Application to the interacting resonant level model

We present further developments of the auxiliary master equation approach (AMEA), a numerical method to simulate many-body quantum systems in as well as out of equilibrium and apply it to the interacting resonant level model to benchmark the new developments. In particular, our results are obtained by employing the stochastic wave functions method to solve the auxiliary open quantum system arising within AMEA. This development allows us to reach extremely low wall times for the calculation of correlation functions with respect to previous implementations of AMEA. An additional significant improvement is obtained by extrapolating a series of results obtained by increasing the number of auxiliary bath sites, N_{B}, used within the auxiliary open quantum system formally to the limit of N_{B}→∞. Results for the current-voltage characteristics and for equilibrium correlation functions are compared with the one obtained by exact and matrix-product states-based approaches. Further, we complement this benchmark by the presentation of spectral functions for higher temperatures where we find different behaviors around zero frequency depending on the hybridization strength.

Nonequilibrium Steady-State Transport in Quantum Impurity Models: A Thermofield and Quantum Quench Approach Using Matrix Product States

The numerical renormalization group (NRG) is tailored to describe interacting impurity models in equilibrium, but it faces limitations for steady-state nonequilibrium, arising, e.g., due to an applied bias voltage. We show that these limitations can be overcome by describing the thermal leads using a thermofield approach, integrating out high energy modes using NRG, and then treating the nonequilibrium dynamics at low energies using a quench protocol, implemented using the time-dependent density matrix renormalization group. This yields quantitatively reliable results for the current (with errors ≲3%) down to the exponentially small energy scales characteristic of impurity models. We present results of benchmark quality for the temperature and magnetic field dependence of the zero-bias conductance peak for the single-impurity Anderson model.

Kardar-Parisi-Zhang equation with short-range correlated noise: Emergent symmetries and nonuniversal observables

We investigate the stationary-state fluctuations of a growing one-dimensional interface described by the Kardar-Parisi-Zhang (KPZ) dynamics with a noise featuring smooth spatial correlations of characteristic range ξ. We employ nonperturbative functional renormalization group methods to resolve the properties of the system at all scales. We show that the physics of the standard (uncorrelated) KPZ equation emerges on large scales independently of ξ. Moreover, the renormalization group flow is followed from the initial condition to the fixed point, that is, from the microscopic dynamics to the large-distance properties. This provides access to the small-scale features (and their dependence on the details of the noise correlations) as well as to the universal large-scale physics. In particular, we compute the kinetic energy spectrum of the stationary state as well as its nonuniversal amplitude. The latter is experimentally accessible by measurements at large scales and retains a signature of the microscopic noise correlations. Our results are compared to previous analytical and numerical results from independent approaches. They are in agreement with direct numerical simulations for the kinetic energy spectrum as well as with the prediction, obtained with the replica trick by Gaussian variational method, of a crossover in ξ of the nonuniversal amplitude of this spectrum.

Keldysh field theory for driven open quantum systems

Recent experimental developments in diverse areas-ranging from cold atomic gases to light-driven semiconductors to microcavity arrays-move systems into the focus which are located on the interface of quantum optics, many-body physics and statistical mechanics. They share in common that coherent and driven-dissipative quantum dynamics occur on an equal footing, creating genuine non-equilibrium scenarios without immediate counterpart in equilibrium condensed matter physics. This concerns both their non-thermal stationary states and their many-body time evolution. It is a challenge to theory to identify novel instances of universal emergent macroscopic phenomena, which are tied unambiguously and in an observable way to the microscopic drive conditions. In this review, we discuss some recent results in this direction. Moreover, we provide a systematic introduction to the open system Keldysh functional integral approach, which is the proper technical tool to accomplish a merger of quantum optics and many-body physics, and leverages the power of modern quantum field theory to driven open quantum systems.

Background field functional renormalization group for absorbing state phase transitions

We present a functional renormalization group approach for the active to inactive phase transition in directed percolation-type systems, in which the transition is approached from the active, finite density phase. By expanding the effective potential for the density field around its minimum, we obtain a background field action functional, which serves as a starting point for the functional renormalization group approach. Due to the presence of the background field, the corresponding nonperturbative flow equations yield remarkably good estimates for the critical exponents of the directed percolation universality class, even in low dimensions.

Driven Markovian Quantum Criticality

We identify a new universality class in one-dimensional driven open quantum systems with a dark state. Salient features are the persistence of both the microscopic nonequilibrium conditions as well as the quantum coherence of dynamics close to criticality. This provides a nonequilibrium analogue of quantum criticality, and is sharply distinct from more generic driven systems, where both effective thermalization as well as asymptotic decoherence ensue, paralleling classical dynamical criticality. We quantify universality by computing the full set of independent critical exponents within a functional renormalization group approach.

Modeling the metastable dynamics of correlated structures

Metastable quantum dynamics of an asymmetric triangular cluster that is coupled to a reservoir is investigated. The dynamics is governed by bath-mediated transitions, which in part require a thermal activation process. The decay rate is controlled by tuning the excitation spectrum of the frustrated cluster. We use the master equation approach and construct transition operators in terms of many-body states. We analyze dynamics of observables and reveal metastability of an excited state and of a magnetically polarized ground state.

A perturbative renormalization group approach to driven quantum systems

We use a perturbative momentum shell renormalization group (RG) approach to study the properties of a driven quantum system at zero temperature. To illustrate the technique, we consider a bosonic ϕ(4) theory with an arbitrary time dependent interaction parameter λ(t) = λ f(ω0t), where ω0 is the drive frequency and we derive the RG equations for the system using a Keldysh diagrammatic technique. We show that the scaling of ω0 is analogous to that of temperature for a system in thermal equilibrium and its presence provides a cutoff scale for the RG flow. We analyze the resultant RG equations, derive an analytical condition for such a drive to take the system out of the gaussian regime, and show that the onset of the non-gaussian regime occurs concomitantly with the appearance of non-perturbative mode coupling terms in the effective action of the system. We supplement the above-mentioned results by obtaining them from equations of motion of the bosons and discuss their significance for systems near critical points described by time-dependent Landau-Ginzburg theories.

Kondo memory in driven strongly correlated quantum dots

We investigate the real-time current response of strongly correlated quantum dot systems under sinusoidal driving voltages. By means of an accurate hierarchical equations of motion approach, we demonstrate the presence of prominent memory effects induced by the Kondo resonance on the real-time current response. These memory effects appear as distinctive hysteresis line shapes and self-crossing features in the dynamic current-voltage characteristics, with concomitant excitation of odd-number overtones. They emerge as a cooperative effect of quantum coherence-due to inductive behavior-and electron correlations-due to the Kondo resonance. We also show the suppression of memory effects and the transition to classical behavior as a function of temperature. All these phenomena can be observed in experiments and may lead to novel quantum memory applications.

Correlations in nonequilibrium Luttinger liquid and singular Fredholm determinants

We study interaction-induced correlations in Luttinger liquid with multiple Fermi edges. Many-particle correlation functions are expressed in terms of Fredholm determinants det(1+ÂB[over ^]), where A(ε) and B(t) have multiple discontinuities in energy and time spaces. We propose a general asymptotic formula for this class of determinants and provide analytical and numerical support to this conjecture. This allows us to establish nonequilibrium Fermi-edge singularities of many-particle correlation functions. As an example, we calculate a two-particle distribution function characterizing genuinely nonequilibrium quantum correlations between left- and right-moving fermions that have left the interaction region.

Nonequilibrium dynamical mean-field theory: an auxiliary quantum master equation approach

We introduce a versatile method to compute electronic steady-state properties of strongly correlated extended quantum systems out of equilibrium. The approach is based on dynamical mean-field theory (DMFT), in which the original system is mapped onto an auxiliary nonequilibrium impurity problem imbedded in a Markovian environment. The steady-state Green's function of the auxiliary system is solved by full diagonalization of the corresponding Lindblad equation. The approach can be regarded as the nontrivial extension of the exact-diagonalization-based DMFT to the nonequilibrium case. As a first application, we consider an interacting Hubbard layer attached to two metallic leads and present results for the steady-state current and the nonequilibrium density of states.

Physics in one dimension: theoretical concepts for quantum many-body systems

Various sophisticated approximation methods exist for the description of quantum many-body systems. It was realized early on that the theoretical description can simplify considerably in one-dimensional systems and various exact solutions exist. The focus in this introductory paper is on fermionic systems and the emergence of the Luttinger liquid concept.

Current-induced decoherence in the multichannel Kondo problem

The properties of a local spin S=1/2 coupled to K independent wires is studied in the presence of bias voltages which drive the system out of thermal equilibrium. For K≫1, a perturbative renormalization group approach is employed to construct the voltage-dependent scaling function for the conductance and the T matrix. In contrast to the single-channel case, the Kondo resonance is split even by bias voltages small compared to the Kondo temperature T(K), V≪T(K). Besides the applied voltage V, the current-induced decoherence rate Γ≪V controls the physical properties of the system. While the presence of V changes the structure of the renormalization group considerably, decoherence turns out to be very effective in prohibiting the flow towards new nonequilibrium fixed points even in variants of the Kondo model where currents are partially suppressed.

Energy relaxation and thermalization of hot electrons in quantum wires

We develop a theory of energy relaxation and thermalization of hot carriers in clean quantum wires. Our theory is based on a controlled perturbative approach for large excitation energies and emphasizes the important roles of the electron spin and finite temperature. Unlike in higher dimensions, relaxation in one-dimensional electron liquids requires three-body collisions and is much faster for particles than holes which relax at nonzero temperatures only. Moreover, comoving carriers thermalize more rapidly than counterpropagating carriers. Our results are quantitatively consistent with a recent experiment.

Continuous-time quantum Monte Carlo and maximum entropy approach to an imaginary-time formulation of strongly correlated steady-state transport

Recently, Han and Heary [Phys. Rev. Lett. 99, 236808 (2007)] proposed an approach to steady-state quantum transport through mesoscopic structures, which maps the nonequilibrium problem onto a family of auxiliary quantum impurity systems subject to imaginary voltages. We employ continuous-time quantum Monte-Carlo solvers to calculate accurate imaginary time data for the auxiliary models. The spectral function is obtained from a maximum entropy analytical continuation in both Matsubara frequency and complexified voltage. To enable the analytical continuation we construct a kernel which is compatible with the analytical structure of the theory. While it remains a formidable task to extract reliable spectral functions from this unbiased procedure, particularly for large voltages, our results indicate that the method in principle yields results in agreement with those obtained by other methods.

Charge transport through single molecules, quantum dots and quantum wires

We review recent progress in the theoretical description of correlation and quantum fluctuation phenomena in charge transport through single molecules, quantum dots and quantum wires. Various physical phenomena are addressed, relating to cotunneling, pair-tunneling, adiabatic quantum pumping, charge and spin fluctuations, and inhomogeneous Luttinger liquids. We review theoretical many-body methods to treat correlation effects, quantum fluctuations, non-equilibrium physics, and the time evolution into the stationary state of complex nanoelectronic systems.

Relaxation versus decoherence: spin and current dynamics in the anisotropic Kondo model at finite bias and magnetic field

Using a nonequilibrium renormalization group method we study the real-time evolution of spin and current in the anisotropic Kondo model (both antiferromagnetic and ferromagnetic) at a finite magnetic field h(0) and bias voltage V. We derive analytic expressions for all times in the weak-coupling regime max{V, h(0),1/t} >> T(c) (T(c) is the strong-coupling scale). We find that all observables decay both with the spin relaxation and decoherence rates Gamma(1/2). Various V-dependent logarithmic, oscillatory, and power-law contributions are predicted. The low-energy cutoff of logarithmic terms is generically identified by the difference of transport decay rates. For small times t << max{V, h(0)}(-1), we obtain universal dynamics for spin and current.

Coupling-geometry-induced temperature scales in the conductance of Luttinger liquid wires

We study electronic transport through a one-dimensional, finite-length quantum wire of correlated electrons (Luttinger liquid) coupled at arbitrary position via tunnel barriers to two semi-infinite, one-dimensional as well as stripe-like (two-dimensional) leads, thereby bringing theory closer towards systems resembling set-ups realized in experiments. In particular, we compute the temperature dependence of the linear conductance G of a system without bulk impurities on the temperature T. The appearance of new temperature scales introduced by the lengths of overhanging parts of the leads and the wire implies a G(T) which is much more complex than the power-law behavior described so far for end-coupled wires. Depending on the precise set-up the wide temperature regime of power-law scaling found in the end-coupled case is broken up into up to five fairly narrow regimes interrupted by extended crossover regions. Our results can be used to optimize the experimental set-ups designed for a verification of Luttinger liquid power-law scaling.

Nonequilibrium Luttinger liquid: zero-bias anomaly and dephasing

A one-dimensional system of interacting electrons out of equilibrium is studied in the framework of the Luttinger liquid model. We analyze several setups and develop a theory of tunneling into such systems. A remarkable property of the problem is the absence of relaxation in energy distribution functions of left and right movers yet the presence of the finite dephasing rate due to electron-electron scattering, which smears zero-bias-anomaly singularities in the tunneling density of states.