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arXiv:1503.09139 · Physical Review Letters

Steady-State Dynamics and Effective Temperature for a Model of Quantum Criticality in an Open System

Pedro Ribeiro, Farzaneh Zamani, Stefan Kirchner

Published Nov 25, 2015• 13 citations
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Abstract

We study the thermal and nonthermal steady-state scaling functions and the steady-state dynamics of a model of local quantum criticality. The model we consider, i.e., the pseudogap Kondo model, allows us to study the concept of effective temperatures near fully interacting as well as weak-coupling fixed points. In the vicinity of each fixed point we establish the existence of an effective temperature-different at each fixed point-such that the equilibrium fluctuation-dissipation theorem is recovered. Most notably, steady-state scaling functions in terms of the effective temperatures coincide with the equilibrium scaling functions. This result extends to higher correlation functions as is explicitly demonstrated for the Kondo singlet strength. The nonlinear charge transport is also studied and analyzed in terms of the effective temperature.

DOI
10.1103/physrevlett.115.220602

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We study the thermal and nonthermal steady-state scaling functions and the steady-state dynamics of a model of local quantum criticality. The model we consider, i.e., the pseudogap Kondo model, allows us to study the concept of effective temperatures near fully interacting as well as weak-coupling fixed points. In the vicinity of each fixed point we establish the existence of an effective temperature-different at each fixed point-such that the equilibrium fluctuation-dissipation theorem is recovered. Most notably, steady-state scaling functions in terms of the effective temperatures coincide with the equilibrium scaling functions. This result extends to higher correlation functions as is explicitly demonstrated for the Kondo singlet strength. The nonlinear charge transport is also studied and analyzed in terms of the effective temperature.
We study the out-of-equilibrium steady-state properties of the Bose-Fermi-Kondo model, describing a local magnetic moment coupled to two ferromagnetic leads that support bosonic (magnons) and fermionic (Stoner continuum electrons) low-energy excitations. This model describes the destruction of the Kondo effect as the coupling to the bosons is increased. Its phase diagram comprises three non-trivial fixed points. Using a dynamical large- N approach on the Keldysh contour, we study two different non-equilibrium setups: a) a finite bias voltage and b) a finite temperature gradient, imposed across the leads. The scaling behavior of the charge and energy currents is identified and characterized for all fixed points. We report the existence of a fixed-point–dependent effective temperature, defined though the fluctuation-dissipation relations of the local spin-susceptibility in the scaling regime, which permits to recover the equilibrium behavior of both dynamical and static spin-susceptibilities.
We study the out-of-equilibrium steady-state properties of the Bose-Fermi-Kondo model, describing a local magnetic moment coupled to two ferromagnetic leads that support bosonic (magnons) and fermionic (Stoner continuum electrons) low-energy excitations. This model describes the destruction of the Kondo effect as the coupling to the bosons is increased. Its phase diagram comprises three non-trivial fixed points. Using a dynamical large-N approach on the Keldysh contour, we study two different non-equilibrium setups: a) a finite bias voltage and b) a finite temperature gradient, imposed across the leads. The scaling behavior of the charge and energy currents is identified and characterized for all fixed points. We report the existence of a fixed-point-dependent effective temperature, defined though the fluctuation-dissipation relations of the local spin-susceptibility in the scaling regime, which permits to recover the equilibrium behavior of both dynamical and static spin-susceptibilities. Copyright (c) EPLA, 2013
Abstract Quantum criticality has attracted considerable attention both theoretically and experimentally as a way to describe part of the phase diagram of strongly correlated systems. A scale‐invariant fluctuation spectrum at a quantum critical point (QCP) implies the absence of any intrinsic scale. Any experimental probe may therefore create an out‐of‐equilibrium setting; the system would be in a non‐linear response regime, which violates the fluctuation–dissipation theorem (FDT). Here, we study this violation and related out‐of equilibrium phenomena in a single‐electron transistor (SET) with ferromagnetic leads, which can be tuned through a quantum phase transition. We review the breakdown of the FDT and study the universal behavior of the fluctuation–dissipation relation of various correlators in the quantum critical regime. In particular, we explore the concept of effective temperature as a means to extend the FDT into the non‐linear regime.
Quantum critical systems out of equilibrium are of extensive interest, but are difficult to study theoretically. We consider here the steady-state limit of a single-electron transistor with ferromagnetic leads. In equilibrium (i.e., bias voltage V = 0), this system features a continuous quantum phase transition with a critical destruction of the Kondo effect. We construct an exact quantum Boltzmann treatment in a dynamical large-N limit, and determine the universal scaling functions of both the nonlinear conductance and fluctuation-dissipation ratios. We also elucidate the decoherence properties as encoded in the local spin response.
In this thesis I investigate several aspects of local quantum criticality, a concept of key importance in a number of physical contexts ranging from critical heavy fermion compounds to quantum dot systems. Quantum critical points are associated with second order phase transitions at zero temperature. In contrast to their finite-temperature counterparts, the zero-point motion cannot be neglected near a quantum critical point. As a result, the incorporation of quantum dynamics leads to an effective dimension larger than the spatial dimension of the system for the order parameter fluctuations within the Ginzburg-Landau-Wilson treatment of criticality. This so-called quantum-to-classical mapping works well for the critical properties in insulating systems but apparently fails in systems containing gapless fermions. This has been experimentally most clearly been demonstrated within a particular class of intermetallic compounds called heavy fermions. A particular way in which the Ginzburg-Landau-Wilson paradigm fails is for critical Kondo destruction that seems to underlie the unconventional quantum criticality seen in the heavy fermions. I focus on studying the properties of critical Kondo destruction and the emergence of energy-over-temperature-scaling in systems without spatial degrees of freedom, i.e., so-called quantum impurity systems. In particular, I employ large-N techniques to address critical properties of this class of quantum phase transitions in and out of equilibrium. As quantum critical systems are characterized by a scale-invariant spectrum with many low-lying excitations, it may appear that any perturbation can lead to a response beyond the linear response regime. Understanding what governs the non-linear response regime near quantum criticality is an interesting area. Here, I first present a path integral version of the Schrieffer-Wolff transformation which relates the functional integral form of the partition function of the Anderson model to that of its effective low-energy model. The equivalence between the low-energy sector of the Anderson model in the Kondo regime and the spin-isotropic Kondo model is usually established via a canonical transformation performed on the Hamiltonian, followed by a projection. The resulting functional integral assumes the form of a spin path integral and includes a geometric phase factor, i.e. a Berry phase. The approach stresses the underlying symmetries of the model and allows for a straightforward generalization of the transformation to more involved models. As an example of the efficiency of the approach I apply it to a single electron transistor attached to ferromagnetic leads and derive the effective low-energy model of such a magnetic transistor. As Kondo screening is a local phenomenon, it and its criticality can be studied using the appropriate impurity model. A general impurity model to study critical Kondo destruction is the pseudogap Bose-Fermi Kondo model. Here, I concentrate on the multi-channel version of the model using the dynamical large-N study. This model allows to study the non-trivial interplay between two different mechanisms of critical Kondo destruction. The interplay of two processes that can each by itself lead to critical Kondo destruction. The zero-temperature residual entropy at various fixed points for the model is also discussed. The two channel Anderson model exhibits several continuous quantum phase transitions between weak- and strong-coupling phases. The non-crossing approximation (NCA) is believed to give reliable results for the standard two-channel Anderson model of a magnetic impurity in a metal. I revisit the reliability of the NCA for the standard two channel Anderson model (constant conduction electron density of states) and investigate its reliability for the two-channel pseudogap Anderson model. This is done by comparing finite-temperature, finite-frequency solutions of the NCA equations and asymptotically exact zero-temperature NCA solutions with numerical renormalization-group calculations. The phase diagram of this model is well established. The focus here will be on the dynamical scaling properties obtained within the NCA. Finally, I study the thermal and non-thermal steady state scaling functions and the steady-state dynamics of the pseudogap Kondo model. This model allows us to study the concept of effective temperatures near fully interacting as well as weak-coupling fixed points and compare the out-of-equilibrium scaling properties of critical Kondo destruction to those of the traditional spin-density wave (SDW) scenario. The differences I identify can be experimentally probed. This may be helpful in identifying the nature of the quantum critical points observed in certain heavy fermion compounds.
Nonequilibrium properties of correlated quantum matter are being intensively investigated because of the rich interplay between external driving and the many-body correlations. Of particular interest is the nonequilibrium behavior near a quantum critical point (QCP), where the system is delicately balanced between different ground states. We present both an analytical calculation of the nonequilibrium steady-state current in a critical system and experimental results to which the theory is compared. The system is a quantum dot coupled to resistive leads: a spinless resonant level interacting with an ohmic dissipative environment. A two channel Kondo-like QCP occurs when the level is on resonance and symmetrically coupled to the leads, conditions achieved by fine-tuning using electrostatic gates. We calculate and measure the nonlinear current as a function of bias (III-VVV curve) at the critical values of the gate voltages corresponding to the QCP. The quantitative agreement between the experimental data and the theory, with no fitting parameter, is excellent. As our system is fully accessible to both theory and experiment, it provides an ideal setting for addressing nonequilibrium phenomena in correlated quantum matter.