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arXiv:1005.2414 · Physical Review B

Coarsening of disordered quantum rotors under a bias voltage

Camille Aron, Giulio Biroli, Leticia F. Cugliandolo

Published Nov 15, 2010• 2 citations
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Abstract

We solve the dynamics of an ensemble of interacting rotors coupled to two leads at different chemical potential letting a current flow through the system and driving it out of equilibrium. We show that at low temperature the coarsening phase persists under the voltage drop up to a critical value of the applied potential that depends on the characteristics of the electron reservoirs. We discuss the properties of the critical surface in the temperature, voltage, strength of quantum fluctuations, and coupling to the bath phase diagram. We analyze the coarsening regime finding, in particular, which features are essentially quantum mechanical and which are basically classical in nature. We demonstrate that the system evolves via the growth of a coherence length with the same time dependence as in the classical limit, $R(t)\ensuremath{\simeq}{t}^{1/2}$---the scalar curvature-driven universality class. We obtain the scaling function of the correlation function at late epochs in the coarsening regime and we prove that it coincides with the classical one once a prefactor that encodes the dependence on all the parameters is factorized. We derive a generic formula for the current flowing through the system and we show that, for this model, it rapidly approaches a constant that we compute.

DOI
10.1103/physrevb.82.174203

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We solve the dynamics of an ensemble of interacting rotors coupled to two leads at different chemical potential letting a current flow through the system and driving it out of equilibrium. We show that at low temperature the coarsening phase persists under the voltage drop up to a critical value of the applied potential that depends on the characteristics of the electron reservoirs. We discuss the properties of the critical surface in the temperature, voltage, strength of quantum fluctuations, and coupling to the bath phase diagram. We analyze the coarsening regime finding, in particular, which features are essentially quantum mechanical and which are basically classical in nature. We demonstrate that the system evolves via the growth of a coherence length with the same time dependence as in the classical limit, R(t)\ensuremath≃t1/2R(t)\ensuremath{\simeq}{t}^{1/2}R(t)\ensuremath≃t1/2---the scalar curvature-driven universality class. We obtain the scaling function of the correlation function at late epochs in the coarsening regime and we prove that it coincides with the classical one once a prefactor that encodes the dependence on all the parameters is factorized. We derive a generic formula for the current flowing through the system and we show that, for this model, it rapidly approaches a constant that we compute.
We study the driven dynamics of quantum coarsening. We analyze models of M-component rotors coupled to two electronic reservoirs at different chemical potential that generate a current threading through the system. In the large M limit, we derive the dynamical phase diagram as a function of temperature, strength of quantum fluctuations, voltage, and coupling to the leads. We show that the slow relaxation in the ordering phase is universal. On large time and length scales, the dynamics are analogous to stochastic classical ones, even for the quantum system driven out of equilibrium at zero temperature. We argue that our results apply to generic driven quantum coarsening.
We investigate bias-driven non-equilibrium quantum phase transitions in a paradigmatic quantum-transport setup: an interacting quantum dot coupled to non-interacting metallic leads. Using the Random Phase Approximation, which is exact in the limit of a large number of dot levels, we map out the zero-temperature non-equilibrium phase diagram as a function of interaction strength and applied bias. We focus our analysis on the behavior of the charge susceptibility and the current noise in the vicinity of the transition. Remarkably, despite the intrinsically non-equilibrium nature of the steady state, critical charge fluctuations admit an effective-temperature description, Teff(T,V)T_{\text{eff}}(T,V)Teff​(T,V), that collapses the steady-state behavior onto its equilibrium form. In sharp contrast, current fluctuations exhibit genuinely non-equilibrium features: the fluctuation-dissipation ratio becomes negative in the ordered phase, corresponding to a negative effective temperature for the current degrees of freedom. These results establish current noise as a sensitive probe of critical fluctuations at non-equilibrium quantum phase transitions and open new directions for exploring voltage-driven critical phenomena in quantum transport systems.
The nonequilibrium dynamics of a system that is located in the vicinity of a quantum critical point is affected by the critical slowing down of order-parameter correlations with the potential for novel out-of-equilibrium universality. After a quantum quench, i.e., a sudden change of a parameter in the Hamiltonian, such a system is expected to almost instantly fall out of equilibrium and undergo aging dynamics, i.e., dynamics that depends on the time passed since the quench. Investigating the quantum dynamics of an NNN-component ${\ensuremath{\varphi}}^{4}$ model coupled to an external bath, we determine this universal aging and demonstrate that the system undergoes a coarsening, governed by a critical exponent that is unrelated to the equilibrium exponents of the system. We analyze this behavior in the large-NNN limit, which is complementary to our earlier renormalization-group analysis, allowing in particular the direct investigation of the order-parameter dynamics in the symmetry-broken phase and at the upper critical dimension. By connecting the long-time limit of fluctuations and response, we introduce a distribution function that shows that the system remains nonthermal and exhibits quantum coherence even on long time scales.
We present a dynamical study of the disordered quantum p=2p=2p=2 spherical model at long times. Its phase behavior as a function of spin-bath coupling, strength of quantum fluctuations and temperature is characterized, and we identify different paramagnetic and coarsened regions. A quantum critical point at zero temperature in the limit of vanishing dissipation is also found. Furthermore we show analytically that the fluctuation-dissipation theorem is obeyed in the stationary regime.
Correlations are a central ingredient of modern condensed matter physics. With the advent of mesoscopic physics, it became possible to study correlated systems in non-equilibrium situations, which can be reached by e.g. driving a constant current through a quantum dot. A basic question is how correlations on the one hand and a non-equilibrium current on the other hand influence each other. Here this interplay is studied for the model system of a quantum dot in the Kondo regime using the real time renormalization group (RTRG) method. This method is a functional renormalization group (RG) method allowing a non-perturbative description of correlated systems in non-equilibrium. For the Kondo quantum dot the method is generalized to the case of double vertices. The RTRG method not only provides equations for the calculation of the renormalized coupling between the quantum dot spin and the reservoir electrons, but also equations for the calculation of the non-equilibrium decoherence rate and the non-equilibrium current.The RG equations are derived to second order. This approximation is expected to be valid as long as the quantum dot does not enter the strong coupling regime, i.e. the effective coupling strength between the quantum dot spin and the electrons does not become too large. It is found that this is the case if the bias voltage applied to the dot or the temperature is much larger than the Kondo temperature.It is shown that the RG equations lead to coupling functions depending on the energy of the incident and the scattered electron. This dependence is simplified to the dependence on the average energy of the incident and the scattered electron in the weak coupling regime.A closer inspection of the RG equations for the couplings shows that they not only contain terms leading to a logarithmic increase of the coupling strength but also subleading terms which are unimportant in the weak coupling regime. The RG equations are solved analytically for the case where the subleading terms are neglected. It is found that the coupling functions develop resonances at energies associated with the chemical potentials of the reservoirs. The height of these resonances is determined by the decoherence rate. However, for bias voltages much larger than the Kondo temperature these resonances do not affect the conductance for a two terminal setup, so the conductance does not provide a measure for the decoherence rate in this case. It is found that the conductance for the two terminal setup is a universal function of voltage/Kondo temperature and temperature/Kondo temperature in the weak coupling regime. This function is calculated analytically in the limits of vanishing voltage or temperature. Furthermore, it is found that for zero temperature the decoherence rate is proportional to the non-equilibrium current flowing through the system. This supports the physical picture that a non-equilibrium current is associated with a decoherence rate, which cuts the logarithmic scaling of the coupling strength.Finally, the RG equations are solved numerically for zero temperature including the subleading corrections found before. As expected, the subleading corrections have no impact on the decoherence rate and the conductance in the weak coupling regime. On the other hand, it is found that they lead to strong deviations from the results of the analytic leading logarithmic approximation if the bias voltage approaches the Kondo temperature. Generally, it is found that the subleading corrections lead to a saturation of the decoherence rate and the conductance to some finite value. However, for the numerical solution a numerical regularization parameter has to be introduced. It turns out that the saturation value found depends on its value, so no statements on the precise value of the decoherence rate and the current can be made in this regime.
We study the nonequilibrium dynamics of the q-state Potts model following a quench from the high-temperature disordered phase to zero temperature. The time-dependent two-point correlation functions of the order parameter field satisfy dynamic scaling with a length scale L(t)\ensuremath{\sim}t1/2{\mathit{t}}^{1/2}t1/2. In particular, the autocorrelation function decays as [L(t)${]}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}(\mathit{q})}$. We illustrate these properties by solving exactly the kinetic Potts model in d=1. We then analyze a Langevin equation of an appropriate field theory to compute these correlation functions for general q and d. We establish a correspondence between the two-point correlations of the q-state Potts model and those of a kinetic Ising model evolving with a fixed magnetization (2/q-1). The dynamics of this Ising model is solved exactly in the large q limit and in the limit of a large number of components n for the order parameter. For general q and in any dimension, we introduce a Gaussian closure approximation and calculate within this approximation the scaling functions and the exponent \ensuremath{\lambda}(q). These are in good agreement with the direct numerical simulations of the Potts model as well as the kinetic Ising model with fixed magnetization. We also discuss the existing and possible experimental realizations of these models.