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Journal of Physics A Mathematical and Theoretical

Conformal invariance in conditioned stochastic particle systems

Gunter Markus Schutz

Published Jun 21, 2017• 6 citations
Publisher

Abstract

Abstract We consider space-time correlations in generic one-dimensional stochastic interacting particle systems with short-range interactions that undergo a fluctuation with an atypically activity of particle jumps or reactions or spin flips. We briefly review the approach in the framework of the quantum Hamiltonian formalism and present examples where the dynamics during such large fluctuations is governed not by the typical stationary dynamics, but by ballistic universality classes with dynamical exponent z = 1 that are described unitary conformally invariant field theories with central charge c . For reaction–diffusion and spin flip dynamics we identify critical points (a) in the Ising universality class with c = 1 / 2 , and (b) in the universality class of the three-states Potts model with c = 4 / 5 . For the Ising universality class we obtain a universal scaling form for the generating function of cumulants of the jump activity. For repulsive driven diffusive systems with one conservation law the regime of an atypically high current or hopping activity is generically conformally invariant with central charge c = 1 .

DOI
10.1088/1751-8121/aa7ab2

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Abstract We consider space-time correlations in generic one-dimensional stochastic interacting particle systems with short-range interactions that undergo a fluctuation with an atypically activity of particle jumps or reactions or spin flips. We briefly review the approach in the framework of the quantum Hamiltonian formalism and present examples where the dynamics during such large fluctuations is governed not by the typical stationary dynamics, but by ballistic universality classes with dynamical exponent z = 1 that are described unitary conformally invariant field theories with central charge c . For reaction–diffusion and spin flip dynamics we identify critical points (a) in the Ising universality class with c = 1 / 2 , and (b) in the universality class of the three-states Potts model with c = 4 / 5 . For the Ising universality class we obtain a universal scaling form for the generating function of cumulants of the jump activity. For repulsive driven diffusive systems with one conservation law the regime of an atypically high current or hopping activity is generically conformally invariant with central charge c = 1 .
We consider the problem of correlation functions in the stationary states of one-dimensional stochastic models having conformal invariance. If one considers the space dependence of the correlators, the novel aspect is that although one considers systems with periodic boundary conditions, the observables are described by boundary operators. From our experience with equilibrium problems one would have expected bulk operators. Boundary operators have correlators having critical exponents being half of those of bulk operators. If one studies the space-time dependence of the two-point function, one has to consider one boundary and one bulk operators. The Raise and Peel model has conformal invariance as can be shown in the spin 1/2 basis of the Hamiltonian which gives the time evolution of the system. This is an XXZ quantum chain with twisted boundary condition and local interactions. This Hamiltonian is integrable and the spectrum is known in the finite-size scaling limit. In the stochastic base in which the process is defined, the Hamiltonian is not local anymore. The mapping into an SOS model, helps to define new local operators. As a byproduct some new properties of the SOS model are conjectured. The predictions of conformal invariance are discussed in the new framework and compared with Monte Carlo simulations.
Abstract We propose a general formalism, within large-deviation theory, giving access to the exact statistics of fluctuations of ballistically transported conserved quantities in homogeneous, stationary states. The formalism is expected to apply to any system with an Euler hydrodynamic description, classical or quantum, integrable or not, in or out of equilibrium. We express the exact scaled cumulant generating function (or full counting statistics) for any (quasi-)local conserved quantity in terms of the flux Jacobian. We show that the “extended fluctuation relations” of Bernard and Doyon follow from the linearity of the hydrodynamic equations, forming a marker of “freeness” much like the absence of hydrodynamic diffusion does. We show how an extension of the formalism gives exact exponential behaviours of spatio-temporal two-point functions of twist fields, with applications to order-parameter dynamical correlations in arbitrary homogeneous, stationary state. We explain in what situations the large-deviation principle at the basis of the results fail, and discuss how this connects with nonlinear fluctuating hydrodynamics. Applying the formalism to conformal hydrodynamics, we evaluate the exact cumulants of energy transport in quantum critical systems of arbitrary dimension at low but nonzero temperatures, observing a phase transition for Lorentz boosts at the sound velocity.
We show that the dynamics of a quantum impurity subject to a stochastic drive on one side and coupled to a quantum critical system on the other display a universal behavior inherited from the quantum critical scaling. Using boundary conformal field theory, we formulate a generic ansatz for the dynamical scaling form of the typical Loschmidt echo and corroborate it with exact numerical calculations in the case of a spin impurity driven by shot noise in a quantum Ising chain. We find that due to rare events the dynamics of the mean echo can follow very different dynamical scaling than the typical echo for certain classes of drives. Our results are insensitive to irrelevant perturbations of the bulk critical model and apply to all the microscopic models in the same universality class.
Non-universal dynamics is shown to occur in a one-dimensional non-equilibrium system of hard-core particles. The stochastic processes included are pair creation and annihilation (with rates and ) and symmetric hopping rates which alternate from one bond to the next ( , ). A dynamical scaling relation between the relaxation time and the correlation length in the steady state is derived in a simple way for the case . We find that the dynamical exponent takes the non-universal value . For the special condition , where the stochastic system is in principle soluble by reduction to a free fermion system, the model is mapped to the Glauber dynamics of an Ising chain with alternating ferromagnetic bonds of values and , in contact with a quantum thermal bath. The full time dependence of the space-dependent magnetization and of the equal time spin-spin correlation function are studied by writing the master equation for this system in the quantum Hamiltonian formalism. In particular, we obtain the dispersion relations and rigorously confirm the results obtained for the correlation length and for the dynamical exponent.
We define a simple model of conformal field theory in random space-time environments, which we refer to as stochastic conformal field theory. This model accounts for the effects of dilute random impurities in strongly interacting critical many-body systems. On one hand, surprisingly, although impurities are separated by macroscopic distances, we find that the infinite-time steady state is factorized on microscopic lengths, a signature of the emergence of localization. The stationary state also displays vanishing energy current and strong uncorrelated spatial fluctuations of local observables. On the other hand, at finite times, the transient shows a crossover from ballistic to diffusive energy propagation. In this regime and a Markovian limit, concentrating on current-generating initial states with a temperature imbalance, we show that the energy current and density satisfy simple dissipative hydrodynamic equations. We describe the space-time scales at which nonequilibrium currents exist. We show that a light-cone effect subsists in the presence of impurities although a momentum burst propagates transiently on a diffusive scale only.
Abstract Exact results are presented for conditioned dynamics in a system of interacting random walks in one dimension that annihilate immediately when two particles meet on the same site and where pairs of particles are deposited randomly on neighbouring sites. For an atypical hopping activity one finds dynamical nonequilibrium phase transitions analogous to the zero-temperature equilibrium phase transitions that appear in the spin-1/2 quantum XY spin chain in a transverse magnetic field. Along the critical line the approach of the particle density to its stationary value is algebraic with an unexpected mean field exponent. The time-dependent local stationary density correlations are universal with dynamical exponent z = 1. Inside the disordered phase spatially oscillating correlations appear below the typical activity.