Searcharxiv⌕ Search

arXiv subjects

Vladimir Rittenberg

Publications and source records attributed to Vladimir Rittenberg.

35 records · Page 2Linked to original sources
arXiv2003

The raise and peel model of a fluctuating interface

We propose a one-dimensional nonlocal stochastic model of adsorption and desorption depending on one parameter, the adsorption rate. At a special value of this parameter, the model has some interesting features. For example, the spectrum is given by conformal field theory, and the stationary non-equilibrium probability distribution is given by the two-dimensional equilibrium distribution of the ice model with domain wall type boundary conditions. This connection is used to find exact analytic expressions for several quantities of the stochastic model. Vice versa, some understanding of the ice model with domain wall type boundary conditions can be obtained by the study of the stochastic model. At the special point we study several properties of the model, such as the height fluctuations as well as cluster and avalanche distributions. The latter has a long tail which shows that the model exhibits self organized criticality. We also find in the stationary state a special phase transition without enhancement and with a crossover exponent $ϕ=2/3$. Furthermore, we study the phase diagram of the model as a function of the adsorption rate and find two massive phases and a scale invariant phase where conformal invariance is broken.

cond-mat.stat-mech↗
arXiv2002

Stochastic processes and conformal invariance

We discuss a one-dimensional model of a fluctuating interface with a dynamic exponent $z=1$. The events that occur are adsorption, which is local, and desorption which is non-local and may take place over regions of the order of the system size. In the thermodynamic limit, the time dependence of the system is given by characters of the $c=0$ conformal field theory of percolation. This implies in a rigorous way a connection between CFT and stochastic processes. The finite-size scaling behavior of the average height, interface width and other observables are obtained. The avalanches produced during desorption are analyzed and we show that the probability distribution of the avalanche sizes obeys finite-size scaling with new critical exponents.

cond-mat.stat-mech↗
arXiv2002

Temperley-Lieb Stochastic Processes

We discuss one-dimensional stochastic processes defined through the Temperley-Lieb algebra related to the Q=1 Potts model. For various boundary conditions, we formulate a conjecture relating the probability distribution which describes the stationary state, to the enumeration of a symmetry class of alternating sign matrices, objects that have received much attention in combinatorics.

math-ph↗
arXiv2001

Critical Q=1 Potts Model and Temperley-Lieb Stochastic Processes

We consider the groundstate wave function and spectra of the $L$-site XXZ $U_q[s\ell(2)]$ invariant quantum spin chain with $q=\exp(πi/3)$. This chain is related to the critical Q=1 Potts model and exhibits $c=0$ conformal invariance. We show that the problem is related to Hamiltonians describing one-dimensional stochastic processes defined on a Temperley-Lieb algebra. The bra groundstate wave function is trivial and the ket groundstate wave function gives the probabilty distribution of the stationary state. The stochastic processes can be understood as interface RSOS growth models with nonlocal rates. Allowing defects which can hop on the interface one obtains stochastic models having the same stationary state and spectra (but not degeracies) as the XXZ chain.

cond-mat.stat-mech↗
arXiv2001

Spontaneous Breaking of Translational Invariance and Spatial Condensation in Stationary States on a Ring: II. The Charged System and the Two-component Burgers Equations

We further study the stochastic model discussed in Ref.[2] in which positive and negative particles diffuse in an asymmetric, CP invariant way on a ring. The positive particles hop clockwise, the negative counter-clockwise and oppositely-charged adjacent particles may swap positions. We extend the analysis of this model to the case when the densities of the charged particles are not the same. The mean-field equations describing the model are coupled nonlinear differential equations that we call the two-component Burgers equations. We find roundabout weak solutions of these equations. These solutions are used to describe the properties of the stationary states of the stochastic model. The values of the currents and of various two-point correlation functions obtained from Monte-Carlo simulations are compared with the mean-field results. Like in the case of equal densities, one finds a pure phase, a mixed phase and a disordered phase.

cond-mat.stat-mech↗
arXiv2000

The "topological" charge for the finite XX quantum chain

It is shown that an operator (in general non-local) commutes with the Hamiltonian describing the finite XX quantum chain with certain non-diagonal boundary terms. In the infinite volume limit this operator gives the "topological" charge.

cond-mat.stat-mech↗
arXiv2000

Spontaneous Breaking of Translational Invariance and Spatial Condensation in Stationary States on a Ring: I. The Neutral System

We consider a model in which positive and negative particles with equal densities diffuse in an asymmetric, CP invariant way on a ring. The positive particles hop clockwise, the negative counter-clockwise and oppositely-charged adjacent particles may swap positions. The model depends on two parameters. Analytic calculations using quadratic algebras, inhomogeneous solutions of the mean-field equations and Monte-Carlo simulations suggest that the model has three phases. A pure phase in which one has three pinned blocks of only positive, negative particles and vacancies and in which translational invariance is broken. A mixed phase in which the current has a linear dependence on one parameter but is independent of the other one and of the density of the charged particles. In this phase one has a bump and a fluid. The bump (condensate) contains positive and negative particles only, the fluid contains charged particles and vacancies uniformly distributed. The mixed phase is separated from the disordered phase by a second-order phase-transition which has many properties of the Bose-Einstein phase-transition observed in equilibrium. Various critical exponents are found.

cond-mat.stat-mech↗
arXiv1997

Spontaneous Breaking of Translational Invariance in One-Dimensional Stationary States on a Ring

We consider a model in which positive and negative particles diffuse in an asymmetric, CP-invariant way on a ring. The positive particles hop clockwise, the negative counterclockwise and oppositely-charged adjacent particles may swap positions. Monte-Carlo simulations and analytic calculations suggest that the model has three phases; a "pure" phase in which one has three pinned blocks of only positive, negative particles and vacancies, and in which translational invariance is spontaneously broken, a "mixed" phase with a non-vanishing current in which the three blocks are positive, negative and neutral, and a disordered phase without blocks.

cond-mat.stat-mech↗
arXiv1997

First-order phase transitions in one-dimensional steady states

The steady states of the two-species (positive and negative particles) asymmetric exclusion model of Evans, Foster, Godreche and Mukamel are studied using Monte Carlo simulations. We show that mean-field theory does not give the correct phase diagram. On the first-order phase transition line which separates the CP-symmetric phase from the broken phase, the density profiles can be understood through an unexpected pattern of shocks. In the broken phase the free energy functional is not a convex function but looks like a standard Ginzburg-Landau picture. If a symmetry breaking term is introduced in the boundaries the Ginzburg-Landau picture remains and one obtains spinodal points. The spectrum of the hamiltonian associated with the master equation was studied using numerical diagonalization. There are massless excitations on the first-order phase transition line with a dynamical critical exponent z=2 as expected from the existence of shocks and at the spinodal points where we find $z=1$. It is for the first time that this value which characterizes conformal invariant equilibrium problems appears in stochastic processes.

cond-mat.stat-mech↗
arXiv1997

Stochastic Models on a Ring and Quadratic Algebras. The Three Species Diffusion Problem

The stationary state of a stochastic process on a ring can be expressed using traces of monomials of an associative algebra defined by quadratic relations. If one considers only exclusion processes one can restrict the type of algebras and obtain recurrence relations for the traces. This is possible only if the rates satisfy certain compatibility conditions. These conditions are derived and the recurrence relations solved giving representations of the algebras.

cond-mat.stat-mech↗
arXiv1996

Universality properties of the stationary states in the one-dimensional coagulation-diffusion model with external particle input

We investigate with the help of analytical and numerical methods the reaction A+A->A on a one-dimensional lattice opened at one end and with an input of particles at the other end. We show that if the diffusion rates to the left and to the right are equal, for large x, the particle concentration c(x) behaves like As/x (x measures the distance to the input end). If the diffusion rate in the direction pointing away from the source is larger than the one corresponding to the opposite direction the particle concentration behaves like Aa/sqrt(x). The constants As and Aa are independent of the input and the two coagulation rates. The universality of Aa comes as a surprise since in the asymmetric case the system has a massive spectrum.

cond-mat.stat-mech↗
arXiv1996

Representations of the quadratic Algebra and Partially Asymmetric Diffusion with Open Boundaries

We consider the one-dimensional partially asymmetric exclusion model with open boundaries. The model describes a system of hard-core particles that hop stochastically in both directions with different rates. At both boundaries particles are injected and extracted. By means of the method of Derrida, Evans, Hakim and Pasquier the stationary probability measure can be expressed as a matrix-product state involving two matrices subject to a quadratic algebra. We obtain the representations of this algebra and use the two-dimensional one to derive exact expressions for the density profile and correlation functions. Using the correspondence between the stochastic model and a quantum spin chain, we obtain exact correlation functions for a spin-$\frac{1}{2}$ Heisenberg XXZ chain with non-diagonal boundary terms. Generalizations to other reaction-diffusion models are discussed.

cond-mat↗
arXiv1993

Reaction-Diffusion Processes, Critical Dynamics and Quantum Chains

The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schrödinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions. Since many one-dimensional quantum chains are integrable, this opens a new field of applications. At the same time physical intuition and probabilistic methods bring new insight into the understanding of the properties of quantum chains. A simple example is the asymmetric diffusion of several species of particles which leads naturally to Hecke algebras and $q$-deformed quantum groups. Many other examples are given. Several relevant technical aspects like critical exponents, correlation functions and finite-size scaling are also discussed in detail.

hep-th↗
arXiv1993

Quantum Groups, Correlation Functions and Infrared Divergences

We show in two simple examples that for one-dimensional quantum chains with quantum group symmetries, the correlation functions of local operators are, in general, infrared divergent. If one considers, however, correlation functions invariant under the quantum group, the divergences cancel out.

hep-th↗
arXiv1992

The Pokrovski-Talapov Phase Transitions and Quantum Groups

We show that the XY quantum chain in a magnetic field is invariant under a two parameter deformation of the $SU(1/1)$ superalgebra. One is led to an extension of the braid group and the Hecke algebras which reduce to the known ones when the two parameter coincide. The physical significance of the two parameters is discussed. When both are equal to one, one gets a Pokrovski-Talapov phase transition. We also show that the representation theory of the quantum superalgebras indicates how to take the appropriate thermodynamical limits.

hep-th↗