Factorization of correlation functions and the replica limit of the Toda lattice equation

Exact microscopic spectral correlation functions are derived by means of the replica limit of the Toda lattice equation. We consider both Hermitian and non-Hermitian theories in the Wigner–Dyson universality class (class A) and in the chiral universality class (class AIII). In the Hermitian case we rederive two-point correlation functions for class A and class AIII as well as several one-point correlation functions in class AIII. In the non-Hermitian case the average spectral density of non-Hermitian complex random matrices in the weak non-Hermiticity limit is obtained directly from the replica limit of the Toda lattice equation. In the case of class A, this result describes the spectral density of a disordered system in a constant imaginary vector potential (the Hatano–Nelson model) which is known from earlier work. New results are obtained for the average spectral density in the weak non-Hermiticity limit of a quenched chiral random matrix model at non-zero chemical potential. These results apply to the ergodic or ϵ domain of the quenched QCD partition function at non-zero chemical potential. Our results have been checked against numerical results obtained from a large ensemble of random matrices. The spectral density obtained is different from the result derived by Akemann for a closely related model, which is given by the leading order asymptotic expansion of our result. In all cases, the replica limit of the Toda lattice equation explains the factorization of spectral one- and two-point functions into a product of a bosonic (non-compact integral) and a fermionic (compact integral) partition function. We conclude that the fermionic partition functions, the bosonic partition functions and the supersymmetric partition function are all part of a single integrable hierarchy. This is the reason that it is possible to obtain the supersymmetric partition function, and its derivatives, from the replica limit of the Toda lattice equation.     

Are bosonic replicas faulty?

Motivated by the ongoing discussion about a seeming asymmetry in the performance of fermionic and bosonic replicas, we present an exact, nonperturbative approach to both fermionic and bosonic zero-dimensional replica field theories belonging to the broadly interpreted beta=2 Dyson symmetry class. We then utilize the formalism developed to demonstrate that the bosonic replicas do correctly reproduce the microscopic spectral density in the QCD-inspired chiral Gaussian unitary ensemble. This disproves the myth that the bosonic replica field theories are intrinsically faulty.     

Replica field theories,painlevé transcendents,and exact correlation functions

Exact solvability is claimed for nonlinear replica sigma models derived in the context of random matrix theories. Contrary to other approaches reported in the literature, the framework outlined does not rely on traditional "replica symmetry breaking" but rests on a previously unnoticed exact relation between replica partition functions and Painlevé transcendents. While expected to be applicable to matrix models of arbitrary symmetries, the method is used to treat fermionic replicas for the Gaussian unitary ensemble (GUE), chiral GUE (symmetry classes A and AIII in Cartan classification) and Ginibre's ensemble of complex non-Hermitian random matrices. Further applications are briefly discussed.     

Stationary Correlations for the 1D KPZ Equation

We study exact stationary properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation by using the replica approach. The stationary state for the KPZ equation is realized by setting the initial condition the two-sided Brownian motion (BM) with respect to the space variable. Developing techniques for dealing with this initial condition in the replica analysis, we elucidate some exact nature of the height fluctuation for the KPZ equation. In particular, we obtain an explicit representation of the probability distribution of the height in terms of the Fredholm determinants. Furthermore from this expression, we also get the exact expression of the space-time two-point correlation function.     

Supersymmetry versus replica

It is shown that already in the simplest case of the averaged one-point Green function of the gaussian unitary ensemble (Wigner model) the replica and the supersymmetric approaches provide different results. In particular, replica is likely to suppress oscillatory behaviour in the 1/N expansion.     

Unitary Transformation in Quantum Teleportation

In the well-known treatment of quantum teleportation, the receiver should convert the state of his EPR particle into the replica of the unknown quantum state by one of four possible unitary transformations. However, the importance of these unitary transformations must be emphasized. We will show in this paper that the receiver cannot transform the state of his particle into an exact replica of the unknown state which the sender wants to transfer if he has not a proper implementation of these unitary transformations. In the procedure of converting state, the inevitable coupling between EPR particle and environment which is needed by the implementation of unitary transformations will reduce the accuracy of the replica.     

Unitary Transformation in Quantum Teleportation

In the well-known treatment of quantum teleportation, the receiver should convert the state of his EPR particle into the replica of the unknown quantum state by one of four possible unitary transformations. However, the importance of these unitary transformations must be emphasized. We will show in this paper that the receiver cannot transform the state of his particle into an exact replica of the unknown state which the sender wants to transfer if he has not a proper implementation of these unitary transformations. In the procedure of converting state, the inevitable coupling between EPR particle and environment which is needed by the implementation of unitary transformations will reduce the accuracy of the replica.     

Current Distribution and Random Matrix Ensembles for an Integrable Asymmetric Fragmentation Process

We calculate the time-evolution of a discrete-time fragmentation process in which clusters of particles break up and reassemble and move stochastically with size-dependent rates. In the continuous-time limit the process turns into the totally asymmetric simple exclusion process (only pieces of size 1 break off a given cluster). We express the exact solution of the master equation for the process in terms of a determinant which can be derived using the Bethe ansatz. From this determinant we compute the distribution of the current across an arbitrary bond which after appropriate scaling is given by the distribution of the largest eigenvalue of the Gaussian unitary ensemble of random matrices. This result confirms universality of the scaling form of the current distribution in the KPZ universality class and suggests that there is a link between integrable particle systems and random matrix ensembles.     

Anderson localization from the replica formalism

We study Anderson localization in quasi-one-dimensional disordered wires within the framework of the replica sigma model. Applying a semiclassical approach (geodesic action plus Gaussian fluctuations) recently introduced within the context of supersymmetry by Lamacraft, Simons, and Zirnbauer, we compute the exact density of transmission matrix eigenvalues of superconducting wires (of symmetry class CI.) For the unitary class of metallic systems (class A) we are able to obtain the density function, save for its large transmission tail.     

Quantum Lattice-Gas Model for the Burgers Equation

A quantum algorithm is presented for modeling the time evolution of a continuous field governed by the nonlinear Burgers equation in one spatial dimension. It is a microscopic-scale algorithm for a type-II quantum computer, a large lattice of small quantum computers interconnected in nearest neighbor fashion by classical communication channels. A formula for quantum state preparation is presented. The unitary evolution is governed by a conservative quantum gate applied to each node of the lattice independently. Following each quantum gate operation, ensemble measurements over independent microscopic realizations are made resulting in a finite-difference Boltzmann equation at the mesoscopic scale. The measured values are then used to re-prepare the quantum state and one time step is completed. The procedure of state preparation, quantum gate application, and ensemble measurement is continued ad infinitum. The Burgers equation is derived as an effective field theory governing the behavior of the quantum computer at its macroscopic scale where both the lattice cell size and the time step interval become infinitesimal. A numerical simulation of shock formation is carried out and agrees with the exact analytical solution.     

Formal exact solution for the Heisenberg spin system in a time-dependent magnetic field and Aharonov-Anandan phase

We study the time evolution of the Heisenberg spin system in a time-dependent magnetic field. By a unitary transformation we obtain the formal exact solutions of the Schrödinger equation for this system. Based on the formal exact solutions, the Aharonov-Anandan phases are worked out.     

Is there a glass transition in planar vortex systems?

The criteria for the existence of a glass transition in a planar vortex array with quenched disorder are studied. Applying a replica Bethe ansatz, we obtain for self-avoiding vortices the exact quenched average free energy and effective stiffness which is found to be in excellent agreement with recent numerical results for the related random bond dimer model [C. Zeng, P. L. Leath, and T. Hwa, Phys. Rev. Lett. 83, 4860 (1999)] Including a repulsive vortex interaction and a finite vortex persistence length xi, we find that for xi-->0 the system is at all temperatures in a glassy phase; a glass transition exists only for finite xi. Our results indicate that planar vortex arrays in superconducting films are glassy at presumably all temperatures.     

Kinetics of a frictional granular motor

Within the framework of a Boltzmann-Lorentz equation, we analyze the dynamics of a granular rotor immersed in a bath of thermalized particles in the presence of a frictional torque on the axis. In numerical simulations of the equation, we observe two scaling regimes at low and high bath temperatures. In the large friction limit, we obtain the exact solution of a model corresponding to asymptotic behavior of the Boltzmann-Lorentz equation. In the limit of large rotor mass and small friction, we derive a Fokker-Planck equation for which the exact solution is also obtained.     

与不变量有关的幺正变换方法和含时均匀电场中的量子Dirac场的演化

在推广的不变量理论的基础上,运用与不变量有关的幺正变换方法研究了含时均匀电场下的量子Dirac场的演化,求解了Dirac场的泛函Schrdinger方程,得到了方程的精确解以及对应的总相位,总相位包括动力学相和几何相(Aharonov-Anandan phase). 关键词：     

The thermodynamic limit for an imperfect Boson gas

We give a rigorous treatment in the infinite volume limit of a model Hamiltonian representing an imperfect Boson gas. In particular we obtain the exact expression for the mean particle density in the infinite volume limit as a function of the chemical potential, and show that the density function has a singularity at the critical density for Bose-Einstein condensation. We prove that, unlike the ideal Boson gas, the imperfect Boson gas has the same behaviour in the infinite volume limit for the grand canonical ensemble as for the canonical ensemble, and is moreover stable under small perturbations. We finally exhibit the possibility of ordinary condensation and prove that a system in an intermediate situation between two pure phases consists of a simple mixture of the two phases involved.     

Universal Nature of Replica Symmetry Breaking in Quantum Systems with Gaussian Disorder

We study quantum spin systems with quenched Gaussian disorder. We prove that the variance of all physical quantities in a certain class vanishes in the infinite volume limit. We study also replica symmetry breaking phenomena, where the variance of an overlap operator in the other class does not vanish in the replica symmetric Gibbs state. On the other hand, it vanishes in a spontaneous replica symmetry breaking Gibbs state defined by applying an infinitesimal replica symmetry breaking field. We prove also that the finite variance of the overlap operator in the replica symmetric Gibbs state implies the existence of a spontaneous replica symmetry breaking.     

Renormalization group limit cycles and first-order phase transitions in superconductors and abelian Higgs models

Renormalization group methods are used to investigate the thermodynamic behaviour of a multicomponent abelian Higgs model. Interpreted as a Ginzburg-Landau-type model for superconductivity, the model interpolates, via the replica trick, between superconductors with and without quenched random impurities. For a range of values of the numbers of complex order parameter components and of replicas, the renormalization group trajectories exhibit a stable focus, surrounded by an unstable limit cycle. By evaluating the free energy and the equation of state, we find that flows within the limit cycle correspond to a type of critical behaviour with oscillatory modulations. Outside the limit cycle, there is a region of the parameter space in which runaway of the trajectories may be interpreted in terms of a fluctuation-induced first-order phase transition, but in a third region, no clear interpretation is possible.     

Physical solutions for unitary matrix models

We study the simplest double scaling limit of the integral over a unitary matrix, shown by Periwal and Shevitz to admit an exact solution in terms of the mKdV hierarchy. We show that there is a unique non-perturbative solution of the string equation corresponding to the true double scaling limit of the integral, which interpolates smoothly between weak and strong coupling regimes.     

Decay of a thermofield-double state in chaotic quantum systems

Scrambling in interacting quantum systems out of equilibrium is particularly effective in the chaotic regime. Under time evolution, initially localized information is said to be scrambled as it spreads throughout the entire system. This spreading can be analyzed with the spectral form factor, which is defined in terms of the analytic continuation of the partition function. The latter is equivalent to the survival probability of a thermofield double state under unitary dynamics. Using random matrices from the Gaussian unitary ensemble (GUE) as Hamiltonians for the time evolution, we obtain exact analytical expressions at finite N for the survival probability. Numerical simulations of the survival probability with matrices taken from the Gaussian orthogonal ensemble (GOE) are also provided. The GOE is more suitable for our comparison with numerical results obtained with a disordered spin chain with local interactions. Common features between the random matrix and the realistic disordered model in the chaotic regime are identified. The differences that emerge as the spin model approaches a many-body localized phase are also discussed.     

The pressure of fermions with gravitational interaction

A system of nonrelativistic fermions with nonrelativistic gravitational interaction is described in the grand canonical ensemble. It is shown rigorously that in the appropriate thermodynamic limit the pressure converges to the mean hydrostatic pressure and the corresponding Thomas-Fermi equation becomes exact.