A solvable tensor field theory

Letters in Mathematical Physics - We solve the closed Schwinger–Dyson equation for the 2-point function of a tensor field theory with a quartic melonic interaction, in terms of...     

A pedagogical review on solvable irrelevant deformations of 2D quantum field theory

This is a pedagogical review on ■ deformation of two dimensional quantum field theories. It is based on three lectures which the author gave at ITP-CAS in December 2018. This review consists of four parts. The first part is a general introduction to ■ deformation. Special emphasises are put on the deformed classical Lagrangian and the exact solvability of the spectrum. The second part focuses on the torus partition sum of the ■ deformed conformal field theories and modular invariance/covariance. In the third part, different perspectives of ■ deformation are presented, including its relation to random geometry, 2D topological gravity and holography. We summarize more recent developments until January 2021 in the last part.     

A stochastic theory for clustering of quenched-in vacancies—2. A solvable model

The model introduced for clustering of quenched-in vacancies in the first part of this series of papers is considered. Using a generating function, the rate equations are converted into a first order partial differential equation for the generating function coupled to a differential equation for the rate of change of the concentration of single vacancy units. A decoupling scheme is effected which gives an exponentially decaying solution with a very short time constant for the concentration of single vacancy units. The differential equation for the generating function is solved for times larger than the time required for the concentration of single vacancy units to reach its asymptotic value. The distribution for the size of the clusters is obtained by inverting the solution thus obtained. Several results that follow are shown to be in reasonably good agreement with the experimental results.     

Exactly solvable models of conformal-invariant quantum field theory in D-dimensional space

The method for exact solution of a certain class of models of conformal quantum field theory in D-dimensional Euclidean space is proposed. The method allows one to derive closed differential equations for all the Green functions and also algebraic equations to scale dimensions of all field. A scalar field P of a scale dimension d_p = D − 2 is needed for nontrivial solutions to exist. At D ≠ 2 this field is converted to a constant that coincides with the central charge of two-dimensional theories. A new class of D = 2 models has been obtained, where the infinite-parametric symmetry is not manifest. The two-dimensional Wess-Zumino model is used to illustrate the method of solution.     

A solvable six-vertex model with defects

The general six-vertex model with defective (missing) horizontal bonds leads to a 22-vertex model. A special case, which is isomorphic to a ten-vertex model amendable to the quantum inverse scattering method, is solved in closed form.     

Dynamical perturbation for classical fluids: A solvable model

We investigate on a one-dimensional model the perturbation to the timedependent correlations in a classical fluid when a small interaction is added to a hard core. Various formulas have already been proposed for this correction. We verify on this model, for which everything can be calculated explicitly, that the expressions proposed by Frisch and Berne yield strongly divergent time integrals for the diffusion coefficient. On the contrary, when all corrections are accounted for, the correction to the velocity time correlation is shown to decay like (Int)/t ₂ at large times, yielding a finite first-order correction to the diffusion coefficient. The extension of this calculation to a gas of hard rods in the case of a perturbation with an infinite range is discussed.     

Test of the non-closed shell many-electron theory in a solvable model

The non-closed shell many-electron theory of Sinano?lu is applied to the two-particle problem in a solvable model called the pseudoatom. The results for overlaps and electric dipolar transitions probabilities are discussed for the approximate and exact solutions.     

Quantum dimer model on the kagome lattice: solvable dimer-liquid and ising gauge theory

We introduce quantum dimer models on lattices made of corner-sharing triangles. These lattices include the kagome lattice and can be defined in arbitrary geometry. They realize fully disordered and gapped dimer-liquid phase with topological degeneracy and deconfined fractional excitations, as well as solid phases. Using geometrical properties of the lattice, several results are obtained exactly, including the full spectrum of a dimer liquid. These models offer a very natural-and maybe the simplest possible-framework to illustrate general concepts such as fractionalization, topological order, and relation to Z2 gauge theories.     

A solvable model exhibiting a glass-like transition

A nonlinear equation of motion of an overdamped oscillator exhibiting a glass-like transition at a critical coupling constant _c is presented and solved exactly. Below _c , in the fluid phase, the oscillator coordinatex(t) decays to zero, while above _c , in the amorphous phase, it decays to a nonzero infinite time limit. Near _c the motion is slowed down by a nonlinear feedback mechanism andx(t) decays exponentially to its long time limit with a relaxation time diverging as (1 – / _c )^–3/2 and (/ _c –1)^–1 for < _c and > _c respectively. At _c x(t) exhibits a power law decay proportional tot ^– with exponent -1/2.     

The harmonically bound electron in the blackbody radiation field. A one-dimensional exactly solvable model

An exactly solvable model for a harmonically bound electron in the blackbody radiation field is discussed. A twofold renormalization leads to cutoff independent exact results. A first order ground state shift is found and a T²-dependent dynamic energy level shift is rederived. Comparisons with previous results are made.     

Charged rods in a periodic background: A solvable model

The one-component Coulomb system with logarithmic potential in a periodic background is considered. In one dimension, when the background has the same period as the average interparticle spacing, the system is exactly solvable for three values of the coupling constant. The exact solution exhibits insulating-conducting phase transitions. An heuristic argument is presented which predicts the phase diagram for this system.     

Phonon dispersion in small-polaron conductivity problem: A solvable model

The stochastic properties of harmonically coupled oscillators studied by Ford, Kac and Mazur are used to investigate the role of the phonon dispersion in the phonon-assisted small-polaron hopping. It is shown that a particular choice of the vibrational frequency spectrum allows one to express the effect of a weak phonon dispersion as an effective damping of independent local oscillators. Therefore, this particular choice makes it possible to employ literally the theory developed in a previous paper. The dependence on the phonon-dispersion width of the small polaron hopping conductivity determined in this way is illustrated by numerical calculations.     

Analytic properties of the Green's functions with respect to the coupling constant in an exactly solvable model of field theory

It is found in an exactly solvable model of field theory that Green's functions for boson fields are analytic in the coupling constant, except for a cut along the real semiaxis. In the case of fermion fields, Green's functions are analytic in the coupling constant near zero.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 28–30, November, 1978.     

Cascade evolution of spherically symmetric pulsons in multivacuum field theory models

Spherically symmetric oscillating solitons of a new type have been observed; their stability and instability regions have been found to be defined by the “potential relief” structure.     

Test of the many-electron theory of atoms in an exactly solvable model

The many-electron theory of ground states of Sinano?lu is tested on an exactly solvable model. The many electron theory wave functions and energies reproduce closely the exact solutions.