Zero-variance of perturbation Hamiltonian density in perturbed spin systems

We study effects of perturbation Hamiltonian to quantum spin systems which can include quenched disorder. Model-independent inequalities are derived, using an additional artificial disordered perturbation. These inequalities enable us to prove that the variance of the perturbation Hamiltonian density vanishes in the infinite volume limit even if the artificial perturbation is switched off. This theorem is applied to spontaneous symmetry breaking phenomena in a disordered classical spin model, a quantum spin model without disorder and a disordered quantum spin model.Comment: 15 pages, final versio

Coleman's theorem on physical assumptions for no Goldstone bosons in two dimensions

Thirty years ago, Coleman proved that no continuous symmetry is broken spontaneously in a two-dimensional relativistic quantum field theory. In his argument, however, it is difficult to understand the physical meaning of the assumption of no infrared divergence. I derive the same result directly from the cluster property of a local field regarded as a physically acceptable assumption.Comment: 7 pages, no figures, corrected serious error

Three loop renormalization group for a marginally perturbed SU(2) WZW model

Employing a simple calculation method obtained by M.-H. Kato, we calculate the three loop renormalization group in the $su(2)$ coset conformal field theory with a slightly relevant perturbation and the $su(2)$ Wess-Zumino-Witten model with a particular invariant marginal perturbation. Zamolodchikov's $c$-theorem, exact data of the perturbation operator and a known exact form of the operator product coefficient enable us to calculate the beta function, the gamma function and the $c$-function to three loop order. This result gives the logarithmic finite size correction to the ground state energy and the low temperature behavior of the specific heat in the Heisenberg antiferromagnetic chain with high accuracy. We describe the consistency with results obtained by several authors on the basis of its exact solvability. We discuss an experiment of the specific heat and the suceptibility recently observed.Comment: 12 pages latex, several mistakes (especially in the beta function in section 2) have been correcte

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.Comment: 19 page

Spin Transmutation in (2+1) Dimensions

We study a relativistic anyon model with a spin-$j$ matter field minimally coupled to a statistical gauge potential governed by the Chern-Simons dynamics with a statistical parameter $\alpha$. A spin and statistics transmutation is shown in terms of a continuous random walk method. An integer or odd-half-integer part of $\alpha$ can be reabsorbed by change of $j$. We discuss the equivalence of a large class of (infinite number) Chern-Simons matter models for given $j$ and $\alpha$.Comment: 21 pages in latex, IFT-488-UNC/NUP-A-94-3, some typos are detected, a paragraph is added to the discussion section, and the references are update

Renormalization group for renormalization-group equations toward the universality classification of infinite-order phase transitions

We derive a new renormalization group to calculate a non-trivial critical exponent of the divergent correlation length which gives a universality classification of essential singularities in infinite-order phase transitions. This method resolves the problem of a vanishing scaling matrix. The exponent is obtained from the maximal eigenvalue of a scaling matrix in this renormalization group, as in the case of ordinary second-order phase transitions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well-known Berezinski\u\i-Kosterlitz-Thouless transition.Comment: 19 pages, 10 eps figure

Diagrammatic Method for Wide Correlators in Gaussian Orthogonal and Symplectic Random Matrix Ensembles

We calculate connected correlators in time dependent Gaussian orthogonal and symplectic random matrix ensembles by a diagrammatic method. We obtain averaged one-point Green's functions in the leading order O(1) and wide two-level and three-level correlators in the first nontrivial order by summing over twisted and untwisted planer diagrams.Comment: 15 pages LaTex including 7 figure

Absence of replica symmetry breaking in the transverse and longitudinal random field Ising model

It is proved that replica symmetry is not broken in the transverse and longitudinal random field Ising model. In this model, the variance of spin overlap of any component vanishes in any dimension almost everywhere in the coupling constant space in the infinite volume limit. The weak Fortuin-Kasteleyn-Ginibre property in this model and the Ghirlanda-Guerra identities in artificial models in a path integral representation based on the Lie-Trotter-Suzuki formula enable us to extend Chatterjee's proof for the random field Ising model to the quantum model.Comment: 17 pages, to appear in J. Stat. Phy

General properties of overlap operators in disordered quantum spin systems

We study short-range quantum spin systems with Gaussian disorder. We obtain quantum mechanical extensions of the Ghirlanda-Guerra identities. We discuss properties of overlap spin operators with these identities.Comment: 12 page

Renormalization group flow in one- and two-matrix models

Large-$N$ renormalization group equations for one- and two-matrix models are derived. The exact renormalization group equation involving infinitely many induced interactions can be rewritten in a form that has a finite number of coupling constants by taking account of reparametrization identities. Despite the nonlinearity of the equation, the location of fixed points and the scaling exponents can be extracted from the equation. They agree with the spectrum of relevant operators in the exact solution. A linearized $\beta$-function approximates well the global phase structure which includes several nontrivial fixed points. The global renormalization group flow suggests a kind of $c$-theorem in two-dimensional quantum gravity.Comment: 34 pages in LaTeX, 4 eps figures included in uufiled form, with a few minor but helpful correction