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On the fermion-boson correspondence for correlation functions of disorder operators

Delfino, G, Grinza, P, Mussardo, G

Published Mar 7, 2002• 5 citations
OpenAlex

Abstract

When a quantum field theory possesses topological excitations in a phase with spontaneously broken symmetry, these are created by operators which are non-local with respect to the order parameter. Due to non-locality, such disorder operators have non-trivial correlation functions even in free massive theories. In two dimensions, these correlators can be expressed exactly in terms of solutions of non-linear differential equations. The correlation functions of the one-parameter family of non-local operators in the free charged bosonic and fermionic models are the inverse of each other. We point out a simple derivation of this correspondence within the form factor approach

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When a quantum field theory possesses topological excitations in a phase with spontaneously broken symmetry, these are created by operators which are non-local with respect to the order parameter. Due to non-locality, such disorder operators have non-trivial correlation functions even in free massive theories. In two dimensions, these correlators can be expressed exactly in terms of solutions of non-linear differential equations. The correlation functions of the one-parameter family of non-local operators in the free charged bosonic and fermionic models are the inverse of each other. We point out a simple derivation of this correspondence within the form factor approach
Disorder operators are a type of non-local observables for quantum many-body systems, measuring the fluctuations of symmetry charges inside a region. It has been shown that disorder operators can reveal global aspects of many-body states that are otherwise difficult to access through local measurements. We study the disorder operator for U(1) (charge or spin) symmetry in two-dimensional Fermi and non-Fermi liquid states, using the multidimensional bosonization formalism. For a region AAA, the logarithm of the charge disorder parameter in a Fermi liquid with isotropic interactions scales asymptotically as lAln⁡lAl_A\ln l_AlA​lnlA​, with lAl_AlA​ being the linear size of the region AAA. We calculate the proportionality coefficient in terms of Landau parameters of the Fermi liquid theory. We then study models of the Fermi surface coupled to gapless bosonic fields realizing non-Fermi liquid states. In a simple spinless model, where the fermion density is coupled to a critical scalar, we find that at the quantum critical point the scaling behavior of the charge disorder operators is drastically modified to lAln⁡2lAl_A \ln^2 l_AlA​ln2lA​. We also consider the composite Fermi liquid state and argue that the charge disorder operator scales as lAl_AlA​.
Abstract Using the idea of the quantum inverse scattering method, we introduce the operators B ( x ), C ( x ) and B ̃ ( x ) , C ̃ ( x ) corresponding to the off-diagonal entries of the monodromy matrix T for the phase model and i -boson model in terms of bc fermions and neutral fermions respectively, thus giving alternative treatment of the KP and BKP hierarchies. We also introduce analogous operators B *( x ) and C *( x ) for the charged free boson system and show that they are in complete analogy to those of bc fermionic fields. It is proved that the correlation function ⟨0| C ( x N )⋯ C ( x 1 ) B ( y 1 )⋯ B ( y N )|0⟩ in the bc fermionic fields is the inverse of the correlation function ⟨0| C *( x N )⋯ C *( x 1 ) B *( y 1 )⋯ B *( y N )|0⟩ in the charged free bosons.
We develop a general method of solving non-trivial conformal field theories by coset imbeddings into free boson models. In particular, we consider ZN parafermion models represented by the coset model constructions su(N1)⊕su(N)1/su(N)2, and bosonize the su(N)1 models. The calculation of correlation functions of these parafermion theories is formulated so that detailed information from the diagonal su(N)2 theory is not required. A differential equation for the four-point functionof order operators σ1 is derived as an example. We also derive differential equations with respect to the modular parameter τ for the parafermion characters in terms of the su(N)1 characters only. In particular, the equations for the Z2 and Z3 characters are first and second order respectively.
We extend the recent approach of M. Jimbo, K. Miki, T. Miwa, and A. Nakayashiki to derive an integral formula for the N-point correlation functions of arbitrary local operators of the antiferromagnetic spin-1 XXZ model. For this, we realize the quantum affine symmetry algebra Uq(su(2)2)U_q(su(2)_2)Uq​(su(2)2​) of level 2 and its corresponding type I vertex operators in terms of a deformed bosonic field free of a background charge, and a deformed fermionic field. Up to GSO type projections, the Fock space is already irreducible and therefore no BRST projections are involved. This means that no screening charges with their Jackson integrals are required. Consequently, our N-point correlation functions are given in terms of usual classical integrals only, just as those derived by Jimbo et al in the case of the spin-1/2 XXZ model through the Frenkel-Jing bosonization of Uq(su(2)1)U_q(su(2)_1)Uq​(su(2)1​).