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A. H. Bougourzi

Publications and source records attributed to A. H. Bougourzi.

16 recordsLinked to original sources
arXiv2025

Revisiting the Exact Dynamical Structure Factor of the Heisenberg Antiferromagnetic Model

We revisit our initial derivations of the exact 2-spinon $S_2$ and 4-spinon $S_4$ dynamical structure factors (DSF). First we show that the latter derivations had normalization factors that are twice and quadriple the correct ones respectively. This means that $S_2$ contributes not 72% as was previously thought but 36% to the total DSF. We also calculated the contribution of $S_4$ to be between 18% and 20% and not 27% as was calculated by Caux and Hagemans. In fact we show that the latter reference had also the normalization factor twice the correct value and had it done the numeric integrations correctly it would have obtained a contribution between 36% and 40% for $S_4$. Furthermore, we prove that its claim that our initial derivation of $S_4$ was also incorrect in its dependency on the spectral parameters is incorrect because fixing the momentun transfer $k$ up to $2π$ as the latter reference did to justify its claim only amounts to mutltiplying the overall factor by 2 because as we will prove in this paper $S_4$ is periodic in $k$ with period $2π$. Also in this paper we derive $S_n$ for general $n$ by following a different approach compared to our initial derivation of $S_4$. Although for $S_4$ both the new derivation and the initial one lead to equivalent formulas that are expressed differently, the new form presented in this paper is much more elegant and compact and also reveals new hidden and nontrivial symmetries which substantially simplify the numeric evaluation of $S_4$ and its sum rules. Moreover based on the results of this paper we propose a simple approximation to the total DSF of the Heisenberg model. Finally we comment on how our work might resolve the discrepancy between the exact theoretic results and experiemntal data as reported by Zaliznyak et al.

cond-mat.stat-mech↗
arXiv1998

Exact four-spinon dynamical correlation function in isotropic Heisenberg model

We discuss some properties of the exact four-spinon dynamical correlation function $S_4$ in the antiferromagnetic spin 1/2 $XXX$-model, the expression of which we derived recently. We show that the region in which it is not identically zero is different from and larger than the spin-wave continuum. We describe its behavior as a function of the neutron momentum transfer $k$ for fixed values of the neutron energy $ω$ and compare it to the one corresponding to the exact two-spinon dynamical correlation function $S_2$. We show that the overall shapes are quite similar, even though the expression of $S_4$ is much more involved than that of $S_2$. We finish with concluding remarks.

cond-mat↗
arXiv1997

Bosonization of quantum affine groups and its application to the higher spin Heisenberg model

In this paper, we present a detailed analysis of the diagonalization of the higher spin Heisenberg model using its quantum affine symmetry $U_q(\hat{sl(2)})$. In particular, we describe the bosonizations of the latter algebra, its highest weight representations, vertex operators and screening operators. Finally, we use this bosonization method to compute the vacuum-to-vacuum expectation values and the form factors of any local operator.

q-alg↗
arXiv1997

Exact four-spinon dynamical correlation function of the Heisenberg model

In this paper we derive the exact expression of the four-spinon contribution to the dynamical correlation function of the spin S= 1/2 anisotropic (XXZ) Heisenberg model in the antiferromagnetic regime. We extensively study its isotropic (XXX) limit and derive perturbatively the Ising one. Our method relies on the quantum affine symmetry of the model, which allows for a systematic diagonalization of the Hamiltonian in the thermodynamic limit and for an exact calculation of matrix elements of local spin operators. In fact, we argue that the familiar criticism of this method related to the complication of these matrix elements is not justified. First, we give, in the form of contour integrals, an exact expression for the n-spinon contribution. After we compile recently found results concerning the two-spinon contribution, we specialize the n-spinon formula to the new case n=4. Then we give an explicit series representation of this contribution in the isotropic limit. Finally, after we show that this representation is free of divergences, we discuss the Ising limit in which a simple expression is found up to first order in the anisotropy parameter.

hep-th↗
arXiv1997

A Hopf algebra isomorphism between two realizations of the quantum affine algebra $U_q(\widehat{gl(2)})$

We consider the algebra isomorphism found by Frenkel and Ding between the RLL and the Drinfeld realizations of $U_q(\widehat{gl(2)})$. After we note that this is not a Hopf algebra isomorphism, we prove that there is a unique Hopf algebra structure for the Drinfeld realization so that this isomorphism becomes a Hopf algebra isomorphism. Though more complicated, this Hopf algebra structure is also closed, just as the one found previously by Drinfeld.

q-alg↗
arXiv1996

Exact two-spinon dynamical correlation function of the Heisenberg model

We derive the exact contribution of two spinons to the dynamical correlation function of the spin-1/2 Heisenberg model. For this, we use the isotropic limits of the exact form factors that have been recently computed through the quantum affine symmetry of the anisotropic Heisenberg model $XXZ$

q-alg↗
arXiv1996

Wakimoto construction in the principal gradation

It is well known that the bosonized version of the Wakimoto construction allows the explicit realization of any affine algebra $\widehat{g}$, with arbitrary level $k$ in the homogeneous gradation, in terms of $dim(g)$ free bosonic fields. In this paper, we show in the case of the simplest affine algebra $\widehat{sl(2)}$, that the bosonized Wakimoto realization can be extended to the principal gradation only when $k$ is equal to the critical level, i.e., -2. In this case, this construction can be achieved in terms of arbitrary number (larger than 1) of free bosonic fields.

q-alg↗
arXiv1994

N-point matrix elements of dynamical vertex operators of the higher spin XXZ model

We extend the concept of conjugate vertex operators, first introduced by Dotsenko in the case of the bosonization of the $su(2)$ conformal field theory, to the bosonization of the dynamical vertex operators (type II in the classification of the Kyoto school) of the higher spin XXZ model. We show that the introduction of the conjugate vertex operators leads to simpler expressions for the N-point matrix elements of the dynamical vertex operators, that is, without redundant Jackson integrals that arise from the insertion of screening charges. In particular, the two-point matrix element can be represented without any integral.

hep-th↗
arXiv1994

The Dynamical Correlation Function of the XXZ Model

We perform a spectral decomposition of the dynamical correlation function of the spin $1/2$ XXZ model into an infinite sum of products of form factors. Beneath the four-particle threshold in momentum space the only non-zero contributions to this sum are the two-particle term and the trivial vacuum term. We calculate the two-particle term by making use of the integral expressions for form factors provided recently by the Kyoto school. We evaluate the necessary integrals by expanding to twelfth order in $q$. We show plots of $S(w,k)$, for $k=0$ and $π$ at various values of the anisotropy parameter, and for fixed anisotropy at various $k$ around $0$ and $π$.

hep-th↗
arXiv1994

A Free Field Representation of the Screening Currents of $U_q(\widehat{sl(3)})

We construct five independent screening currents associated with the $U_q(\widehat{sl(3)})$ quantum current algebra. The screening currents are expressed as exponentials of the eight basic deformed bosonic fields that are required in the quantum analogue of the Wakimoto realization of the current algebra. Four of the screening currents are `simple', in that each one is given as a single exponential field. The fifth is expressed as an infinite sum of exponential fields. For reasons we discuss, we expect that the structure of the screening currents for a general quantum affine algebra will be similar to the $U_q(\widehat{sl(3)})$ case.

hep-th↗
arXiv1993

N-point Correlation Functions of the Spin-1 XXZ Model

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 $U_q(su(2)_2)$ 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 $U_q(su(2)_1)$.

hep-th↗
arXiv1993

Vertex Operators and Matrix Elements of $U_q(su(2)_k)$ via Bosonization

We construct bosonized vertex operators (VOs) and conjugate vertex operators (CVOs) of $U_q(su(2)_k)$ for arbitrary level $k$ and representation $j\leq k/2$. Both are obtained directly as two solutions of the defining condition of vertex operators - namely that they intertwine $U_q(su(2)_k)$ modules. We construct the screening charge and present a formula for the n-point function. Specializing to $j=1/2$ we construct all VOs and CVOs explicitly. The existence of the CVO allows us to place the calculation of the two-point function on the same footing as $k=1$; that is, it is obtained without screening currents and involves only a single integral from the CVO. This integral is evaluated and the resulting function is shown to obey the q-KZ equation and to reduce simply to both the expected $k=1$ and $q=1$ limits.

hep-th↗
arXiv1993

Uniqueness of the bosonization of the $U_q(su(2)_k)$ quantum current algebra

Four apparently different bosonizations of the $U_q(su(2)_k)$ quantum current algebra for arbitrary level $k$ have recently been proposed in the literature. However, the relations among them have so far remained unclear except in one case. Assuming a special standard form for the $U_q(su(2)_k)$ quantum currents, we derive a set of general consistency equations that must be satisfied. As particular solutions of this set of equations, we recover two of the four bosonizations and we derive a new and simpler one. Moreover, we show that the latter three, and the remaining two bosonizations which cannot be derived directly from this set of equations since by construction they do not have the standard form, are all related to each other through some redefinitions of their Heisenberg boson oscillators.

hep-th↗