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arXiv:1003.4557 · Journal of Statistical Mechanics Theory and Experiment

The thermodynamic limit of particle–hole form factors in the masslessXXZHeisenberg chain

N Kitanine, Karol Kajetan Kozlowski, J M Maillet, N A Slavnov, V. Terras

Published May 31, 2011• 66 citations
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Abstract

We study the thermodynamic limit of the particle–hole form factors of the XXZ Heisenberg chain in the massless regime. We show that, in this limit, such form factors decrease as an explicitly computed power law in the system size. Moreover, each corresponding amplitude can be obtained as a product of a 'smooth' and a 'discrete' part: the former depends continuously on the rapidities of the particles and holes, whereas the latter has an additional explicit dependence on the set of integer numbers that label each excited state in the associated logarithmic Bethe equations. We also show that special form factors corresponding to zero-energy excitations lying on the Fermi surface decrease as a power law in the system size with the same critical exponents as in the long-distance asymptotic behavior of the related two-point correlation functions. The methods that we develop in this paper are rather general and can be applied to other massless integrable models associated with the six-vertex R -matrix and having determinant representations for their form factors.

DOI
10.1088/1742-5468/2011/05/p05028

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We study the thermodynamic limit of the particle–hole form factors of the XXZ Heisenberg chain in the massless regime. We show that, in this limit, such form factors decrease as an explicitly computed power law in the system size. Moreover, each corresponding amplitude can be obtained as a product of a 'smooth' and a 'discrete' part: the former depends continuously on the rapidities of the particles and holes, whereas the latter has an additional explicit dependence on the set of integer numbers that label each excited state in the associated logarithmic Bethe equations. We also show that special form factors corresponding to zero-energy excitations lying on the Fermi surface decrease as a power law in the system size with the same critical exponents as in the long-distance asymptotic behavior of the related two-point correlation functions. The methods that we develop in this paper are rather general and can be applied to other massless integrable models associated with the six-vertex R -matrix and having determinant representations for their form factors.
We consider the problem of computing form factors of the massless XXZ Heisenberg spin-1/2 chain in a magnetic field in the (thermodynamic) limit where the size M of the chain becomes large. For that purpose, we take the particular example of the matrix element of the operator σz between the ground state and an excited state with one particle and one hole located at the opposite ends of the Fermi interval (umklapp-type term). We exhibit its power-law decrease in terms of the size of the chain M and compute the corresponding exponent and amplitude. As a consequence, we show that this form factor is directly related to the amplitude of the leading oscillating term in the long-distance asymptotic expansion of the correlation function ⟨σ1zσm+1z⟩.
This work constructs a well-defined and operational form factor expansion in a model having a massless spectrum of excitations. More precisely, the dynamic two-point functions in the massless regime of the XXZ spin-1/2 chain are expressed in terms of a properly regularised series of multiple integrals. These series are obtained by taking, in an appropriate way, the thermodynamic limit of the finite volume form factor expansions. The series are structured in a way allowing one to identify directly the contributions to the correlator stemming from the conformal-type excitations on the Fermi surface and those issuing from the massive excitations (deep holes, particles, and bound states). The obtained form factor series opens up the possibility of a systematic and exact study of asymptotic regimes of dynamical correlation functions in the massless regime of the XXZ spin 1/2 chain. Furthermore, the assumptions on the microscopic structure of the model’s Hilbert space that are necessary so as to write down the series appear to be compatible with any model—not necessarily integrable—belonging to the Luttinger liquid universality class. Thus, the present analysis also provides the phenomenological structure of form factor expansions in massless models belonging to this universality class.
We propose a form factor approach for the computation of the large distance asymptotic behavior of correlation functions in quantum critical (integrable) models. In the large distance regime we reduce the summation over all excited states to one over the particle/hole excitations lying on the Fermi surface in the thermodynamic limit. We compute these sums, over the so-called critical form factors, exactly. Thus we obtain the leading large distance behavior of each oscillating harmonic of the correlation function asymptotic expansion, including the corresponding amplitudes. Our method is applicable to a wide variety of integrable models and yields precisely the results stemming from the Luttinger liquid approach, the conformal field theory predictions and our previous analysis of the correlation functions from their multiple-integral representations. We argue that our scheme applies to a general class of non-integrable quantum critical models as well.
Since a long-time, the quantum integrable systems have remained an area where modern mathematical methods have given an access to interesting results in the study of physical systems. The exact computations, both numerical and asymptotic, of the correlation function is one of the most important subject of the theory of the quantum integrable models. In this context an approach based on the calculation of form factors has been proved to be a more effective one. In this thesis, we develop a new method based on the algebraic Bethe ansatz for the computation of the form-factors in thermodynamic limit. It is both applied to and described in the context of isotropic XXX Heisenberg chain, which is one of the examples of an interesting case of critical models where the Fermi-zone is non-compact. In a particular case of two-spinon form-factors, we obtain an exact result in a closed-form which matches the previous result obtained from an approach based on qqq-vertex operator algebra. This method is then generalised to form-factors in higher spinon sectors where we find a reduced determinant representation for the form-factors, in which a higher-level structure for the form-factors is revealed.
We consider the longitudinal dynamical two-point function of the XXZ quantum spin chain in the antiferromagnetic massive regime. It has a series representation based on the form factors of the quantum transfer matrix of the model. The nth summand of the series is a multiple integral accounting for all n-particle–n-hole excitations of the quantum transfer matrix. In previous works, the expressions for the form factor amplitudes appearing under the integrals were either again represented as multiple integrals or in terms of Fredholm determinants. Here, we obtain a representation which reduces, in the zero-temperature limit, essentially to a product of two determinants of finite matrices whose entries are known special functions. This will facilitate the further analysis of the correlation function.
We study the large-volume- L limit of form factors of the longitudinal spin operators for the XXZ spin-1/2 chain in the massive regime. We find that the individual form factors decay as L − n , n being an even integer counting the number of physical excitations—the holes—that constitute the excited state. Our expression allows us to derive the form-factor expansion of two-point spin–spin correlation functions in the thermodynamic limit L → +∞. The staggered magnetisation appears naturally as the first term in this expansion. We show that all other contributions to the two-point correlation function are exponentially small in the large-distance regime.