Dynamical correlations and quantum phase transition in the quantum Potts model

Rapp, Ákos; Zaránd, Gergely

doi:10.1103/physrevb.74.014433

Cited by 26 publications

(66 citation statements)

References 59 publications

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“…We study in detail the two-particle spectrum of the q = 3 state quantum Potts chain using the powerful numerical method of density matrix renormalization group (DMRG). We find that the finite size spectra are indeed in complete agreement with the theory of [3] and an asymptotic (i.e. k → 0 momentum) S-matrix of the exchange form.…”

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confidence: 76%

Asymptotic scattering and duality in the one-dimensional three-state quantum Potts model on a lattice

et al. 2013

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We determine numerically the single-particle and the two-particle spectrum of the three-state quantum Potts model on a lattice by using the density matrix renormalization group method, and extract information on the asymptotic (small momentum) S-matrix of the quasiparticles. The low energy part of the finite size spectrum can be understood in terms of a simple effective 8 In the continuum limit, the properties of the Potts model are usually described by the so-called scaling Potts field theory, which is a perturbation of the fixed point CFT, uniquely determined by the symmetries. In this approach, the cut-off (lattice spacing) is removed, and only the leading relevant operator is kept. The application of the machinery known as the Smatrix bootstrap [13] yields a diagonal quasiparticle S-matrix for low energy particles [14][15][16][17][18][19] and implies that the internal quantum numbers of two colliding particles are conserved during a scattering process (see figure 1(a)). The bootstrap S-matrix and the perturbed CFT yield a fully consistent picture [20].Recent perturbative calculations as well as renormalization group arguments showed [3], on the other hand, that rather than being diagonal, the asymptotic S-matrix of the lattice Potts model assumes the 'universal' form, also emerging in various spin models [11], as well as in the sine-Gordon model [13]:Ŝ → −X , withX the exchange operator (see figure 1(b)).Although the arguments of [3] are very robust, the results of [3] were met by some skepticism. On the one hand, the lattice results seemed to conflict with results obtained within the scaling Potts theory [13,[15][16][17][18][19]. On the other hand, various thermodynamical properties of the two dimensional (2D) classical Potts model (on a lattice) such as critical exponents [7] or universal amplitude ratios [38,39] also seem to agree with the predictions of the scaling Potts model (perturbed C = 4/5 minimal model). We must emphasize that the structure of the asymptotic S-matrix has important physical consequences: an S-matrix of the exchange form yields diffusive finite temperature spin-spin correlation functions at intermediate times [3,21,22], while a diagonal S-matrix would result in exponentially damped correlations [12,23,24].The purpose of the present paper is to investigate and possibly resolve this apparent controversy. We study in detail the two-particle spectrum of the q = 3 state quantum Potts chain using the powerful numerical method of density matrix renormalization group (DMRG). We find that the finite size spectra are indeed in complete agreement with the theory of [3] and an asymptotic (i.e. k → 0 momentum) S-matrix of the exchange form. However, our analysis also reveals the emergence of a new momentum scale, p * , below which this exchange S-matrix dominates. By approaching the critical point, this scale vanishes faster than the Compton momentum, mc, suggesting that the new scale and the corresponding exchange scattering is generated by some dangerously irrelevant operator, usually neglected...

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Asymptotic scattering and duality in the one-dimensional three-state quantum Potts model on a lattice

et al. 2013

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“…(33) coincides with the correlation function obtained from the recent semiclassical analysis in Refs. [6,21]. Both these papers explicitely assume that the two-particle S-matrix is equal to the permutation operator [22].…”

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confidence: 99%

Low temperature correlation functions in integrable models: Derivation of the large distance and time asymptotics from the form factor expansion

Altshuler¹,

Konik²,

Tsvelik³

2006

Nuclear Physics B

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We propose an approach to the problem of low but finite temperature dynamical correlation functions in integrable one-dimensional models with a spectral gap. The approach is based on the analysis of the leading singularities of the operator matrix elements and is not model specific.We discuss only models with well defined asymptotic states. For such models the long time, large distance asymptotics of the correlation functions fall into two universality classes. These classes differ primarily by whether the behavior of the two-particle S matrix at low momenta is diagonal or corresponds to pure reflection. We discuss similarities and differences between our results and results obtained by the semi-classical method suggested by Sachdev and Young, Phys. Rev. Lett. 78, 2220Lett. 78, (1997.

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“…For these results to have a greater use one has to establish their relationship with the formfactor approach. A step in this direction was made in [8] where the semiclassical results [5], [6], [7] were reproduced by summing up the leading singularities in the formfactor expansion. Such summation was restricted to the leading order in the soliton density n ∼ exp(−M/T ) (M is the spectral gap).…”

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“…However, for models of free fermions (such as the XY model or the Quantum Ising model), there are alternative means to calculate the correlation functions which allow to bypass the above problems. These alternative approaches include the determinant representation of the correlation functions [3], [4] and the semiclassical method [5] (which may have much wider application, see [6], [7]). For these results to have a greater use one has to establish their relationship with the formfactor approach.…”

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Finite-temperature correlation function for the one-dimensional quantum Ising model: The virial expansion

Reyes

Tsvelik

2006

Phys. Rev. B

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We rewrite the exact expression for the finite temperature two-point correlation function for the magnetization as a partition function of some field theory. This removes singularities and provides a convenient form to develop a virial expansion (the expansion in powers of soliton density).To calculate correlation functions in strongly correlated systems is not an easy task, even if the corresponding models happen to be integrable. For models with dynamically generated spectral gaps the most powerful technique is the formfactor approach pioneered by Karowski et. al.[1] and perfected by Smirnov [2]. This approach works wonderfully for zero temperature, but encounters difficulties at T = 0. These difficulties are related to singularities in the operator matrix elements (formfactors). These singularities exist for operators nonlocal with respect to solitons, they originate from forward scattering processes and their treatment requires careful infrared regularization. Despite long efforts a correct regularization has not been yet found. However, for models of free fermions (such as the XY model or the Quantum Ising model), there are alternative means to calculate the correlation functions which allow to bypass the above problems. These alternative approaches include the determinant representation of the correlation functions [3], [4] and the semiclassical method [5] (which may have much wider application, see [6], [7]). For these results to have a greater use one has to establish their relationship with the formfactor approach. A step in this direction was made in [8] where the semiclassical results [5], [6], [7] were reproduced by summing up the leading singularities in the formfactor expansion. Such summation was restricted to the leading order in the soliton density n ∼ exp(−M/T ) (M is the spectral gap). In this paper we describe a formfactor-based representation for correlation functions which, though in its present form is valid only for models of free fermions, is rather suggestive and may give rise to useful generalizations in the future. For the Quantum Ising (QI) model, which is the main object of this paper, this procedure naturally gives rise to a virial expansion of the dynamical spin susceptibility. The QI model Hamiltonian is

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Dynamical correlations and quantum phase transition in the quantum Potts model

Cited by 26 publications

References 59 publications

Asymptotic scattering and duality in the one-dimensional three-state quantum Potts model on a lattice

Asymptotic scattering and duality in the one-dimensional three-state quantum Potts model on a lattice

Low temperature correlation functions in integrable models: Derivation of the large distance and time asymptotics from the form factor expansion

Finite-temperature correlation function for the one-dimensional quantum Ising model: The virial expansion

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