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

Abstract

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...

Excited state TBA and renormalized TCSA in the scaling Potts model

“…A detailed DMRG analysis has shown that the observed discrepancy between the factorized S matrix and the low-energy scattering of quasi-particles in the discrete spin chain persists even non-perturbatively [31], and it was speculated that this was due to the presence of an irrelevant operator that has a large effect on the low-energy limit of the scattering amplitudes away from the scaling limit. In this connection, first of all we note that the raw TCSA phase-shifts in figures 5.3, 5.4 and 5.5 show a characteristic deviation from the field theory predictions at low energies which is very similar to that observed in the DMRG results of [31]. In contrast to the DMRG study, in this paper we are in a position to identify the source of this deviation: it originates from the cut-off dependence introduced by the operators that appear in the OPE The counter term from the identity I is the universal contribution shown explicitly in (3.69), which only renormalizes the bulk energy density and thus makes no contribution to the extracted phase-shifts.…”

Relaxation and entropy generation after quenching quantum spin chains

“…The quantum Potts spin chain is not integrable apart from the critical point. Nevertheless, its spectrum can easily be guessed, and subsequently verified using perturbation theory for g far away from the critical value [35]. In the ferromagnetic phase, the elementary excitations are domain walls with dispersion relation…”

Quasi-particle spectrum and entanglement generation after a quench in the quantum Potts spin chain

“…For more information regarding the quasi-particle spectrum of the chain we refer the interested reader to the work [49] and references therein.…”