Exchange-versus-bootstrap S-matrix crossover in the near-critical three-state quantum Potts chain: the scale p* and its dangerously irrelevant origin

Density-matrix renormalization group (DMRG) studies of the one-dimensional three-state quantum Potts chain near its self-dual critical point show that the low-momentum two-quasiparticle S-matrix has an exchange form, in which quasiparticles swap internal $Z_3$ quantum numbers as they scatter, while the integrable scaling Potts field theory predicts a diagonal (transmission-only) S-matrix from factorized-scattering bootstrap, recovered on the lattice only in the singlet part of the finite-size spectrum. The two regimes are separated by a momentum scale $p^*<mc$ that appears to vanish faster than the Compton scale $mc$ as the critical coupling is approached, pointing to a dangerously irrelevant operator. The open problem is to turn this scenario into a controlled, quantitative description: establish whether the triplet phase shift converges to the bootstrap value at fixed rapidity in the continuum limit, and identify the integrability-breaking irrelevant operator that generates $p^*$ and its scaling law.

Background

The q-state quantum Potts chain is a one-dimensional lattice model in which each site carries a spin with q internal states, coupled ferromagnetically to its neighbours and subjected to a transverse field whose relative strength is a dimensionless coupling g. For q = 3 the model is self-dual under $g\leftrightarrow 1/g$ and undergoes a second-order quantum phase transition at the self-dual point $g_c=1$ , with the quasiparticle gap vanishing as $|g-1|^{5/6}$ . The critical point is described by the three-state Potts conformal field theory (CFT), i.e. the D4 modular invariant (a particular non-diagonal consistent choice of the operator spectrum) of the unitary c = 4/5 minimal model, and the massive scaling limits on either side of the transition are described by perturbing this CFT with its unique $S_3$ -invariant relevant (thermal) operator. This scaling Potts field theory is integrable — it possesses infinitely many conserved charges — and its exact two-particle S-matrix was found by the factorized-scattering bootstrap (the method that fixes two-particle amplitudes exactly from unitarity, crossing symmetry and factorization of multi-particle scattering into two-body processes; Chim and Zamolodchikov). In the ferromagnetic phase the excitations are kinks, i.e. domain walls between pairs of the three degenerate vacua, while in the paramagnetic phase they are massive quasiparticles A and $\bar A$ carrying conjugate $Z_3$ charges; the bootstrap S-matrix is diagonal in the internal $S_3$ space, meaning quasiparticles transmit through one another without exchanging internal quantum numbers.

The lattice chain, in contrast, is believed to be non-integrable for every fixed $g\neq 1$ (only the critical point is integrable), and this mismatch has observable consequences. A perturbative renormalization-group and numerical study (Rapp and Zaránd, 2006) found that the long-distance scattering of quasiparticles in the chain is not described by the factorized amplitudes, and a companion truncated-conformal-space study (Takács, 2011) confirmed that the bootstrap S-matrix does describe the finite-volume spectrum of the scaling field theory itself — sharpening rather than resolving the conflict. (The truncated conformal space approach, TCSA, is a numerical method that diagonalizes the perturbed-CFT Hamiltonian in a finite volume keeping conformal levels below an ultraviolet cutoff; the thermodynamic Bethe ansatz, TBA, recovers the same finite-volume energies from the exact S-matrix via integral equations.) The 2013 DMRG study that poses the present question (Rapp, Schmitteckert, Takács and Zaránd) — DMRG, the density matrix renormalization group, is a variational method for low-lying eigenstates of one-dimensional quantum systems — extracted the two-quasiparticle spectrum on large lattices and analysed it with Lüscher-type finite-size quantization, in which two-particle energy levels on a ring of circumference L determine the scattering phase shifts at momenta $k\approx 2\pi n/L$ . Symmetry restricts the two-body S-matrix of the $Z_3$ -charged quasiparticles to eigenvalues $s_{1,2}(k)$ and $s_3(k)$ ; in the sector containing the pairs {+−} and {−+} the relevant combinations are the singlet (antisymmetric) and triplet (symmetric) eigenvalues $s_s=s_1-s_2$ and $s_t=s_1+s_2$ with phase shifts $\delta_{s,t}(k)$ . For one-dimensional interacting spinless bosons any local interaction is relevant and drives the triplet phase shift to $\delta_t(k\to 0)=\pm\pi/2$ , so the generic low-momentum S-matrix has exchange form, $S(k\to 0)\to -\hat X$ , with $\hat X$ the internal-state exchange operator; the bootstrap S-matrix instead predicts a diagonal low-energy limit with $\delta_t(k\to 0)=0$ . The DMRG data are consistent with exchange scattering below a momentum scale $p^*<mc$ (m the quasiparticle mass, c the limiting velocity of the low-energy dispersion), with the bootstrap amplitudes describing only the singlet part of the finite-size spectrum; the scale $p^*$ appears to vanish faster than the Compton scale $mc$ as $g\to g_c$ , the hallmark of a dangerously irrelevant operator — an operator that flows to zero at the fixed point but whose effects on finite-distance observables vanish only slowly, or non-analytically, in the distance to criticality. The same study found that the scattering lengths $a_{s,t}(g)$ , i.e. the linear coefficients of the low-momentum phase shifts, $\delta_t(k) = -\mathrm{sgn}(k)\,\pi/2 + a_t k + \dots$ and $\delta_s(k) = -a_s k + \dots$ , diverge at the transition, and verified the duality relations between the two phases. A subsequent TCSA/TBA analysis with renormalized counterterms (Lencsés and Takács, 2014) showed that the analogous low-energy deviations of the TCSA phase shifts are ultraviolet-cutoff effects sourced by the irrelevant operators appearing in the operator-product expansion (the short-distance expansion of a product of two local fields in local operators) of the thermal perturbing field (the leading field $\Phi_{3,1}$ and the subleading stress-tensor composite $T\bar T$ ), removable by counterterms — strong support for the same mechanism operating on the lattice, but without a quantitative matching.

Problem Statement

Determine how the exact S-matrix of the integrable scaling Potts field theory and the exchange scattering observed on the lattice coexist in the continuum limit $g\to g_c$ of the three-state quantum Potts chain, by resolving two linked facets of one objective:

(i) Convergence to the bootstrap amplitude: establish whether the triplet phase shift $\delta_t(\beta)$ extracted from the lattice approaches the scaling Potts (bootstrap) value at any fixed finite rapidity $\beta$ , i.e. at fixed $k/(m(g)c(g))$ with momentum parametrized as $k = m(g)c(g)\sinh\beta$ ( $m$ the quasiparticle mass, $c$ the limiting velocity of the low-energy dispersion), in the strict continuum limit $g\to g_c$ — i.e. whether the bootstrap S-matrix is recovered for $|k|\gg p^*$ as $p^*/(mc)\to 0$ . Existing DMRG data are inconclusive: singlet deviations are already explained by the ultraviolet cutoff, while triplet deviations remain considerably larger than the bootstrap prediction.

(ii) The crossover scale: identify which dangerously irrelevant operator of the perturbed-CFT expansion generates the new low-momentum scale $p^*(g)$ , and determine the precise scaling law by which $p^*$ vanishes relative to the Compton scale as $g\to g_c$ .

These facets share one scientific objective — a controlled continuum account of the near-critical scattering of the lattice three-state Potts chain. The source study suggests, as a possible candidate mechanism, the leading integrability-breaking irrelevant operator of perturbed CFT, in which case one expects the diagonal S-matrix to be recovered at any fixed finite rapidity as $g\to g_c$ , with $p^*$ separating the scaling-Potts regime ( $|k|\gg p^*$ ) from the exchange-scattering regime ( $|k|\ll p^*$ ); no particular scheme or demonstration route is prescribed for establishing the convergence and the scaling law.

References

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