Analytic control of the non-compact continuum limit of two coupled Potts models (the $a_3^{(2)}$ model in regime III): logarithmic finite-size amplitudes, density of states, energy exponents, and the underlying conformal field theory

The integrable critical line of two coupled antiferromagnetic $Q$-state Potts models ($2<Q\le 4$) maps onto the $a_3^{(2)}$ vertex model in regime III, whose continuum limit contains two compact bosons and one non-compact boson. The non-compact direction produces a continuum of critical exponents, discrete states emerging under twists, and strong $1/(\log L)^2$ finite-size corrections whose leading amplitude $A(\gamma)=10\gamma(\pi-\gamma)/(\pi-4\gamma)^2$ and subleading functions $B_{n_1,n_2,j}(\gamma)$ rest on numerical evidence only. The problem is to establish analytic control of this continuum limit: an exact finite-size treatment of the twisted Bethe ansatz equations (for instance via non-linear integral equations) deriving the logarithmic amplitudes, the density of states of the non-compact sector, and the energy-type critical exponents, together with a complete identification of the underlying non-compact conformal field theory.

Background

Two-dimensional $Q$-state Potts models — lattice models in which each site carries a spin with $Q$ possible values — are a laboratory of critical phenomena. While a single Potts model is well understood at its ferromagnetic critical point, new physics appears when replicas are coupled: in the random-bond Potts model the effective inter-replica interaction is the product of the local energy operators (the operators measuring whether neighbouring spins are equal), and this coupling is relevant for $Q>2$. For $N=2$ ferromagnetic replicas with $Q>2$ the coupled model flows to a massive phase, but there exists a special integrable line — first found by direct solution of the star-triangle (Yang-Baxter) equation and established in the form used here by Martins and Nienhuis — on which two coupled antiferromagnetic $Q$-state Potts models remain critical throughout $2<Q\le 4$. Fendley and Jacobsen analysed this critical line via level-rank duality (an equivalence exchanging the rank and the level of the underlying current algebras), identifying the continuum limit with a coset conformal field theory — a conformal field theory built as the quotient of two current-algebra theories, denoted $G/H$ — namely $SO(k)_3\times SO(k)_1/SO(k)_4\cong SU(2)_k\times SU(2)_k/SU(2)_{2k}$ of central charge $c=3k^2/((k+1)(k+2))$, with $\sqrt Q=2\cos(\pi/(k+2))$.

Vernier, Jacobsen and Saleur (2014) showed that the transfer matrix of these two coupled Potts models coincides with that of the $a_3^{(2)}$ vertex model — the vertex model built on the fundamental representation of the twisted affine quantum group $U_q(sl_4^{(2)})$ — in its 'regime III' ($0\le\gamma<\pi/4$ with $\gamma=\pi/(k+2)$), and used this to study the model by the Bethe ansatz: eigenvalues are parameterised by the roots of Bethe ansatz equations depending on two twist angles $\phi_1,\phi_2$ (each twist reweights the non-contractible loops of one Potts model, i.e. one colour in the dense two-colour loop representation). Numerical solution for systems up to $L\approx 100$ sites, combined with finite-size scaling — for a critical system of width $L$, $E_0(L)=E_\infty-v_{\rm F}\pi c/(6L^2)$ and gaps $=v_{\rm F}\,2\pi x/L^2$, yielding the central charge $c$ and scaling dimensions $x$ — established the structure of the continuum limit: two compact bosons (the magnetisations of the two Potts models) and one non-compact boson (the quantum number $j$ generated by their coupling). A compact boson takes values on a circle, so its zero modes are quantised; a non-compact boson takes values on a line, and its presence forces a continuous part in the spectrum of critical exponents, with discrete states emerging from — and re-dissolving into — the continuum as the total twist crosses the sector-dependent threshold $\phi_1+\phi_2=(|n_1|+|n_2|+2j+1)\gamma$, where $n_1,n_2$ are the two compact zero-mode numbers and $j$ the non-compact quantum number.

Such non-compact criticality is not an isolated curiosity. The rank-one member of the same family, the $a_2^{(2)}$ (Izergin-Korepin) model in regime III, has a continuum limit identified with the $SL(2,\mathbb{R})_k/U(1)$ Euclidean black hole conformal field theory — the two-dimensional sigma model with cigar-shaped target introduced by Witten — which is also the continuum limit of the antiferromagnetic Potts model and of the staggered six-vertex model. Lattice regularisations of non-compact theories are the proposed route to field theories with continuous spectra of scaling dimensions, invoked for the multifractal scaling of the integer quantum Hall plateau transition, and they supply finite, exactly solvable analogues of non-compact target spaces. The $a_3^{(2)}$ model is the first family member beyond the black-hole case, with a three-boson continuum limit (two compact, one non-compact).

The non-compact degree of freedom leaves a sharp quantitative fingerprint. The finite-size data for the effective central charges $c_{n_1,n_2,j}=c-12x_{n_1,n_2,j}$ in each sector are not described by $1/L^2$ scaling alone but require strong logarithmic corrections,

$-\frac{c_{n_1,n_2,j}}{12}=\frac{\gamma}{2\pi}(n_1+n_2)^2+\left(N_{n_1,n_2,j}\right)^2\frac{A(\gamma)}{\left[B_{n_1,n_2,j}(\gamma)+\log L\right]^2},$

with $N_{n_1,n_2,j}=1+(\text{number of Bethe roots of imaginary part }\pi/2)$ — the count runs over the $\lambda$-type roots, one of the two coupled Bethe-root families — and $v_{\rm F}=\pi/(\pi-4\gamma)$, together with the conjecture

$A(\gamma)=\frac{10\,\gamma(\pi-\gamma)}{(\pi-4\gamma)^2}.$

The numerical support for the functional dependence of $A(\gamma)$ on $\gamma$ was described as very strong, that for the proportionality factor 10 as only moderate, and the subleading functions $B_{n_1,n_2,j}(\gamma)$ could not be resolved within the numerical accuracy. The authors state that progress would presumably require setting up non-linear integral equations (NLIE) — the standard analytic continuation of Bethe ansatz methods to finite size — for this model.

What is already available: the closed-form conjectures for the magnetic watermelon exponents $x_{2n_1,2n_2}$ (the scaling dimensions of operators inserting $2n_1$ and $2n_2$ propagating through-lines of the two colours), which invalidated the earlier transfer-matrix estimates of Fendley and Jacobsen (all of them except $c$ and $x_{2,0}$, the two cases without logarithmic corrections); and the precise twist angles of discrete-state emergence. What is missing is everything analytic. A companion study of the general $a_{N-1}^{(2)}$ family (Vernier-Jacobsen-Saleur 2016) related regime III to the gauged Wess-Zumino-Witten (WZW) cosets $SO(N)/SO(N-1)$ — for $a_3^{(2)}$ this involves $SO(4)/SO(3)$ and parafermionic perturbations of $SU(3)/SO(3)$ — but explicitly left open the sigma-model realisation, the density of states of the continuous spectrum, and the projection onto discrete states, remarking that completing this program 'should take quite a while'. As of 2023, specialists state that for the translation-invariant $A_n^{(2)}$ series in regime III 'a reliable description of the conformal spectra in these models has not been possible' (Frahm et al., JHEP 2023).

Problem Statement

The problem is to replace the numerically supported conjectures about the continuum limit of the $a_3^{(2)}$ model in regime III — equivalently, of the integrable critical line of two coupled antiferromagnetic $Q$-state Potts models with $2<Q\le 4$, or of the dense two-colour loop model at $\sqrt Q=2\cos(\pi/(k+2))$, $0\le\gamma<\pi/4$ — by exact results, and to complete the resulting description of the critical spectrum. A full solution comprises four linked components.

(1) Exact finite-size solution. Construct an exact analytic treatment of the twisted $a_3^{(2)}$ Bethe ansatz equations at finite $L$ — for example non-linear integral equations (NLIE), a thermodynamic Bethe ansatz (integral equations for finite-size energies derived from the infinite-length limit), an analytic Bethe ansatz (closed-form eigenvalue expressions parameterised by Bethe roots), or another exact method of comparable power (such as the ODE/IM correspondence, which identifies spectral problems of certain ordinary differential equations with integrable models) — from which the scaling spectrum can be derived rather than merely computed numerically.

(2) Logarithmic amplitudes and density of states. From this solution, derive the leading logarithmic amplitude $A(\gamma)$ (confirming or correcting the conjecture $A(\gamma)=10\gamma(\pi-\gamma)/(\pi-4\gamma)^2$), the explicit $\gamma$-dependence of the subleading functions $B_{n_1,n_2,j}(\gamma)$ in each sector $(n_1,n_2,j)$, governing the $1/(\log L)^2$ corrections, and the density of states of the continuous (non-compact) part of the spectrum, including the mechanism by which discrete states emerge when the total twist crosses the sector threshold $\phi_1+\phi_2=(|n_1|+|n_2|+2j+1)\gamma$ (whose ground-state-sector case is $(2j+1)\gamma$).

(3) Energy-type exponents. Derive the closed-form energy (thermal) critical exponents of the coupled Potts models — the sector complementary to the already-conjectured watermelon exponents — completing the operator content.

(4) Identification of the continuum theory. Identify the non-compact conformal field theory underlying regime III through an explicit construction (a sigma-model or gauged WZW realisation, for instance via $SO(4)/SO(3)$ or the parafermionic perturbation of $SU(3)/SO(3)$, or another realisation) that reproduces the central charge $c=3k^2/((k+1)(k+2))$, the twist-dependent spectrum, the discrete states and their emergence conditions, and the density of states, and that matches the exact lattice results.

Components (2) and (3) are the quantitative output of component (1), and component (4) is the field-theoretic formulation of the same continuum limit. Extensions to other geometries (open or integrable boundary conditions, boundary Temperley-Lieb algebras) and to the dilute two-colour loop counterpart related to truncated quantum Hall plateau transitions are recognised future directions in the source but constitute distinct problems; they are not part of this problem's completion boundary.

References

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