Massive integrable structure of the off-critical logarithmic minimal models LM(p,p';t)

The off-critical logarithmic minimal models $\mathcal{LM}(p,p';t)$ — the integrable $\varphi_{1,3}$ perturbations of the critical logarithmic minimal models $\mathcal{LM}(p,p')$, realized on the lattice as the logarithmic limit of the Forrester–Baxter restricted solid-on-solid models in Regime III — were constructed by Pearce and Seaton using corner transfer matrices, which yield the finitized quasi-rational Kac characters, the full band of critical exponents, and the exact singular free-energy amplitude. The massive theory itself, however, remains undetermined: the lattice correlation length and mass scale in the logarithmic limit (and hence universal amplitude ratios such as the ratio of the singular free energy to the square of the mass), the off-critical functional relations ($T$- and $Y$-systems) and the massive thermodynamic Bethe Ansatz equations, the logarithmic limit of double-row transfer matrices with integrable $(r,s)$ boundary conditions and of periodic/torus geometries off criticality, and the extension to the $\varphi_{2,1}$ and $\varphi_{1,2}$ perturbations via non-unitary dilute-$A$ lattice models are all open. The problem is to complete the massive integrable solution of $\mathcal{LM}(p,p';t)$.

Background

Two-dimensional critical phenomena are organized by the Virasoro minimal models $\mathcal{M}(m,m')$ of central charge $c=1-6(m-m')^2/(mm')$ with coprime integers $2\le m<m'$; the family is unitary on the diagonal $m'=m+1$ and non-unitary otherwise. Each minimal model admits an integrable massive perturbation by the thermal field $\varphi_{1,3}$, giving the off-critical theories $\mathcal{M}(m,m';t)$ with $t=e^{-\varepsilon}$ measuring the departure from criticality. Their lattice realization is the Forrester–Baxter restricted solid-on-solid (RSOS) model on the square lattice — a model whose heights take values $1,\dots,m'-1$ with adjacent heights differing by $\pm 1$, and whose crossing parameter is $\lambda=(m'-m)\pi/m'$ — whose ferromagnetic Regime III flows to $\mathcal{M}(m,m';t)$ in the continuum scaling limit.

The critical logarithmic minimal models $\mathcal{LM}(p,p')$ are the Yang–Baxter integrable Temperley–Lieb loop models — lattice models of non-crossing loops on the square lattice — whose crossing parameter $\lambda=(p'-p)\pi/p'$ is a rational fraction of $\pi$; they include bond percolation $\mathcal{LM}(2,3)$ and critical dense polymers $\mathcal{LM}(1,2)$. Their Kac tables of conformal weights extend infinitely, and their representations are reducible but indecomposable: the Virasoro action is non-diagonalizable, so that a descendant field is proportional to the logarithm of the worldsheet coordinate times its primary — the hallmark of Jordan cells and logarithmic conformal field theory. Pearce and Seaton construct the off-critical logarithmic minimal models $\mathcal{LM}(p,p';t)$ — the $\varphi_{1,3}$ perturbation of $\mathcal{LM}(p,p')$ — by the logarithmic limit $m,m'\to\infty$ through coprime pairs with $m/m'\to p/p'$, taken after the thermodynamic limit; in practice, quantities for $\mathcal{LM}(p,p';t)$ are approximated by $\mathcal{M}(m,m';t)$ at a rational crossing parameter with huge numerator and denominator.

Their principal tool is Baxter's corner transfer matrix (CTM): a transfer matrix for one quadrant of the infinite lattice whose diagonalization reduces one-point functions to one-dimensional configurational sums. In the logarithmic limit these sums reproduce the finitized (truncated) quasi-rational characters of the Kac representations of the critical theory, a lattice instance of the Kyoto school's correspondence between solvable lattice models and representation data. Off-critical observables $O_{r,s}$ yield critical exponents that produce all conformal weights in the band $(ps-p'r)^2<8p(p'-p)$ of the infinitely extended Kac table. By the inversion relation method — solving the exact functional relations satisfied by the partition function under inversion of the spectral parameter — and Poisson summation, the singular part of the free energy at the isotropic point carries the amplitude $A_f=-4\tan(\pi m/(2(m'-m)))$, consistent with the field-theory value of Lukyanov and Zamolodchikov for the $\varphi_{1,3}$ perturbation, $f^{\mathrm{sing}}/M^2=-(1/4)\tan(\pi m/(2(m'-m)))$, where the mass $M$ is proportional to the inverse correlation length. However, the lattice correlation length $\xi$ of the Forrester–Baxter models had not been calculated, so the universal amplitude ratio could not be completed on the lattice; for the (non-logarithmic) minimal and Potts models such ratios had previously been obtained from the lattice by Seaton. (Bianchini and Ravanini later computed $\xi$ for the Forrester–Baxter models at fixed $(m,m')$ by extending the O'Brien–Pearce calculation for the Andrews–Baxter–Forrester RSOS models, but the logarithmic-limit determination of the mass scale and of the amplitude ratio remains open.) Two structural facts frame what remains. First, the conjugate-modulus (modular) transformation of characters — the standard device for extracting critical exponents from one-point functions — breaks down in the logarithmic limit: the modular $\mathcal{S}$-matrix entries of the minimal models carry a common prefactor $\sqrt{8/(mm')}$ that vanishes as $m,m'\to\infty$, so the infinite sums over quasi-rational Kac characters cannot be transformed term by term. Second, the degeneracies that generate the Jordan cells of the critical theory are lifted by the $\varphi_{1,3}$ perturbation, so the off-critical models are not expected to exhibit indecomposable representation structure.

For orientation, both flanks of the problem are largely solved. For the minimal models, the $\varphi_{1,3}$ perturbations are massive integrable quantum field theories — the RSOS scattering theories — for which Zamolodchikov derived thermodynamic Bethe Ansatz (TBA) equations: non-linear integral equations whose solution yields the exact free energy, and the logarithmic limit of these systems is the natural but uncontrolled target. For the critical $\mathcal{LM}(p,p')$, the Melbourne program derived the fusion hierarchies and the $T$- and $Y$-systems (the bilinear and closed functional equations satisfied by the fused single-row and double-row transfer matrices), implemented the integrable $(r,s)$ Kac boundary conditions (boundary conditions labelled by Kac-table indices) and Robin boundary conditions (a one-parameter family interpolating between fixed and free boundaries) on the strip, converted the percolation $Y$-system into a closed $D_3$ TBA system, and conjectured modular-covariant torus partition functions for the dense and dilute loop realizations (the dilute realizations are the dilute-$A$ RSOS models, in which adjacent heights may also be equal, so that the associated loops are dilute rather than dense). What does not exist is the corresponding massive machinery for $\mathcal{LM}(p,p';t)$.

Problem Statement

Complete the massive integrable solution of the off-critical logarithmic minimal models $\mathcal{LM}(p,p';t)$ — the $\varphi_{1,3}$-perturbed critical logarithmic minimal models, realized as the logarithmic limit of the Forrester–Baxter RSOS models in Regime III — to the standard of completeness attained for the critical $\mathcal{LM}(p,p')$ and for the massive minimal models $\mathcal{M}(m,m';t)$. The common objective is the determination of the massive integrable structure of this single family; its components, as posed by Pearce and Seaton, are:

(a) Derive the off-critical functional relations — the fusion hierarchy and the $T$- and $Y$-systems — for the single-row and double-row transfer matrices in the logarithmic limit, and use them to derive the massive thermodynamic Bethe Ansatz equations for $\mathcal{LM}(p,p';t)$, replacing the conjugate-modulus technology that fails on the infinite family of quasi-rational Kac characters.

(b) Apply the logarithmic limit off criticality to the double-row transfer matrices on a strip with Yang–Baxter integrable $(r,s)$ Kac boundary conditions, obtaining the corresponding off-critical finitized characters and boundary spectra.

(c) Apply the logarithmic limit off criticality to the periodic row transfer matrices and torus partition functions, determining the massive modular structure of $\mathcal{LM}(p,p';t)$.

(d) Determine on the lattice the correlation length $\xi$ and mass scale $M$ of $\mathcal{LM}(p,p';t)$ in the logarithmic limit, and thereby the universal amplitude ratios such as $f^{\mathrm{sing}}/M^2$, at least for the prototypical cases (off-critical percolation $\mathcal{LM}(2,3;t)$ and dense polymers $\mathcal{LM}(1,2;t)$), with comparison against the field-theory value $f^{\mathrm{sing}}/M^2=-(1/4)\tan(\pi m/(2(m'-m)))=(1/4)\cot(\pi^2/(2\lambda))$, whose logarithmic-limit value at $\lambda\to(p'-p)\pi/p'$ vanishes for $\mathcal{LM}(2,3;t)$ and diverges for $\mathcal{LM}(1,2;t)$, and against the Lukyanov–Zamolodchikov normalization convention for $M$.

(e) Extend the construction to the $\varphi_{2,1}$ and $\varphi_{1,2}$ integrable off-critical perturbations in the logarithmic limit, which requires developing the logarithmic limit of the non-unitary dilute-$A$ lattice models.

An answer may resolve all of (a)–(e) for general coprime $0<p<p'$, or a subset with an explicit statement of what remains; partial answers restricted to specific models are admissible provided they address the massive regime itself rather than only its critical limit.

References

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