Full $1/N$ finite-size expansion of the lattice involutive charges of critical lattice models and the generality of the lattice-continuum correspondence of the Baxter Q-operator integrable structure

For critical dense polymers, the loop model whose continuum limit is the logarithmic conformal field theory with central charge $c=-2$ , Nigro (J. Stat. Mech. 2009 P10008) constructed the Baxter Q operator as a Temperley-Lieb-valued lattice object, defined local, nonlocal and dual nonlocal involutive charges directly on the lattice, and introduced a lattice $\Psi(\nu)$ whose leading continuum limit reproduces the Bazhanov-Lukyanov-Zamolodchikov (BLZ) master function, with Q-operator eigenvalues reproducing the spectral determinant of the harmonic oscillator. The paper itself left open the completion of this lattice-continuum correspondence: the full $1/N$ finite-size expansion of the local and nonlocal lattice charges, each term expected to be uniquely expressible in terms of conformally invariant continuum quantities; the proof (or refutation) of this conformally-invariant-coefficient property for arbitrary integrable lattice models near criticality with a transfer-matrix description; the extension of the lattice Q, $\Psi(\nu)$ and charge construction to other central charges, concretely the critical Ising model, including the exact eigenvalue formulae of the $c=1/2$ Psi function and their numerical test against spectral zeta functions of the BLZ Schrodinger operators; and the massive flow of the nonlocal charges under a bulk thermal perturbation from the $c=1/2$ ultraviolet fixed point to the infrared $c=0$ theory.

Background

Exactly solved two-dimensional lattice models are built from a transfer matrix $T(u)$ , a commuting family of matrices depending on a spectral parameter $u$ whose eigenvalues encode the thermodynamics. Baxter's solution method introduces a second commuting family, the Q operator, related to $T$ by functional equations of TQ type, whose eigenvalue zeroes are the Bethe roots (the auxiliary rapidities parametrizing the exact spectrum in the Bethe ansatz, the standard exact-solution technique for quantum lattice Hamiltonians). In the continuum, conformal field theory (CFT) possesses an analogous integrable structure found by Bazhanov, Lukyanov and Zamolodchikov (BLZ): the enveloping algebra of the Virasoro algebra contains a maximal commuting (involutive) family of conserved quantities - local integrals of motion $I_{2n-1}$ of odd spin $2n-1$ , nonlocal charges $H_n$ , and dual nonlocal charges - all generated by a single analytic function $\Psi_{BLZ}(\nu)$ of a complex variable $\nu$ , whose values at special imaginary points give the charges; the eigenvalues of these charges are expressible through spectral determinants of one-dimensional Schrodinger operators, the origin of the ODE/IM correspondence (the identification of the eigenvalues of the continuum integrals of motion with spectral quantities of certain ordinary differential equations). Critical dense polymers is the loop model $\mathcal{LM}(1,2)$ (first member of the logarithmic minimal model family): dense nonintersecting loops of fugacity zero, whose continuum limit is a logarithmic CFT with central charge $c=-2$ , describable by symplectic fermions, a pair of fermionic fields of integer spin whose Virasoro representations are reducible but indecomposable (logarithmic). The local lattice operators of such models are organized by the Temperley-Lieb algebra, the diagram algebra generated by elements $e_i$ with relations $e_i^2=\beta\,e_i$ , $e_ie_{i\pm1}e_i=e_i$ and $e_ie_j=e_je_i$ for $|i-j|\ge2$ , where $\beta$ is the loop fugacity ( $\beta=0$ for polymers); Koo and Saleur had earlier shown how Virasoro generators can be discretized as Temperley-Lieb elements, initiating the study of lattice realizations of continuum chiral symmetry algebras. In this setting Nigro built, for critical dense polymers, an explicit Temperley-Lieb-valued lattice Baxter Q operator, together with a lattice matrix-valued function $\boldsymbol{\Psi}(\nu)$ defined by a Mellin-type integral transform of $\log\mathbf{Q}$ ; evaluated at special imaginary $\nu$ this produces lattice local charges $\mathbf{I}_{2n-1}$ , nonlocal charges $\mathbf{H}_n$ and dual nonlocal charges in mutual involution. The companion paper computed, at eigenvalue level, the all-order finite-size ( $1/N$ , $N$ the system size) expansion of the local-charge eigenvalues, finding every coefficient proportional to an eigenvalue of a continuum BLZ charge, with all proportionality constants given by exact closed-form expressions; the source paper then showed that in the scaling limit the lattice $Q$ , $\boldsymbol{\Psi}(\nu)$ and charges reproduce the BLZ continuum objects at leading order, and that the Q eigenvalues reproduce the spectral determinant of the harmonic oscillator. What the paper explicitly declines to carry out is the full $1/N$ expansion of the lattice involutive charges in the spirit of the companion paper - also for the nonlocal charges - to find out that, once again, each term in this expansion can be uniquely expressed in terms of conformally invariant quantities, together with the analogous construction at other central charges (Ising via a decomposition in the Clifford algebra of fermionic gamma-matrix generators in place of the Temperley-Lieb one) and the fate of the nonlocal charges away from criticality. The paper's Discussion poses these as future directions of one research program: establishing, order by order in $1/N$ , that the BLZ integrable structure of the continuum theory is encoded in the lattice Q-operator formalism, and asking whether this is a general feature of integrable lattice models near criticality.

Problem Statement

Complete the lattice-continuum correspondence for the integrable structure of critical lattice models initiated by the cited construction for critical dense polymers, along the open directions posed in its Discussion. As posed by the source: (1) Compute the full $1/N$ finite-size expansion, at operator level (that is, as an identity in the Temperley-Lieb algebra, not merely at the level of the eigenvalues of the charges), of the lattice involutive charges of critical dense polymers - the local $\mathbf{I}_{2n-1}$ , the nonlocal $\mathbf{H}_n$ and the dual nonlocal charges - in the spirit of the companion paper (arXiv:0903.5051), extending its all-order analysis in particular to the nonlocal charges, and establish that each term of this expansion can be uniquely expressed in terms of conformally invariant continuum quantities, with the proportionality constants of the correspondence fixed. (2) Settle the general conjecture raised there: for an integrable lattice model close to its critical point that admits a transfer-matrix description, is every term of the $1/N$ expansion of its lattice involutive charges uniquely expressible through conformally invariant continuum quantities? A proof for general central charge, or an explicit counterexample, is required; other values of the central charge (the Ising model in its restricted solid-on-solid (RSOS) lattice formulation) are the prescribed test cases. (3) Extend the lattice construction of $\mathbf{Q}$ , $\boldsymbol{\Psi}(\nu)$ and the involutive charges beyond $c=-2$ , in particular to the critical Ising model ( $c=1/2$ ) via the Clifford-algebra decomposition of the lattice operators suggested in the source: build the Ising lattice Q operator and $\boldsymbol{\Psi}(\nu)$ , establish exact eigenvalue formulae for the BLZ $\Psi$ function at $c=1/2$ (no reliable published conjecture currently exists - the sole proposal was withdrawn), and test them numerically against the spectral determinants and spectral zeta functions (zeta functions built from the eigenvalue spectrum of an operator) of the BLZ Schrodinger operators. (4) In the Ising case, obtain the massive (off-critical) flow of the nonlocal charges under a bulk thermal perturbation, following them from the ultraviolet $c=1/2$ Ising fixed point to the infrared $c=0$ fixed point. These items share one scientific objective - establishing that the BLZ integrable structure of the continuum theory is encoded in the lattice Q-operator formalism order by order in system size. A complete resolution requires all four directions, and no proper subset suffices: they are not interchangeable, since (1) is the concrete all-order computation for critical dense polymers with the proportionality constants fixed, whereas (2) states the general proof-or-counterexample property about the conformally-invariant coefficients and is therefore not implied by (1), nor does it supply the explicit expansion and constants that (1) demands; a result on any single direction is partial progress and does not by itself resolve the problem.

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