Analytic theory of the corner transfer matrix spectrum and deformed corner Hamiltonian for lattice models on hyperbolic lattices

Ueda, Krcmar, Gendiar and Nishino (JPSJ 76, 084004 (2007)) adapted the corner transfer matrix renormalization group (CTMRG) method to the ferromagnetic Ising model on the regular hyperbolic {5,4} lattice and observed that the corner transfer matrix (CTM) / density-matrix eigenvalue spectrum decays extremely fast even at the bulk critical point, so that about ten retained states already yield converged thermodynamics — a 'hyperbolic regularization' of CTMRG at criticality. They left the analytic origin of this effect open: no closed-form or asymptotic characterization of the CTM spectrum at criticality on negatively curved lattices exists, their one-parameter deformed corner Hamiltonian $H_C(\Lambda)$ with hyperbolic sinh-weights remains a formal ansatz, no controlled renormalization-group framework explains the regularization, and their suggested generalization to vertex (e.g. eight-vertex) models on hyperbolic lattices was never carried out. The present problem asks for an analytic theory of the corner environment of hyperbolic-lattice models: the spectrum at criticality and its curvature dependence, the microscopic status of $H_C(\Lambda)$, a controlled RG regularization at criticality together with the associated classical–quantum correspondence, and the phase structure and ordered-state classification of vertex models on hyperbolic lattices.

Background

A regular hyperbolic lattice is an infinite tiling of the hyperbolic plane — the two-dimensional surface of constant negative Gaussian curvature — by congruent $p$-sided polygons meeting $q$ at each vertex, denoted by the Schläfli symbol $\{p,q\}$ and hyperbolic whenever $(p-2)(q-2)>4$; the pentagonal $\{5,4\}$ lattice is the case studied in the source paper. Negative curvature makes the number of sites within graph distance $r$ of a chosen center grow exponentially in $r$ (instead of polynomially), so even in the thermodynamic limit the outer boundary carries a non-negligible fraction of all sites and boundary conditions remain physically relevant. Ferromagnetic spin models on such lattices, for example the Ising model with coupling $J$ on $\{5,4\}$, undergo mean-field-like phase transitions: numerically one finds the critical exponents $\alpha=0$, $\beta=1/2$, $\delta=3$, and the bulk correlation length stays finite at the transition temperature.

The corner transfer matrix (CTM), introduced by Baxter in his variational approximations for square-lattice models, is the transfer matrix of one quadrant of a two-dimensional classical lattice model: the product of four CTMs with the appropriate half-row transfer matrices reproduces the partition function, and the normalized CTM plays the role of the reduced density matrix of a half-system when the classical model is reinterpreted as a one-dimensional quantum object along the radial direction. Writing $C = \exp(-H_C)$ defines the corner Hamiltonian; for the flat square-lattice Ising model $H_C$ is a sum of local terms with linearly growing weights, $H_C = h_1 + 2h_2 + 3h_3 + \ldots$, where $h_i$ couples the $i$-th pair of spins along the quadrant edge. For exactly solvable flat models away from criticality the CTM eigenvalue spectrum decays geometrically, which is what makes the CTM a powerful low-rank representation; at criticality on flat lattices the spectrum decays only slowly, and a general analytic form is not known even there.

The corner transfer matrix renormalization group (CTMRG) method of Nishino and Okunishi merges Baxter's CTM construction with the truncation philosophy of the density matrix renormalization group (DMRG): the corner and half-row transfer matrices are grown by one lattice step and then truncated to the $m$ dominant eigenvectors of the reduced density matrix. Ueda, Krcmar, Gendiar and Nishino (2007) modified the CTMRG iteration so that the system expands along geodesics — the straight lines of the hyperbolic plane — and applied it to the Ising model on the $\{5,4\}$ lattice, finding $T_C = 2.799$ in units of $J$, mean-field-like exponents, and — the observation at the origin of this problem — that the density-matrix (CTM) eigenvalue spectrum decays extremely fast even at the transition temperature, so that $m \approx 10$ retained states already give converged magnetization and bond energy, in sharp contrast with the slow spectral decay that cripples flat-lattice CTMRG at criticality. This 'hyperbolic regularization' suggests that negative curvature regularizes the CTM at criticality, but the observation has remained numerical. The natural analytic target is the corner Hamiltonian: the source paper proposed the one-parameter deformation $H_C(\Lambda) = h_1 + \frac{\sinh 2\Lambda}{\sinh\Lambda}\, h_2 + \frac{\sinh 3\Lambda}{\sinh\Lambda}\, h_3 + \ldots,$ which reduces to the flat corner Hamiltonian as $\Lambda \to 0$ and satisfies the recursive structure required by Okunishi's real-space renormalization group for the corner Hamiltonian, an RG acting directly on $H_C$ that had earlier been developed for flat and one-dimensional systems, where a cutoff on the corner-Hamiltonian spectrum acts as a regulator (as in Okunishi's Wilson-like RG for the XXZ chain in the critical regime). The ansatz $H_C(\Lambda)$ was, however, left purely formal: no derivation from the microscopic transfer matrices of a hyperbolic lattice, no identification of $\Lambda$ with the lattice curvature or tiling data, and no resulting spectral prediction exist. In the opposite (quantum) direction, Ueda and Nishino later introduced one-dimensional quantum lattice Hamiltonians whose nearest-neighbor couplings grow as $\cosh(j\lambda)$ along the chain, with $j$ the site index and $\lambda \ge 0$ a deformation parameter — a 'hyperbolic deformation' whose spatial translation is induced by corner Hamiltonians — establishing that hyperbolic deformation and corner-Hamiltonian structure are linked, but again without touching the classical hyperbolic CTM itself.

Two later developments define the current context. First, CTMRG was reformulated with asymmetric (non-Hermitian) corner and half-row transfer matrices, which made triangular-tiled $\{3,q\}$ hyperbolic lattices — tilings not composed of geodesics, which the original geodesic-based scheme could not handle — accessible to the method; on these lattices the density-matrix spectrum again decays exponentially at the transition for $q \ge 7$ (a power law appears only for the Euclidean $\{3,6\}$ lattice at its critical point), and it was conjectured that the correlation length is proportional to the radius of Gaussian curvature in the large-$q$ limit. Second, hyperbolic-lattice tensor networks are now studied holographically, by analogy with correspondences between boundary and bulk physics in anti-de Sitter / conformal field theory duality: boundary spin correlation functions of the hyperbolic-lattice Ising model show power-law decay with quasi-periodic oscillation while bulk correlations always decay exponentially, and boundary-sensitive phases appear because a finite fraction of sites sits on the boundary. None of this supplies the missing analytic theory of the corner environment, which is the subject of the present problem.

Problem Statement

Establish an analytic theory of the corner transfer matrix — equivalently, of the half-system reduced density matrix — for classical lattice models on regular hyperbolic lattices, replacing the numerical observations of Ueda–Krcmar–Gendiar–Nishino (2007) and their successors by derivations. The common objective is to understand and control the corner environment of negatively curved lattices. A complete resolution of the present problem addresses all four of the following requested, scientifically linked results; each result is nonetheless resolvable on its own, being settled by establishing or refuting the claim it states.

(1) Spectrum at criticality. Characterize analytically the eigenvalue spectrum of the CTM / density matrix of the Ising model (and, where feasible, related classical spin models) on regular hyperbolic lattices such as $\{5,4\}$ and $\{3,q\}$ in the thermodynamic limit at the bulk transition point: its precise asymptotic decay form (whether exactly geometric, as in Baxter's off-critical flat-lattice results, or of another form), its degeneracy structure, and the dependence of the decay constant on the Schläfli symbol and on the Gaussian curvature of the tiling — including the status of the conjecture that the correlation length, and hence the spectral scale, is proportional to the radius of Gaussian curvature in the large-$q$ limit.

(2) Deformed corner Hamiltonian. Determine the microscopic status of the corner Hamiltonian on hyperbolic lattices: derive — or refute — the deformation $H_C(\Lambda) = h_1 + \frac{\sinh 2\Lambda}{\sinh\Lambda}\, h_2 + \frac{\sinh 3\Lambda}{\sinh\Lambda}\, h_3 + \ldots$ from the exact corner transfer matrix of a hyperbolic lattice; fix the relation between the deformation parameter $\Lambda$ and the tiling/curvature; and decide whether the spectrum of $H_C(\Lambda)$ reproduces the spectrum characterized in (1), with the flat limit $\Lambda \to 0$ recovering the known flat-lattice corner Hamiltonian.

(3) Controlled RG and classical–quantum correspondence. Construct — or rule out — a controlled real-space renormalization group acting on the corner Hamiltonian, of the type introduced by Okunishi for flat and one-dimensional systems, that explains the observed rapid eigenvalue decay at criticality as a regularization effect and supports systematic, convergent approximations for hyperbolic-lattice models at their transition; and clarify the associated classical–quantum correspondence between the two-dimensional classical hyperbolic model and the one-dimensional quantum chain with hyperbolically deformed couplings proportional to $\cosh(j\lambda)$.

(4) Vertex-model generalization. Using the same corner-environment machinery, determine the phase structure and classify the ordered states of vertex models on regular hyperbolic lattices, taking as the primary case the eight-vertex model — the square-lattice model whose degrees of freedom live on the edges, with Boltzmann weights assigned to the allowed configurations of the edges meeting at each vertex, solved exactly by Baxter in the flat case. On such lattices the boundary carries a finite fraction of the sites, so duality transformations — maps of a model onto an equivalent dual model, such as the Kramers–Wannier high-/low-temperature duality of the Ising model — act non-trivially on boundary conditions.

References

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