Error-controlled tensor-network computation of corner magnetization and corner critical exponents

The Corner Transfer Matrix Renormalization Group (CTMRG) was recently extended with dedicated boundary and corner tensors to compute the magnetization of the two-dimensional Ising model at free edges and at the right-angle corner of the square lattice, together with its temperature derivative obtained by automatic differentiation, from which critical exponents are extracted through a ratio method (Krcmar, Genzor, Gendiar and Nishino, arXiv:2506.17194). That work reached fifth-to-sixth-decimal accuracy for bulk and boundary magnetization against exact formulas, and its corner estimate $\beta_c = 1.00018(8)$ agrees with the long-established free-edge corner exponent $\beta_c = 1$ (Barber, Peschel and Pearce's angle law $\beta_c = \pi/(2\theta)$ for opening angle $\theta$). Two methodological issues were left unresolved: (i) it is unknown whether the bulk/boundary-level precision carries over to corner observables, since the corner error was assessed only by mutual comparison of two truncation schemes and qualitative convergence in the number of kept states $m$, never against the exact corner magnetization or a systematic error model; (ii) automatic-differentiation derivative estimates carry errors about two orders of magnitude larger than the magnetization errors, with the origin of this amplification and its reduction unexplained. The protocol is demonstrated only for the right-angle corner of the square lattice. The problem is to place tensor-network corner observables on controlled footing: quantify the truncation error and convergence of corner magnetizations and the exponents extracted from them, and diagnose and suppress the derivative-error amplification.

Background

Near its critical point, the spontaneous magnetization of the two-dimensional Ising model vanishes as $M \sim |T-T_c|^{\beta}$ with the bulk exponent $\beta = 1/8$ (Yang 1952). At a free (open) boundary the decay is governed instead by the surface exponent $\beta_b = 1/2$, a classic result of boundary critical behavior (McCoy and Wu 1967). When two free edges meet at a corner of opening angle $\theta$, the spin sitting at the corner has its own local exponent $\beta_c$. Cardy's conformal treatment of edge and surface critical behavior predicts geometry-dependent local exponents, and Barber, Peschel and Pearce (1984), combining corner-transfer-matrix and fermionic transfer-matrix calculations with numerics on finite wedges, established that for isotropic systems with free edges $\beta_c = \pi/(2\theta)$; the right-angle corner ($\theta = \pi/2$) of the square lattice therefore has $\beta_c = 1$, twice the boundary and eight times the bulk value. Corner magnetizations at other opening angles were verified numerically for Ising and three-state Potts models by transfer-operator techniques (Karevski, Lajkó and Turban 1997), and the spontaneous magnetization at a right-angle corner of a rectangular planar Ising ferromagnet is known exactly (Abraham and Latrémolière 1995); earlier corner-transfer-matrix studies of corner magnetization include Peschel (1985), Kaiser and Peschel (1989), and Davies and Peschel (1991). The benchmark corner-exponent values for the Ising model are thus known, and what is at stake in the present problem is the reliability of a modern computational route to them.

The corner transfer matrix (CTM), introduced by Baxter, is the product of Boltzmann weights over one quadrant of an infinite lattice; its spectrum determines observables near the corner. The Corner Transfer Matrix Renormalization Group (CTMRG) of Nishino and Okunishi iteratively enlarges the CTM environment while truncating it to the $m$ dominant eigenstates of a reduced density matrix, where $m$ is the number of kept states. Standard CTMRG is extremely precise for bulk observables but progressively suppresses boundary degrees of freedom. Krcmar, Genzor, Gendiar and Nishino (2025) modified CTMRG by growing the system from one edge in a half-plane fashion and introducing dedicated boundary tensors (an unrenormalized half-row tensor $H_0$ and a corner tensor $C_b$ whose second leg is never projected), and, in parallel, adapted the Boundary Tensor Renormalization Group (BTRG) of Iino, Morita and Kawashima — a tensor-renormalization-group variant carrying explicit boundary tensors for open systems — to a corner geometry. With these constructions they computed the magnetization of the square-lattice Ising model in the bulk ($M$), at the middle of a free edge ($M_b$), and at the corner ($M_c$), using a magnetic field on one edge to seed the symmetry breaking. To extract exponents sharply they computed the temperature derivative $M'$ by automatic differentiation — the technique of evaluating derivatives of a numerical program to machine precision by applying the chain rule through its elementary operations — and formed the ratio $R = M/M'$, whose leading behavior $R(T) = -(T_c-T)/\beta + \ldots$ isolates the exponent; fits near $T_c$ gave $\beta = 0.125007(2)$, $\beta_b = 0.500020(7)$, and $\beta_c = 1.00018(8)$ at $m = 1000$. Two limitations of this corner protocol motivate the problem. First, the corner accuracy was assessed only by comparing the two truncation schemes with each other and by qualitative convergence in $m$, never against an exact corner reference or a systematic error model; the authors state they are "not sure if this accuracy will translate into our results for the calculations at the corner." Second, the automatically differentiated derivative $M'$ carries errors roughly two orders of magnitude larger than those of $M$, for reasons the authors could not identify ("For some reason, our automatic differentiation produces a larger error for the derivative of the magnetization"). The construction is moreover specific to the right-angle corner of the square lattice, although the source quotes the general-angle relation $\beta_c = (\pi/\theta)\beta_b$ as motivation.

Problem Statement

Establish tensor-network computation of corner observables — the magnetization at a lattice corner, its temperature derivative, and the corner critical exponent extracted from them — on a footing of controlled, quantified accuracy. Concretely, for two-dimensional classical lattice models with a corner formed by two free edges (the square-lattice Ising model with a free-edge right-angle corner being the benchmark, with its exact bulk and boundary magnetizations, the exact corner spontaneous magnetization of Abraham and Latrémolière, and the established corner-exponent law as references; results for this benchmark instance suffice, and generalization to other opening angles or lattice models is not required by the deliverables below):

1. Corner truncation-error control. Quantify the error of the corner magnetization $M_c$, of its temperature derivative $M_c'$, and of the corner exponent extracted from the ratio $R_c = M_c/M_c'$ as a function of the number of retained states $m$ and of the iteration count, including how truncation errors propagate through the boundary and corner tensors of the modified CTMRG construction, and determine whether the bulk/boundary-level accuracy (errors at the fifth decimal place or better) is attainable for corner observables. The quantification must be validated against exact corner references where they exist, not only by mutual comparison of two truncation schemes.

2. Derivative-error amplification. Identify the origin of the roughly two-orders-of-magnitude error amplification in the automatically differentiated temperature derivative $M'$ relative to $M$ (for instance, conditioning of the CTMRG fixed-point iteration, differentiation through truncated eigenvector projectors and non-symmetric boundary tensors, or cancellation within the differentiated network), and suppress the amplification.

These two components jointly support one objective — a corner-observable tensor-network protocol whose accuracy is quantified and controlled — and a complete answer addresses both. The problem is methodological: the benchmark values for the Ising corners are known, so success is measured by controlled and verified accuracy, not by the numerical values themselves.

References

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