Implicit versus explicit truncation in corner-transfer-matrix and corner-tensor simulation of infinite quantum lattice systems: which truncation style is more accurate per computational cost?

Orús (Phys. Rev. B 85, 205117 (2012)) built algorithms for infinite quantum lattice systems from Baxter's corner transfer matrix (CTM) and its higher-dimensional generalization, the corner tensor, and left a precise methodological question open: is 'implicit' truncation, which truncates in the spectrum of the CTM or corner tensor itself by keeping its $\chi$ largest singular values, or 'explicit' truncation, which diagonalizes a reduced density matrix built from the corner tensors and re-truncates at every renormalization step, more accurate by construction for a given computational budget — the CTM analogue of single-site versus conventional two-site DMRG? The companion benchmark question — how much accuracy per $\chi$ the 'full' one-, two-, and three-directional 2D corner-tensor methods (explicit truncation) gain over the symmetry-adapted simplified scheme (implicit truncation) — was deferred to a follow-up paper that never appeared. The question remains open: no published work establishes the comparative accuracy by construction or delivers the promised benchmarks.

Background

Tensor network methods approximate many-body quantum states by networks of small tensors whose connecting indices (bond dimensions) bound the entanglement that can be represented. In one spatial dimension the reference method is the Density Matrix Renormalization Group (DMRG), which iteratively optimizes a Matrix Product State (MPS); its infinite-size variants iDMRG and iTEBD (time-evolving block decimation) exploit translation invariance. In two dimensions the analogous state is the Projected Entangled Pair State (PEPS), with infinite-size variant iPEPS; computing expectation values in an iPEPS requires contracting an infinite two-dimensional network, which is computationally hard, so the environment of each tensor must be approximated. The corner transfer matrix (CTM), introduced by Baxter in 1968 in classical statistical mechanics, is the transfer matrix of one quadrant of a two-dimensional lattice: the partition function factorizes into four CTMs, and the rapid decay of the CTM eigenvalue spectrum makes keeping the $\chi$ dominant eigenvectors a controllable approximation. The Corner Transfer Matrix Renormalization Group (CTMRG) of Nishino and Okunishi turns this into an iterative coarse-graining algorithm, and CTMRG with explicit truncation has become the workhorse environment method for modern iPEPS calculations. Orús (Phys. Rev. B 85, 205117 (2012)) recast these ideas for infinite quantum lattice systems by working in the space-time ('classical') representation: in one dimension the evolution operator is applied as a Matrix Product Operator (MPO, a matrix-shaped operator network acting on an MPS), and in two dimensions as a Projected Entangled Pair Operator (PEPO, its 2D analogue); the corner object generalizing the CTM is then a corner tensor, which, unlike a matrix, has no unique eigenvalue decomposition. Two truncation styles run through these algorithms. Implicit truncation truncates in the spectrum of the CTM or corner tensor itself, keeping its $\chi$ largest singular values, as in the 'simplified' symmetry-adapted schemes; in the simplified 1D method the rank $\chi$ is chosen at initialization and never grows, so this truncation is automatic, whereas in the simplified 2D scheme the corner-tensor indices proliferate at every move and the truncation isometries are recomputed at every move from singular value decompositions of the current corner and transfer tensors, truncated to the $\chi$ largest singular values. Explicit truncation recomputes, at every renormalization step, the relevant reduced density matrix (a contraction of corner tensors, weighting each basis state by its contribution to the approximate state, as in DMRG's density-matrix step) and keeps its $\chi$ largest-magnitude eigenvalues, which maximizes the fidelity between the unrenormalized and renormalized reduced density matrices; the 'full' 2D one-, two-, and three-directional methods differ in how many lattice directions are renormalized this way and in operations per step by orders of magnitude. The paper shows that the two styles coincide for 1D systems with sufficient spatial symmetry, benchmarks only the simplest implicit scheme (spin-1/2 transverse-field Ising model in 1D and 2D), and notes an analogy to single-site versus conventional two-site DMRG, where two-site DMRG re-optimizes the environment basis at each step and single-site DMRG keeps it fixed. Understanding which truncation style is intrinsically more accurate, and how much accuracy the full multi-directional methods buy at fixed $\chi$, directly informs how the standard 2D contraction algorithm should be implemented and extrapolated.

Problem Statement

Determine, for corner-transfer-matrix and corner-tensor simulation of infinite quantum lattice systems, whether implicit truncation (truncation in the spectrum of the CTM or corner tensor itself, keeping its $\chi$ largest singular values, as in the symmetry-adapted simplified schemes) or explicit truncation (reduced density matrices contracted from corner tensors, diagonalized and re-truncated at every renormalization step, keeping the $\chi$ largest-magnitude eigenvalues) is the more accurate scheme per unit of computational cost. Concretely, the question has two parts: (1) a structural characterization of when the two styles are equivalent and when they differ — the source already establishes equivalence for 1D systems with sufficient symmetry, so the target is the non-symmetric and 2D corner-tensor settings — including, where they differ, an argument establishing whether either style is more accurate by construction and, if so, which (analogous to the single-site versus conventional two-site DMRG distinction the author invokes); and (2) a controlled numerical benchmark quantifying, as a function of $\chi$, the accuracy and cost of the full one-, two-, and three-directional 2D corner-tensor methods relative to each other and to the symmetry-adapted simplified scheme, on explicitly defined models, including the spin-1/2 transverse-field Ising model on the square lattice (where the critical point is known exactly) and Heisenberg-type models, with an explicit cost metric since the schemes differ by orders of magnitude in operations per step. The same source poses three adjacent research directions — 3D corner-tensor generalizations, CTM-based real-time evolution stable at long times, and corner-method strategies for periodic boundary conditions; these lie outside this problem's completion boundary because each is an open-ended program whose satisfaction criteria are not fixed by the source.

References

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