McCulloch's infinite-system DMRG wave-function prediction and Baxter's corner transfer matrix (CTM) renormalization both build truncated bases for 2D classical vertex models and the 1D quantum chains they map to, and the two constructions coincide in the uniform thermodynamic limit, where the DMRG density-matrix spectrum equals the square of the CTM spectrum. Ueda, Gendiar and Nishino (J. Phys. Soc. Jpn. 79, 044001 (2010)) showed how the half-column transfer matrix used in wave-function prediction can be built from the singular-value-decomposition (SVD) factors of neighboring CTMs, but observed a puzzle at finite size: the matrix product state (MPS) produced by finite-system DMRG sweeping cannot, in general, be reconstructed by SVD-decomposing neighboring CTMs independently, and the structural reason is not understood. The surviving open core is the finite-size structural relation itself: characterize which features of finite-system DMRG (targeting, boundary conditions, global sweeping optimization) are responsible for the mismatch, determine when the CTM-derived and DMRG-optimized MPS representations coincide, and explain the finite-size relation between the DMRG density-matrix spectrum Λi\Lambda_i and the CTM-SVD spectrum Ωi\Omega_i beyond the established bulk limit Λi∼Ωi2\Lambda_i \sim \Omega_i^2.

Background

The density matrix renormalization group (DMRG) is a variational method, originally for one-dimensional quantum lattice systems, that optimizes a matrix product state (MPS) — a wave function written as a chain of rank-3 tensors whose contracted bond indices reproduce every configuration amplitude, the maximal bond dimension controlling the accuracy. Its infinite-system variant (iDMRG) grows the chain from the middle; McCulloch (arXiv:0804.2509, 2008) introduced 'wave function prediction', which initializes the tensors added at each iDMRG step from the leading part of the half-infinite-chain wave function, greatly improving stability. The same machinery applies to two-dimensional classical vertex models — lattice models whose Boltzmann weights are attached to the edges meeting at each vertex — via the transfer matrix, the operator whose successive powers build up the partition function row by row. Baxter's corner transfer matrix (CTM) method expresses the partition function and half-plane wave functions through products of four corner matrices CiC_i; the corner transfer matrix renormalization group (CTMRG) then truncates the spectra of these corners. The finite-system DMRG of Steven White (who introduced the method), by contrast, optimizes the MPS by sweeping back and forth over the chain, at each bond truncating the singular value decomposition (SVD) of the current target state to retain the dominant Schmidt vectors — the vectors appearing in the bipartite (Schmidt) decomposition of the state across that bond, i.e. the leading eigenvectors of the reduced density matrix with eigenvalues Λ\Lambda.

Ueda, Gendiar and Nishino (2010) asked how these pictures connect. Constructing the approximate half-column transfer matrix (HCTM) Pi=Ci−1−1CiP_i = C_{i-1}^{-1}C_i from the SVD factors of neighboring corners, they showed that in the uniform thermodynamic limit the CTM-SVD spectrum Ωi\Omega_i and the DMRG spectrum obey Λi∼Ωi2\Lambda_i \sim \Omega_i^2 — a relation proven exactly for integrable models (models whose weights satisfy the Yang-Baxter consistency equation, guaranteeing commuting transfer matrices) — so the two constructions generate the same MPS in the bulk. Their derivation, however, assumes left-right (and up-down) symmetric vertex weights: the reflected half-column tensor Pˉi\bar{P}_i is used in place of Pi+1P_{i+1} only under that symmetry, and only 'when the system size is large enough'. At finite size the paper's conclusions state the puzzle: the finite-size wave function can be written as Ci−1Ci+1TC_{i-1}C_{i+1}^{T}, but the MPS of the state optimized by finite-system DMRG 'cannot be obtained from the SVD applied to C3C_3 and C5C_5 independently', with only a hedged remark that this 'is something to do with' the targeting scheme for asymmetric vertex models and with the determination of optimal renormalization transformations in real-time DMRG (DMRG applied to real-time evolution), where the density matrix is time dependent.

The finite-size mismatch is the gap this problem addresses: the DMRG basis is White's truncation-optimal low-rank approximation of the globally swept target state, while the CTM construction factorizes the same information locally, and how the sweep's global optimization, boundary conditions and targeting modify the local CTM SVD tensors is not understood.

Problem Statement

For a two-dimensional classical vertex model with left-right and up-down symmetric weights (or the equivalent one-dimensional quantum chain), characterize the structural relation at finite system size between the two MPS constructions: (a) the CTM/HCTM construction, in which the finite-size wave function Ψ=Ci−1Ci+1T\Psi = C_{i-1}C_{i+1}^{T} is factorized by applying the SVD to the neighboring corner transfer matrices Ci−1C_{i-1} and Ci+1C_{i+1} independently, and (b) the MPS produced by finite-system DMRG sweeping with White's density-matrix truncation. Ueda, Gendiar and Nishino (2010) establish that (b) cannot in general be reconstructed from (a) at finite size and that the two coincide in the uniform thermodynamic limit. The problem is to identify precisely which features of the finite-system DMRG procedure (for example the targeting scheme — the choice of which state or states the reduced density matrix is built from — the boundary conditions, or the global character of the sweeping optimization) are responsible for the mismatch; to give a criterion — necessary and sufficient conditions on the model, the system size, or the state — under which the two constructions coincide; and to explain the relation between the DMRG density-matrix spectrum Λi\Lambda_i and the CTM-SVD spectrum Ωi\Omega_i at finite size, beyond the established bulk limit Λi∼Ωi2\Lambda_i \sim \Omega_i^2. These three parts are independent required deliverables, none subsumed by another. (The asymmetric-weights extension of the construction and the time-dependent-density-matrix generalization are outside the scope of this task.)

References

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