Transfer-matrix functions for algebraically decaying interactions in variational infinite matrix product states

• URL: http://arxiv.org/abs/2606.20522v1
• Date: Thu, 18 Jun 2026 17:37:58 GMT
• Title: Transfer-matrix functions for algebraically decaying interactions in variational infinite matrix product states
• Abstract summary: We formulate the fixed-$D$ variational energy without introducing a finite-pole sum-of-exponentials surrogate.<n>A long-range Ising-chain calculation illustrates a practical consequence of avoiding a finite-pole Hamiltonian representation.
• Score: 2.3895943129711212
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Variational infinite matrix product state (iMPS) calculations usually make Hamiltonians with algebraically decaying interactions compatible with standard MPO algorithms by first replacing the target Hamiltonian with a finite-pole sum-of-exponentials surrogate, thereby introducing a Hamiltonian-representation residual. We formulate the fixed-$D$ variational energy without introducing such a surrogate. For a fixed finite-$D$ MPS, the algebraic tail can be summed directly through the connected transfer matrix: the tail $e^{\mathrm{i} Qr}/r^α$ is represented by the matrix function $F_{α,Q}(\widetilde{T}_A)$, with $F_{α,Q}(z)=\operatorname{Li}_α(e^{\mathrm{i} Q}\,z)/z$. We evaluate the resulting matrix-function action using a Krylov method and obtain stable gradients by combining a Fréchet adjoint with implicit fixed-point differentiation. Benchmarks on long-range free fermions and the inverse-square Heisenberg family, including the Haldane--Shastry point, validate the transfer-matrix-function formulation. A long-range Ising-chain calculation illustrates a practical consequence of avoiding a finite-pole Hamiltonian representation. At a fixed, independently known critical field, finite-pole surrogate Hamiltonians can bias a critical diagnostic away from criticality, whereas the matrix-function calculation retains the expected critical signatures of the target algebraic Hamiltonian.

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