代数衰减相互作用在变分无限矩阵乘积态中的转移矩阵函数
Transfer-matrix functions for algebraically decaying interactions in variational infinite matrix product states
AI总结：
提出一种无需有限极点指数和替代的变分无限矩阵乘积态方法，通过转移矩阵函数直接处理代数衰减相互作用，在长程自由费米子和反平方海森堡模型上验证了有效性。
摘要：
变分无限矩阵乘积态（iMPS）计算通常通过首先用有限极点指数和替代目标哈密顿量，使具有代数衰减相互作用的哈密顿量与标准MPO算法兼容，从而引入哈密顿量表示残差。我们无需引入此类替代即可制定固定$D$的变分能量。对于固定的有限$D$ MPS，代数尾部可以通过连接的转移矩阵直接求和：尾部$e^{\mathrm{i} Qr}/r^\alpha$由矩阵函数$F_{\alpha,Q}(\widetilde{T}_A)$表示，其中$F_{\alpha,Q}(z)=\operatorname{Li}_\alpha(e^{\mathrm{i} Q}\\,z)/z$。我们使用Krylov方法评估所得的矩阵函数作用，并通过结合Fréchet伴随与隐式不动点微分获得稳定梯度。对长程自由费米子和反平方海森堡族（包括Haldane-Shastry点）的基准测试验证了转移矩阵函数公式。长程伊森链计算说明了避免有限极点哈密顿量表示的实际后果。在固定且独立已知的临界场下，有限极点替代哈密顿量可能使临界诊断偏离临界性，而矩阵函数计算保留了目标代数哈密顿量的预期临界特征。
英文摘要：
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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