Probing the Dynamical Structure Factor of Quantum Spin Chains via Low-Temperature Gibbs States with Matrix Product State Subspace Expansion

TL;DR

The paper tackles the difficulty of simulating finite-temperature, low-temperature dynamics in quantum spin chains with tensor networks. It introduces a generating-function matrix product state (GFMPS) framework that directly computes a large set of Bloch-type excited states to form a Gibbs state for low-energy subspaces. The approach provides a concrete algorithm: construct a translationally invariant ground-state MPS with tensor A, generate excited states using an impurity tensor B, obtain a GFMPS via differentiation to build |G_φ(λ,B)⟩, solve a generalized eigenproblem for B, and evaluate dynamical quantities through derivatives of G_DSF to compute the DSF, followed by Lorentzian broadening. Benchmark results show strong agreement with exact diagonalization and experiments, indicating a scalable path to low-temperature dynamical properties in 1D quantum materials and potential extensions to higher dimensions.

Abstract

Studying finite-temperature properties with tensor networks is notoriously difficult, especially at low temperatures, due to the rapid growth of entanglement and the complexity of thermal states. Existing methods like purification and minimally entangled typical thermal states offer partial solutions but struggle with scalability and accuracy in low-temperature regime. To overcome these limitations, we propose a new approach based on generating-function matrix product states (GFMPS). By directly computing a large set of Bloch-type excited states, we construct Gibbs states that moderate the area-law constraint, enabling accurate and efficient approximation of low-temperature thermal behavior. Our benchmark results show magnificent agreement with both exact diagonalization and experimental observations, validating the accuracy of our approach. This method offers a promising new direction for overcoming the longstanding challenges of studying low-temperature properties within the tensor network framework. We also expect that our method will facilitate the numerical simulation of quantum materials in comparison with experimental observations.

Figures (1)

• Figure 1: Overview of our algorithm. (1) The translationally invariant ground state is represented by an MPS with a single tensor $A$ forming a ring under periodic boundary conditions. Excited states are constructed as Bloch states with a wave number $k$ by inserting an impurity tensor $B$. (2) After optimizing $A$ via automatic differentiation Liao2019, excited states are generated by defining the GFMPS $|G_{\phi}(\lambda, B)\rangle$ and taking its derivative with respect to $\lambda$. (3) The tensor $B$ is obtained from a generalized eigenvalue problem, where the effective Hamiltonian $\bm{H}_{\mu\nu}$ and effective norm $\bm{N}_{\mu\nu}$ follow from differentiating with respect to $B$ and its complex conjugate, $\bar{B}$. The indices $\mu$ and $\nu$ label the components contracted with $B$ and $\bar{B}$, respectively, and their dimensionality matches the size of the Hilbert space spanned by the accessible quantum states. This procedure yields an orthogonal set of $B$ tensors, and translational invariance ensures that the GFMPS needs to be evaluated only in the ket layer. (4) Matrix elements, $\left\langle \Psi^{m} \left| \hat{S}^{z}_k \right| \Psi^n \right\rangle$, require three derivatives of $G_{\text{DSF}}(\lambda_1,\lambda_2, \lambda_3)$. At zero temperature, one fewer derivative is needed, as shown in Eq. (\ref{['eq:DSF_0K']}). Applying Lorentzian broadening gives the final DSF results.