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arXiv:2410.22411 · Physical review. B./Physical review. B

Zero-point energy of tensor fluctuations on the matrix product state manifold

Sebastian Leontica, Andrew G. Green

Published May 28, 2025• 0 citations
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Abstract

This work presents a method to quantize low energy excitations of systems whose ground state is best represented by a matrix product state (MPS) such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) model. The approach is similar to the spin-wave model of excitations in typical magnetic materials, but includes a more general set of fluctuations in entanglement structure. We argue that the quasilocal action of tensor fluctuations facilitates a description in terms of bosonic modes. We apply this approach to compute fluctuation corrections to the bilinear-biquadratic Heisenberg model, whose ground state we expect to be close to the exact bond dimension 2 AKLT state in a certain parameter range. Our results show significant improvements in energy approximations, highlighting both the qualitative and quantitative potential of this paradigm for studying complex entangled systems. This approach paves the way for new insights into the low-energy physics of correlated materials and the development of effective field theories beyond traditional semiclassical methods.

DOI
10.1103/f38t-ldgq

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This work presents a method to quantize low energy excitations of systems whose ground state is best represented by a matrix product state (MPS) such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) model. The approach is similar to the spin-wave model of excitations in typical magnetic materials, but includes a more general set of fluctuations in entanglement structure. We argue that the quasilocal action of tensor fluctuations facilitates a description in terms of bosonic modes. We apply this approach to compute fluctuation corrections to the bilinear-biquadratic Heisenberg model, whose ground state we expect to be close to the exact bond dimension 2 AKLT state in a certain parameter range. Our results show significant improvements in energy approximations, highlighting both the qualitative and quantitative potential of this paradigm for studying complex entangled systems. This approach paves the way for new insights into the low-energy physics of correlated materials and the development of effective field theories beyond traditional semiclassical methods.
An alternative method of writing down the path integral for spin-1 Heisenberg antiferromagnetic chain is introduced. In place of the conventional coherent-state basis that leads to the nonlinear $\ensuremath{\sigma}$ model, we use a basis called the fluctuating matrix product states which embodies intersite entanglement from the outset. It forms an overcomplete set spanning the entire Hilbert space of the spin-1 chain. Saddle-point analysis performed for the bilinear-biquadratic spin model predicts the Affleck-Kennedy-Lieb-Tasaki (AKLT) state as the ground state in the vicinity of the AKLT Hamiltonian. Quadratic effective action derived by gradient expansion around the saddle point is free from constraints that plagued the nonlinear $\ensuremath{\sigma}$ model and exactly solvable. The obtained excitation modes agree precisely with the single-mode approximation result for the AKLT Hamiltonian. Excitation spectra for other bilinear-biquadratic Hamiltonians are obtained as well by diagonalizing the quadratic action.
We propose a simple variational wave function that captures the correct ground-state energy of the spin-1 Heisenberg chain model to within 0.04%. The wave function is written in the matrix product state (MPS) form with the bond dimension D=8D=8D=8 and is characterized by three fugacity parameters. The proposed MPS generalizes the Affleck-Kennedy-Lieb-Tasaki state by dressing it with dimers, trimers, and general qqq-mers. The fugacity parameters control the number and the average size of the qqq-mers. Furthermore, the D=8D=8D=8 variational MPS state captures the ground states of the entire family of the bilinear-biquadratic Hamiltonian belonging to the Haldane phase to high accuracy. The 2-4-2 degeneracy structure in the entanglement spectrum of our MPS state is found to match well the results of the density matrix renormalization group (DMRG) calculation, which is computationally much heavier. Spin-spin correlation functions also find an excellent fit with those obtained by DMRG.
We exactly calculate the reduced density matrix of matrix product states (MPS). Our compact result enables one to perform analytic studies of entanglement in MPS. In particular, we consider the MPS ground states of two anisotropic spin chains. One is a q -deformed Affleck-Kennedy-Lieb-Tasaki (AKLT) model and the other is a general spin-1 quantum antiferromagnet with nearest-neighbor interactions. Our analysis shows how anisotropy affects entanglement on different continuous parameter manifolds. We also construct an effective boundary spin model that describes a block of spins in the ground state of the q -deformed AKLT Hamiltonian. The temperature of this effective model is given in terms of the deformation parameter q .
Matrix-product states (MPS) can be understood as an extension of product states able to capture area laws of entanglement in 1D. In this thesis, I investigate the representation power of the MPS manifold as a whole. I represent the state of quantum systems, not by the usual approximation as an MPS, but by constructing wavelets and distributions on the entire MPS space. First, I introduce the MPS-Husimi distribution, a classical representation of arbitrary (possibly volume law) quantum states on the over-complete basis of MPS. When states exceed the entanglement limit of the manifold, the additional entanglement is captured by emerging classical correlations between the tensors. By constructing a path integral on the manifold and analyzing the resulting bosonic action at quadratic order, I show that the linear growth of entanglement after a quench is governed by the classical Kolmogorov-Sinai rate of the projected dynamics. A direct treatment overestimates the number of independent degrees of freedom, an artifact that we resolve by properly accounting for the geometry of the manifold. This rigorously establishes the Lyapunov spectrum of time-dependent variational principle (TDVP) trajectories as a probe of entanglement growth and a possible diagnostic for quantum chaos. Second, I argue that tensor fluctuations are bosonic in nature when separated by distances larger than the correlation length. To account for the additional curvature when they are close together, I introduce the bosonic Gram operator. The resulting bosonic problem captures zero-point energies and dispersion of “tensor spin-waves” at quadratic order, improving ground-state energies for benchmark S = 1 models (e.g. the bilinear-biquadratic chain near the AKLT point) using low bond dimensions. Third, I construct a distribution over the MPS space that is uniform with respect to the Fubini-Study metric and show that its properties differ drastically from the typical approach of sampling the tensors independently. Together, these results (i) bridge classical chaos and quantum entanglement through flow equations on the MPS manifold, (ii) provide a systematic route to incorporate quantum fluctuations above mean-field tensor network reference points and (iii) introduce and investigate the maximum uncertainty distribution on MPS space. The thesis concludes with two ongoing directions: constructing a path integral over MPS and formulating a thermodynamic theory of area-law entanglement using the formalism developed here.
Using exact diagonalization and tensor network techniques, we compute the gap for the Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian in one and two spatial dimensions. Tensor network methods are used to extract physical properties directly in the thermodynamic limit, and we support these results using finite-size scalings from exact diagonalization. Studying the AKLT Hamiltonian perturbed by an external field, we show how to obtain an accurate value of the gap of the original AKLT Hamiltonian from the field value at which the ground state verifies e0<0{e}_{0}<0e0​<0, which is a quantum critical point. With the tensor network renormalization group methods we provide direct evidence of a finite gap in the thermodynamic limit for the AKLT models in the one-dimensional chain and two-dimensional hexagonal and square lattices. This method can be applied generally to Hamiltonians with rotational symmetry, and we also show results beyond the AKLT model.
Matrix Product States (MPS) and Tensor Networks provide a general framework for the construction of solvable models. The best-known example is the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, which is the ground state of a 2-body nearest-neighbor parent Hamiltonian. We show that such simple parent Hamiltonians for MPS models are, in fact, much more prevalent than hitherto known: The existence of a single example with a simple Hamiltonian for a given choice of dimensions already implies that any generic MPS with those dimensions possesses an equally simple Hamiltonian. We illustrate our finding by discussing a number of models with nearest-neighbor parent Hamiltonians, which generalize the AKLT model on various levels.