Matrix product states and variational methods applied to critical quantum field theory

TL;DR

This work demonstrates that matrix product state methods, particularly uniform MPS with TDVP and a variational conjugate-gradient approach, can faithfully capture the critical behavior of (1+1)-dimensional φ^4 theory on an infinite lattice. By combining ground-state searches, excitation ansatzes, and finite-entanglement scaling, the authors extract the continuum critical parameter, critical exponents (β=1/8, ν=1), and the central charge ($c\approx 0.5$) consistent with the transverse Ising universality class. They also reveal practical baselines via mean-field theory and analyze spectral densities, showing the method’s ability to access dispersion relations and kink excitations directly in the thermodynamic limit. Overall, the results establish MPS techniques as a competitive, scalable toolkit for critical quantum field theories, complementing Monte Carlo approaches and enabling direct access to spectral and conformal data. The study highlights the importance of basis choices and entanglement control (via bond dimension $D$ and cut-offs) for accurate continuum extrapolations.

Abstract

We study the second-order quantum phase-transition of massive real scalar field theory with a quartic interaction ($φ^4$ theory) in (1+1) dimensions on an infinite spatial lattice using matrix product states (MPS). We introduce and apply a naive variational conjugate gradient method, based on the time-dependent variational principle (TDVP) for imaginary time, to obtain approximate ground states, using a related ansatz for excitations to calculate the particle and soliton masses and to obtain the spectral density. We also estimate the central charge using finite-entanglement scaling. Our value for the critical parameter agrees well with recent Monte Carlo results, improving on an earlier study which used the related DMRG method, verifying that these techniques are well-suited to studying critical field systems. We also obtain critical exponents that agree, as expected, with those of the transverse Ising model. Additionally, we treat the special case of uniform product states (mean field theory) separately, showing that they may be used to investigate non-critical quantum field theories under certain conditions.

Figures (22)

• Figure 1: Feynman diagram of the one-loop correction to the free particle propagator in $\phi^4$-theory.
• Figure 2: The classical effective potential in $\phi^4$-theory illustrated for $\mu_0^2 \ge 0$ (red) and $\mu_0^2 < 0$ (blue dashed) showing the two possible ground states for the latter case.
• Figure 3:
• Figure 4: Convergence of the field expectation value $\braket{\phi}$ with CPU time for the conjugate gradient (CG) method versus imaginary-time evolution via Euler integration of the TDVP flow equations and gradient descent (GD | stepping along the gradient as with the TDVP, but using a line-search to determine the size of each step by minimizing the energy). The model is $\phi^4$-theory (as defined in section \ref{['sec:qft_phi4_lattice']}) with parameters $\tilde{\lambda} = 0.2$ and $\tilde{\lambda} / \tilde{\mu}_R^2 = 69$. The bond-dimension is $D = 64$ and the stopping criterion is $\eta < 10^{-6}$. The line ("CG final") indicates the final value taken from the CG curve. We use the same line-search algorithm for both the CG and GD methods. The discontinuities in the GD curve are large jumps that could occasionally be made in a particular direction.
• Figure 5: Illustration of the field shift needed to center fluctuations about zero.