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arXiv:1302.5582 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

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

Ashley Milsted, Jutho Haegeman, Tobias J. Osborne

Published Oct 24, 2013• 87 citations
arXivHTMLPDF

Abstract

We study the second-order quantum phase transition of massive real scalar field theory with a quartic interaction in ($1+1$) dimensions on an infinite spatial lattice using matrix product states. We introduce and apply a naive variational conjugate gradient method, based on the time-dependent variational principle 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 density matrix normalization group 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 noncritical quantum field theories under certain conditions.

DOI
10.1103/physrevd.88.085030

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We study the second-order quantum phase transition of massive real scalar field theory with a quartic interaction in (1+11+11+1) dimensions on an infinite spatial lattice using matrix product states. We introduce and apply a naive variational conjugate gradient method, based on the time-dependent variational principle 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 density matrix normalization group 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 noncritical quantum field theories under certain conditions.
We study critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. The critical point, exponents, and central charge are determined by optimizing them to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling Ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ with mass parameter ${\ensuremath{\mu}}^{2}$ and lattice spacing aaa, we demonstrate a double data collapse for the correlation length $\ensuremath{\delta}\ensuremath{\xi}(\ensuremath{\mu},\ensuremath{\lambda},D)=\stackrel{\texttildelow{}}{\ensuremath{\xi}}((\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}_{c})(\ensuremath{\delta}/a{)}^{\ensuremath{-}1/\ensuremath{\nu}})$ with DDD the bond dimension, $\ensuremath{\delta}$ the gap between eigenvalues of the transfer matrix, and ${\ensuremath{\alpha}}_{c}={\ensuremath{\mu}}_{R}^{2}/\ensuremath{\lambda}$ the parameter which fixes the critical quantum field theory.
We investigate the critical behavior and real-time scattering dynamics of the interacting 𝜙4 quantum field theory in (1+1) dimensions using uniform matrix product states (uMPS) and the time-dependent variational principle (TDVP). A finite-entanglement scaling analysis at 𝜆=0.8 bounds the critical mass-squared to 𝜇2𝑐∈]−0.2595,−0.2594[ and provides a quantitative map of the symmetric, near-critical, and spontaneously broken regimes. Using these ground states as asymptotic vacua, we simulate two-particle collisions in a sandwich geometry and extract the elastic scattering probability 𝑃11→11⁡(𝐸) and Wigner time delay Δ⁢𝑡⁡(𝐸) using a sandwich geometry protocol. We find strongly inelastic scattering in the symmetric phase (𝑃11→11≃0.712, Δ⁢𝑡≃−158 for 𝜇2=+0.2) and almost perfectly elastic collisions in the spontaneously broken phase (𝑃11→11≃1, Δ⁢𝑡≃−108 for 𝜇2=−0.1 and 𝑃11→11≃1, Δ⁢𝑡≃−177.781 for 𝜇2=−0.5). Crucially, the scattering protocol exhibits a distinctive divergence near the critical coupling; we show that this behavior serves as a dynamical signature of the quantum critical point, arising directly from the closing of the mass gap. These results demonstrate that TDVP-based uMPS can effectively probe nonperturbative scattering and critical dynamics in lattice field theories with controlled entanglement truncation.
We investigate the critical behavior and real-time scattering dynamics of the interacting φ4φ^4φ4 quantum field theory in (1+1)-dimensions using uniform matrix product states (uMPS) and the time-dependent variational principle (TDVP). A finite-entanglement scaling analysis at λ=0.8λ= 0.8λ=0.8 bounds the critical mass-squared to μc2∈]−0.2595,−0.2594[μ_c^2 \in ]-0.2595,-0.2594[μc2​∈]−0.2595,−0.2594[ and provides a quantitative map of the symmetric, near-critical, and spontaneously broken regimes. Using these ground states as asymptotic vacua, we simulate two-particle collisions in a sandwich geometry and extract the elastic scattering probability P11→11(E)P_{11\to 11}(E)P11→11​(E) and Wigner time delay Δt(E)Δt(E)Δt(E) using a sandwich geometry protocol. We find strongly inelastic scattering in the symmetric phase (P11→11≃0.712P_{11\to 11} \simeq 0.712P11→11​≃0.712, Δt≃−158Δt \simeq -158Δt≃−158 for μ2=+0.2μ^2 = +0.2μ2=+0.2) and almost perfectly elastic collisions in the spontaneously broken phase (P11→11≃1P_{11\to 11} \simeq 1P11→11​≃1, Δt≃−108Δt \simeq -108Δt≃−108 for μ2=−0.1μ^2=-0.1μ2=−0.1 and P11→11≃1P_{11\to 11} \simeq 1P11→11​≃1, Δt≃−177.781Δt \simeq -177.781Δt≃−177.781 for μ2=−0.5μ^2=-0.5μ2=−0.5). Crucially, the scattering protocol exhibits a distinctive divergence near the critical coupling; we show that this behavior serves as a dynamical signature of the quantum critical point, arising directly from the closing of the mass gap. These results demonstrate that TDVP-based uMPS can effectively probe nonperturbative scattering and critical dynamics in lattice field theories with controlled entanglement truncation.
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We demonstrate an efficient method that allows for simultaneous determination of the ground state, low energy excitation properties and excitation gap in quantum many body systems. To this aim we first use the \textit{ab-initio} optimization principle of tensor networks (TN), to show that the infinite density matrix renormalization group (iDMRG) in the real space is associated in a natural manner to the infinite time-evolving block decimation (iTEBD) implemented on a continuous matrix product state (MPS), and defined in imaginary time. We illustrate this association showing that the (imaginary) time matrix product state (MPS) in iTEBD reproduces accurately the properties of the two-dimensional (2D) classical Ising model, verifying in this way that the time MPS corresponds to a well-defined physical state. We apply then our scheme to the one-dimensional (1D) quantum Ising chain, where the time MPS is defined in continuous imaginary time. It is found that the time MPS at or close to the critical point is always maximally entangled, but gives essentially different correlations than those obtained from iDMRG; we argue that these correlations describe the dynamic correlations for the ground state.
The thesis consists of five major parts or chapters, the first of which is an introduction and review relating the field of investigation to problems in quantum field theory and statistical mechanics.Although the thesis generally deals with eigenvalue bounding techniques, these methods are applied with a view to calculating the critical properties of infinite systems [7].The approach advocated in the thesis intends the use of approximate methods to calculate the thermodynamics of large, but finite lattices.Finite size scaling [3] can then be used to estimate the bulk system thermodynamics, especially critical phenomena, from the approximate large lattice data.The approximations used include projected mean fields [5], for both ground and excited states, correlated product wave functions [6], which are extended to kink states, and blocking approximations [2].Part of this work is motivated by the promise these wave functions have in their application to variationally guided Monte Carlo methods for finite lattices [4], Also, it is shown that the critical indices predicted by the finite size rescaling formalism, are left invariant by the application of an approximate (blocking) renormalisation group to the finite lattices involved.Moreover, this invariance of the finite size rescaling equations is used to define a sequence transformation suitable for improving