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arXiv:2501.09797 · Journal of High Energy Physics

A relativistic continuous matrix product state study of field theories with defects

Karan Tiwana, Edoardo Lauria, Antoine Tilloy

Published May 13, 2025• 1 citation
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Abstract

Abstract We propose a method to compute expectation values in 1+1-dimensional massive Quantum Field Theories (QFTs) with line defects using Relativistic Continuous Matrix Product State (RCMPS). Exploiting Euclidean invariance, we use a quantization scheme where (imaginary) time runs perpendicularly to the defect. With this choice, correlation functions of local operators in the presence of the defect can be computed as expectation values of extended operators in the no-defect vacuum, which can be approximated by a homogeneous RCMPS. We demonstrate the effectiveness of this machinery by computing correlation functions of local bulk and defect operators in ϕ 4 theory with a magnetic line defect, in perturbative, strong coupling, critical, and symmetry-broken regimes.

DOI
10.1007/jhep05(2025)097

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Abstract We propose a method to compute expectation values in 1+1-dimensional massive Quantum Field Theories (QFTs) with line defects using Relativistic Continuous Matrix Product State (RCMPS). Exploiting Euclidean invariance, we use a quantization scheme where (imaginary) time runs perpendicularly to the defect. With this choice, correlation functions of local operators in the presence of the defect can be computed as expectation values of extended operators in the no-defect vacuum, which can be approximated by a homogeneous RCMPS. We demonstrate the effectiveness of this machinery by computing correlation functions of local bulk and defect operators in ϕ 4 theory with a magnetic line defect, in perturbative, strong coupling, critical, and symmetry-broken regimes.
Strongly coupled quantum field theories in (1+1)(1+1)(1+1) dimensions are notoriously hard to solve non-perturbatively. Variational methods, despite their success for quantum many-body physics on the lattice, have long lacked a natural ansatz adapted to the relativistic setting. This monograph explains the intuition behind relativistic continuous matrix product states (RCMPS), a variational ansatz tailored to (1+1)(1+1)(1+1)-dimensional QFT, and reports on several years of progress in developing and applying this approach. Using Riemannian optimization on the manifold of RCMPS, we obtain competitive non-perturbative approximations to ground state energies and local observables in the φ4φ^4φ4, Sine-Gordon, and Sinh-Gordon models, including in strongly coupled regimes where perturbation theory fails. We then describe extensions to models with several interacting fields. Beyond energy density and local observables, we show how the framework can be used to evaluate non-local observables (defects) and, through an original linear programming approach, to extract spectral data such as particle masses. We close by discussing the current limitations of the method and the most promising directions for future work.
Relativistic continuous matrix product states (RCMPS) are a powerful variational ansatz for quantum field theories of a single field. However, they inherit a property of their non-relativistic counterpart that makes them divergent for models with multiple fields, unless a regularity condition is satisfied. This has so far restricted the use of RCMPS to toy models with a single self-interacting field. We address this long standing problem by introducing a Riemannian optimization framework, that allows to minimize the energy density over the regular submanifold of multi-field RCMPS, and thus to retain purely variational results. We demonstrate its power on a model of two interacting scalar fields in 1+11+11+1 dimensions. The method captures distinct symmetry-breaking phases, and the signature of a Berezinskii-Kosterlitz-Thouless (BKT) transition along an O(2)O(2)O(2)-symmetric parameter line. This makes RCMPS usable for a far larger class of problems than before.
The Relativistic Continuous Matrix Product State (RCMPS) ansatz introduced in arXiv:2102.07741 is a translation-invariant variational ansatz intended to approximate the ground states of gapped one-dimensional relativistic Quantum Field Theories directly in the continuum limit and in infinite volume. In this work, we develop an ansatz, referred to as the Relativistic Quasiparticle (RQP) ansatz, for the low-lying one-particle excitations of the RCMPS vacuum, as well as for solitons interpolating between two distinct degenerate vacua. The corresponding energies of these states can be obtained as the eigenvalues of an effective Hamiltonian. The RQP ansatz generalizes constructions used in the Matrix Product State and Continuous Matrix Product State literature. We apply the RQP ansatz to the sine-Gordon model and find good agreement with results from integrability for both the soliton and breather masses. We also apply the RQP ansatz to the ϕ4ϕ^4ϕ4 theory, in both its Z2\mathbb Z_2Z2​-preserving and Z2\mathbb{Z}_2Z2​-broken phases, and determine the spectrum of particles and solitons. This enables us to determine the parameter regime for which multiple stable particles exist.
We extend the recently introduced continuous matrix product state (cMPS) variational class to the setting of (1 + 1)-dimensional relativistic quantum field theories.This allows one to overcome the difficulties highlighted by Feynman concerning the variational procedure applied to relativistic theories, and provides a new way to regularize quantum field theories.A fermionic version of the continuous matrix product state is introduced which is manifestly free from fermion doubling and sign problems.We illustrate the power of the formalism with the simulation of free massive Dirac fermions, the Gross-Neveu model, and the Casimir effect.We find that cMPS can capture chiral symmetry breaking with absolute scaling of the chiral parameter, and that boundary effects can be accommodated with modest computational effort.The many faces of QCD November 1-5,
Théories quantiques des champs utilisant les RCMPS L’étude non perturbative des théories quantiques des champs (QFT) relativistes constitue un défi majeur de la physique théorique. Les approches conventionnelles se heurtent souvent à la complexité numérique, au problème de signe ou aux limitations inhérentes à la discrétisation sur réseau. Cette thèse développe et applique un cadre variationnel fondé sur les Relativistic Continuous Matrix Product States (RCMPS) pour la simulation des QFT en 1 + 1 dimensions, opérant directement dans le continuum et à la limite thermodynamique.L’ansatz RCMPS est conçu pour satisfaire aux critères de Feynman pour une simulation viable des QFT. Il intègre le comportement à courte distance requis en incorporant la loi d’échelle universelle de l’intrication ultraviolette (divergente) des théories libres en 1 + 1 dimensions. Ce cadre garantit l'extensivité du système et permet une évaluation efficace des observables. Nous détaillons le formalisme numérique de l’ansatz : les valeurs moyennes des opérateurs locaux y sont ramenées à une hiérarchie d’équations différentielles ordinaires (EDO), tandis que les états fondamentaux sont déterminés par optimisation riemannienne sur la variété RCMPS.La pertinence de la méthode est d’abord éprouvée sur la théorie φ⁴ à une composante, pour laquelle nous obtenons des résultats de référence concernant l’énergie de l’état fondamental. Nous étendons ensuite le formalisme aux théories à plusieurs composantes en développant une nouvelle méthode de projection de gradient. Celleci permet de traiter les contraintes géométriques imposées par les conditions de régularité multiespèces, nous permettant ainsi d'explorer le diagramme de phases d’un modèle scalaire couplé présentant une symétrie D4.Ce cadre est ensuite généralisé aux systèmes fermioniques (fRCMPS). Nous formulons le modèle de Schwinger massif et identifions les conditions de régularité UV et IR cruciales pour assurer un problème variationnel bien posé. Enfin, nous démontrons la polyvalence des RCMPS en étendant le calcul à une classe d’observables non locales, ouvrant la voie à l’étude des défauts de ligne dans le modèle φ⁴. Ce travail établit le RCMPS comme un outil systématiquement améliorable et numériquement performant pour une vaste classe de QFT en 1 + 1 dimensions.
I study the Sine-Gordon (SG) and Sinh-Gordon (ShG) quantum field theories with a recently introduced variational method, the relativistic continuous matrix product states (RCMPS). The main advantage is to work directly in the thermodynamic limit, and without any UV regulator. The SG model is well understood and integrable, which provides a convenient benchmark for the variational method and serves as a warm-up. RCMPS approximate the ground state of the SG model arbitrary well up to the free Fermion point [coupling β=4πβ=\sqrt{4π}β=4π​ in equal-time quantization convention, or b=1/2b=1/\sqrt{2}b=1/2​ in CFT convention], where the ground energy collapses to −∞-\infty−∞, and some renormalized ansatz would be needed. The ShG model, while integrable, is less understood and its strong coupling regime β≈1β\approx 1β≈1 is subject to some controversy. RCMPS also fit the ground state of the ShG model up to approximately b=1/2b=1/\sqrt{2}b=1/2​, after which their predictions start to deviate substantially from the "exact" results. This is more puzzling as nothing is expected to happen physically for the ShG model at that point (eg, the ground energy density does not diverge). Either the "exact" ShG results are not exact (the analytic continuation of the SG Bethe Ansatz solution is unwarranted), or, more likely, the physical structure of the ShG ground state changes in such a way that it becomes out of reach of the RCMPS manifold for reasonable bond dimensions.