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arXiv:2508.16363 · Journal of High Energy Physics

Infinite matrix product states for (1 + 1)-dimensional gauge theories

Ross Dempsey, Anna-Maria E. Glück, Silviu S. Pufu, Benjamin T. Søgaard

Published Mar 19, 2026• 2 citations
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Abstract

A bstract We present a matrix product operator construction that allows us to represent the lattice Hamiltonians of (abelian or non-abelian) gauge theories in a local and manifestly translation-invariant form. In particular, we use symmetric matrix product states and introduce link-enhanced matrix product operators (LEMPOs) that can act on both the physical and virtual spaces of the matrix product states. This construction allows us to study Hamiltonian lattice gauge theories on infinite lattices. As examples, we show how to implement this method to study the massless and massive one-flavor Schwinger model and adjoint QCD 2 .

DOI
10.1007/jhep03(2026)181

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A bstract We present a matrix product operator construction that allows us to represent the lattice Hamiltonians of (abelian or non-abelian) gauge theories in a local and manifestly translation-invariant form. In particular, we use symmetric matrix product states and introduce link-enhanced matrix product operators (LEMPOs) that can act on both the physical and virtual spaces of the matrix product states. This construction allows us to study Hamiltonian lattice gauge theories on infinite lattices. As examples, we show how to implement this method to study the massless and massive one-flavor Schwinger model and adjoint QCD 2 .
We present a general method for simulating lattice gauge theories in low dimensions using infinite matrix product states (iMPS). A central challenge in Hamiltonian formulations of gauge theories is the unbounded local Hilbert space associated with gauge degrees of freedom. In one spatial dimension, Gauss's law permits these gauge fields to be integrated out, yielding an effective Hamiltonian with long-range interactions among matter fields. We construct efficient matrix product operator (MPO) representations of these Hamiltonians directly in the thermodynamic limit. Our formulation naturally includes background fields and θθθ-terms, requiring no modifications to the standard iDMRG algorithm. This provides a broadly applicable framework for 1+1D gauge theories and can be extended to quasi-two-dimensional geometries such as infinite cylinders, where tensor-network methods remain tractable. As a benchmark, we apply our construction to the Schwinger model, reproducing expected features including confinement, string breaking, and the critical behavior at finite mass. Because the method alters only the MPO structure, it can be incorporated with little effort into a wide range of iMPS and infinite-boundary-condition algorithms, opening the way to efficient studies of both equilibrium and non-equilibrium gauge dynamics.
Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems. The matrix product states (MPS) are one particular case of TNS and are used for the simulation of 1+1 dimensional systems. In [1] we considered the MPS formalism for the simulation of the Hamiltonian lattice gauge formulation of 1+1 dimensional one flavor quantum electrodynamics, also known as the massive Schwinger model. We deduced the ground state and lowest lying excitations. Furthermore, we performed a full quantum real-time simulation for a quench with a uniform background electric field. In this proceeding we continue our work on the Schwinger model. We demonstrate the advantage of working with gauge invariant MPS by comparing with MPS simulations on the full Hilbert space, that includes numerous non-physical gauge variant states. Furthermore, we compute the chiral condensate and recover the predicted UV-divergent behavior.
The matrix product state formalism is used to simulate Hamiltonian lattice gauge theories. To this end, we define matrix product state manifolds which are manifestly gauge invariant. As an application, we study (1+1)-dimensional one flavor quantum electrodynamics, also known as the massive Schwinger model, and are able to determine very accurately the ground-state properties and elementary one-particle excitations in the continuum limit. In particular, a novel particle excitation in the form of a heavy vector boson is uncovered, compatible with the strong coupling expansion in the continuum. We also study full quantum nonequilibrium dynamics by simulating the real-time evolution of the system induced by a quench in the form of a uniform background electric field.
Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems.The matrix product states (MPS) are one particular case of TNS and are used for the simulation of 1+1 dimensional systems.In [1] we considered the MPS formalism for the simulation of the Hamiltonian lattice gauge formulation of 1+1 dimensional one flavor quantum electrodynamics, also known as the massive Schwinger model.We deduced the ground state and lowest lying excitations.Furthermore, we performed a full quantum real-time simulation for a quench with a uniform background electric field.In this proceeding we continue our work on the Schwinger model.We demonstrate the advantage of working with gauge invariant MPS by comparing with MPS simulations on the full Hilbert space, that includes numerous non-physical gauge variant states.Furthermore, we compute the chiral condensate and recover the predicted UV-divergent behavior.
Matrix Product State (MPS) wavefunctions have many applications in quantum information and condensed matter physics. One application is to represent states in the thermodynamic limit directly, using a small set of position independent matrices. For this infinite MPS ansatz to be useful it is necessary to be able to calculate expectation values, and we show here that a large class of observables, including operators transforming under lattice translations as eigenstates of arbitrary momentum kkk, can be represented in the Schur form of a lower or upper triangular matrix and we present an algorithm for evaluating such expectation values in the asymptotic limit. The sum or the product of two such Schur operators is also a Schur operator, and is easily constructed to give a simple method of constructing arbitrary polynomial combinations of operators. Some simple examples are the variance ⟨(H−E)2⟩\langle (H-E)^2\rangle⟨(H−E)2⟩ of an infinite MPS, which gives a simple method of evaluating the accuracy of a numerical approximation to a eigenstate, or a vertex operator ⟨ck1†ck2†ck4ck3⟩\langle c^\dagger_{k_1} c^\dagger_{k_2} c_{k_4} c_{k_3}\rangle⟨ck1​†​ck2​†​ck4​​ck3​​⟩. This approach is a step towards improved algorithms for the calculation of dynamical properties and excited states.
The term Tensor Network States (TNS) refers to a number of families of states that represent different ansätze for the efficient description of the state of a quantum many-body system.Matrix Product States (MPS) are one particular case of TNS, and have become the most precise tool for the numerical study of one dimensional quantum many-body systems, as the basis of the Density Matrix Renormalization Group method.Lattice Gauge Theories (LGT), in their Hamiltonian version, offer a challenging scenario for these techniques.While the dimensions and sizes of the systems amenable to TNS studies are still far from those achievable by 4-dimensional LGT tools, Tensor Networks can be readily used for problems which more standard techniques, such as Markov chain Monte Carlo simulations, cannot easily tackle.Examples of such problems are the presence of a chemical potential or out-of-equilibrium dynamics.We have explored the performance of Matrix Product States in the case of the Schwinger model, as a widely used testbench for lattice techniques.Using finite-size, open boundary MPS, we are able to determine the low energy states of the model in a fully non-perturbative manner.The precision achieved by the method allows for accurate finite size and continuum limit extrapolations of the ground state energy, but also of the chiral condensate and the mass gaps, thus showing the feasibility of these techniques for gauge theory problems.