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arXiv:2502.14175 · Physical review. D/Physical review. D.

Monomer-dimer tensor-network basis for qubit-regularized lattice gauge theories

Shailesh Chandrasekharan A, Rui Xian Siew, Tanmoy Bhattacharya

Published May 12, 2025• 4 citations
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Abstract

Traditional SU ( N ) lattice gauge theories (LGTs) can be formulated using an orthonormal basis constructed from the irreducible representations (irreps) V λ of the SU ( N ) gauge symmetry. On a lattice, the elements of this basis are tensor networks comprising dimer tensors on the links labeled by a set of irreps { λ ℓ } and monomer tensors on sites labeled by { λ s } . These tensors naturally define a local site Hilbert space, H s g , on which gauge transformations act. Gauss’s law introduces an additional index α s = 1 , 2 , … , D ( H s g ) that labels an orthonormal basis of the gauge-invariant subspace of H s g . This monomer-dimer tensor-network (MDTN) basis, | { λ s } , { λ ℓ } , { α s } ⟩ , of the physical Hilbert space enables the construction of new qubit-regularized SU ( N ) gauge theories that are free of sign problems while preserving key features of traditional LGTs. Here, we investigate finite-temperature confinement-deconfinement transitions in a simple qubit-regularized SU(2) and SU(3) gauge theory in d = 2 and d = 3 spatial dimensions, formulated using the MDTN basis, and show that they reproduce the universal results of traditional LGTs at these transitions. Additionally, in d = 1 , we demonstrate using a plaquette chain that the string tension at zero temperature can be continuously tuned to zero by adjusting a model parameter that plays the role of the gauge coupling in traditional LGTs.

DOI
10.1103/gns9-l3gk

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Traditional SU ( N ) lattice gauge theories (LGTs) can be formulated using an orthonormal basis constructed from the irreducible representations (irreps) V λ of the SU ( N ) gauge symmetry. On a lattice, the elements of this basis are tensor networks comprising dimer tensors on the links labeled by a set of irreps { λ ℓ } and monomer tensors on sites labeled by { λ s } . These tensors naturally define a local site Hilbert space, H s g , on which gauge transformations act. Gauss’s law introduces an additional index α s = 1 , 2 , … , D ( H s g ) that labels an orthonormal basis of the gauge-invariant subspace of H s g . This monomer-dimer tensor-network (MDTN) basis, | { λ s } , { λ ℓ } , { α s } ⟩ , of the physical Hilbert space enables the construction of new qubit-regularized SU ( N ) gauge theories that are free of sign problems while preserving key features of traditional LGTs. Here, we investigate finite-temperature confinement-deconfinement transitions in a simple qubit-regularized SU(2) and SU(3) gauge theory in d = 2 and d = 3 spatial dimensions, formulated using the MDTN basis, and show that they reproduce the universal results of traditional LGTs at these transitions. Additionally, in d = 1 , we demonstrate using a plaquette chain that the string tension at zero temperature can be continuously tuned to zero by adjusting a model parameter that plays the role of the gauge coupling in traditional LGTs.
We construct simple qubit-regularized Hamiltonian lattice gauge theories formulated in the monomer--dimer--tensor-network (MDTN) basis that are free of sign problems in the pure gauge sector. These models naturally realize both confined and deconfined phases. Using classical Monte Carlo methods, we investigate the associated finite-temperature phase transitions and show that they exhibit the expected universality classes of conventional SU(N) lattice gauge theories in various spacetime dimensions. Furthermore, we argue that second-order quantum phase transitions separating the confined and deconfined phases are likely to exist. Such critical points would provide a nonperturbative route to defining continuum limits of qubit-regularized gauge theories, potentially allowing Yang--Mills theory and related continuum gauge theories to emerge from finite-dimensional lattice constructions.
We construct simple qubit-regularized Hamiltonian lattice gauge theories formulated in the monomer--dimer--tensor-network (MDTN) basis that are free of sign problems in the pure gauge sector. These models naturally realize both confined and deconfined phases. Using classical Monte Carlo methods, we investigate the associated finite-temperature phase transitions and show that they exhibit the expected universality classes of conventional SU(N) lattice gauge theories in various spacetime dimensions. Furthermore, we argue that second-order quantum phase transitions separating the confined and deconfined phases are likely to exist. Such critical points would provide a nonperturbative route to defining continuum limits of qubit-regularized gauge theories, potentially allowing Yang--Mills theory and related continuum gauge theories to emerge from finite-dimensional lattice constructions.
To study quantum field theories on a quantum computer, we must begin with Hamiltonians defined on a finite-dimensional Hilbert space and then take appropriate limits. This approach can be seen as a new type of regularization for quantum field theories, which we refer to as qubit regularization. A related finite-dimensional regularization, known as the D-theory approach, was proposed long ago as a general framework for all quantum field theories. In this framework, the dimensionality of the local Hilbert space at each spatial point can increase as needed through an additional flavor index. To reproduce asymptotically free QFTs, most studies assume that qubit-regularized theories require extending the local Hilbert space to infinity. However, contrary to this common belief, recent discoveries in (1+1) dimensions have revealed two examples where asymptotic freedom appears to emerge within a strictly finite-dimensional local Hilbert space through a novel renormalization group (RG) flow. These findings motivate further investigation into whether asymptotically free gauge theories could also emerge within a strictly finite-dimensional local Hilbert space. To support these explorations, we propose an orthonormal basis called the monomer-dimer-tensor-network (MDTN) basis and use it to construct new types of qubit-regularized lattice gauge theories.
The development of quantum technology and the advent of quantum computation have renewed interest in Hamiltonian formulations of lattice gauge theories. In contrast to Euclidean Monte Carlo calculations, Hamiltonian simulation requires a direct representation of the physical Hilbert space, an efficient treatment of enforcing Gauss' law as constraints, and a formulation of real-time dynamics in terms of degrees of freedom that can be encoded on quantum or quantum-inspired classical platforms. The starting point corresponds to finding a suitable basis for Hamiltonian lattice gauge theories. The existing literature focuses on several complementary approaches: electric basis, magnetic basis, gauge-fixed, quantum-link, qubit-regularised, finite-group, and orbifold formulations. This article primarily focuses on the prepotential and loop--string--hadron formulations, where non-Abelian Gauss laws are solved locally and the dynamics is written directly in terms of gauge-invariant loop, string and hadron excitations. The discussion covers the SU(2) and SU(3) constructions in one spatial dimension, the role of residual Abelian Gauss laws, and the extension to higher dimensions through point splitting. We conclude by highlighting its direct applications: eliminating redundant gauge degrees of freedom makes exact diagonalisation and tensor network calculations feasible for studying real-time dynamics, provides insights into thermalisation properties, enables mapping of physical degrees of freedom onto analog and digital quantum hardware, and enables quantum simulation of dynamics.
A bstract Tensor networks offer a sign-problem-free approach to study lattice gauge theories, but extracting precise universal information associated with the deconfinement transition remains challenging. In this work, we study the deconfinement transition of (2+1)-dimensional ℤ N lattice gauge theories at finite temperature using a thermal tensor network approach, where the partition functions at finite temperature are formulated as three-dimensional tensor networks. These tensor networks are first contracted in the temporal direction, and the subsequent coarse-graining in the spatial directions yields a renormalized transfer matrix, the spectrum of which directly encodes the universal conformal field theory data. In particular, by numerically extracting the central charge and scaling dimensions, we verify that the universality class of the thermal deconfinement transition matches the prediction of the Svetitsky-Yaffe conjecture for N = 2, 3, 5. Moreover, we show that the ℤ 5 theory at finite temperature exhibits an intermediate phase with an emergent U(1) symmetry. Critical couplings are determined via Gu-Wen ratios and agree with existing Monte Carlo simulations. Finally, extrapolating these critical couplings at finite temperature enables us to determine the deconfinement transition points for N = 2, 3 at zero temperature.
We present a unified framework to describe lattice gauge theories by means of tensor networks: this framework is efficient as it exploits the high local symmetry content native to these systems by describing only the gauge invariant subspace. Compared to a standard tensor network description, the gauge invariant model allows one to increase real and imaginary time evolution up to a factor that is square of the dimension of the link variable. The gauge invariant tensor network description is based on the quantum link formulation, a compact and intuitive formulation for gauge theories on the lattice, which is alternative to and can be combined with the global symmetric tensor network description. We present some paradigmatic examples that show how this architecture might be used to describe the physics of condensed matter and high-energy physics systems. Finally, we present a cellular automata analysis which estimates the gauge invariant Hilbert space dimension as a function of the number of lattice sites that might guide the search for effective simplified models of complex theories.