It is well known that all physically relevant states of gauge theories lie in the sectors of the Hilbert space which satisfy the Gauss law. On the lattice, the manifeslty gauge invariant subspace is known to be exactly spanned by gauged tensor networks. In this work, we demonstrate that the continuum limit of certain types of gauged tensor networks is well defined and leads to a new class of states that may be helpful for the non-perturbative study of gauge theories directly in the continuum.
It is well known that all physically relevant states of gauge theories lie in the sectors of the Hilbert space which satisfy the Gauss law. On the lattice, the manifeslty gauge invariant subspace is known to be exactly spanned by gauged tensor networks. In this work, we demonstrate that the continuum limit of certain types of gauged tensor networks is well defined and leads to a new class of states that may be helpful for the non-perturbative study of gauge theories directly in the continuum.
The Lattice Gauge Theory Hilbert space is divided into gauge-invariant sectors selected by the background charges. Such a projector can be directly embedded in a tensor network ansatz for gauge-invariant states as originally discussed in [Phys. Rev. B 83, 115127 (2011)] and in [Phys. Rev. X 4, 041024 (2014)] in the context of PEPS. The original ansatz is based on sparse tensors, though parts of them are not explicitly symmetric, and thus their actual implementation in numerical simulations has been hindered by the complexity of developing ad hoc libraries. Here we provide a new PEPS tensor network formulation of gauge-invariant theories purely based on symmetric elementary tensors. The new formulation can be implemented in numerical simulation using available state-of-the-art tensor network libraries but also holds interest from a purely theoretical perspective since it requires embedding the original gauge theory with gauge symmetry G into an enlarged globally symmetric theory with symmetry GxG. By revisiting the original ansatz in the modern landscape of i) duality transformations between gauge and spin systems, ii) finite depth quantum circuits followed by measurements that allow generating topologically ordered states, and iii) Clifford enhanced tensor networks, we show that such a new formulation provides a novel duality transformation between lattice gauge theories and specific sectors of globally invariant systems.
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.
Abstract 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.
Lattice gauge theories (LGT) play a central role in modern physics, providing insights into high-energy physics, condensed matter physics, and quantum computation. Due to the nontrivial structure of the Hilbert space of LGT systems, entanglement in such systems is tricky to define. However, when one limits themselves to superselection-resolved entanglement, that is, entanglement corresponding to specific gauge symmetry sectors (commonly denoted as superselection sectors), this problem disappears, and the entanglement becomes well-defined. The study of superselection-resolved entanglement is interesting in LGT for an additional reason: when the gauge symmetry is strictly obeyed, superselection-resolved entanglement becomes the only distillable contribution to the entanglement. In our work, we study the behavior of superselection-resolved entanglement in LGT systems. We employ a tensor network construction for gauge-invariant systems as defined by Zohar and Burrello (2016) and find that, in a vast range of cases, the leading term in superselection-resolved entanglement depends on the number of corners in the partition, that is, corner-law entanglement. To our knowledge, this is the first case of such a corner-law being observed in any lattice system.
A bstract Lattice gauge theories (LGT) play a central role in modern physics, providing insights into high-energy physics, condensed matter physics, and quantum computation. Due to the nontrivial structure of the Hilbert space of LGT systems, entanglement in such systems is tricky to define. However, when one limits themselves to superselection-resolved entanglement, that is, entanglement corresponding to specific gauge symmetry sectors (commonly denoted as superselection sectors), this problem disappears, and the entanglement becomes well-defined. The study of superselection-resolved entanglement is interesting in LGT for an additional reason: when the gauge symmetry is strictly obeyed, superselection-resolved entanglement becomes the only distillable contribution to the entanglement. In our work, we study the behavior of superselection-resolved entanglement in LGT systems. We employ a tensor network construction for gauge-invariant systems as defined by Zohar and Burrello [1] and find that, in a vast range of cases, the leading term in superselection-resolved entanglement depends on the number of corners in the partition — corner-law entanglement. To our knowledge, this is the first case of such a corner-law being observed in any lattice system.
The solution of gauge theories is one of the most promising applications of quantum technologies. Here, we discuss the approach to the continuum limit for U(1) gauge theories regularized via finite-dimensional Hilbert spaces of quantum spin-S operators, known as quantum link models. For quantum electrodynamics (QED) in one spatial dimension, we numerically demonstrate the continuum limit by extrapolating the ground state energy, the scalar, and the vector meson masses to large spin lengths S, large volume N, and vanishing lattice spacing a. By analytically solving Gauss' law for arbitrary S, we obtain a generalized PXP spin model and count the physical Hilbert space dimension analytically. This allows us to quantify the required resources for reliable extrapolations to the continuum limit on quantum devices. We use a functional integral approach to relate the model with large values of half-integer spins to the physics at topological angle Θ=π. Our findings indicate that quantum devices will in the foreseeable future be able to quantitatively probe the QED regime with quantum link models.