We exploit the spin network properties of the magnetic eigenstates of SU(2) Hamiltonian lattice gauge theory and use the Wilson loop operators to obtain a wide class of new identities amongst 3nj Wigner coefficients. We also show that the topological ground states of the SU(2) toric code Hamiltonian lead to Wigner 3nj identities with non-trivial phases. The method is very general and involves only the eigenvalue equations of any gauge invariant operator and their solutions. Therefore, it can be extended to any higher dimensional spin networks as well as larger SU(N) groups.
We exploit the spin network properties of the magnetic eigenstates of SU(2) Hamiltonian lattice gauge theory and use the Wilson loop operators to obtain a wide class of new identities amongst 3nj Wigner coefficients. We also show that the topological ground states of the SU(2) toric code Hamiltonian lead to Wigner 3nj identities with non-trivial phases. The method is very general and involves only the eigenvalue equations of any gauge invariant operator and their solutions. Therefore, it can be extended to any higher dimensional spin networks as well as larger SU(N) groups.
In this work, exact solutions are obtained for a class of generalized gauge-invariant n-chain Ising models (n=1,2,3,4) with arbitrary multi-spin interactions that are invariant under the local Z2 gauge group. On a strip lattice of finite length L and width n with periodic or free boundary conditions, an explicit expression for the partition function is derived using the transfer-matrix method. Two successive transformations are developed: elimination of gauge redundancy and reduction of the original model to an effective n-chain Ising model with all possible interactions between neighboring vertical layers. On the basis of the spectral decomposition of the 2n×2n transfer matrix, general formulas are obtained for gauge-invariant correlation functions and Wilson loops of arbitrary width. For n≤3, explicit expressions are derived in terms of eigenvalues and eigenvectors. A detailed analysis of the behavior of the Wilson loop is performed, which allows us to identify regimes exhibiting area-law (confinement-like) and perimeter-law (deconfinement-like) dependence. For specific Hamiltonians, the string tension is computed and the corresponding phase diagrams are constructed. The results generalize and substantially extend the classical works on the gauge-invariant Ising model.
We construct canonical transformations to reformulate SU(N) Kogut-Susskind lattice gauge theory in terms of a set of fundamental loop & string flux operators along with their canonically conjugate loop & string electric fields. The canonical relations between the initial SU(N) link operators and the final SU(N) loop & string operators, consistent with SU(N) gauge transformations, are explicitly constructed over the entire lattice. We show that as a consequence of SU(N) Gauss laws all SU(N) string degrees of freedom become cyclic and decouple from the physical Hilbert space H. The Kogut-Susskind Hamiltonian rewritten in terms of the fundamental physical loop operators has global SU(N) invariance. There are no gauge fields. We further show that the (1/g) magnetic field terms on plaquettes create and annihilate the fundamental plaquette loop fluxes while the (g) electric field terms describe all their interactions. In the weak coupling (g → 0) continuum limit the SU(N) loop dynamics is described by SU(N) spin Hamiltonian with nearest neighbour interactions. In the simplest SU(2) case, where the canonical transformations map the SU(2) loop Hilbert space into the Hilbert spaces of hydrogen atoms, we analyze the special role of the hydrogen atom dynamical symmetry group SO(4, 2) in the loop dynamics and the spectrum. A simple tensor network ansatz in the SU(2) gauge invariant hydrogen atom loop basis is discussed.
We present a neural network wave function framework for solving non-Abelian lattice gauge theories in a continuous group representation. Using a combination of SU(2) equivariant neural networks alongside an SU(2) invariant, physics-inspired ansatz, we learn a parametrization of the ground state wave function of SU(2) lattice gauge theory in 2+1 and 3+1 dimensions. Our method, performed in the Hamiltonian formulation, has a straightforward generalization to SU(N). We benchmark our approach against a solely invariant ansatz by computing the ground state energy, demonstrating the need for bespoke gauge equivariant transformations. We evaluate the Creutz ratio and average Wilson loop, and obtain results in strong agreement with perturbative expansions. Our method opens up an avenue for studying lattice gauge theories beyond one dimension, with efficient scaling to larger systems, and in a way that avoids both the sign problem and any discretization of the gauge group.
Tensor network states have been a very prominent tool for the study of quantum many-body physics, thanks to their physically relevant entanglement properties and their ability to encode symmetries. In the last few years, the formalism has been extended and applied to theories with local symmetries too---lattice gauge theories. In order to extract physical properties (such as expectation values and correlation functions of physical observables) out of such states, one has to use the so-called transfer operators, the local properties of which dictate the long-range behavior of the state. In this paper we study transfer operators of tensor network states (in particular, projected entangled pair states) of lattice gauge theories, and consider the implications of the local symmetry on their structure and properties. In particular, we study the implications on the computation of the Wilson loop---a nonlocal, gauge-invariant observable which is central to pure gauge theories, the long-range decay behavior of which probes the confinement or deconfinement of static charges. Using the symmetry, we show how to simplify the tensor contraction required for computing Wilson loop expectation values for such states, eliminate nonphysical parts of the tensors, and formulate conditions relating local properties (that is, of the tensors) to their decay fashion.
Tensor network methods are powerful and efficient tools for studying the properties and dynamics of statistical and quantum systems, in particular in one and two dimensions. In recent years, these methods have been applied to lattice gauge theories, yet these theories remain a challenge in ( 2 + 1 ) dimensions. In this article, we present a new (decorated) tensor network algorithm, in which the tensors encode the lattice gauge amplitude expressed in the fusion basis. This has several advantages—firstly, the fusion basis does diagonalize operators measuring the magnetic fluxes and electric charges associated to a hierarchical set of regions. The algorithm allows therefore a direct access to these observables. Secondly the fusion basis is, as opposed to the previously employed spin network basis, stable under coarse-graining. Thirdly, due to the hierarchical structure of the fusion basis, the algorithm does implement predefined disentanglers. We apply this new algorithm to lattice gauge theories defined for the quantum group SU ( 2 ) k and identify a weak and a strong coupling phase for various levels k . As we increase the level k , the critical coupling g c decreases linearly, suggesting the absence of a deconfining phase for the continuous group SU ( 2 ) . Moreover, we illustrate the scaling behaviour of the Wilson loops in the two phases.
We expand a set of notions recently introduced providing the general setting for a universal representation of the quantum structure on which quantum information stands. The dynamical evolution process associated with generic quantum information manipulation is based on the (re)coupling theory of SU(2) angular momenta. Such scheme automatically incorporates all the essential features that make quantum information encoding much more efficient than classical: it is fully discrete; it deals with inherently entangled states, naturally endowed with a tensor product structure; it allows for generic encoding patterns. The model proposed can be thought of as the non-Boolean generalization of the quantum circuit model, with unitary gates expressed in terms of 3nj coefficients connecting inequivalent binary coupling schemes of n+1 angular momentum variables, as well as Wigner rotations in the eigenspace of the total angular momentum. A crucial role is played by elementary j-gates (6j symbols) which satisfy algebraic identities that make the structure of the model similar to "state sum models", employed in discretizing Topological Quantum Field Theories and quantum gravity. The spin network simulator can thus be viewed also as a Combinatorial QFT model for computation. The semiclassical limit (large j's) is discussed.