Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks

Abstract

Gauge theories are of paramount importance in our understanding of fundamental constituents of matter and their interactions. However, the complete characterization of their phase diagrams and the full understanding of non-perturbative effects are still debated, especially at finite charge density, mostly due to the sign-problem affecting Monte Carlo numerical simulations. Here, we report the Tensor Network simulation of a three dimensional lattice gauge theory in the Hamiltonian formulation including dynamical matter: Using this sign-problem-free method, we simulate the ground states of a compact Quantum Electrodynamics at zero and finite charge densities, and address fundamental questions such as the characterization of collective phases of the model, the presence of a confining phase at large gauge coupling, and the study of charge-screening effects.

Loop-free tensor networks for high-energy physics

“…As an paradigmatic example of the possible analysis that are enabled by TN methods, we report here a result recently presented in [56] by some of the authors, where the first analysis of a threedimensional lattice gauge theory in presence of matter-a compact version of QED-has been reported. Following the steps introduced in §3a, and considering h = c = 1, the Hamiltonian in three-spatial dimensions simplifies to…”

“…Finally, we mention that in [56] an extensive analysis of the model has been reported, including screening effect in a quantum capacitor and the characterization of its ground state properties at zero and finite global chemical potential. We stress that the latter simulations (not reported here) efficiently attack a challenge inefficient to tackle with Monte Carlo methods due to the sign problem arising in that context.…”

“…By quantitatively validating quantum simulators in out-of-equilibrium situations, even if only in lower dimensions, MPS and TN methods play a very important role towards establishing quantum simulators as reliable tools in quantum physics. Finally, recently the applications of TN to higher-dimensional LGTs has start becoming within reach of the available algorithms and numerical resources [51][52][53][54][55][56][57], providing an important stimulus to further develop these techniques and explore their application to LGTs.…”

Exploring Bosonic and Fermionic Link Models on $(3+1)-$d tubes

“…A natural next step would be to address these issues by means of ab-initio calculations, such as a Markov chain Monte Carlo. However, for the fermionic model, sign problems currently preclude the existence of any efficient Monte-Carlo algorithm to our knowledge, and different numerical approaches such as tensor network methods [32][33][34] might be necessary. While the volume scaling of the gap is here only presented for bosonic link variables, we observe qualitatively similar behavior for fermionic links and therefore expect the transition between an ordered phase and a QSL for both models (we return to this point in Sect.…”

Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency

“…In particular, this is carried out by compressing the exponentially large wavefunctions into a network of tensors interconnected through auxiliary indices with bond dimension m. The main Anzätze for the representation of quantum many-body states based on TNs include Matrix Product States (MPS) for 1D systems [30][31][32], Projected Entangled Pair States (PEPS) [33][34][35], Tree Tensor Networks (TTN) [36][37][38][39] and Multiscale Entanglement Renormalization Ansatz (MERA) [40,41] which can be defined in any dimension. However, while for one-dimensional systems MPS are the established TN ansatz, the development of efficient TN algorithms for higher-dimensional systems is still ongoing [42][43][44][45][46][47][48][49].…”