Phase transitions in Zn gauge models: Towards quantum simulations of the Schwinger-Weyl QED

Abstract

We study the ground-state properties of a class of Zn lattice gauge theories in 1 + 1 dimensions, in which the gauge fields are coupled to spinless fermionic matter. These models, stemming from discrete representations of the Weyl commutator for the U(1) group, preserve the unitary character of the minimal coupling, and have therefore the property of formally approximating lattice quantum electrodynamics in one spatial dimension in the large-n limit. The numerical study of such approximated theories is important to determine their effectiveness in reproducing the main features and phenomenology of the target theory, in view of implementations of cold-atom quantum simulators of QED. In this paper we study the cases n = 2 ÷ 8 by means of a DMRG code that exactly implements Gauss' law. We perform a careful scaling analysis, and show that, in absence of a background field, all Zn models exhibit a phase transition which falls in the Ising universality class, with spontaneous symmetry breaking of the CP symmetry. We then perform the large-n limit and find that the asymptotic values of the critical parameters approach the ones obtained for the known phase transition the zero-charge sector of the massive Schwinger model, which occurs at negative mass.

Symmetry-protected topological phases in lattice gauge theories: TopologicalQED2

“…, and we will focus on f (V n ) = 1 2 g 2L2 n [56,69]. As shown in [56], the properties of the massive Schwinger mode with vacuum angle θ = π can be recovered from a large-N scaling of the Z N massive Schwinger model (B1).…”

ZN gauge theories coupled to topological fermions: QED2 with a quantum mechanical θ angle

“…where the main difference with the Kogut-Susskind approach (4) is that the gauge fields are defined through the pair of link operatorsŨ n ,Ṽ n that obey the Z N algebra, fulfilling U N n =Ṽ N n = I, andṼ † nŨnṼn = e i2π/NŨ n [30][31][32]. By using the electric-flux eigenbasisṼ n |v = v |v with v ∈ Z N on each link, one can understand that the link oper-atorŨ n acts as ladder operator that raises the electric flux by one quantumŨ n |v = |v + 1 in a cyclic way, i.e.Ũ n |N = |1 .…”

“…Since the model parameters (ga, ∆) are controlled by atomic parameters in Eqs. (24), (35) and (32), the cold-atom quantum simulator has the potential of realizing the interesting physics of topological QED 2 outlined in previous sections. In particular, by adiabatic state preparation, the coldatom mixture could explore the full phase diagram of Fig.…”

“…We found no adiabatic path connecting these phases, but instead Isingtype quantum phase transitions take place. In this work, we perform a quantitative benchmark of the bosonization predictions by exploring larger values of N, and performing a scaling analysis to extract large-N limit [32], where one expects to recover the predictions for the compact U (1) LGT. We provide a more detailed description of the numerical DMRG simulations presented in [13], and present new numerical results that provide further information about the properties of the topological Schwinger model.…”

Real-time-dynamics quantum simulation of (1+1)-dimensional lattice QED with Rydberg atoms

“…In order to encode the gauge degrees of freedom in a quantum simulator, we need to truncate and discretize the spectrum of the electric field. To this purpose, we replace the continuous-spectrum operatorÛ j with the discrete clock operatorÛ j such thatÛ n j = (Û † j ) n = 1 with n ∈ N. That is, we move from the continuous gauge symmetry group U (1) to Z n [33,34]. We fix n = 3, so that the electric fieldÊ j admits only three possible states {|−1 , |0 , |+1 } and the operatorsÛ j (Û † j ) cyclically permute them clockwise (anti-clockwise) as shown in FIG 1 (a).…”