Dynamical Quantum Phase Transitions in U(1) Quantum Link Models

Quantum link models (QLMs) are extensions of Wilson-type lattice gauge theories which realize exact gauge invariance with finite-dimensional Hilbert spaces. QLMs not only reproduce standard features of Wilson lattice gauge theories in equilibrium, but can also host new phenomena such as crystalline confined phases. The local constraints due to gauge invariance also provide kinetic restrictions that can influence substantially the real-time dynamics in these systems. We aim to characterize the nonequilibrium evolution in lattice gauge theories through the lens of dynamical quantum phase transitions, which provide general principles for real-time dynamics in quantum many-body systems. Specifically, we study quantum quenches for two representative cases, U(1) QLMs in $(1+1)\mathrm{D}$ and $(2+1)\mathrm{D}$, for initial conditions exhibiting long-range order. Finally, we discuss the connection to the high-energy perspective and the experimental feasibility to observe the discussed phenomena in recent quantum simulator settings such as trapped ions, ultracold atoms, and Rydberg atoms.

Figure 1

(a) Schematic plot of the wave function dynamics in Hilbert space of the considered lattice gauge theories. The two symmetry-broken ground states $|{\psi }_{\pm }⟩$ represent extremal points, with maximal values for the order parameters. Starting at $|{\psi }_{-}⟩$, the state explores the Hilbert space. The projection onto the ground-state manifold classifies the state according to whether the state is closer to $|{\psi }_{-}⟩$ or $|{\psi }_{+}⟩$ (blue or red). (b) The dynamics of the dominant rate function $\lambda (t)$ of the full return probability. Blue and red lines represent the dominant components ${\lambda }_{-}(t)$ and ${\lambda }_{+}(t)$, respectively. The vertical dashed lines mark the times when $\lambda (t)$ has kinks and undergoes a DQPT, switching between the two components. We compare $\lambda (t)$ to (c) the dynamics of the order parameter $\mathcal{E}(t)$ and (d) the fermionic matter particle density $n(t)$.

Figure 2

The timescales ${\tau }_{\lambda }$, ${\tau }_{\mathcal{E}}$, and ${\tau }_n$, as defined in Figs. 1, as a function of the coupling $\kappa$ in the final Hamiltonian for two system sizes $L=24$, 48. For $\kappa$ not too close to ${\kappa }_c$ of the underlying quantum phase transition, the timescales ${\tau }_{\lambda }$ and ${\tau }_{\mathcal{E}}$ (in contrast to ${\tau }_n$) are close to each other.

Figure 3

(a) Illustration of the initial product state for the $(1+1)\mathrm{D}$ QLM at $\kappa =0$. The filled (hollow) circle is the particle (antiparticle) sites, the crosses are the gauge-field sites, separated by color blocks. The rows with horizontal arrows are the fermionic representation of the state. The two symmetry-breaking ground states $|{\psi }_{\pm }⟩$ are shown. The lower three rows are the wave function representation using spin states. (b) Particle-antiparticle pairs can propagate upon flipping the intermediate gauge spins. (c) The matter-gauge link interaction couples two shaded blocks. (d) The dynamics of $\lambda (t)$, $\mathcal{E}(t)$, and $n(t)$ upon changing $\kappa$.

Figure 4

DQPT in the $(2+1)\mathrm{D}$ U(1) QLM. The dimensionless time, $t$, in units of $J$. The quench dynamics using Lanczos-based ED for system size $4\times 4$ (32 spins), $4\times 6$ (48 spins), and $6\times 6$ (72 spins). (a) The dynamics of the rate function after quench. (b) The dynamics of the two-component order parameter $(M_A,M_B)$, denoting the plaquette orientation in the two sublattices, $A$ and $B$, of the square lattice. (c) The dynamics of the order parameter after quench in the $(M_A,M_B)$ plane. The two red points represent the order related by charge conjugation symmetry, $\mathcal{C}$, which transform $M_B\to -M_B$ [28]. Under a lattice translation, the order parameter gets reflected about the red dashed line.