Tensor Networks for Lattice Gauge Theories and Atomic Quantum Simulation

We show that gauge invariant quantum link models, Abelian and non-Abelian, can be exactly described in terms of tensor networks states. Quantum link models represent an ideal bridge between high-energy and cold atom physics, as they can be used in cold atoms in optical lattices to study lattice gauge theories. In this framework, we characterize the phase diagram of a $(1+1)\mathrm{D}$ quantum link version of the Schwinger model in an external classical background electric field: the quantum phase transition from a charge and parity ordered phase with nonzero electric flux to a disordered one with a net zero electric flux configuration is described by the Ising universality class.

Figure 1

Ground state of the spin-$\frac{1}{2}$ quantum link model in the limiting cases of $|\mu |\gg |\epsilon |$: in the upper (lower) panel the fermion and the gauge field states are represented for $\mu \ll \epsilon$ ($\mu \gg \epsilon$) resulting in zero electric flux, $\mathcal{E}=0$, and a $C$ and $P$ invariant state (nonzero electric flux, $\mathcal{E}\ne 0$, $C$ and $P$ symmetry broken).

Figure 2

(a) Electric flux $\mathcal{E}$ as a function of $\mu$ for $L=\{40,60,80,100,120,140\}$ from top to bottom, $S=\frac{1}{2}$ and $D=30$. (b) Finite size scaling of the electric flux $\mathcal{E}$ shown in panel (a), resulting in the critical point ${\mu }_c=0.655\pm 0.003$ and critical exponents $\nu \sim 1$ and $\beta \sim 1/8$. (c) Uniform part of the entanglement entropy (green circles, first order approximation, i.e., $u_{x,L}=\frac{1}{2}(u_{x,L}+u_{x+1,L})$, and blue squares, third order approximation [48]). Inset: fit of $u_{x,L}$ as a function of the system size $\log L$: a linear fit results in the central charge $c=0.49\pm 0.01$.

Figure 3

(a) Electric flux $\mathcal{E}$ for $S=1$, $L=\{40,60,80,100\}$, $D=30$ and $\epsilon =0.5$ as a function of $\mu$. The estimated critical point is ${\mu }_c=-0.2173\pm 0.0005$. (b) Phase diagram of the $S=1$ representation. The critical line is fitted via ${\mu }_c\sim {\mu }_0+{\mu }_{\frac{1}{2}}\sqrt{\epsilon }+{\mu }_1\epsilon$ resulting in ${\mu }_0=-0.04\pm 0.03$, ${\mu }_{\frac{1}{2}}=-0.20\pm 0.03$ and ${\mu }_1=-0.113\pm 0.005$.