Symmetry-protected topology and deconfined solitons in a multi-link $\mathbb{Z}_2$ gauge theory

Abstract

With the advent of quantum simulators, exploring exotic collective phenomena in lattice models with local symmetries and unconventional geometries is at reach of near-term experiments. Motivated by recent progress in this direction, we study a $\mathbb{Z}_2$ lattice gauge theory defined on a multi-graph with links that can be visualized as great circles of a spherical shell hosting the $\mathbb{Z}_2$ gauge fields. Elementary Wilson loops along pairs of these bonds allow to identify a dynamical gauge-invariant flux, responsible for Aharonov-Bohm-like interference effects in the tunneling dynamics of charged matter residing on the vertices. Focusing on an odd number of links, we show that this leads to state-dependent tunneling amplitudes underlying a phenomenon analogous to the Peierls instability. We find inhomogeneous phases in which an ordered pattern of the gauge fluxes spontaneously breaks translational invariance, and intertwines with a bond order wave for the gauge-invariant kinetic matter operators. Long-range order is shown to coexist with symmetry protected topological order, which survives the quantum fluctuations of the gauge flux induced by an external electric field. Doping the system above half filling leads to the formation of topological soliton/anti-soliton pairs interpolating between different inhomogeneous orderings of the gauge fluxes. By performining a detailed analysis based on matrix product states, we prove that charge deconfinement emerges as a consequence of charge-fractionalization. Quasiparticles carrying fractional charge and bound at the soliton centers can be arbitrarily separated without feeling a confining force, in spite of the long-range attractive interactions set by the small electric field on the individual integer charges.

Figures (10)

• Figure 1: Multi-link $\mathbb{Z}_2$ gauge theory. (a) Pictorial representation of the multi-link chain: fermions live on the lattice sites (red dots) while gauge fields are spin-$\frac{1}{2}$ Pauli matrices located on its bonds, which are depicted as halves of great circles on a sphere of diameter equal to the lattice spacing $a$. The $\mathbb{Z}_2$ gauge generators $G_i$ have support on each lattice site and on the neighboring bonds belonging to the adjacent spherical shells - see Eq. \ref{['eq:generators']}. The Hamiltonian \ref{['eq:model']} consists of a magnetic term $H_m$ comprising all possible pairwise antiferromagnetic interactions between spins, controlled by a parameter $J$, of the gauge invariant tunneling $H_t$ and the electric field contribution $H_e$, respectively proportional to the parameters $t$ and $h$. Particles can hop along any of the $N_b$ bonds connecting neighboring sites, followed by a flip of the corresponding spin state (in the Hadamard basis, represented by arrows directed along each bond) to fulfill gauge invariance. (b) In the case $N_b=3$, working in the maximal spin $S=\frac{3}{2}$ sector of the total spin operator $\boldsymbol{S}_{i_\ell}$, the eigenstates $S^z_{i_\ell}$ corresponding to $m=\frac{3}{2},\frac{1}{2}$ are illustrated, on the left, in terms of their spin-$\frac{1}{2}$ components on each bond. Each spherical cap subtended by a pair of links is threaded by a gauge-flux $\langle W^{a,b}_{i_\ell,\circlearrowleft}\rangle = \langle \sigma^z_{i_\ell,a} \sigma^z_{i_\ell,b}\rangle$ taking values $0$ or $\pi$, respectively shown in green or red and corresponding to a ferromagnetic or antiferromagnetic ordering of the spins on the two bonds. On the right, these states are represented by shaded blue circles at a fixed value of the z-projection of $\langle \boldsymbol{S}_{i_\ell}\rangle$, illustrated as a red vector originating from the center of the sphere. While the two different eigenstates of $S^z_{i_\ell}$ would result in different tunneling amplitudes in a naive mean-field treatment of Eq. \ref{['eq:totspin']}, gauge invariance forces $\langle S^z_{i_\ell}\rangle=0$ on physical states, emphasizing the need for a different local spin basis to discuss state-dependent tunneling amplitudes - see Eq. \ref{['eq:good_basis']}. (c) Cat state $\ket{\frac{3}{2},-}=\frac{1}{\sqrt{2}}(\ket{\frac{3}{2}} - \ket{-\frac{3}{2}})$ supporting tunneling with the amplitude $t_2 = \frac{3}{2} t$. (d) Gauss-law constraints on the distribution of $\mathbb{Z}_2$ charges and parity cat states of the gauge fields at the multi-links.
• Figure 2: Incompressible regions and LRO at small $h/t$. Panel (a) displays the filling fraction $\nu$ in the ground state of the system as a function of $J/t$ and $\mu/t$, at $h/t = 0.1$. Incompressible regions are stabilized in which the filling fraction is fixed to $0,1/2,2/3$ or $1$, as denoted by the red labels inside each lobe and are delimited by dashed lines. In the regions corresponding to $\nu = 1/2,2/3$ the lattice translational symmetry is spontaneously broken, as manifested by a spatially-periodic modulation of $\langle T_{i_\ell}\rangle$, whose Fourier transform $\tilde{T}(k)$ is peaked at the order wave-vector. Three different orderings appear: in the half-filled region, $\tilde{T}(k)$ has two peaks, a dominant one at the Fermi momentum $k_F(1/2)=\pi$ (b) and a sub-dominant one at $\tilde{k}=\frac{\pi}{2}$ (d); in the $\nu = 2/3$ lobe, $\langle T_{i_\ell} \rangle$ oscillates with a period of three lattice spacings, corresponding to a peak in the Fourier transform at the Fermi momentum $k_F(2/3)=\frac{4\pi}{3}$ (c) - see main text. In panel (e) we show that the structure factor associated to the correlator $C^T_{ij}$, $S_T(k)$, evaluated at $J/t = 0.2$ and $h/t = 0.1$ in the incompressible regions, is peaked at the order wave-vectors, a signature of long-range order - see main text. All the figures correspond to $L=120$ sites.
• Figure 3: BOW structure factor. The BOW structure factor $S_{\rm BOW}(k)$ is displayed for $\nu = \frac{1}{2}$ and $\nu=\frac{2}{3}$ fillings, for a chain of $L=120$ sites and $J/t = 0.2$, $h/t = 0.1$. In both cases, the behaviour of the BOW structure factor parallels that of $S_T(k)$ - see panel (e) of Fig. \ref{['fig:cp_analysis']}, being peaked at the Fermi momentum $k_F=2\pi\nu$ and manifesting LRO. For the half-filled case, the subleading peak at the wave-vector $\tilde{k} = \frac{\pi}{2}$ is here smaller and is partially hidden by the curve at $\frac{2}{3}$ filling.
• Figure 4: Dimerization order parameter. We show the order parameter $\tilde{T}(\pi) = \frac{1}{N_s}\sum_i (-1)^i \, \langle T_{i_\ell} \rangle$ characterizing the long-range inhomogeneous flux ordering, for the system size $L=120$. The points $J_{c,1},J_{c,2}$, marked by the vertical dashed lines, are the critical points of a first-order transition and are reported in Eq. \ref{['eq:fo_crit']}, derived analytically at $h=0$. The black points that delimit the boundaries of the dimerized region are instead critical points of second order phase-transitions, found by the finite-scaling analysis reported at the end of Sec. \ref{['sec:tbow']} and the dashed black curve connecting them has been drawn to lead the eye.
• Figure 5: Local Berry phase. We study the local Berry phase obtained by introducing a local twist in the hopping amplitude accross link $i_\ell$: $t_{i_\ell} \to {\rm e}^{\rm i \theta} t_{i_\ell}$ and considering a cyclic variation of $\theta$ - see text. In panel (a) we show the symmetry broken configuration of $T_{i_\ell}$ stabilized in the ground state of an open chain of $L=120$ sites at $J/t = 0.2$ and $h/t=0.1$, which alternates strong to weak bonds in a tetramerized pattern $S \, W_1 \, S \, W_2$, evidenced by a different box coloring at odd (blue) and even (red) bonds. In panel (b) we plot the absolute value of the local Berry phase obtained by twisting the corresponding bonds, resulting in the repeated pattern $\pi \, 0 \, \pi \, 0$. Panels (c) and (d) respectively show the inhomogeneous patterns of $\langle T_{i_\ell}\rangle$ and $\left|\gamma_{i_\ell}\right|$ when the pinning potential \ref{['eq:pinning']}$V(\lambda = 5 \cdot 10^{-3} \, t)$ is applied to select a topological configuration, demonstrating localized domain walls whose spread is controlled by the value of $h/t$. The Berry phase's pattern $0 \, \pi \, 0 \, \pi$ is realized in the bulk of the chain, while its value is not quantized in the proximity of the localized edge modes.