Quantum simulation of ($1+1$)-dimensional U(1) gauge-Higgs model on a lattice by cold Bose gases

We present a theoretical study of quantum simulations of ($1+1$)-dimensional U(1) lattice gauge-Higgs models, which contain a compact U(1) gauge field and a Higgs matter field, by using ultracold bosonic gases on a one-dimensional optical lattice. Starting from the extended Bose-Hubbard model with on-site and nearest-neighbor interactions, we derive the U(1) lattice gauge-Higgs model as a low-energy effective theory. The derived gauge-Higgs model exhibits nontrivial phase transitions between the confinement and Higgs phases, and we discuss the relation with the phase transition in the extended Bose-Hubbard model. Finally, we study the real-time dynamics of an electric flux by the Gross-Pitaevskii equations and the truncated Wigner approximation. The dynamics is governed by a bosonic analog of the Schwinger mechanism—i.e., the shielding of an electric flux by a condensation of Higgs fields, which occurs differently in the Higgs and the confinement phases. These results, together with the obtained phase diagrams, shall guide experimentalists in designing quantum simulations of the gauge-Higgs models by using cold gases.

Figure 1

Optical and gauge lattices. Each site $a$ of the optical lattice corresponds to a link of the gauge lattice as $a\leftrightarrow (r,r+1)$. On the gauge link $(r,r+1)$ sits the spatial component ${\theta }_{r,1}$ of the gauge field.

Figure 2

Mean-field result for the 1D ground-state configuration of the average density ${\rho }_0$. It shows that the homogeneous configuration ($\delta \rho \equiv {\rho }_{\mathrm{even}}-{\rho }_{\mathrm{odd}}=0$) is stable for $g^2>0$ as expected. However, a homogeneous state exists even for $g^2<0$ for moderate values of negative $g^2$. The inhomogeneous configuration stands for the density-wave pattern ($\delta \rho >0$). Detailed study on this point is given in the Appendix.

Figure 3

Thermodynamic quantities of the GHM of Eq. (10) and their scaling behavior for $c_2=c_3=2.0$: (a) Specific heat $C_{\mathrm{GH}}$ of Eq. (13). (b) Susceptibility density $\chi /L^2$ of Eq. (15). (c) Scaled susceptibility density $\overline{\chi }$ of Eq. (16) with the choice of critical values $c_1=c_{1c}\sim 1.22$ and $\eta (c_{1c})\sim 0.68$. These values are determined so that the three curves of $\overline{\chi }$ for $L=16$, 32, 64 merge at a single point (see text). A large value of $\eta (c_{1c})$ reflects the fluctuations of ${\theta }_{x,0}$ in $A_{\mathrm{I}}$, which suppress the correlations among $\{{\sigma }_x\}$. (d) Average critical index $\eta (c_1)$ and deviation $\mathrm{\Delta }\eta (c_1)$ from a triple intersection (see the text for their definitions). $\mathrm{\Delta }\eta$ decreases as $c_1$ increases and reaches its minimum $\mathrm{\Delta }\eta =0$ at $c_{1c}$ (three curves merge there) and keeps small values. Thus, $c_{1c}$ is interpreted as a BKT-type critical point dividing the disordered phase ($c_1<c_{1c}$) and quasi-ordered phase ($c_1>c_{1c}$), as Eq. (16) shows.

Figure 4

Phase diagram of the short-range GHM of Eq. (10) (corresponding to EBHM for homogeneous large fillings) obtained by the path-integral MC simulation. In the Higgs phase, fluctuations of the gauge field $\mathrm{\Delta }\theta$ are small, while in the confinement phase, $\mathrm{\Delta }\theta$ is large and fluctuations of the electric field $\mathrm{\Delta }E$ are small.

Figure 5

Phase diagram obtained by the path-integral MC simulations for $L=16$, 32, and 64, where $L^2$ is the size of the space-time lattice. There are two phases, SF (superfluid) and MI (Mott insulator). We show the gauge-theoretical interpretation of each phase. Experimental setups within the parameter regions labeled “Higgs” and “confinement” are suitable for an atomic quantum simulation of the U(1) GHMs. The blue broken line represents the vanishing gauge coupling, $g^2(\equiv U-2V)=0$. The red dotted line is the phase boundary obtained in Sec. 2 of the Appendix. From the viewpoint of the gauge theory, for $g^2>0$ and large $U$, the ground state is the confinement phase, in which the system has no phase coherence; whereas for $g^2>0$ and small $U$, the ground state is Higgs phase, in which the system has strong phase coherence—i.e, the SF phase.

Figure 6

Solution of the GP equation (19) for $U/J=10^3$, $V/J=1.0$, and $g^2/J=0.2$. An electric flux string is stable because the effect of the Higgs field is negligibly small.

Figure 7

Real-time dynamics of a single electric flux. (a) Global phase diagram of the EBHM by the path integral MC. (See the Appendix and Fig. 11). The black crosses in the phase diagram indicate the locations at which the string dynamics is observed as in the three panels at top right. (b)–(d) The time evolution of a single electric flux of length $R=60$ in a $L=200$ lattice. All time evolutions have the same gauge coupling $g^2/J=0.2$. For $V/J=0$ (b) and $V/J=0.5$ (c), the initial electric flux gradually spreads out along the time evolution, and the pure-gauge Gauss law, div $E=0$, does not work. For $V/J=1$ (d), the shape of the initial electric flux is not broken. This indicates that the pure-gauge Gauss law works for the initial flux. For all cases (b)–(d), the electric fields oscillate, changing their signs. This phenomenon is explained by the string-antistring oscillation that is also observed in the QED real-time simulation in Refs. [22, 32]. (e) and (f) For the case (c), the electric field averaged over the central ten sites, $⟨E{⟩}_{\text{center}}$. The oscillation period is $\sim 3.1[\mathrm{kHz}]$ in our time scale, and the period depends only on the value of $J$.

Figure 8

Mean value of the electric field per link out of the initial electric flux, $⟨E^2{⟩}_{\mathrm{out}}$. As the value of $V$ decreases, the electric field spreads out of its initial string position. $g^2/J=0.2$.

Figure 9

Schematic picture of string-antistring oscillation. This phenomenon comes from the Schwinger mechanism that was first suggested for QED [36]. The red circle represents a positive-charge particle, and the blue one is a negative-charge particle. The black arrow represents an electric flux tube between a pair of opposite static charges.

Figure 10

Real-time dynamics of the flux string in the confinement region with random distributions of the wave-function phase in the initial condition. These results were obtained by averaging 100 samples with different initial phase distribution. For ${\mathrm{\Delta }}_m=0.01$, the stable evolution of the string is observed, whereas for ${\mathrm{\Delta }}_m=0.05$, only a small portion of the initial string configuration remains.

Figure 11

(a) Phase diagram of the effective model in Eq. (a6) obtained by the MC simulations. This MC calculation includes the update of the variational parameter $\{{\rho }_{0a}\}$. The phase boundaries are determined by the scaling of $\chi =\sum _{\ell }{G}_{\mathrm{B}}(\ell )$. (b) Specific heat $C_{\mathrm{EBH}}$ for $V/J=1$. (c) Susceptibility density $\chi /L^2$ in three different lattice sizes $L=16$, 32, and 64. (d) Scaled susceptibility density $\overline{\chi }=L^{\eta -2}\chi$. To determine the detailed phase boundary of the SF phase, we use $\overline{\chi }$. By choosing the exponent $\eta$ and $U/J$ suitably, we find that three curves of $\overline{\chi }$ merge at a point, which is the transition point (see Sec. 3). Through this analysis, we verify that the SF-MI transition is a BKT-type phase transition. For $V/J={\rho }_0$, the critical point is $U/J\sim 1.17{\rho }_0$, and the exponent is $\eta \sim 0.24$. MC simulation of the effective model in Eq. (a6) underestimates the SF in the vicinity $g^2\approx 0$, as explained in the text.

Figure 12

Phase diagram of the effective model in Eq. (a6) obtained by the MC simulations for $L=16$. Average filling is 10 per site (${\rho }_0=10$), and we employ the canonical ensemble. This MC calculation includes the update of the variational parameter $\{{\rho }_{0a}\}$. The target regime is large $V$. The specific regime of the phase boundary between SF and MI, $5\lesssim V/J\lesssim 10$, is not clear.