Quantum XX model with competing short- and long-range interactions: Phases and phase transitions in and out of equilibrium

We consider the quantum XX model in the presence of competing nearest-neighbor and global-range interactions, which is equivalent to a Bose-Hubbard model with cavity mediated global-range interactions in the hard-core boson limit. Using fermionic techniques the problem is solved exactly in one dimension in the thermodynamic limit. The ground state phase diagram consists of two ordered phases: ferromagnetic (F) and antiferromagnetic (AF), as well as an XY phase having quasi-long-range order. We have also studied quantum relaxation after sudden quenches. Quenching from the AF phase to the XY region remanent AF order is observed below a dynamical transition line. In the opposite quench, from the XY region to the AF phase beyond a static metastability line AF order arises on top of remanent XY quasi-long-range order, which corresponds to dynamically generated supersolid state in the equivalent Bose-Hubbard model with hard-core bosons.

Figure 1

The $k_m$ dependence of the energy at different points of the phase diagram. The vertical red line shows the position of ${\tilde{k}}_m$. For $k<{\tilde{k}}_m$ ($k>{\tilde{k}}_m$) the staggered magnetization is $x=0$ ($x>0$). Upper panel $\varepsilon /J=1$: $h/J=0.5$ AF ground state; $h/J=1.09$ coexistence between the XY and AF ground states; $h/J=1.5$ XY ground state; $h/J=2.$ continuous transition from the XY to the F ground state; $h/J=2.5$ F ground state. Middle panel $\varepsilon /J=1.7145738$: $h/J=1.5$ AF ground state; $h/J=2.$ tricritical point, coexistence between the F, XY, and AF ground states; $h/J=2.5$ F ground state. Lower panel $\varepsilon /J=2$: $h/J=1.5$ AF ground state; $h/J=2.2$ coexistence between the F and AF ground states; $h/J=2.5$ F ground state.

Figure 2

Phase diagram of the quantum XX model with cavity-induced global-range interactions. In the upper panel $J=1$, in the lower panel $\varepsilon =1$. The color codes indicate the value of the staggered magnetization $x$ and that of the longitudinal magnetiz- ation $m^z$.

Figure 3

Phase diagrams calculated in the Ising basis with the Hamiltonian in Eq. (1). From top to bottom $L=10$, 12, 14, and 16.

Figure 4

Inset: Time dependence of the staggered magnetization after a quench protocol with ${\epsilon }_0/J_0=1$ to $\epsilon /J={(\epsilon /J)}_c(1+\delta )$ with ${(\epsilon /J)}_c=1/2$ and $\delta =0.25$, 0.125, 0.0625, 0.03125, and 0, from up to down. Main panel: Stationary value of the staggered magnetization after a quench from the AF phase with ${\epsilon }_0/J_0=1$ to $\epsilon /J<{\epsilon }_0/J_0$. There is a the dynamical phase transition at ${(\epsilon /J)}_c=1/2$, at which point $x_{\text{st}}\simeq 1.06\delta$.

Figure 5

Nonequilibrium spin-spin correlation functions after a quench protocol with ${\epsilon }_0/J_0=1$ to $\epsilon /J=0.125$, 0.25, 0.375, and 0.5, at time $t=20$, compared with the equilibrium value at $t=0$. The decays are exponential and the correlation length increases with $\epsilon /J$. Inset: The same for quenches to the dynamically generated AF phase with $\epsilon /J=0.625$, 0.75, and 0.875. Notice that $G_t(j+r,j)$ changes sign and for odd or even distance between the spins there is an alternation, which is a sign of the dynamical AF order.

Figure 6

Time dependence of the staggered magnetization after a quench protocol: ${\epsilon }_0/J_0=1$ and $\epsilon /J={(\epsilon /J)}_c(1+\delta )$, with ${(\epsilon /J)}_c=1/2$ and $\delta =0$, 0.001, 0.002, 0.004, 0.008, 0.016, and 0.0032 from down to up in log-log scale. The straight line with slope $-3/2$ indicates the asymptotic behavior at the critical point, see Eq. (39). Inset: Scaling plot of the staggered magnetization curves in the main panel using the relations in Eqs. (40) and (42).

Figure 7

Position $t_{\text{min}}$ (upper set of points) and value $x_{\text{min}}$ (lower set of points) of the minimum of the dynamical staggered magnetization as a function of $\delta$ in log-log scale. The straight lines with slopes $-1$ and 1.5 indicate the respective expected asymptotic behaviors in Eq. (40). Inset: Amplitude of the oscillations in the stationary region, measured as the difference between two consecutive extremal (maxima and minima) values, as a function of time in log-log scale for $\delta =0.032$ (upper set of points) and $\delta =0.016$ (lower set of points). The straight lines with slope $1/2$ represent the conjectured behavior in Eq. (41).

Figure 8

Left inset: Time dependence of the staggered magnetization after a quench from the XY phase from ${\epsilon }_0/J_0=0.5$ with a perturbation $\mathrm{\Delta }x={10}^{-10}$ to different values of the reduced control parameter $\delta =0.009$, 0.004, 0.001, 0.0005, and 0.00025, from up to down. Right inset: Scaling plot using Eq. (44) and keeping the plot of $\delta =0.001$ unscaled. Main panel: $\epsilon /J$ dependence of $x_{\text{max}}$.

Figure 9

$\delta$ dependence of the (first) maximum $x_{\text{max}}$ (right scale) and the timescale $t_{\text{max}}$ (left scale) in Fig. 8 in log-log scale. The straight lines with slopes 0.5 and $-0.5$, respectively, represent the expected asymptotic relations in Eq. (45).

Figure 10

Quench from the XY phase with $h_0=h=1$, $J_0=J=1$, ${\epsilon }_0=0.5$, and $\delta =0.001$ for different values of the perturbation $\mathrm{\Delta }x={10}^{-6},{10}^{-7},\cdots ,{10}^{-11}$, from left to right.

Figure 11

The equilibrium spin-spin correlation function $G_0(j+r,j)$ and its nonequilibrium counterpart $G_t(j+r,j)$ after a quench protocol from ${\epsilon }_0/J_0=0.5$ to $\epsilon /J=1.5$ at $t=3.8$ and $x\left(t\right)=0.328$. The straight line with slope $-1/2$ indicates the exact asymptotic behavior. The algebraic decay of $G_0(j+r,j)$ is preserved under the dynamical phase transition, but for $x\left(t\right)>0$ the prefactor is different for even and odd distances between the spins. Inset: Ratio of $G_t(j+r,j)$ and $G_0(j+r,j)$ at different times after a quench protocol from ${\epsilon }_0/J_0=0.5$ to $\epsilon /J=1.5$. The difference between the ratios of different parity is larger for larger values of $x\left(t\right)$.

Figure 12

Phase diagram of the one-dimensional Bose-Hubbard model with cavity mediated global-range interaction in the hard-core limit obtained by using the equivalence with the XX model analyzed in this work. $\rho$ is particle density and $x$ is the occupancy imbalance. MI = Mott insulator, SF = superfluid, and DW = density wave.