Quantum geometric tensor and quantum phase transitions in the Lipkin-Meshkov-Glick model

We study the quantum metric tensor and its scalar curvature for a particular version of the Lipkin-Meshkov-Glick model. We build the classical Hamiltonian using Bloch coherent states and find its stationary points. They exhibit the presence of a ground-state quantum phase transition where a bifurcation occurs, showing a change in stability associated with an excited-state quantum phase transition. Symmetrically, for a sign change in one Hamiltonian parameter, the same phenomenon is observed in the highest-energy state. Employing the Holstein-Primakoff approximation, we derive analytic expressions for the quantum metric tensor and compute the scalar and Berry curvatures. We contrast the analytic results with their finite-size counterparts obtained through exact numerical diagonalization and find excellent agreement between them for large sizes of the system in a wide region of the parameter space except in points near the phase transition where the Holstein-Primakoff approximation ceases to be valid.

Figure 1

(a) The two lower-energy surfaces $e_2$ (green) and $e_3$ (orange) as functions of the coupling parameters ${\mathrm{\Omega }}_x$ and ${\xi }_y<0$. (b) The two higher-energy surfaces $e_1$ (red) and $e_4$ (blue) as functions of the coupling parameters ${\mathrm{\Omega }}_x$ and ${\xi }_y>0$.

Figure 2

Critical energies plotted as a function of the coupling parameter ${\mathrm{\Omega }}_x$ for (a) ${\xi }_y=-2.3$ and (b) ${\xi }_y=2.3$. The color code is $e_1$ (red), $e_2$ (green), $e_3$ (orange), and $e_4$ (blue).

Figure 3

Comparison of the expectation values of different observables taken in the highest-energy eigenstate (dashed blue) and in spin coherent states (solid orange) for $j=128$ and ${\xi }_y=2.3$.

Figure 4

Lyapunov exponent for the stationary point ${\mathbf{x}}_1$ as a function of the coupling parameters ${\mathrm{\Omega }}_x$ and ${\xi }_y>0$. The black zone indicates a null Lyapunov exponent.

Figure 5

Energy surfaces for different values of the parameters $\mathrm{\Omega }$ with ${\mathrm{\Omega }}_{xc}=\sqrt{4{\xi }_y^2-1}$ with ${\xi }_y=2$. Green points are stable center points ${\mathbf{x}}_2$, the blue ones are unstable center points: ${\mathbf{x}}_4$ and ${\mathbf{x}}_4^{\prime }$ in (a) and (b), and in (c) and (d), the red point is the stationary point with positive Lyapunov exponent ${\mathbf{x}}_1$, only present in (a) and (b).

Figure 6

Density of states when ${\xi }_y=2$ and ${\mathrm{\Omega }}_x=0.2{\mathrm{\Omega }}_{xc}$ for $j=256$. The red dashed line indicates the classical energy $e_1=\sqrt{1+{\mathrm{\Omega }}_x^2}=1.265$ where the ESQPT takes place.

Figure 7

QMT and its scalar curvature for the ground state when $j=120$ and ${\xi }_y=2.3$.

Figure 8

Scalar curvature map for different values of $j$, all in the ground state.

Figure 9

Comparison of the numeric QMT components and the scalar curvature with the analytic results for ${\xi }_y=2.3$ and $j=96$ for the highest-energy state. The inset shows the $g_{11}$ component on the logarithmic scale. The agreement is excellent except near the QPT (dashed gray). Note in the plots of $g_{11}$ the difference between the analytical and the numerical curves as ${\mathrm{\Omega }}_x\to 0$.

Figure 10

QMT components and scalar curvature for the highest-energy state with $j=32$. The cyan line is the separatrix given in Fig. 5 when ${\xi }_y=\frac{\sqrt{1+{\mathrm{\Omega }}_x^2}}{2}$.

Figure 11

Maps of the QMT components and the scalar curvature for the highest-energy state with $j=32$.

Figure 12

QMT components and scalar curvature for ${\xi }_y=2.3$ and $j=32,48,64,96,128$. The inset shows the $g_{11}$ component on the logarithmic scale.

Figure 13

Behavior of the maximum of each metric component and scalar curvature with respect to ${\mathrm{\Omega }}_x$ for ${\xi }_y=2.3$. The points correspond to $j=32,48,64,96,128,160,192,256,300,384,512$.

Figure 14

Maximum of each metric component and the scalar curvature with respect to $j$ for ${\xi }_y=2.3$.

Figure 15

Plots of $g_{11},g_{22}$, and $R$ as functions of $j$ for ${\mathrm{\Omega }}_x=0$ and ${\xi }_y=2.3$.

Figure 16

Maximum of each metric component and the scalar curvature with respect to $j$ for ${\xi }_y=0.5$.

Figure 17

Comparison of the QMT and $R$ obtained with the truncated Holstein-Primakoff approximation (solid black), coherent states (dot-dashed orange), and exact diagonalization (dashed blue). We fixed $j=96$ and ${\xi }_y=2.3$.