Signatures of excited-state quantum phase transitions in quantum many-body systems: Phase space analysis

Using the Husimi quasiprobability distribution, we investigate the phase space signatures of excited-state quantum phase transitions (ESQPTs) in the Lipkin-Meshkov-Glick and coupled top models. We show that the ESQPT is evinced by the dynamics of the Husimi function, that exhibits a distinct time dependence in the different ESQPT phases. We also discuss how to identify the ESQPT signatures from the long-time averaged Husimi function and its associated marginal distributions. Moreover, from the calculated second moment and Wherl entropy of the long-time averaged Husimi function, we estimate the critical points of the ESQPT in both models, obtaining a good agreement with analytical (mean field) results. We provide a firm evidence that phase space methods are both a new probe for the detection and a valuable tool for the study of ESQPTs.

Figure 1

(a) Energy spectrum (arbitrary units) of the LMG model as a function of $\kappa$ with $j=N/2=20$. (b) Rescaled density of states, ${\tilde{\omega }}_{\text{LMG}}\left(E\right)={\omega }_{\text{LMG}}\left(E\right)/j$, for the LMG model with $\kappa =0.4$ and $j=N/2=500$. The red solid line is the semiclassical result and the horizontal blue dashed line denotes the critical energy $E/\left(2j\right)=E_c/\left(2j\right)=0$. (c) Energy spectrum (arbitrary units) of the CT model as a function of $\xi$ with $j=7$. (d) Rescaled density of states, ${\tilde{\omega }}_{\text{CT}}\left(E\right)={\omega }_{\text{CT}}\left(E\right)/j^2$, of the CT model with $\xi =3$ and $j=70$. The red solid line denotes the semiclassical result. (e) Derivative of ${\tilde{\omega }}_{\text{CT}}\left(E\right)$ for the CT model with $\xi =3$ and $j=70$. Two vertical green dot-dashed lines in panels (d) and (e) indicate the critical energy values $E/j=E_c/j=\pm 2$ for the ESQPTs in the CT model. The axes in all panels are dimensionless.

Figure 2

Schematic demonstration of the LMG model excited state phase diagram. The system is initialized in the ${\widehat{H}}_{\text{LMG}}$ model ground state with $0<\kappa <4/5$, and an external magnetic field along the $z$ axis with $\eta$ strength is suddenly applied to the model. This sudden quantum quench process gives rise to two different phases, separated by the critical line ${\eta }_c=2-5\kappa /2$, plotted with a white dashed line: For $\eta <{\eta }_c$ ($\eta >{\eta }_c$), the quenched system energy is less (greater) than the critical ESQPT energy, ${\mathcal{E}}_{\text{LMG}}(\kappa ,\eta )<E_c/j=0$ $[{\mathcal{E}}_{\text{LMG}}(\kappa ,\eta )>E_c/j=0]$. The quenched system energy equals the ESQPT critical energy along the critical line. In addition, two phases are also manifested in the classical dynamics of a particle in a double-well potential, which exhibits either a localized ($\eta <{\eta }_c$) or a delocalized ($\eta >{\eta }_c$) behavior. Further details are given in the main text and the Appendices pp2 and pp3. The axes in the figure are dimensionless.

Figure 3

Snapshots of the rescaled spin Husimi function ${\mathcal{Q}}_t(p,q)=Q_t(p,q)/Q_t^m$, with $Q_t^m$ being the maximum value of $Q_t(p,q)$, at different time steps for the LMG model with $\eta =0.4,1,1.7$ (from top to bottom). All panels: $j=N/2=200$ and $\kappa =0.4$. The axes in all panels are dimensionless.

Figure 4

Rescaled long-time averaged Husimi function ${\overline{\mathcal{Q}}}_{\text{LMG}}(p,q)={\overline{Q}}_{\text{LMG}}(p,q)/{\overline{Q}}_{\text{LMG},m}$ of the LMG model for (a) $\eta =0.4$, (b) $\eta =1$, and (c) $\eta =1.8$ with $j=N/2=200$ and $\kappa =0.4$. Here ${\overline{Q}}_{\text{LMG},m}$ denotes the maximum value of ${\overline{Q}}_{\text{LMG}}(p,q)$. Panels (a–c) use the same color scale employed in Fig. 3. Marginal distributions of ${\overline{Q}}_{\text{LMG}}(p,q)$, ${\overline{Q}}_{\text{LMG}}\left(q\right)$, and ${\overline{Q}}_{\text{LMG}}\left(p\right)$, for several values of $\eta$ are plotted in panels (d) and (e), respectively. All quantities are unitless.

Figure 5

(a) Second moment of ${\overline{Q}}_{\text{LMG}}(p,q)$ as a function of $\eta$ for several $\kappa$ with $j=N/2=200$ (solid lines) and $j=N/2=400$ (dashed lines). Inset: Critical second moment, ${\overline{M}}_{2,\text{LMG}}^c={\overline{M}}_{2,\text{LMG}}\left({\eta }_c\right)$, as a function of $N$ for ${\eta }_c=1.5$ (triangles), ${\eta }_c=1$ (squares), and ${\eta }_c=0.5$ (diamonds). Red dashed lines are of the form ${\overline{M}}_{2,\text{LMG}}^c\sim N^{-{\gamma }_M}$. (b) Wehrl entropy of ${\overline{Q}}_{\text{LMG}}(p,q)$ as a function of $\eta$ for various $\kappa$ with $j=N/2=200$ (solid lines) and $j=N/2=400$ (dashed lines). Inset: Critical Wehrl entropy, ${\overline{W}}_{\text{LMG}}^c={\overline{W}}_{\text{LMG}}\left({\eta }_c\right)$, as a function of $N$ for ${\eta }_c=1.5$ (triangles), ${\eta }_c=1$ (squares), and ${\eta }_c=0.5$ (diamonds). Red dotted lines are of the form ${\overline{W}}_{\text{LMG}}^c\sim {\gamma }_{W}ln\left(N\right)$. (c) Values of the finite-size scaling exponents of ${\overline{M}}_{2,\text{LMG}}^c$ and ${\overline{W}}_{\text{LMG}}^c$ for several ${\eta }_c$ with $j=N/2=200$. (d) Critical values ${\eta }_{c,M},{\eta }_{c,W}$ estimated from the extrema of ${\overline{M}}_{2,\text{LMG}}$ and ${\overline{W}}_{\text{LMG}}$, respectively, for different values of $\kappa$ with $j=N/2=400$. The dot-dashed is the analytical result from Eq. (15). The axes in all figures are dimensionless.

Figure 6

Second moment (upper panel) and Wehrl entropy (bottom panel) of the marginal distributions of ${\overline{Q}}_{\text{LMG}}(p,q)$ as a function of $\eta$ for several $\kappa$ values with $j=N/2=200$. The vertical green dashed lines in both panels mark the critical values ${\eta }_c$ for each corresponding $\kappa$. The axes in all panels are dimensionless.

Figure 7

Schematic illustration of the excited state phase diagram of the CT model. The ground state of ${\widehat{H}}_{\text{CT}}$ at ${\xi }_0>1$ is quenched to ${\xi }_1$, which results in two different phases with critical line takes place at ${\xi }_1^c=2{\xi }_0/({\xi }_0+1)$. For ${\xi }_1>{\xi }_1^c$, the quenched system has energy less than the critical energy of ESQPT, ${\mathcal{E}}_{\text{CT}}({\xi }_0,{\xi }_1)<E_c/j=-2$, while the quenched energy is greater than $E_c/j=-2$ when ${\xi }_1>{\xi }_1^c$. At the critical quench ${\xi }_1={\xi }_1^c$, the quenched energy equals to the critical energy of ESQPT. In addition, two phases are also identified in the different classical dynamics of a classical particle in a double well potential, which exhibits the localized behaviors for ${\xi }_1>{\xi }_1^c$, whereas it displays delocalized behaviors as long as ${\xi }_1>{\xi }_1^c$. The white dashed line indicates the critical line ${\xi }_1^c$. Further details are discussed in the main text. The axes in the figure are dimensionless.

Figure 8

Snapshots of the rescaled spin Husimi function ${\mathcal{Q}}_t(p_1,q_1)=Q_t(p_1,q_1)/Q_t^m(p_1,q_1)$ with $Q_t^m(p_1,q_1)$ being the maximum value of $Q_t(p_1,q_1)$, at different time steps for the CT model with ${\xi }_1=2.5,1.5,0.5$ (from top to bottom). Other parameters are: $j=30$ and ${\xi }_0=3$. The color scale of Fig. 3 has been used. The axes in all figures are dimensionless.

Figure 9

Rescaled long-time averaged Husimi function ${\overline{\mathcal{Q}}}_{\text{CT}}(p_1,q_1)={\overline{Q}}_{\text{CT}}(p_1,q_1)/{\overline{Q}}_{\text{CT},m}$ of the CT model for (a) ${\xi }_1=2.5$, (b) ${\xi }_1=1.5$, and (c) ${\xi }_1=0.5$. By ${\overline{Q}}_{\text{CT},m}$ we denote the maximum value of ${\overline{Q}}_{\text{CT}}(p_1,q_1)$. The color scale in Fig. 3 has been employed for panels (a–c). Marginal distributions ${\overline{Q}}_{\text{CT}}\left(q_1\right)$ (d) and ${\overline{Q}}_{\text{CT}}\left(p_1\right)$ (e) of ${\overline{Q}}_{\text{CT}}(p_1,q_1)$ for several values of ${\xi }_1$ (see legend). All panels: $j=30$ and ${\xi }_0=3$. The axes in all panels are dimensionless.

Figure 10

(a) Second moment of ${\overline{Q}}_{\text{CT}}(p_1,q_1)$ as a function of ${\xi }_1$ for different system sizes, $j$, with ${\xi }_0=3$ and ${\xi }_1^c=1.5$ [see Eq. (20)]. Inset: ${\overline{M}}_{2,\text{CT}}$ as a function of $j$ at ${\xi }_1=0.3$ (blue squares), ${\xi }_1=1.5$ (red circles), and ${\xi }_1=2.7$ (green diamonds). Dashed lines are functions of the form ${\overline{M}}_{2,\text{CT}}\sim j^{-{\nu }_M}$. (b) Wehrl entropy of ${\overline{Q}}_{\text{CT}}(p_1,q_1)$ as a function of ${\xi }_1$ for different $j$ with ${\xi }_0=3$. Inset: ${\overline{W}}_{\text{CT}}$ as a function of $j$ for ${\xi }_1=0.3$ (blue squares), ${\xi }_1=1.5$ (red circles), and ${\xi }_1=2.7$ (green diamonds). Dotted lines have a dependence ${\overline{W}}_{\text{CT}}\sim {\nu }_{W}ln\left(j\right)$. (c) Finite-size scaling exponents ${\nu }_M$ and ${\nu }_W$ as a function of ${\xi }_1$ with ${\xi }_0=3$. (d) Estimated critical points ${\xi }_{1,M}^c,{\xi }_{1,W}^c$, obtained from the minima in $d{\nu }_{M\left(W\right)}/d{\xi }_1$, as a function of ${\xi }_0$. The dot-dashed line denotes the analytical result from Eq. (20). All quantities are unitless.

Figure 11

Second moment (upper panel) and Wehrl entropy (bottom panel) of the marginal distributions of ${\overline{Q}}_{\text{CT}}(p_1,q_1)$ as a function of ${\xi }_1$ for different $j$ with ${\xi }_0=3$ and ${\xi }_1^c=1.5$. All quantities are unitless.

Figure 12

In the classical limit, both LMG and CT models can be described as a particle (green dot) in a double well potential ${\mathcal{V}}^c\left(x\right)$ with $x=q$ (for the LMG model) and $x=q_1$ (for the CT model). The particle's energy is indicated by a violet dashed line. If the quantum quench leads to a particle with an energy lower than the top of the potential barrier [panel (a)], then the particle stays confined into either the left or right well. In the critical quench case [panel (b)], the energy of the particle equals the height of the central barrier, and the particle is able to explore the origin for the first time. In case that after the quench the particle's energy is larger than the central barrier [panel (c)], the particle can surpass the central barrier and it moves periodically between the two wells. Panels (d)–(f): Snapshots of the classical dynamics of the LMG model for different $\eta$, a maximum time $t=200$, and $\kappa =0.4$. Blue dots correspond to an initial condition with $q<0$, while red dots correspond to an initial condition with $q>0$. Panels (g–i): Snapshots of the classical dynamics of the CT model for different ${\xi }_1$, a maximum time $t=700$, and ${\xi }_0=3$. As in the LMG model, blue and red dots in each panel correspond to different initial conditions, $q_1<0$ (blue) and $q_1>0$ (red). All quantities are unitless.