From thermal to excited-state quantum phase transition: The Dicke model

We study the thermodynamics of the full version of the Dicke model, including all the possible values of the total angular momentum $j$, with both microcanonical and canonical ensembles. We focus on both the excited-state quantum phase transition, appearing in the microcanonical description of the maximum angular momentum sector, $j=N/2$, and the thermal phase transition, which occurs when all the sectors are taken into account. We show that two different features characterize the full version of the Dicke model. If the system is in contact with a thermal bath and is described by means of the canonical ensemble, the parity symmetry becomes spontaneously broken at the critical temperature. In the microcanonical ensemble, and despite that all the logarithmic singularities which characterize the excited-state quantum phase transition are ruled out when all the $j$ sectors are considered, there still exists a critical energy (or temperature) dividing the spectrum into two regions: one in which the parity symmetry can be broken, and another in which this symmetry is always well defined.

Figure 1

$J_z$ for the microcanonical ensemble (solid green points), and the canonical ensemble (dashed red line). The vertical dashed line shows the energy of the ESQPT.

Figure 2

$d{J}_z/dE$ for the microcanonical ensemble (solid green points) and the canonical ensemble (dashed red line). The vertical dashed line shows the energy of the ESQPT.

Figure 3

$J_x$ for the microcanonical ensemble (solid green points). The vertical dashed line shows the energy of the ESQPT.

Figure 4

Value of $y_0$ and expected values of $J_z$ and $J_x$, both without the symmetry-breaking term (thick line, red online) and with the symmetry-breaking term, taking the limit $\epsilon \to 0$ (thin line, green online). Critical temperature ${\beta }_c$ is marked with a dotted vertical line.

Figure 5

Microcanonical calculation for fixed and different values of $j$, for $N=1\times {10}^5$: (a) density of states, $\rho \left(E\right)$; (b) derivative of the density of states, ${\rho }^{\prime }\left(E\right)$; (c) third component of the angular momentum, $〈J_z〉$; (d) derivative of the third component of the angular momentum, $〈d{J}_z/dE〉$; (e) temperature, $\beta$; and (f) first component of the angular momentum, $〈J_x〉$. Different colors show different values of $j, j=2N/16,3N/16,...,8N/16$.

Figure 6

Temperature $\beta$ versus energy $〈E〉/N$ obtained by means of the microcanonical (solid green line) and the canonical (dashed red line) ensemble. The vertical dashed line shows the critical energy $〈E_c〉/N$. The inset shows the same results around this critical value.

Figure 7

$J_z$ versus energy $〈E〉/N$ obtained by means of the microcanonical (solid green line) and the canonical (dashed red line) ensemble. The vertical dashed line shows the critical energy $〈E_c〉/N$. The inset shows the same results around this critical value.

Figure 8

Derivative of $J_z$ versus energy $〈E〉/N$ obtained by means of the microcanonical (solid green line) and the canonical (dashed red line) ensemble. The vertical dashed line shows the critical energy $〈E_c〉/N$. The inset shows the same results around this critical value.

Figure 9

$J_x$ versus energy $〈E〉/N$ obtained by means of the microcanonical ensemble (solid green line) and the canonical ensemble with the symmetry-breaking term $\epsilon J_x$ (dashed red line). The vertical dashed line shows the critical energy $〈E_c〉/N$. The inset shows the same results around this critical value.

Figure 10

Exact numerical results (solid red circles) and microcanonical calculation (solid green curve) for the expected value of $J_z/N$, in a system with $N=50$ atoms.

Figure 11

Exact numerical results (solid red circles) and microcanonical calculation (solid green curves) for the expected value of $J_x/N$, in a system with $N=50$ atoms. Numerical results have been obtained after introducing a small symmetry-breaking term, $\epsilon J_x$, with $\epsilon =1\times {10}^{-6}$. For the analytical result, the symmetry-breaking term is not introduced, and the two disjoint regions of the phase space below the critical energy are integrated separately, to obtain the two branches of the theoretical curve.

Figure 12

Finite-size scaling for the critical energy, obtained with $J_z$ (left) and $J_x$ (right). Both cases are depicted in a double logarithmic scale. The solid lines represent the least-square fits to straight lines, showing a power-law scaling with the system size.

Figure 13

Finite-size scaling for the critical energy value of $J_z$, in a double logarithmic scale. The solid lines represent the least-square fits to straight lines, showing a power-law scaling with the system size.