Thermal phase transitions for Dicke-type models in the ultrastrong-coupling limit

We consider the Dicke model in the ultrastrong-coupling limit to investigate thermal phase transitions and their precursors at finite particle numbers $N$ for bosonic and fermionic systems. We derive partition functions with degeneracy factors that account for the number of configurations and derive explicit expressions for the Landau free energy. This allows us to discuss the difference between the original Dicke (fermionic) and the bosonic case. We find a crossover between these two cases that shows up, for example, in the specific heat.

Figure 1

The specific heat per particle ($k_B=1$) as a function of ${\alpha }^{-2}\equiv \omega /\left(g^2\beta \right)$ (proportional to the temperature) in the Dicke case of $N$ localized spin-$\frac{1}{2}$s, [derived from the fermionic partition sum Eq. (5)], for $N=20,75,400$, and 1000 particles (dotted line, $N\to \infty$). (Upper inset) Shift of peak from critical point ${\alpha }^{*}=1/\sqrt{2}$ for all $N⩽1000$. (Lower inset) Peak height at maximum. (Dashed lines) Fit of the numerical data with $f_{\mathrm{upper}}^1\equiv 1.64-\frac{3.9}{\sqrt{N}}$ and $f_{\mathrm{lower}}^1\equiv 1.98-\frac{3.1}{\sqrt{N}}$.

Figure 2

The specific heat per particle as in Fig. 1 but for the bosonic partition sum Eq. (7), for $N=2,5,12,20$, and 40 particles. (Dashed lines) Linear fit $f_{\mathrm{upper}}^2\equiv 0.6+0.2N$ for maximum position (upper inset); $f_{\mathrm{lower}}^2\equiv \frac{1.1}{N}$ fit for maximum (lower inset).

Figure 3

Crossover between bosonic and Dicke case in the Landau free energy, Eq. (12), for $N=400$ bosons and coupling strengths $\alpha =0.2,0.4,0.6,1.0$ within each graph. (Upper left) $N_s=1$ ($N_s$, number of two-level sites); this corresponds to the simple bosonic case Eq. (9) [and Eq. (11) for $N\to \infty$], that is, the absence of a phase transition, since the number of minima remains two irrespective of $\alpha$. The phase transition is re-established for macroscopic degeneracy (upper right, $N_s=N/4$; lower left, $N_s=N$), since a transition from a double- to single-minimum occurs. The lower right shows the Dicke case ${\Phi }^{\mathrm{D}}\left(y\right)$, Eq. (8), with the phase transition occurring at ${\alpha }^{*}=1/\sqrt{2}\approx 0.71$.

Figure 4

The specific heat per particle at fixed particle number $N=400$ for different number of two-level sites $N_s=1$ (bosonic case, almost coincides with $x$ axis), $N_s=100$, $N_s=400$ (Dicke case), and $N_s=600,5000$ as obtained from Eqs. (4) and  (6). (Upper inset) With increasing $N_s$ the peak position in $C/N$ shifts towards the critical point ${\alpha }^{-2}=2$ of the Dicke case. (Lower inset) Increase of maximum of $C/N$ with $N_s$. (Dashed lines) Fit of the numerical data with $f_{\mathrm{upper}}^4\equiv 1.82-\frac{275}{N_s}$ and $f_{\mathrm{lower}}^4\equiv \frac{1.5N_s}{458.6+N_s}$.