Topology of the Fermi surface and universality of the metal-metal and metal-insulator transitions: $d$-dimensional Hatsugai-Kohmoto model as an example

Abstract

The earlier theory [1] of the quantum phase transitions related to the change of the Fermi Surface Topology (FST) is advanced. For such transitions the Fermi surface as a quantum critical manifold determined by the Lee-Yang zeros, the order parameter $\mathcal{P}$ as the $d$-volume of the Fermi sea, and the special FST universality class were introduced in [1]. The exactly solvable Hatsugai-Kohmoto (HK) $d$-dimensional ($d=1,2,3$) model of interacting fermions is analyzed. We explore the relation between the Lee-Yang zeros, the Luttinger and the plateau (Oshikawa) theorems. The validity of the Luttinger theorem in the HK model is confirmed. It is shown that the order parameter $\mathcal{P}$ and the FST universality class describe the transitions between metal and band/Mott insulators, as well as the Lifshitz and van Hove gapless-to-gapless transitions. The gapless phases are established to be the Landau Fermi liquids (metals). In addition to the conventional paradigm with a continuous order parameter, we apply the homology theory to analyze the FST transitions. They are critical points of the Morse function. To quantify FST we use the Euler characteristic, which is calculated for each phase of the HK model. We claim that the FST universality class is robust with respect to interactions and other model details, under the condition that the critical points are non-degenerate.

Figures (13)

• Figure 1: (a) The distribution function $n_{\mathbf{k}\sigma}(\xi)$ at $T=0$ given by Eq. \ref{['nksig0']}; (b) The ground-state values $n_{\mathbf{k}\sigma}(\xi)=\{0,\frac{1}{2},1\}$, where $\xi=\varepsilon-\mu$, shown in the plane $(\varepsilon,\mu)$. The bold solid lines are $\mu =\varepsilon \pm U/2$, the energy band $\varepsilon \in [-d,d]$.
• Figure 2: The Matsubara Green's function $G_\sigma (\xi, 0)$\ref{['GF']} at $T=0$. Note that $n_{\mathbf{k}\sigma}(\xi)=1/2$ at $\xi=0$, see Fig. \ref{['Nk-AB']}a.
• Figure 3: (a,b) One-dimensional spectra $\varepsilon_\pm(k)$ in the cases $U<U_c$ and $U>U_c$, respectively. The critical values of chemical potential: (a): ($\mu_{c,1},\mu_{L,1},\mu_{L,2},\mu_{c,2}$), from bottom to the top; (b): ($\mu_{c,1},\mu_{M,1},\mu_{M,2},\mu_{c,2}$), from bottom to the top. (c,d) The interacting chemical potential $\mu$ as a function of $\mu_\circ$ in the cases $U<U_c$ and $U>U_c$, respectively. The cusp of $\mu$ at the Lifshitz point seen in panel (c) becomes a discontinuity at the Mott transition, panel (d). The plots (a) and (c) are done for $U=0.8$, $\mu_{L,\sharp}=\pm 0.6$, $\mu_{c,\sharp}=\pm 1.4$ ; (b) and (d) for $U=3.6$, $\mu_{M,\sharp}=\pm 0.8$, $\mu_{c,\sharp}=\pm 2.8$.
• Figure 4: $d=1$ ($U_c=2$): (a,b) -- filling $\bar{n}(\mu)$; (c,d) -- compressibility $\chi(\mu)$ in the cases $U<U_c$ and $U>U_c$. The plots (a) and (c) are done for $U=0.8$ with the critical values of chemical potential: $\mu_{L,\sharp}=\pm 0.6$, $\mu_{c,\sharp}=\pm 1.4$ . For the plots (b) and (d): $U=3.6$, $\mu_{M,\sharp}=\pm 0.8$, $\mu_{c,\sharp}=\pm 2.8$. The Euler characteristics $\chi_E$ for each phase are shown in parentheses, see panels (c,d).
• Figure 5: $d=2$ ($U_c=4$): (a,b) -- filling $\bar{n}(\mu)$; (c,d) -- compressibility $\chi(\mu)$ in the cases $U<U_c$ and $U>U_c$. The plots (a) and (c) are done for $U=2.6$ with the critical values of chemical potential: $\mu_{L,\sharp}=\pm 0.7$, $\mu_{c,\sharp}=\pm 3.3$. The van Hove critical points correspond to $\mu_{vH,\sharp}=\pm 1.3$ and $\bar{n}=1/4,3/4$. For the plots (b) and (d): $U=8$, $\mu_{M,\sharp}=\pm 2$, $\mu_{c,\sharp}=\pm 6$, $\mu_{vH,\sharp}=\pm 4$. The Euler characteristics $\chi_E$ for each phase are shown in parentheses, see panels (c,d).