Detecting phase transitions and crossovers in Hubbard models using the fidelity susceptibility

A generalized version of the fidelity susceptibility of single-band and multiorbital Hubbard models is systematically studied using single-site dynamical mean-field theory in combination with a hybridization expansion continuous-time quantum Monte Carlo impurity solver. We find that the fidelity susceptibility is extremely sensitive to changes in the state of the system. It can be used as a numerically inexpensive tool to detect and characterize a broad range of phase transitions and crossovers in Hubbard models, including (orbital-selective) Mott metal-insulator transitions, magnetic phase transitions, high-spin to low-spin transitions, Fermi-liquid to non-Fermi-liquid crossovers, and spin-freezing crossovers.

Figure 1

Mott metal-insulator transition in the single-band Hubbard model on a Bethe lattice ($\beta =100.0,t=0.5$). (a) Interaction-driven transition at half-filling, i.e., $\mu =U/2.0$. (b) Doping-driven transition for $U=4.0$. The two panels share the same legend. Here, ${\tilde{\chi }}_{\text{FS}}={\tilde{\chi }}_{\text{FS}}^{11},{\kappa }_{\text{L}}^1={\kappa }_{\text{L}}^{1\uparrow }+{\kappa }_{\text{L}}^{1\downarrow }$, and ${\kappa }_{\text{R}}^1={\kappa }_{\text{R}}^{1\uparrow }+{\kappa }_{\text{R}}^{1\downarrow }$. The definition for $\mathrm{\Delta }{\chi }_{\text{loc}}$ can be found in Eq. (9), and the growth of this quantity indicates the emergence of local magnetic moments (see Sec. 3e for more explanations). The dashed line shows the total occupation number $N$ ($=n_{1\downarrow }+n_{1\uparrow }$). The green region indicates the Mott insulating phase.

Figure 2

Orbital-selective Mott metal-insulator transition in the two-band Hubbard model on a Bethe lattice ($\beta =100.0,t_1=0.5,t_2=1.0,\text{and}J=U/4.0$). Here, ${\tilde{\chi }}_{\text{FS}}={\tilde{\chi }}_{\text{FS}}^{11}+{\tilde{\chi }}_{\text{FS}}^{22}+{\tilde{\chi }}_{\text{FS}}^{12}+{\tilde{\chi }}_{\text{FS}}^{21},{\kappa }_{\text{L}}^a={\kappa }_{\text{L}}^{a\uparrow }+{\kappa }_{\text{L}}^{a\downarrow }$, and ${\kappa }_{\text{R}}^a={\kappa }_{\text{R}}^{a\uparrow }+{\kappa }_{\text{R}}^{a\downarrow }$ ($a=1,2$). The abbreviation “OSMP” means the orbital-selective Mott phase.

Figure 3

Antiferromagnetic phase transition in the half-filled single-band Hubbard model on a bipartite Bethe lattice ($t=0.5\text{and}U=4.0$). For these model parameters, both the antiferrromagnetic (AFM) and paramagnetic (PM) phases are (Mott) insulating. Here, ${\tilde{\chi }}_{\text{FS}}={\tilde{\chi }}_{\text{FS}}^{11}$ and $M=|\langle n_{1\uparrow }-n_{1\downarrow }\rangle |$ is the staggered magnetization.

Figure 4

(a) High-spin to low-spin transition in the two-band Hubbard model on a Bethe lattice ($\beta =50.0,t=1.0,{\mathrm{\Delta }}_{\text{cf}}=2.5,\text{and}J=U/4.0$). The dashed line shows the occupation number for the high-lying band 1. (b) Fermi-liquid to non-Fermi-liquid and spin-freezing crossovers in the same model [zoomed view of the $U\in [4.0,7.0]$ region in (a)]. Here ${\tilde{\chi }}_{\text{FS}}={\tilde{\chi }}_{\text{FS}}^{11}+{\tilde{\chi }}_{\text{FS}}^{22}+{\tilde{\chi }}_{\text{FS}}^{12}+{\tilde{\chi }}_{\text{FS}}^{21}$. The low-energy scattering rate ${\gamma }_{\alpha }$ is defined via Eq. (8). The definition for $\mathrm{\Delta }{\chi }_{\text{loc}}$, which can be used to locate the spin-freezing crossover, can be found in Eq. (9). See main text for more explanations. The ${\tilde{\chi }}_{\text{FS}}$ and $\mathrm{\Delta }{\chi }_{\text{loc}}$ data are rescaled for a better visualization.

Figure 5

Spin-freezing crossover in the two-band Hubbard model on a Bethe lattice ($\beta =50.0,t=1.0,\text{and}J=U/6.0$). (a) $U=12.0$. (b) $U=6.0$. (c) $U=3.0$. The three panels share the same legend. Here ${\tilde{\chi }}_{\text{FS}}={\tilde{\chi }}_{\text{FS}}^{11}+{\tilde{\chi }}_{\text{FS}}^{22}+{\tilde{\chi }}_{\text{FS}}^{12}+{\tilde{\chi }}_{\text{FS}}^{21}$. The definition for $\mathrm{\Delta }{\chi }_{\text{loc}}$ can be found in Eq. (9). The dashed line shows the total occupation number $N={\sum }_{\alpha ,\sigma }{n}_{\alpha ,\sigma }$. The abbreviation “FL” means the Fermi-liquid state, and “Fro. Mom.” the spin-frozen moment regime. In (b) and (c), ${\tilde{\chi }}_{\text{FS}}$ and $\langle S_z(\beta /2)S_z\left(0\right)\rangle$ are rescaled for a better visualization.

Figure 6

High-spin to low-spin transition and spin-freezing crossover in the two-band Hubbard model on a Bethe lattice ($\beta =50.0,t=1.0,{\mathrm{\Delta }}_{\text{cf}}=2.5,\text{and}J=U/4.0$). (a) Generalized variance ${\tilde{\chi }}_{\kappa }$, which is defined in Eq. (A1). The data are rescaled for a better visualization. (b) Histograms of perturbation expansion order $\kappa$ for selected $U$ parameters. (c) Kurtosis analysis of histograms of perturbation expansion order $\kappa$ for selected $U$ parameters. The kurtosis ${\gamma }_{\kappa }$ is defined in Eq. (A3). See text for more explanations.