Variational cluster approach to thermodynamic properties of interacting fermions at finite temperatures: A case study of the two-dimensional single-band Hubbard model at half filling

We formulate a finite-temperature scheme of the variational cluster approximation (VCA) particularly suitable for an exact-diagonalization cluster solver. Based on the analytical properties of the single-particle Green's function matrices, we explicitly show the branch-cut structure of logarithm of the complex determinant functions appearing in the self-energy-functional theory (SFT) and whereby construct an efficient scheme for the finite-temperature VCA. We also derive the explicit formulas for entropy and specific heat within the framework of the SFT. We first apply the method to explore the antiferromagnetic order in a half-filled Hubbard model by calculating the entropy, specific heat, and single-particle excitation spectrum for different values of on-site Coulomb repulsion $U$ and temperature $T$. We also calculate the $T$ dependence of the single-particle excitation spectrum in the strong coupling region, and discuss the overall similarities to and the fine differences from the spectrum obtained by the spin-density-wave mean-field theory at low temperatures and the Hubbard-I approximation at high temperatures. Moreover, we show a necessary and sufficient condition for the third law of thermodynamics in the SFT. On the basis of the thermodynamic properties, such as the entropy and the double occupancy, calculated via the $T$ and/or $U$ derivative of the grand potential, we obtain a crossover diagram in the $(U,T)$ plane, which separates a Slater-type insulator and a Mott-type insulator. Next, we demonstrate the finite-temperature scheme in the cluster-dynamical-impurity approximation (CDIA), i.e., the VCA with noninteracting bath orbitals attached to each cluster, and study the paramagnetic Mott metal-insulator transition in the half-filled Hubbard model. Formulating the finite-temperature CDIA, we first address a subtle issue regarding the treatment of the artificially introduced bath degrees of freedom which are absent in the originally considered Hubbard model. We then apply the finite-temperature CDIA to calculate the finite-temperature phase diagram in the $(U,T)$ plane. Metallic, insulating, coexistence, and crossover regions are distinguished from the bath-cluster hybridization-variational-parameter dependence of the grand-potential functional. We find that the Mott transition at low temperatures is discontinuous, and the coexistence region of the metallic and insulating states persists down to zero temperature. The result obtained here by the finite-temperature CDIA is complementary to the previously reported zero-temperature CDIA phase diagram.

Figure 1

Complex $z$ plane for the contour integral appearing in Eqs. (19) and (23). The fermionic Matsubara frequencies are denoted by solid dots on the imaginary axis, where the Fermi-distribution function $n_{\mathrm{F}}\left(z\right)$ displays singularities. The contour $C_R$, on which the complex frequency is represented as $z=R{\mathrm{e}}^{\mathrm{i}\theta }$, is shown by a red solid line with arrow. The branch cuts and branch points of $lndet[\mathit{I}-\mathit{V}\left(\tilde{\mathbf{k}}\right){\mathit{G}}^{\prime }\left(z\right)]$ (see Sec. 3c) are also indicated by red dotted lines and crosses on the real axis, respectively.

Figure 2

The branch cuts of $lndet\left[\mathit{I}-\mathit{V}\left(\tilde{\mathbf{k}}\right){\mathit{G}}^{\prime }\left(z\right)\right]$ with $N_{\mathrm{pole}}=4$ in the complex $z$ plane [see Eq. (25)]. The crosses represent the poles of $det{\mathit{G}}^{\prime }\left(z\right)$ and $det\tilde{\mathit{G}}(\tilde{\mathbf{k}},z)$. The thick red dotted lines represent the branch cuts across which $\mathrm{Im}lndet\left[\mathit{I}-\mathit{V}\left(\tilde{\mathbf{k}}\right){\mathit{G}}^{\prime }\left(z\right)\right]$ changes discontinuously by $\pm 2\pi$. The branch cuts are all on the real-frequency axis in a finite range bounded by the largest and smallest poles of $det{\mathit{G}}^{\prime }\left(z\right)$ and $det\tilde{\mathit{G}}(\tilde{\mathbf{k}},z)$.

Figure 3

Intensity plots of (a) ${\varphi }_1\left(z\right)=\mathrm{Im}lndet{\mathit{G}}_{\sigma }^{\prime }\left(z\right)$, (b) ${\varphi }_2\left(z\right)=\mathrm{Im}lndet{\tilde{\mathit{G}}}_{\sigma }(\tilde{\mathbf{k}},z)$, and (c) ${\varphi }_3\left(z\right)=\mathrm{Im}lndet\left[\mathit{I}-{\mathit{V}}_{\sigma }\left(\tilde{\mathbf{k}}\right){\mathit{G}}_{\sigma }^{\prime }\left(z\right)\right]$ in the complex $z$ plane for the Hubbard model on the square lattice with $L_{\mathrm{c}}=2\times 2$ and $\tilde{\mathbf{k}}=(0,0)$. The other parameters are $U/t=8$, $\mu /t=4$, and $T/t=0.1$. (d) Enlarged figure of (c) near the real axis. The phases ${\varphi }_1\left(z\right)$, ${\varphi }_2\left(z\right)$, and ${\varphi }_3\left(z\right)$ are plotted in the range of $-\pi <{\varphi }_i\le \pi$, as indicated by the color bar.

Figure 4

Clusters used in this study are indicated by solid lines. The size of each cluster is also indicated. The primitive translational vectors of each cluster is summarized in Table 1. The red (blue) circles represent sites on sublattice $A$ ($B$) of the square lattice.

Figure 5

The Néel temperature $T_{\mathrm{N}}$ obtained by the VCA with various clusters indicated in the figure (also see Fig. 4 and Table 1).

Figure 6

The grand-potential functional $\mathrm{\Omega }$ as a function of variational parameter $h^{\prime }$ for $U/t=8$ at temperatures $T/t=0.01,0.02,\cdots ,0.34$, and 0.35 (from violet to red lines). The chemical potential is set at $\mu =U/2$. The clusters used are (a) $2\times 2$, (b) $4\times 2$, and (c) 10 sites. Each dot indicates the variational parameter $h^{\prime *}$ where the stationary condition is satisfied with the lowest grand potential for a given temperature. The solution with $h^{\prime *}\ne 0$ indicates the antiferromagnetic state, while $h^{\prime *}=0$ corresponds to the paramagnetic state. Notice that the grand-potential functionals for $T/t\le 0.04$ are almost degenerate in the cases studied here.

Figure 7

Temperature dependence of (a) entropy $S\left(T\right)$ and (b) specific heat $C\left(T\right)$ at $U/t=8$ for the paramagnetic solution and the antiferromagnetic solution with the clusters of $L_{\mathrm{c}}=2\times 2$, $4\times 2$, and 10 sites. In (b), $C\left(T\right)$ for the antiferromagnetic solution just below $T_{\mathrm{N}}$ is connected to that for the paramagnetic solution just above $T_{\mathrm{N}}$. All the results are obtained for the insulating phase in the sense that the single-particle gap at the Fermi level is finite (see also Fig. 8)

Figure 8

Single-particle excitation spectrum $\mathcal{A}(\mathbf{k},\omega )$ [(a)–(e)] and the imaginary part $\mathcal{S}(\mathbf{k},\omega )$ of the self-energy [(f)–(j)] for the half-filled Hubbard model with $U/t=8$ calculated using [(a) and (f)] the self-consistent SDW mean-field theory at $T/t=0$ in the antiferromagnetic state, [(b) and (g)] the VCA at $T/t=0.001$ in the antiferromagnetic state, [(c) and (h)] the VCA at $T/t=0.3$ in the paramagnetic state, [(d) and (i)] the VCA at $T/t=0.5$ in the paramagnetic state, and [(e) and (j)] the Hubbard-I approximation in the paramagnetic state. The horizontal line at $\omega =0$ denotes the Fermi level. The Lorentzian broadening of $\eta /t=0.2$ is used. The $L_{\mathrm{c}}=10$ cluster is used for the VCA in (b)–(d) and (g)–(i). The results for the Hubbard-I approximation in (e) and (j) are independent of the temperature. Note that different figures use different intensity scales as indicated in the color bars.

Figure 9

Temperature dependence of (a) the entropy $S\left(T\right)$ and (b) the specific heat $C\left(T\right)$ for $U/t=1,2,4,8,16$, and 32. The horizontal dashed lines in (a) represent $S\left(T\right)=ln2=0.693$ and $S\left(T\right)=ln4=1.386$. (c) and (d) are enlarged plots of (a) and (c) for $0\le T/t\le 1$, respectively. The results are for the paramagnetic solution with the cluster of $L_{\mathrm{c}}=4\times 2$ sites.

Figure 10

The contour plots of (a) the entropy $S/ln2$, (b) the double occupancy $\langle \widehat{D}\rangle$, (c) the specific heat $C$, (d) the double-occupancy susceptibility ${\chi }_D$, and (e) the mixed derivative $-{\partial }_U{\partial }_T\mathrm{\Omega }$ in the $(U,T)$ plane with a range of $0\le U/t\le 16$ and $0\le T/t\le 3$. The results are obtained for the paramagnetic state with the cluster of $L_{\mathrm{c}}=4\times 2$ sites. The black lines with open circles in (a), (b), and (e) indicate the contours on which ${\partial }_U{\partial }_T\mathrm{\Omega }=0$. The black dashed line in (a) indicates $S(U,T)=ln2$. The black line with open triangles (squares) in (c) indicates the peak of $C(U,T)$ at low (high) temperatures, and the black line with inverted open triangles indicates the dip of $C(U,T)$. The black line with open diamonds in (d) indicates the contour on which ${\chi }_D$ takes the maximum at finite $U$.

Figure 11

The double occupancy $\langle \widehat{D}\rangle$ as a function of $T$ for $U/t=2,4,6,8,10,12,14$, and 16 (from top to bottom). The results are for the paramagnetic state with the cluster of $L_{\mathrm{c}}=4\times 2$ sites. The increment of $T/t$ is set to be 0.01. For comparison, the double occupancy, $\langle \widehat{D}\rangle =1/\left[2+2exp(U/2T)\right]$, of the single-site Hubbard model (i.e, in the atomic limit) at half-filling is also shown by dashed lines for the same values of $U/t$.

Figure 12

A finite-temperature crossover diagram for the half-filled Hubbard model in the paramagnetic state obtained by the VCA with the cluster of $L_{\mathrm{c}}=4\times 2$ sites. $T_{\mathrm{spin}}$ (red line with solid triangles) denotes the temperature where $C\left(T\right)$ shows the low-temperature peak, $T_{\mathrm{charge}}$ (orange line with solid squares) denotes the temperature where $C\left(T\right)$ shows the high-temperature peak, and $T_{\mathrm{dip}}$ (grey line with solid inverted triangles) denotes the temperature where $C\left(T\right)$ shows the dip between $T_{\mathrm{spin}}$ and $T_{\mathrm{charge}}$. The violet dashed line indicates $T$ and $U$, where $S(T,U)=ln2$. The blue line with open diamonds denotes the value of $U$ on which ${\chi }_D$ takes a maximum with increasing $U$ for a given $T$. The green lines with circles denote the parameters across which $\partial S/\partial U$ changes the sign. The antiferromagnetic correlations are expected to develop below $T_{\mathrm{spin}}$. The large spin fluctuations are expected in the region around the blue line with open diamonds. Local moments are formed in the region below the green line with open triangles for $U\gtrsim 6$.

Figure 13

Schematic figures of (a) original, (b) auxiliary, and (c) reference systems considered in the CDIA. (a) The original system consists of the correlated sites (blue circles). (b) The auxiliary system consists of the correlated sites and bath orbitals (red squares) but they are decoupled. (c) The reference system is composed of a collection of the disconnected small clusters, each of which consists of the correlated sites and bath orbitals with hybridization (solid lines between blue circles and red squares). The corresponding actions of the original system $\mathcal{S}$ in Eq. (115), the auxiliary system ${\mathcal{S}}_{\mathrm{aux}}$ in Eq. (125), and the reference system ${\mathcal{S}}_{\mathrm{ref}}$ in Eq. (122) are also indicated.

Figure 14

The grand-potential functional ${\mathrm{\Omega }}_{\mathrm{aux}}\left(V\right)$ per site of the auxiliary system as a function of the variational parameter $V^{\prime }$ for (a) $U/t=5.4$ in the metallic phase, (b) $U/t=5.8$ in the vicinity of the Mott metal-insulator transition, and (c) $U/t=6.2$ in the paramagnetic Mott insulating phase at temperatures $T/t=0.001,0.005,0.01,0.015,\cdots ,0.055$, and 0.06 (from violet to red lines). The cluster composing $L_{\mathrm{c}}=2\times 2$ correlated sites connected to four bath orbitals is used. Dots (crosses) indicate the solutions of nonzero $V^{\prime }$ with the lowest (second-lowest) grand potential satisfying the stationary condition in Eq. (148).

Figure 15

The finite-temperature phase diagram for the paramagnetic Mott metal-insulator transition obtained by the finite-temperature CDIA. The cluster composing $L_{\mathrm{c}}=2\times 2$ correlated sites connected to four bath orbitals is used. The three lines with dots represent $U_{\mathrm{c}1}\left(T\right)$, $U_{\mathrm{c}}\left(T\right)$, and $U_{\mathrm{c}2}\left(T\right)$, which terminate at a critical point $(U^{*}/t,T^{*}/t)=(5.95,0.061)$. Black dots denote the first-order transition boundary separating the metallic and insulating phases. The metallic and insulating phases coexist in the region surrounded by blue dots.

Figure 16

Single-particle excitation spectrum $\mathcal{A}(\mathbf{k},\omega )$ [(a)–(c)] and the imaginary part $\mathcal{S}(\mathbf{k},\omega )$ of the self-energy [(d)–(f)] for the half-filled Hubbard model at [(a) and (d)] $U/t=5.4$ and $T/t=0.02$ in the metallic phase, [(b) and (e)] $U/t=5.95$ and $T/t=0.061$ at the critical point, and [(c) and (f)] $U/t=6.2$ and $T/t=0.02$ in the paramagnetic Mott insulating phase. The horizontal line at $\omega =0$ denotes the Fermi level. The Lorentzian broadening of $\eta /t=0.2$ is used. The CDIA is employed with the cluster of $L_{\mathrm{c}}=2\times 2$ correlated sites connected to four bath orbitals. Note that different figures use different intensity scales as indicated in the color bars.

Figure 17

Contour ${\mathrm{\Gamma }}^{\prime }$ (dashed lines) encloses the fermionic Matsubara frequencies (solid circles) in a clockwise manner. Contour $\mathrm{\Gamma }$ (solid lines) enclosing the real axis is obtained by deforming contour ${\mathrm{\Gamma }}^{\prime }$. The branch cuts and the branch points of $lndet\left[\mathit{I}-\mathit{V}\left(\tilde{\mathbf{k}}\right){\mathit{G}}^{\prime }\left(z\right)\right]$ are indicated by magenta dotted lines and black crosses on the real axis, respectively (see also Figs. 1 and 2).

Figure 18

Ground-state energy $E_{\mathrm{ED}}$ per site for (a) $U/t=4$ and (b) $U/t=8$ at half filling calculated by the exact-diagonalization method for different size of clusters under the open-boundary conditions. The pluses, crosses, and dots represent the energies plotted with respect to $1/L_{\mathrm{c}}$, $1/\sqrt{L_{\mathrm{c}}}$, and $1-Q$, respectively. The blue horizontal line is the ground-state energy in the thermodynamic limit obtained by the auxiliary-field QMC method taken from Ref. [197].

Figure 19

Ground-state energy $E$ per site for (a) $U/t=4$ and (b) $U/t=8$ at half-filling. The empty and the filled circles denote $E\left(h^{\prime *}\right)$ and $E(h^{\prime *},\delta t^{\prime *})$, respectively. The dots are obtained by the exact-diagonalization method for the clusters under the open-boundary conditions, as shown also in Fig. 18. The blue horizontal line is the ground-state energy in the thermodynamic limit obtained by the auxiliary-field QMC method taken from Ref. [197].