Topological interpretation of the Luttinger theorem

Based solely on the analytical properties of the single-particle Green's function of fermions at finite temperatures, we show that the generalized Luttinger theorem inherently possesses topological aspects. The topological interpretation of the generalized Luttinger theorem can be introduced because (i) the Luttinger volume is represented as the winding number of the single-particle Green's function and, thus, (ii) the deviation of the theorem, expressed with a ratio between the interacting and noninteracting single-particle Green's functions, is also represented as the winding number of this ratio. The formulation based on the winding number naturally leads to two types of the generalized Luttinger theorem. Exploring two examples of single-band translationally invariant interacting electrons, i.e., simple metal and Mott insulator, we show that the first type falls into the original statement for Fermi liquids given by Luttinger, where poles of the single-particle Green's function appear at the chemical potential, while the second type corresponds to the extended one for nonmetallic cases with no Fermi surface such as insulators and superconductors generalized by Dzyaloshinskii, where zeros of the single-particle Green's function appear at the chemical potential. This formulation also allows us to derive a sufficient condition for the validity of the Luttinger theorem of the first type by applying the Rouche's theorem in complex analysis as an inequality. Moreover, we can rigorously prove in a nonperturbative manner, without assuming any detail of a microscopic Hamiltonian, that the generalized Luttinger theorem of both types is valid for generic interacting fermions as long as the particle-hole symmetry is preserved. Finally, we show that the winding number of the single-particle Green's function can also be associated with the distribution function of quasiparticles, and therefore the number of quasiparticles is equal to the Luttinger volume. This implies that the fundamental hypothesis of the Landau's Fermi-liquid theory, the number of fermions being equal to that of quasiparticles, is guaranteed if the Luttinger theorem is valid since the theorem states that the number of fermions is equal to the Luttinger volume. All these general statements are made possible because of the finding that the Luttinger volume is expressed as the winding number of the single-particle Green's function at finite temperatures, for which the complex analysis can be readily exploited.

Figure 1

Schematic figure of the diagonal element of the single-particle Green's function $G_{\alpha \alpha }\left(\omega \right)={\sum }_m{\left|Q_{\alpha m}\right|}^2/(\omega -{\omega }_m^{\left(\alpha \right)})$ (thick solid lines) on the real-frequency axis $\omega$. This toy Green's function $G_{\alpha \alpha }\left(\omega \right)$ has five poles at ${\omega }_m^{\left(\alpha \right)}=-2,-1,0,1,$ and 2 (indicated by dashed vertical lines) with spectral weight $|Q_{\alpha m}{|}^2=\frac{1}{5}$ for $m=1,2,\cdots ,5$.

Figure 2

(a) Contour $\mathrm{\Gamma }$ in complex $z$ plane. (b) Contours ${\mathrm{\Gamma }}_{<},{\mathrm{\Gamma }}_0$, and ${\mathrm{\Gamma }}_{>}$ in complex $z$ plane. Filled dots on the imaginary axis represent the Matsubara frequencies $i{\omega }_{\nu }=(2\nu +1)i\pi T$ with $\nu$ integer. The origin is indicated by an open dot in each figure.

Figure 3

Schematic figure for the contour of $det{\mathit{G}}^{-1}\left(\mathcal{C}\right)$ in Eq. (43), parametrized by $z\in \mathcal{C}$, on the complex $det{\mathit{G}}^{-1}$ plane. The arrowheads indicate the direction of the trajectory in $det\mathit{G}{\left(z\right)}^{-1}$ with $z\in \mathcal{C}$ where $\mathcal{C}(={\mathrm{\Gamma }}_{<},{\mathrm{\Gamma }}_0,\mathrm{and}{\mathrm{\Gamma }}_{>})$ is shown in Fig. 2. The winding number of $det{\mathit{G}}^{-1}$ around the origin corresponds to $n_{det{\mathit{G}}^{-1}}\left(\mathcal{C}\right)$ defined in Eq. (43). The winding number in this figure is $n_{det{\mathit{G}}^{-1}}\left(\mathcal{C}\right)=2$.

Figure 4

Schematic figure to explain the relation between $D\left(z\right)=det{\mathit{G}}_0\left(z\right)/det\mathit{G}\left(z\right)$ and the generalized Luttinger theorem. The red solid and blue dashed lines represent the integral contours $D\left(\mathcal{C}\right)$ in Eq. (55) parametrized by $z\in \mathcal{C}(={\mathrm{\Gamma }}_{<}\mathrm{and}{\mathrm{\Gamma }}_0)$. The generalized Luttinger theorem of type I with Eq. (57) is valid (violated) when $D\left(\mathcal{C}\right)$ does not (does) enclose the origin of complex $D$ plane, i.e., zero (nonzero) winding number of $D\left(z\right)$ around the origin, as indicated by red solid (blue dashed) line. The red dot on the positive real axis at $D\left(z\right)=1$ represents the noninteracting limit.

Figure 5

Schematic figures of $A_{\mathbf{k}}\left(\omega \right)=-\mathrm{Im}{G}_{\mathbf{k}}(\omega +i{\delta }^{+})/\pi$ (blue shaded region) and $G_{\mathbf{k}}\left(\omega \right)$ (red solid lines) at momentum $\mathbf{k}$ (a) inside, (b) on, and (c) outside the Fermi surface for a simple metal. Here, $\omega =0$ corresponds to the Fermi energy. The poles of $G_{\mathbf{k}}\left(\omega \right)$ are indicated by dots along with dashed vertical lines.

Figure 6

$A_{\mathbf{k}}\left(\omega \right)=-\mathrm{Im}{G}_{\mathbf{k}}(\omega +i{\delta }^{+})/\pi$ (blue solid lines) and $S_{\mathbf{k}}\left(\omega \right)=-\mathrm{Im}{\mathrm{\Sigma }}_{\mathbf{k}}(\omega +i{\delta }^{+})/\pi$ (red dashed lines) of the one-dimensional single-band Hubbard model at half-filling for (a) $U=\mu =0$, (b) $U=2\mu =6t$, and (c) same as (b) but only for $k=\pi /2$. ${\delta }^{+}/t=0.05$ is set for all calculations. Note that different figures use different intensity scales. For visibility, $S_{\mathbf{k}}\left(\omega \right)$ is divided by ${(U/t)}^2$ in (b) and (c). The results for $k=\pi /2$ in (a) and (b) are indicated by thick lines.

Figure 7

Schematic figures of $A_{\mathbf{k}}\left(\omega \right)=-\mathrm{Im}{G}_{\mathbf{k}}(\omega +i{\delta }^{+})/\pi$ (blue shaded region) and $G_{\mathbf{k}}\left(\omega \right)$ (red solid lines) with momentum $\mathbf{k}$ (a) inside, (b) on, and (c) outside the Luttinger surface for a Mott insulator. Here, $\omega =0$ corresponds to the chemical potential in the zero-temperature limit. The poles of $G_{\mathbf{k}}\left(\omega \right)$ are indicated by dots along with dashed vertical lines.

Figure 8

Contours ${\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_{<}$, and ${\mathrm{\Gamma }}_{>}$ in the complex $z$ plane, parametrized by $\varphi$ ($-\pi <\varphi \le \pi$), which correspond to those in Fig. 2.

Figure 9

Trajectory of $G_{\mathbf{k}}^{-1}\left(z\right)$ in the complex $G_{\mathbf{k}}^{-1}$ plane when $z$ moves along contour $\mathcal{C}$ ($={\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_{<}$, and ${\mathrm{\Gamma }}_{>}$) given in Eq. (95) and also in Fig. 8 for three representative momenta, i.e., $k=0$ (inside the Luttinger surface), $k=\pi /2$ (on the Luttinger surface), and $k=\pi$ (outside the Luttinger surface). The CPT is employed for the one-dimensional one-band Hubbard model defined in Eq. (87) with $U/t=6$ at half-filling. By directly counting how many times and which direction (indicated by arrows) the trajectory winds around the origin in the complex $G_{\mathbf{k}}^{-1}$ plane, we find that (a) $n_{G_{k=0}^{-1}}\left({\mathrm{\Gamma }}_{<}\right)=1$, (b) $n_{G_{k=0}^{-1}}\left({\mathrm{\Gamma }}_0\right)=0$, (c) $n_{G_{k=0}^{-1}}\left({\mathrm{\Gamma }}_{>}\right)=0$, (d) $n_{G_{k=\pi /2}^{-1}}\left({\mathrm{\Gamma }}_{<}\right)=1$, (e) $n_{G_{k=\pi /2}^{-1}}\left({\mathrm{\Gamma }}_0\right)=-1$, (f) $n_{G_{k=\pi /2}^{-1}}\left({\mathrm{\Gamma }}_{>}\right)=1$, (g) $n_{G_{k=\pi }^{-1}}\left({\mathrm{\Gamma }}_{<}\right)=0$, (h) $n_{G_{k=\pi }^{-1}}\left({\mathrm{\Gamma }}_0\right)=0$, and (i) $n_{G_{k=\pi }^{-1}}\left({\mathrm{\Gamma }}_{>}\right)=1$. These are exactly the same as those obtained by counting the number of poles and zeros of the single-particle Green's function $G_{\mathbf{k}}\left(z\right)$ given in Eqs. (92, 93, 94) (see also Table 3).

Figure 10

Trajectory of $D_{\mathbf{k}}\left(z\right)=G_{0\mathbf{k}}\left(z\right)/G_{\mathbf{k}}\left(z\right)$ in the complex $D_{\mathbf{k}}$ plane when $z$ moves along contour $\mathcal{C}$ ($={\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_{<}$, and ${\mathrm{\Gamma }}_{>}$) given in Eq. (95) and also in Fig. 8 for three representative momenta, i.e., $k=0$ (inside the Luttinger surface), $k=\pi /2$ (on the Luttinger surface), and $k=\pi$ (outside the Luttinger surface). The CPT is employed for the one-dimensional one-band Hubbard model defined in Eq. (87) with $U/t=6$ at half-filling. By directly counting how many times and which direction (indicated by arrows) the trajectory winds around the origin in the complex $D_{\mathbf{k}}$ plane, we find that (a) $n_{D_{k=0}}\left({\mathrm{\Gamma }}_{<}\right)=0$, (b) $n_{D_{k=0}}\left({\mathrm{\Gamma }}_0\right)=0$, (c) $n_{D_{k=0}}\left({\mathrm{\Gamma }}_{>}\right)=0$, (d) $n_{D_{k=\pi /2}}\left({\mathrm{\Gamma }}_{<}\right)=1$, (e) $n_{D_{k=\pi /2}}\left({\mathrm{\Gamma }}_0\right)=-2$, (f) $n_{D_{k=\pi /2}}\left({\mathrm{\Gamma }}_{>}\right)=1$, (g) $n_{D_{k=\pi }}\left({\mathrm{\Gamma }}_{<}\right)=0$, (h) $n_{D_{k=\pi }}\left({\mathrm{\Gamma }}_0\right)=0$, and (i) $n_{D_{k=\pi }}\left({\mathrm{\Gamma }}_{>}\right)=0$. Notice that the trajectory winds twice around the origin in (e). Therefore, we find that $n_D\left({\mathrm{\Gamma }}_0\right)=-2n_D\left({\mathrm{\Gamma }}_{<}\right)=-4$, satisfying the condition of type II for the generalized Luttinger theorem in Eq. (59).

Figure 11

(a) The available $\mathbf{k}$ points (red solid circles) for the $3\times 3$ unit-cell cluster of the honeycomb lattice with periodic boundary conditions. The black solid lines represent the boundaries of the Brillouin zone and the blue solid lines indicate the momentum path along which the single-particle energy-dispersion relations are shown in (b) and (c). Four high-symmetric momenta are denoted as $\mathrm{\Gamma }$: $(0,0),K$: $(2\pi /3,2\pi /\sqrt{3})$, $M$: $(0,2\pi /\sqrt{3})$, and $K^{\prime }$: $(-2\pi /3,2\pi /\sqrt{3})$. (b) The single-particle energy-dispersion relation for $U=0$. (c) Same as (b) but for $U=5t$ obtained within the Hubbard-I approximation at half-filling. The red vertical lines in (b) and (c) represent the available $\mathbf{k}$ points for the $3\times 3$ unit-cell cluster shown in (a).

Figure 12

Arguments of $det{\mathit{G}}^{-1}\left(z\right)$ and $D\left(z\right)$ when $z$ moves along contour $\mathcal{C}(={\mathrm{\Gamma }}_0\mathrm{and}{\mathrm{\Gamma }}_{<}$ in Fig. 8) for the Hubbard model on the honeycomb lattice with $3\times 3$ unit cells at half-filling. (a)–(d): $arg[det{\mathit{G}}_0^{-1}\left(z\right)]$ for $z$ along contour ${\mathrm{\Gamma }}_0$ (a) and ${\mathrm{\Gamma }}_{<}$ (b) with $U=0$, and $arg[det{\mathit{G}}^{-1}\left(z\right)]$ for $z$ along contour ${\mathrm{\Gamma }}_0$ (c) and ${\mathrm{\Gamma }}_{<}$ (d) with $U=5t$. (e), (f): $arg\left[D\right(z\left)\right]$ for $z$ along contour ${\mathrm{\Gamma }}_0$ (e) and ${\mathrm{\Gamma }}_{<}$ (f) with $U=5t$. Notice that the arguments are divided by 2, 9, or 11 (indicated in the figures), for clarity. The determinants of the single-particle Green's functions, $det{\mathit{G}}_0\left(z\right)$ and $det\mathit{G}\left(z\right)$, are analytically given in Eqs. (B6) and (B12), respectively, and $D\left(z\right)=det{\mathit{G}}_0\left(z\right)/det\mathit{G}\left(z\right)$. By directly counting how many times and which directions these quantities wind around the origin while $\varphi$ varies from $-\pi$ to $\pi$, we find that (a) $n_{det{\mathit{G}}_0^{-1}}\left({\mathrm{\Gamma }}_0\right)/2=4$, (b) $n_{det{\mathit{G}}_0^{-1}}\left({\mathrm{\Gamma }}_{<}\right)/2=7$, (c) $n_{det{\mathit{G}}^{-1}}\left({\mathrm{\Gamma }}_0\right)/9=-4$, (d) $n_{det{\mathit{G}}^{-1}}\left({\mathrm{\Gamma }}_{<}\right)/9=4$, (e) $n_D\left({\mathrm{\Gamma }}_0\right)/11=-4$, and (f) $n_D\left({\mathrm{\Gamma }}_{<}\right)/11=2$.

Figure 13

Diagrammatic representation for the Luttinger volumes of a noninteracting system $V_{\mathrm{L}}^0$ (left) and an interacting system $V_{\mathrm{L}}$ (right). Here, $\mathrm{Tr}[\cdots ]=T{\sum }_{\nu =-\infty }^{\infty }{e}^{i{\omega }_{\nu }{0}^{+}}\mathrm{tr}[\cdots ]$ and ${\mathbf{\Gamma }}_0=\frac{\partial {\mathit{G}}_0^{-1}\left(z\right)}{\partial z}=\mathit{I}$ (unit matrix). The thin line with an arrow represents ${\mathit{G}}_0$, the dot ${\mathbf{\Gamma }}_0$, the double line with an arrow $\mathit{G}$, and the circle $\mathbf{\Gamma }$. The Luttinger theorem equates these two quantities at zero temperature.