Characterization of two-dimensional fermionic insulating states

Inspired by the duality picture between superconductivity (SC) and insulator in two spatial dimension (2D), we conjecture that the order parameter, suitable for characterizing 2D fermionic insulating state, is the disorder operator, usually known in the context of statistical transformation. Namely, the change of the phase of the disorder operator along a closed loop measures the particle density accommodating inside this loop. Thus, identifying this (doped) particle density with the dual counterpart of the magnetic induction in 2D SC, we can naturally introduce the disorder operator as the dual order parameter of 2D insulators. The disorder operator has a branch cut emitting from this “vortex” to the single infinitely far point. To test this conjecture against arbitrary two-dimensional lattice models, we have chosen this branch cut to be compatible with the periodic boundary condition and obtain a general form of its expectation value for noninteracting metal and/or insulator wave function, including gapped mean-field order wave function. Based on this expression, we observed analytically that it indeed vanishes for a wide class of band metals in the thermodynamic limit. On the other hand, it takes a finite value in insulating states, which is quantified by the localization length or the real-valued gauge invariant two form dubbed as the quantum metric tensor. When successively applied along a closed loop, our disorder operator plays the role of twisting the boundary condition of a periodic system. We argue this point by highlighting the Aharonov-Bohm phase associated with this nonlocal operator.

Figure 1

(Color online) 2D lattice models. $(x,y)$ and $(x^{\prime },y^{\prime })$ specify a lattice point (filled circle mark). A unit cell is depicted as the bold line in every figure and we use different colors (shapes) in order to distinguish $N_b$ number of inequivalent lattice points within it. Crossed mark represents a dual lattice site on which $(x+x_0,y+y_0)$ is located. (From top to down): Triangle and square lattice with $N_b=1$, honeycomb and checkerboard lattice with $N_b=2$, and kagome lattice with $N_b=3$. Note that $(x+x_0,y+y_0)$ (crossed mark) never coincides with the original lattice sites on which a fermion operator is defined.

Figure 2

(Color online) $\mathrm{Arg}\ln [-i(e^{ix}-1)+(e^{iy}-1)]$ as a function of $x$ and $y$, where we only show the principle branch which ranges from $-\frac{\pi }{4}$ to $-\frac{9\pi }{4}$ in this picture and those below.

Figure 3

A branch cut runs from the vortex $[(x,y)=(0,0)]$ to the antivortex $[(x,y)=(\frac{\pi }{2},-\frac{\pi }{2})]$. $\mathrm{Arg}\ln [-i(e^{ix}-1)+(e^{iy}-1)]$ takes $-\frac{\pi }{4}-$ in the shaded side of this branch cut, while takes $-\frac{9\pi }{4}+$ in the other side.

Figure 4

(a) Type-O Fermi surface with uneven boundary and annulus. (b) Type-$A$ or $B$ Fermi surface. The strip, which is enclosed by the two dashed straight line, winds the torus along $k_x$ direction. $\left(b^{\prime }\right)$ Another type-$A$ or $B$ Fermi surface. The presence of annulus inside the Fermi sea does not change the type of the Fermi surface. Namely, we have only to replace ${\Delta }_y$ in Eq. (31) by ${\Delta }_x^1+{\Delta }_x^2$. (c) Type-$AB$ FS with a loop which winds around the torus along $k_x$ and $k_y$ direction.

Figure 5

(Color online) (a) A quantized $\left(2\pi \right)$ flux tube associated with the disorder operator (DOP) $\eta \left(\vec{r}\right)$ is shown as a red (thin) arrows on the $(x^{\prime },y^{\prime })$ plane. Corresponding to the phase inconsistency of $\ln \zeta ({\vec{r}}^{\prime }-\vec{r})$, the flux tube appears outside the torus at $(x,y)$, i.e., at the vortex, and after traveling above the branch cut, it disappears inside the torus at the antivortex, $(x+L∕4,x-L∕4)$, forming a loop. (b) The path $E$ is a straight blue (dotted) line. Flux tubes associated with $\eta \left(\vec{r}\right)$ are superposed with respect to $\vec{r}$ over this path $E$. In this summation over $\vec{r}$ cancellations between flux tubes heading outside and inside the system occur on the 2D plane (shadowed region). As a result, those flux which penetrate the system completely disappear, leaving two bundles of flux tubes above and beneath the system in parallel with $E$ [see also Fig. 6a].

Figure 6

(Color online) (a) Two flux bundles and the periodic 2D flat surface (= our system). The red (solid) line represents a bundle of $N∕4$ quantized flux tubes which flows above (↔ outside the torus) and in parallel with our 2D plane. The red (broken) line is the other bundle which flows beneath (↔ inside the torus) and in parallel with the plane. (b) Two flux bundles and the torus (= our system). The former pierces the handle of the torus encircled by the $y^{\prime }$ axis, whereas the latter penetrates the interior of the torus wound by the $x^{\prime }$ axis.

Figure 7

(Left): A $x^{\prime }\text{−}{y}^{\prime }$ plane, where branch cut of $\mathrm{Im}\ln \zeta ({\vec{r}}^{\prime }-\vec{r})$ runs from $(x,y)$ to $(x+\frac{L}{4},y-\frac{L}{4})$. The shaded side of it is for $-\frac{\pi }{4}-$ while the other side is for $-\frac{9\pi }{4}+$. (Right): One can easily convince oneself that Eq. (57) is independent from a specific choice of the branch cut.

Figure 8

Two integral paths with respect to $z$, $C_{<}$ and $C_{>}$.

Figure 9

Dotted line emitting from $z_1$ to infinity is the branch cut of $\ln [z^{\prime }-z_1]$ which we have chosen in Eq. (A4). $\arg \ln [z^{\prime }-z_1]$ takes $+0$ in the shaded side of this dotted line while $2\pi$ is in the other side.

Figure 10

(Left): A ${\theta }_x-{\theta }_y$ plane, where branch cut of $\ln [i+e^{i{\theta }_x}-1+e^{i{\theta }_y}]$ runs from $(-\frac{\pi }{2},0)$ to $(0,-\frac{\pi }{2})$. The shaded side of it is for $-\frac{\pi }{4}-$ while the other side is for $-\frac{9\pi }{4}+$. (Right): One can easily convince oneself that Eqs. (45, A12) are independent from a specific choice of the branch cut.