Quantized Charge Polarization as a Many-Body Invariant in $(2+1)\mathrm{D}$ Crystalline Topological States and Hofstadter Butterflies

We show how to define a quantized many-body charge polarization $\overrightarrow{𝒫}$ for $(2+1)\mathrm{D}$ topological phases of matter, even in the presence of nonzero Chern number and magnetic field. For invertible topological states, $\overrightarrow{𝒫}$ is a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$, ${\mathbb{Z}}_3$, ${\mathbb{Z}}_2$, or ${\mathbb{Z}}_1$ topological invariant in the presence of ($M=2$, 3, 4, or 6)-fold rotational symmetry, lattice (magnetic) translational symmetry, and charge conservation. $\overrightarrow{𝒫}$ manifests in the bulk of the system as (i) a fractional quantized contribution of $\overrightarrow{𝒫}·\overrightarrow{b}\text{mod}1$ to the charge bound to lattice disclinations and dislocations with Burgers vector $\overrightarrow{b}$, (ii) a linear momentum for magnetic flux, and (iii) an oscillatory system size dependent contribution to the effective 1D polarization on a cylinder. We study $\overrightarrow{𝒫}$ in lattice models of spinless free fermions in a magnetic field. We derive predictions from topological field theory, which we match to numerical calculations for the effects (i)–(iii), demonstrating that these can be used to extract $\overrightarrow{𝒫}$ from microscopic models in an intrinsically many-body way. We show how, given a high symmetry point o, there is a topological invariant, the discrete shift ${𝒮}_{\mathrm{o}}$, such that $\overrightarrow{𝒫}$ specifies the dependence of ${𝒮}_{\mathrm{o}}$ on o. We derive colored Hofstadter butterflies, corresponding to the quantized value of $\overrightarrow{𝒫}$, which further refine the colored butterflies from the Chern number and discrete shift.

Popular Summary

Since the discovery of superconductors and topological insulators, there has been spectacular progress in our understanding of symmetry-protected topological invariants, properties of a system that do not change when perturbed thanks to certain symmetries of the system. However, a complete understanding of topological invariants arising from crystalline symmetries is lacking. Recent theoretical work has provided a way to classify topological invariants for certain symmetries and predicts the existence of a quantized charge polarization, a redistribution of charge in response to an electric field. Here, we show how such a polarization can be defined in a certain insulator, and we explore its physical manifestations.

Specifically, we establish that the charge polarization is well defined in an insulator with a magnetic field and a nonzero quantized Hall conductance—a discrete conductance that emerges in some 2D systems under strong magnetic fields. That finding stands in contrast to over a decade of conventional wisdom. We show that in the presence of crystalline point group symmetry, in which rotations or reflections of a 2D space around a point keep the space invariant, one can define a quantized charge polarization as an intrinsically 2D many-body topological invariant. We further reveal its deep connection to the discrete shift, another crystalline topological invariant. Second, we explore in detail its physical manifestations in topological phases of matter with crystalline symmetry. Notably, we predict that the discrete shift and charge polarization manifest in the form of fractionally quantized charge bound to crystal defects.

Excitingly, the physical-response properties we discuss could potentially be accessible in Chern insulators, which are insulating in the interior but conducting on the surface, providing a deeper characterization of these crystalline topological phases and broadening our understanding of their properties.

Figure 1

A menagerie of butterflies in the spinless square lattice Hofstadter model. $\alpha$ and $\beta$ represent a plaquette center and vertex, respectively. For any $C_4$ symmetric origin o, ${𝒮}_{\mathrm{o}}$ has a ${\mathbb{Z}}_4$ classification, while ${\overline{𝒫}}_{\mathrm{o}}$ has a ${\mathbb{Z}}_2$ classification. We empirically find that $\{{𝒮}_{\beta },{𝒮}_{\alpha },{\overline{𝒫}}_{\beta },{\overline{𝒫}}_{\alpha }\}$ follow Eqs. (41), (42), (52), and (53), respectively. Note that ${𝒮}_{\alpha }$ has period $8\pi$ in $\varphi$, ${\overline{𝒫}}_{\beta }$ has period $4\pi$ in $\varphi$, while ${𝒮}_{\beta }$ and ${\overline{𝒫}}_{\beta }$ have period $2\pi$ in $\varphi$. See Secs. 4 and 5 for discussions.

Figure 2

Maximal Wyckoff positions for $C_2$, $C_3$, $C_4$, and $C_6$ symmetries (colored circles). $\times$ marks the unit cell center, which we denote as $\alpha$. The high symmetry points ${\beta }_i$ and ${\gamma }_i$ each belong to a single maximal Wyckoff position ($\beta$ or $\gamma$), but are all inequivalent under lattice translations. The dotted lines show a possible division of the unit cell into $M$ subcells.

Figure 3

Dislocation construction on a square lattice. (a) Drawing a cut in the $\widehat{y}$ direction. (b) Conjugate hoppings. (c) Doing local moves. (d) Reorganizing.

Figure 4

Cut-and-glue procedure of constructing a disclination. (a) Original lattice on an open plane. (b) Cutting. (c) Gluing; this creates new plaquettes ${\zeta }_i$. (d) Reorganizing.

Figure 5

(a) A pure disclination with $\mathrm{\Omega }=(2\pi /3)$ created using ${\tilde{C}}_{3,{\mathrm{o}}^{*}}$ via a cut-and-glue construction. The disclination core ${\mathrm{o}}^{*}$ is marked as $\times$. We can define a “frame,” i.e., an $x$, $y$ basis at every point on the defect lattice (they are plotted at the four plaquette centers near the disclination). The frame rotates by $-2\pi /3$ upon crossing the green dotted branch cut which passes through ${\mathrm{o}}^{*}$ and o. We calculate ${\overrightarrow{b}}_{\mathrm{o}}={\overrightarrow{v}}_1+{\overrightarrow{v}}_2+{\overrightarrow{v}}_3+{\overrightarrow{v}}_4$ for different o. (b) o at a plaquette center. $\{{\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3,{\overrightarrow{v}}_4\}=\{(-1,1),(-1,0),(0,-1),(1,-1)\}$. (c) o at one of the sites. $\{{\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3,{\overrightarrow{v}}_4\}=\{(-2/3,1/3),(-1/3,-1/3),(1/3,-2/3),(2/3,-1/3)\}$. (d) o at the other site. $\{{\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3,{\overrightarrow{v}}_4\}=\{(-1/3,2/3),(-2/3,1/3),(-1/3,-1/3),(1/3,-2/3)\}$.

Figure 6

Pure disclinations with two different o. (a) $\mathrm{o}=\beta$ and (b) $\mathrm{o}=\alpha$. The disclination core ${\mathrm{o}}^{*}$ is marked as $\times$. The weightings $\mathrm{wt}(i)$ within a representative choice of $W$ are marked near each site. When $\mathrm{o}=\alpha$, there is an irregular unit cell at the disclination core. (c) The unit cell choice; black solid lines represent hoppings.

Figure 7

Region $W$ for a square lattice dislocation with $\overrightarrow{b}=(1,0)$. The weightings $\mathrm{wt}(i)$ are labeled on each relevant MWP. Dashed lines represent unit cell boundaries, and colored circles represent the MWPs: $\alpha$ (red), $\beta$ (blue), and $\gamma$ (green).

Figure 8

Gauge choices defined in Eq. (56) on an $L_x\times L_y$ torus. Each blue arrow represents a vector potential $A_{ij}=(2\pi /L_x{L}_y)$. Each red arrow represents a vector potential $A_{ij}=(2\pi /L_y)$. ${\overline{\mathrm{o}}}_x$ and ${\overline{\mathrm{o}}}_y$ mark the distance between (0,0) and the cycle with trivial holonomy in the $x$ and $y$ directions, respectively. The blue region is the partial translation region $D$ which is centered around ${\overline{\mathrm{o}}}_x+(L_x/2)$.

Figure 9

Butterflies showing the constant in $\varphi$ contribution $L_y{𝒫}_{\mathrm{o},y}+K^{\prime }$ of the 1D polarization ${\mathcal{P}}_{\mathcal{O},x}$, for different ${\mathcal{O}}_x$ and $L_y$. Fixing $L_x=20$, the result depends only on the parity of $L_y$. For a fixed $\mathcal{O}$, ${𝒫}_{\mathrm{o},y}$ is the difference between the $\varphi =0$ intercept for $L_y=13$ and $L_y=12$. $\mathcal{O}$ and o satisfy the relation Eq. (75). For ${\mathcal{O}}_x=\frac{1}{2}$ only half of the period $0<\varphi <2\pi$ is plotted.

Figure 10

${𝒮}_{\mathrm{o}}$ and ${\overrightarrow{𝒫}}_{\mathrm{o}}$ in the honeycomb Hofstadter model. The site symmetry group is ${\mathbb{Z}}_{M^{\prime }}$, with $M^{\prime }=6$, 3, 2 for the MWPs $\{\alpha ,\beta ,\gamma \}$, respectively. ${𝒮}_{\alpha },{𝒮}_{{\beta }_1},{\overline{𝒫}}_{\alpha },{\overline{𝒫}}_{{\beta }_2},{\overline{𝒫}}_{{\beta }_1},{\overline{𝒫}}_{{\gamma }_3,x}$ are all calculated from the charge response [Eq. (82)] of suitable lattice defects (here we have not shown ${𝒮}_{\mathrm{o}}$ and ${\overrightarrow{𝒫}}_{\mathrm{o}}$ for all possible o but only some representative ones). The numerical calculations are done on an open disk with a radius $R=20$. The defects are located at the center of the disk. The noisy features appear since the butterfly is numerically calculated on a finite size system rather than analytically derived with an empirical formula as in Fig. 1. In the first three main Landu levels, ${𝒮}_{\alpha }$ quantize to $\{\frac{1}{2}\pm 0.0002,2\pm 0.004,\frac{9}{2}\pm 0.04\}$; other invariants in the first three main Landau levels have similar standard deviations.

Figure 11

Two representative ways of dividing a $C_4$ unit cell into subcells.

Figure 12

Pure disclinations with two different choices of ${\mathrm{o}}^{*}$. (a) ${\mathrm{o}}^{*}=\beta$, (b) ${\mathrm{o}}^{*}=\alpha$. Dashed lines represent unit cell boundary. (c) The $M=4$ unit cell.

Figure 13

Landau gauge on a honeycomb lattice with different gauge origin $\overline{\mathrm{o}}$. Each blue arrow represents a hopping phase $A_{ij}=(\varphi /4)$.

Figure 14

Dislocation construction procedure on a square lattice. (a) Drawing a cut in the $\widehat{y}$ direction. (b) Conjugate hoppings. (c) Doing local moves. (d) Reorganizing.

Figure 15

Dislocation construction procedure on a honeycomb lattice. (a) Draw a cut in the $\widehat{y}$ direction. (b) Conjugate a set of hopping terms. (c) Perform local moves. (d) Reorganize.

Figure 16

Trimming method for a pure disclination on the square lattice. We consider two regions. (a) $W_1$ and (b) $W_2$. $W_1$, $W_2$ correspond to choices of the unit cell shown in (c) and (d), respectively. The origin o is defined to be at a site. Black solid lines represent hoppings. Here $W_1$ is trimmed into $W_2$ by excluding subcells at the boundary.

Figure 17

Trimming method for a dislocation. (a) Honeycomb dislocation with region $W_1$ covering the $\overrightarrow{b}=(0,-1)$ dislocation. The boundary of $W$ is aligned with unit cell boundary of (c), shown as red dashed line. (b) The region $W_2$ after trimming. Now the boundary of $W_2$ is aligned with the hopping, which is also the unit cell boundary of (d), shown as black solid line.

Figure 18

Bare numerical result of (top) angular momentum $l_{\mathrm{o}}$ which follows Eq. (40); both calculated on an $L_x\times L_y=12\times 12$ torus. Bottom: linear momentum $p_{\lambda ,y}$ which follows Eq. (21), calculated on the $L_x\times L_y=19\times 11$ and $L_x\times L_y=20\times 11$ torus, respectively. All panels are plotted in the $0<\varphi <2\pi$ range.

Figure 19

The U(1) phase of a randomly chosen low-energy single-particle eigenstate on a $64\times 64$ torus. The gauge origin is at $\overline{\mathrm{o}}=(51+\frac{1}{2},0)$. The parameters are (left) $m=5$, $C=1$, (right) $m=15$, $C=1$. The variation in phase along $y$ is lowest around $x={\overline{\mathrm{o}}}_x$.

Figure 20

Left: $\overrightarrow{b}=(0,1)$ dislocation of a square lattice with next-nearest-neighbor hopping (red links). The $C_4$ symmetry requires all four colored triangles with different orientations having same flux $\varphi /2$. Right: the choice of unit cell. There are no sites at the MWP $\alpha$ and $\gamma$, and there is only one site at $\beta$. The next-nearest-neighbor hoppings cross each other.

Figure 21

${\overline{𝒫}}_{\alpha }$ as a function of the next-neighbor hopping amplitude $t^{\prime }$; the nearest-neighbor hopping amplitude is $t=1$. The calculation is done on a $L_x\times L_y=41\times 41$ open disk, with a $\overrightarrow{b}=(0,1)$ dislocation placed at the center. The defects are located at the center of the disk. The noisy features appear since the butterfly is numerically calculated on a finite size system rather than analytically derived from an empirical formula as in Fig. 1.