Fractional quantum Hall effect in the interacting Hofstadter model via tensor networks

We show via tensor network methods that the Harper-Hofstadter Hamiltonian for hard-core bosons on a square geometry supports a topological phase realizing the $\nu =\frac{1}{2}$ fractional quantum Hall (FQH) effect on the lattice. We address the robustness of the ground-state degeneracy and of the energy gap, measure the many-body Chern number, and characterize the system using Green functions, showing that they decay algebraically at the edges of open geometries, indicating the presence of gapless edge modes. Moreover, we estimate the topological entanglement entropy by taking a combination of lattice bipartitions that reproduces the topological structure of the original proposals by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and Levin and Wen [Phys. Rev. Lett. 96, 110405 (2006)]. The numerical results show that the topological contribution is compatible with the expected value $\gamma =\frac{1}{2}$. Our results provide extensive evidence that FQH states are within reach of state-of-the-art cold-atom experiments.

Figure 1

Sketch of the system: $N$ hard-core bosons (red balls) are hopping on a $L\times L$ lattice, perpendicular to an external magnetic field. The magnetic field gives rise to a flux $\varphi$ through each plaquette, which in Landau gauge leads to the indicated phase factors in the hopping amplitudes $t$. The fluxes in the topmost row and rightmost column (faded) are only present in case of periodic boundary conditions (PBC). For PBC, additional phase twists ${\theta }_x$ and ${\theta }_y$ can be introduced (see text), allowing for the definition of topological invariants.

Figure 2

Binary tree-tensor network ansatz for a $L\times L$ lattice. The blue dots are the physical sites with local dimension $d$ ($d=2$ for hard-core bosons). Each tensor groups two sites to one virtual site (gray dots), leading to a hierarchical tree structure. In order to capture the 2D lattice geometry, the grouping is performed in the $x$ and $y$ directions, alternating from level to level. The dimension of the virtual sites in the $l\mathrm{th}$ level (counting from below) is $min(d^{2^l},m)$, where $m$ is the bond dimension of the TTN. The additional cyan link at the top left tensor has dimension one and selects the global particle number $N$ [32].

Figure 3

Low-lying spectrum (a) and many-body Chern number (MBCN) (b)–(e) for a system with $L=16$, $N=8$, $\varphi =\frac{1}{16}$ ($\nu =\frac{1}{2}$). (a) Finite-size spectrum as a function of the twist angles of the boundary conditions. The twofold-degenerate GS manifold is separated from the first excited state by a gap $\mathrm{\Delta }\simeq 0.08$. (b), (c) Regions where ${\mathrm{\Lambda }}_{\mathrm{\Phi }}$ and ${\mathrm{\Lambda }}_{{\mathrm{\Phi }}^{\prime }}$, respectively, vanish. The reference multiplet $\mathrm{\Phi }$, ${\mathrm{\Phi }}^{\prime }$ is composed of the two orthogonal GSs at $({\theta }_x,{\theta }_y)=(0.2,0.12)$, $(0.64,0.56)$. (d) Color plot of the argument field ${\mathrm{\Omega }}_{\mathrm{\Phi }\to {\mathrm{\Phi }}^{\prime }}$ whose count of branch vortices gives the MBCN. (e) Location (and sign) of the branch points, obtained by summing up the angle differences $\mathrm{\Delta }\mathrm{\Omega }$ along the nearest neighbors of each grid point, resulting in a MBCN of one.

Figure 4

(a) Green functions along the $y$ direction for PBC (a1) and OBC (a2), for $L=16$, $N=8$, $\varphi =\frac{1}{16}$. In the PBC case, the decay is exponential, irrespective of the choice of $x$. In contrast, for OBC, the decay at the edges ($x\approx 1$ or $y\approx L$) is much slower, signaling the presence of edge modes with algebraic correlations. (b) Current in the $x$ direction. (b1) Comparison between PBC (no current) and OBC (currents at the edges). (b2) OBC current for $L=32$, $\varphi =\frac{1}{32}$, and (b3) for $\varphi =\frac{1}{8}$, $L=16$, each with two different bond dimensions $m$. All panels use the same color code for $x$, which is illustrated in (c). The cartoon in (c) shows the edge current (red arrow) and demonstrates why ${\mathcal{I}}_x$ vanishes in the bulk (dark-shaded background), while at the edges (bright background) it is nonzero whenever $y\approx 1$ or $y\approx L$. Moreover, it is obvious that the sign of ${\mathcal{I}}_x$ at $y\approx 1$ is opposite to the sign of ${\mathcal{I}}_x$ at $y\approx L$.

Figure 5

(a) TEE $\gamma$ obtained from an extrapolation in $m$: lines are linear fits through data points with $\ell >10$, for $L=16$, $N=16$, $\varphi =\frac{1}{8}$. The resulting $\gamma$ is compatible with $\frac{1}{2}$. (b) Arrangement of square partitions for the extraction of $\gamma$ [23] and resulting TEE (c) as a function of the block size $a$. For large enough $a$, the data are compatible with $\gamma =\frac{1}{2}$.

Figure 6

Left column: onsite occupations $\langle n_{x,y}\rangle$ of the lattice sites for different values of $\mu$, showing how the particles gradually localize at the energetically favored sites of the superlattice. Right column: plots of ${\mathrm{\Gamma }}_{\mathrm{\Phi }}$ for different values of $\mu$. For $\mu \gtrsim -1.0$ there exist regions (visible as black areas enclosed by cyan lines) where ${\mathrm{\Gamma }}_{\mathrm{\Phi }}$ vanishes, indicating topologically nontrivial character. System parameters are $L=16$, $N=8$, $\varphi =\frac{1}{16}$.

Figure 7

(a) Three lowest eigenenergies of $\tilde{H}$ as a function of $\mu$, at ${\theta }_x={\theta }_y=0$. (b) Low-energy spectra on the whole boundary condition torus for $\mu =-1.0$ and $-2.0$. System parameters are $L=16$, $N=8$, $\varphi =\frac{1}{16}$.

Figure 8

(a) MBCN as a function of $\mu$, displaying a transition at ${\mu }_c\approx -1$. (b) Determination of MBCN for $\mu =-1.0$, resulting in $\mathrm{MBCN}=1$. System parameters are $L=16$, $N=8$, $\varphi =\frac{1}{16}$.

Figure 9

(a)–(d) Four example configurations for the argument field ${\mathrm{\Omega }}_{\mathrm{\Phi }\to {\mathrm{\Phi }}^{\prime }}$ around a point $({\theta }_x,{\theta }_y)$ on the boundary condition grid. The branch vortex count $1/2\pi {\sum }_{\diamond }\mathrm{\Delta }\mathrm{\Omega }$ is 1 for the configurations shown in (a) and (b), $-1$ for the configuration in (c), while for the one in (d) it is 0. (e) Illustration of the color code for ${\mathrm{\Omega }}_{\mathrm{\Phi }\to {\mathrm{\Phi }}^{\prime }}$, used in Figs. 3 and 8.

Figure 10

GS energy $E_0$ as a function of bond dimension $m$ for two different system sizes $L=16$ (a) and $L=32$ (b). Also shown are two different $m\to \infty$ extrapolations, a simple one linear in $1/m$ and a more versatile one where the exponent of $1/m$ is allowed to be adjusted by the fit. This procedure can be used to estimate the ground-state energies and corresponding errors to be $E_0=-29.015\pm 0.016$ (a) and $E_0=-60.84\pm 0.03$ (b).

Figure 11

Bond dimension convergence of the Green functions from Fig. 4. Different colors denote different bond dimensions, while the two different symbols refer to two different choices of $x$ (circles: $x=2$, crosses: $x=8$). For PBC the different choices of $x$ result in the same curves, as expected for a system with translational invariance; only at $y\approx L/2$ translational invariance is slightly broken, indicating a finite bond dimension error of the order of ${10}^{-2}$.