Weyl-type topological phase transitions in fractional quantum Hall like systems

We develop a method to characterize topological phase transitions for strongly correlated Hamiltonians defined on two-dimensional lattices based on the many-body Berry curvature. Our goal is to identify a class of quantum critical points between topologically nontrivial phases with fractionally quantized Hall (FQH) conductivity and topologically trivial gapped phases through the discontinuities of the many-body Berry curvature in the so-called flux Brillouin zone (fBZ), the latter being defined by imposing all possible twisted boundary conditions. For this purpose, we study the finite-size signatures of several quantum phase transitions between fractional Chern insulators and charge-ordered phases for two-dimensional lattices by evaluating the many-body Berry curvature numerically using exact diagonalization. We observe degeneracy points (nodes) of many-body energy levels at high-symmetry points in the fBZ, accompanied by diverging Berry curvature. We find a correspondence between the number and order of these nodal points, and the change of the topological invariants of the many-body ground states across the transition, in close analogy with Weyl nodes in noninteracting band structures. This motivates us to apply a scaling procedure, originally developed for noninteracting systems, for the Berry curvature at the nodal points. This procedure offers a useful tool for the classification of topological phase transitions in interacting systems harboring FQH like topological order.

Figure 1

Many-body Berry curvature $F({\mathbf{\varphi }}_0^{},{\mu }_{\mathrm{s}}^{})$ for a noninteracting Chern insulator [specifically, the triangular lattice model defined by the noninteracting Hamiltonian (3.3)] for fixed value of flux ${\mathbf{\varphi }}_0^{}=(\pi -0.7,0)$, for a series of system sizes $L\equiv L_1^{}=L_2^{}=3,5,7,9,11,15,21,25$. In the thermodynamic limit, the model has a topological phase transition in which the Chern number changes from 1 (gray line) to 0 at ${\mu }_{\mathrm{s}}^{}=1$ (taking $t=1$ as energy unit). In the thermodynamic limit, $F({\mathbf{\varphi }}_0^{},{\mu }_{\mathrm{s}}^{})$ becomes constant and thus independent of ${\mathbf{\varphi }}_0^{}$ and ${\mu }_{\mathrm{s}}^{}$ except at the phase transition. The many-body Berry curvature $F(\mathbf{\varphi },{\mu }_{\mathrm{s}}^{})$ is singular at ${\mathbf{\varphi }}_{\mathrm{c}}^{}=(\pi ,0)$ and ${({\mu }_{\mathrm{s}}^{}/t)}_{\mathrm{c}}^{}=1$ for $L\equiv L_1^{}=L_2^{}=3,5,7,9,11,15,21,25$.

Figure 2

Schematic definition of (a) triangular-lattice model of Eq. (3.3), (b) checkerboard-lattice model of Eq. (3.4), and (c) honeycomb-lattice model of Eq. (3.5). Hoppings in the direction of an arrow add $\phi$ to the phase of the electron wave function, hopping in the opposite direction subtracts $\phi$ ($\phi =\pi /2$ for triangular-lattice model, $\phi =\pi /4$ for checkerboard-lattice model, $\phi =0.4\pi$ for honeycomb-lattice model); dashed (dotted) lines denote hoppings with a negative (positive) sign; in all models, third-neighbor hoppings are uniform and are omitted for clarity.

Figure 3

Many-body Berry curvatures (top panels) and many-body energy dispersions in the fBZ (bottom panels) for the FCI and CDW states of the fermionic model defined by Eqs. (3.3) on a triangular lattice made of 18 sites at the fermionic density (number of fermions divided by the number of sites) $\rho =\frac{1}{3}$ holding $t_3^{}/t=0.22$ fixed. In (a), $V_1^{}/t=1.11$ (topologically nontrivial $\nu =\frac{2}{3}$ FCI phase). In (b), $V_1^{}/t=1.23$ (close to the transition point). In (c), $V_1^{}/t=1.3$ (topologically trivial CDW phase).

Figure 4

Many-body energy gaps between the FCI and CDW energy levels for the bosonic interacting Haldane model of Ref. [27] on a honeycomb lattice made of 32 sites, as defined in Eq. (3.5). The average density of hard-core bosons per site is $\rho =\frac{1}{4}$ and $V_1^{}=0$. In (a), $V_2^{}$ is varied holding $\mathbf{\varphi }\equiv ({\varphi }_1^{},{\varphi }_2^{})$ fixed to either $\mathbf{\varphi }=(0,0)$ or $\mathbf{\varphi }=(\pi ,0)$. (a) Shows that the many-body gap closes at $V_2^{}\sim 1.446$ and 1.642, respectively. In (b), ${\varphi }_1^{}$ is varied holding $V_2^{}=1.446$ and ${\varphi }_2^{}=0$ fixed. The red line in (b) is the fit ${\beta }_1{\varphi }_1^2$ with ${\beta }_1\simeq \pi /10$.

Figure 5

(a), (c) Berry curvature of the state $|{\mathrm{\Psi }}_1^{}\rangle$ corresponding to first-excited energy level $E_1^{}$ and (b), (d) energy levels $E_1^{}$ and $E_2^{}$ in flux space for a 32-site cluster of the interacting Haldane model of Ref. [27], as defined in Eq. (3.5), at density $\rho =\frac{1}{4}$ hard-core bosons per site with $V_1^{}=0$ and (a), (b) $V_2^{}/t=1.48$, which is close to the critical point, and (c), (d) $V_2^{}/t=1.52$ that is far away from it.

Figure 6

(a), (b) Illustrate the scaling procedure for Assumptions 2.1 (peak-divergence scenario) and 2.2 (ring-divergence scenario), respectively, which demands the red dot to be equal to the orange dot to solve for the ${\mathit{M}}^{\prime }$, as indicated by the dashed line. The profile of Berry curvature evolves from the red lines that have a large divergence to the orange lines that have a small divergence under this procedure, without changing the topological invariant, hence the system flows away from the critical point.

Figure 7

(a) BCRG flow for the triangular-lattice model using $\mathrm{\Delta }\mathit{M}=(\mathrm{\Delta }{t}_3^{},\mathrm{\Delta }{V}_1^{})=(0.001,0.01)$, and, without loss of generality, choosing $b=0$ in Eq. (4.3), with the topological invariant $C$ labeled for each phase. The flow rate in logarithmic scale is shown by the color code. Blue color indicates a low flow rate and orange a high flow rate. Red dots label the phase boundary, and the green dots the line at which ${\mathbf{\nabla }}_{\mathit{M}}^{}{F}_{★}^{}{|}_{{\mathbf{\varphi }}_{\mathrm{c}}^{},\mathit{M}}^{}$ in Eq. (4.4) vanishes. The scaling ansatz (4.2) is made to fit the Berry curvature in the neighborhood of the HSP and of a plateau transition. (b), (c) Show the divergence of $F_{★}^{}({\mathbf{\varphi }}_{\mathrm{c}}^{},t_3^{},V_1^{})$ and ${\xi }_1^{}$ versus $V_1^{}$ at a few selected $t_3^{}$. (d) BCRG flow of the checkerboard model using $\mathrm{\Delta }\mathit{M}=(\mathrm{\Delta }{V}_2^{},\mathrm{\Delta }{\mu }_2^{})=(0.01,0.002)$. (e), (f) Show the divergence of $F_{★}^{}({\mathbf{\varphi }}_{\mathrm{c}}^{},V_2^{},{\mu }_2^{})$ and ${\xi }_1^{}$ versus ${\mu }_2^{}$ at a few selected $V_2^{}$. The small displacement $\mathrm{\Delta }\mathbf{\varphi }=(2\pi /1000,0)$ is used for the calculations. Both models feature the same HSP ${\mathbf{\varphi }}_{\mathrm{c}}^{}=(0,0)$, and the divergence of $F_{★}^{}({\mathbf{\varphi }}_{\mathrm{c}}^{},\mathit{M})$ and ${\xi }_1^{}$ are well fitted by the same exponents $\alpha =2$ and ${\nu }_1^{}=1$, as indicated by the solid lines.

Figure 8

(a) BCRG flow of the tuning parameter $V_2^{}$ for the honeycomb lattice model, obtained using Eq. (4.4) with the choice $b=2$, the grid spacing $\mathrm{\Delta }\mathbf{\varphi }=(3\pi /250,0)$, and $\mathrm{\Delta }{V}_2^{}=0.002$. Plotted in units of $\mathrm{\Delta }{V}_2^{}/\mathrm{\Delta }{\mathbf{\varphi }}^2$, the BCRG flow signals the existence of a quantum critical point at $V_{2\mathrm{c}}^{}\approx 1.446$ in the thermodynamic limit. (b) Schematics of extracting the two correlation lengths ${\xi }_{1,\max }^{}$ and ${\xi }_{1,\mathrm{wid}}^{}$ from the Berry curvature (purple) along the ${\widehat{\mathbf{\varphi }}}_1^{}$ direction. (c) The inverse $1/F_{★,\max }^{}$ of the Berry curvature extremum $F_{★,\max }^{}$ versus $V_2^{}$. The values at $V_2^{}>V_{2\mathrm{c}}^{}$ are enlarged 200 times for readability. The linear behavior near $V_{2\mathrm{c}}^{}$ corresponds to the exponent $\alpha \approx 1$. (d) Inverses $1/{\xi }_{1,\max }^2$ and $1/{\xi }_{1,\mathrm{wid}}^2$ of the squared correlation lengths ${\xi }_{1,\max }^2$ (orange) and ${\xi }_{1,\mathrm{wid}}^2$ (red) versus $V_2^{}$ plotted in units of ${(\pi /500)}^2$. The $1/{\xi }_{1,\mathrm{wid}}^2$ at $V_2^{}>V_{2\mathrm{c}}^{}$ is enlarged by 10 times for readability. The linear behavior near $V_{2\mathrm{c}}$ corresponds to the critical exponent $\nu \approx \frac{1}{2}$.