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arXiv:2107.06902 · SciPost Physics

Topological lattice models with constant Berry curvature

Dániel Varjas, Ahmed Abouelkomsan, Kang Yang, Emil J. Bergholtz

Published Apr 6, 2022• 50 citations
arXivHTMLPDF

Abstract

Band geometry plays a substantial role in topological lattice models. The Berry curvature, which resembles the effect of magnetic field in reciprocal space, usually fluctuates throughout the Brillouin zone. Motivated by the analogy with Landau levels, constant Berry curvature has been suggested as an ideal condition for realizing fractional Chern insulators. Here we show that while the Berry curvature cannot be made constant in a topological two-band model, lattice models with three or more degrees of freedom per unit cell can support exactly constant Berry curvature. However, contrary to the intuitive expectation, we find that making the Berry curvature constant does not always improve the properties of fractional Chern insulator states. In fact, we show that an ``ideal flatband'' cannot have constant Berry curvature, equivalently, we show that the density algebra of Landau levels cannot be realised in any tight-binding lattice system.

DOI
10.21468/scipostphys.12.4.118

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Band geometry plays a substantial role in topological lattice models. The Berry curvature, which resembles the effect of magnetic field in reciprocal space, usually fluctuates throughout the Brillouin zone. Motivated by the analogy with Landau levels, constant Berry curvature has been suggested as an ideal condition for realizing fractional Chern insulators. Here we show that while the Berry curvature cannot be made constant in a topological two-band model, lattice models with three or more degrees of freedom per unit cell can support exactly constant Berry curvature. However, contrary to the intuitive expectation, we find that making the Berry curvature constant does not always improve the properties of fractional Chern insulator states. In fact, we show that an ``ideal flatband'' cannot have constant Berry curvature, equivalently, we show that the density algebra of Landau levels cannot be realised in any tight-binding lattice system.
We investigate the many-body ground states in a family of fractionally filled bands where the Berry curvature fluctuations can be tuned while maintaining ideal quantum geometry. We numerically find that the neutral gap of the fractional Chern insulator (FCI) ground state decreases as the Berry curvature becomes less homogeneous, ultimately driving an instability to a charge density wave. We further extend our analysis to bands perturbed away from the ideal limit and give examples where a less ideal band geometry results in a more stable FCI phase. To explain our findings, we apply the single-mode approximation to the ground-state wave functions of the ideal band, from which we obtain analytic expressions for the FCI many-body gap in the limit of weak Berry curvature variations. Finally, we make a connection between our results and experimentally relevant systems where FCIs have been observed.
Recent numerical simulations of flat-band models with interactions which show clear evidence of fractionalized topological phases in the absence of a net magnetic field have generated a great deal of interest. We provide an explanation for these observations by showing that the physics of these systems is the same as that of conventional fractional quantum Hall phases in the lowest Landau level under certain ideal conditions which can be specified in terms of the Berry curvature and the Fubini-Study or quantum metric of the topological band. In particular, we show that when these ideal conditions hold, the density operators projected to the topological band obey the ${W}_{\ensuremath{\infty}}$ algebra. Our approach leads us to propose a way of testing the suitability of topological bands for hosting fractionalized phases.
With recent developments in fractional quantum mechanics, we introduce a fractional generalization of the Bernevig-Hughes-Zhang (BHZ) model to investigate the effects of fractional dispersion on band topology and quantum geometry. In the low-energy limit, the model reduces to a fractional Dirac Hamiltonian while preserving momentum-space periodicity, thereby ensuring a compact Brillouin zone and a well-defined topological invariant. We further show that the corresponding real-space tight-binding model can be constructed directly through a Fourier-series transformation, providing a simpler and more general alternative to the methods typically employed for fractional lattice systems. This approach readily extends beyond the generalized BHZ model considered here. Using the Bloch eigenstates, we compute the quantum geometric tensor and analyze its components, namely the Berry curvature and quantum metric. We find that fractional tuning redistributes these quantities throughout the Brillouin zone as the dispersion exponent becomes fractional, leading to pronounced modifications of the local band geometry. In contrast, the Chern number remains invariant, demonstrating the robustness of the global topological phase against fractional deformation. We further argue that this invariance persists for a broad class of fractional models.
We devise local lattice models whose ground states are \emph{ideal fractional Chern insulators}---Abelian and non-Abelian topologically ordered states characterized by exact ground state degeneracies at any finite size and infinite entanglement gaps. Most saliently, we construct exact parent Hamiltonians for two distinct families of bosonic lattice generalizations of the Zk\mathcal{Z}_kZk​ parafermion quantum Hall states: (i) color-entangled fractional Chern insulators at band filling fractions ν=k/(C+1)\nu=k/(\mathcal{C}+1)ν=k/(C+1) and (ii) nematic states at ν=k/2\nu=k/2ν=k/2, where C\mathcal{C}C is the Chern number of the lowest band in our models. In spite of a fluctuating Berry curvature, our construction is partially frustration free: the ground states reside entirely within the lowest band and exactly minimize a local (k+1)(k+1)(k+1)-body repulsion term by term. In addition to providing the first known models hosting intriguing states such as higher Chern number generalizations of the Fibonacci anyon quantum Hall states, the remarkable stability and finite-size properties make our models particularly well-suited for the study of novel phenomena involving e.g. twist defects and proximity induced superconductivity, as well as being a guide for designing experiments.
Topological properties lie at the heart of many fascinating phenomena in solid-state systems such as quantum Hall systems or Chern insulators. The topology of the bands can be captured by the distribution of Berry curvature, which describes the geometry of the eigenstates across the Brillouin zone. Using fermionic ultracold atoms in a hexagonal optical lattice, we engineered the Berry curvature of the Bloch bands using resonant driving and show a full momentum-resolved measurement of the ensuing Berry curvature. Our results pave the way to explore intriguing phases of matter with interactions in topological band structures.
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.