Symmetry and Quantum Geometry in Bloch Bands

The gist The central claim of this work is that in order to minimize the variance of Berry curvature or the integrated quantum metric for a given tight-binding model, orbital positions should be chosen to obey all the spatial symmetries that the tight-binding model is compatible with up to gauge transformation and possible time-reversal

Setup and Definitions

A tight-binding model is specified by a set of atomic orbitals connected by some hopping amplitudes. The quantum geometric tensor is defined as Qij = ⟨∂kiu(k)Q(k)∂kj u(k)⟩. Using this, the quantum metric tensor is the real-part of the quantum geometric tensor, i.e. gij = R[Qij]. The Berry curvature is given by the imaginary part of its off-diagonal entry omega = −2I[Qxy]. The integrated quantum metric is given by I ≡ Z BZ d 2k tr[g], and the variance of Berry curvature is defined as σ squared omega ≡ ABZ 4π/2 Z BZ d 2k (omega(k) − omega)¯ squared.

Symmetry Constraints on Orbital Positions

The central claim of this work is that in order to minimize the integrated quantum metric or the variance of Berry curvature, the real-space realization of a tight-binding model must obey all the spatial symmetries that the tight-binding model is compatible with up to gauge transformation and possible time-reversal. The real-space realization involves assigning a particular position rα to each atomic orbital α, which can be written as rm,n,a = ma1 + na2 + ra. A tight-binding model is compatible with a set of spatial symmetries if and only if there exists some embedding rα such that the corresponding real-space model obeys the set of spatial symmetries in the sense defined above.

Minimization Properties

The quantum metric tensor depends on the orbital positions ra,i as a polynomial up to the second order. It is also straightforward to show that Berry curvature depends on the orbital positions only up to linear order, omega(˜k) = omega(k) − X aϵ ij ra i∂kjua(k)2. Since both I and σ squared omega are bounded below, the quadratic part of the polynomial in ra,i must be positive semi-definite, implying there must be a global minimum of either I or σ squared omega in the space of all embeddings.

Symmetry and Gauge Transformations

The discussion extends to include gauge transformations. A real-space model is said to obey a spatial symmetry up to gauge if Eq. 4 is replaced with tσ(α),σ(β) = tα,βe-i(ϕα−ϕβ). The central claim of this work is that in order to minimize the integrated quantum metric or the variance of Berry curvature, the real-space realization of a tight-binding model must obey all the spatial symmetries that the tight-binding model is compatible with up to gauge transformation and possible time-reversal. For a Hofstadter model, the “natural” choice to place the orbitals within a magnetic unit cell is to place them on a regular grid so that magnetic translation symmetries are manifestly obeyed, and this choice actually minimizes the variance of Berry curvature and the integrated quantum metric.

Ideal Bands and Corollaries

An ideal band, as mentioned, is one whose integrated quantum metric saturates a certain lower bound that is determined solely by the Chern number. This naturally implies that out of all possible choices of orbital positions, the choice for which the model is ideal must minimize the integrated quantum metric. Corollaries include that an ideal band’s orbital positions must obey the symmetry that the tight-binding model is compatible with. Furthermore, in the context of fractional Chern insulators, satisfying Eq. 31 necessarily implies that the embedding has been chosen to minimize I.

Conclusion

In this work, we have studied the problem of minimizing the variance of Berry curvature and the integrated quantum metric over real-space orbital positions (embeddings) for a tight-binding model. Our central conclusion is that the optimal choice of orbital positions should obey all the spatial symmetries that the tight-binding model is compatible with, up to gauge transformation and possible time-reversal. For some cases, this symmetry requirement uniquely fixes the optimal positions of the orbitals, and places them at special Wyckoff positions. For other cases, the symmetry requirement does not fully pin down the orbital positions. Instead the symmetry-allowed embeddings form a connected sub-manifold of all possible embeddings. We also investigate the degeneracy of minima for the variance of Berry curvature and integrated quantum metric respectively. Corollaries of our results include that the conventional embedding of the Hofstadter model actually minimizes the variance of Berry curvature and the integrated quantum metric over all possible rearrangement of orbitals within the magnetic unit cell, and that the embedding of an ideal band must obey all the symmetries that the underlying tight-binding model is compatible with. The physical orbital positions would then normally have the same symmetry as the tight-binding model.

How it works

The proof for pure spatial symmetry shows that if a spatial operation generates a new set of orbital positions, the new set of orbital positions give the same quantum geometric quantities, i.e., the same I and σ squared omega. For σ squared omega, if there is no additional null vector of the quadratic part of the dependence, exactly the same argument can be applied to show that it is minimized by embeddings that obey all spatial symmetries.

Gauge Transformation Effects

For gauge transformations, we need to show that gauge transformation also does not alter quantum geometric properties. For a class of gauge transformations given by cˆR,a → cˆR,ae i(k0·R+ϕa), the additional phase is independent of the unit cell R, and such a transformation keeps the quantum geometric tensor Qij invariant. For other gauge transformations that alter periodicity, it was shown that for a system with magnetic translation symmetries T1 and T2, their respective Hamiltonians have the same integrated quantum metric I.

Lattice Geometry Optimization

The discussion on lattice vectors shows that for any choice of lattice vectors, the relative embedding that minimizes the integrated quantum metric (i.e., I) is the same. Furthermore, many symmetries can only be realized if an appropriate Bravais lattice is chosen. We show that the lattice vectors that minimize I (for each choice of lattice vectors, we assume optimal choice of orbital positions within the unit cell) are the ones that are compatible with the symmetry of the model.

Wannier Function Spread

The spread of Wannier function is defined as ω = ⟨r 2⟩ − ⟨r⟩ squared. The gauge-invariant part of the spread is given by ωI = 1/ABZ Z dk tr[g], which is our familiar integrated quantum metric up to a factor. The optimal gauge condition, ∇k · A(k) = 0, leads to the optimal spread being ωmin = 1/ABZ Z dk tr[g] + X R!=0 omega(R)2R2. The maximally localized Wannier orbitals must correspond to an embedding that respects the same symmetry as the hopping model.

1. The improved AI system can optimize tight-binding model parameters by selecting real-space orbital positions that obey all those spatial symmetries that the tight-binding model is compatible with to minimize geometric quantities like the variance of Berry curvature or the integral of the trace of the quantum metric tensor.

2. The system can determine if a given band is an ideal band by checking if its orbital positions are chosen such that they satisfy the symmetry that the tight-binding model is compatible with, which implies minimizing the integrated quantum metric, as discussed in Section VI.

3. The AI can select optimal lattice vectors for a model to minimize the integrated quantum metric by choosing those compatible with the symmetry of the model, as shown in Appendix C, where it is demonstrated that the lattice vectors that minimize I (for each choice of lattice vectors, we assume optimal choice of orbital positions within the unit cell) are the ones that are compatible with the symmetry of the model.

4. The system can evaluate gauge-invariant properties by demonstrating that for a class of gauge transformations, the integrated quantum metric is unchanged, which allows for calculations where gauge transformations do not alter quantum geometric properties like I and σ2omega.

5. The AI can optimize the spread of maximally localized Wannier functions by choosing an embedding that respects the hopping model's symmetry, as it claims the maximally localized Wannier orbitals must correspond to an embedding that respects the same symmetry as the hopping model.

Quantum geometric quantities have featured heavily in the discussion of the properties of quantum systems in recent years. Among quantities most commonly discussed is the variance of Berry curvature and the integral of the trace of the quantum metric tensor. Despite their usefulness, it is known that they suffer from one significant complication: for a tight-binding model, the quantum geometric quantities depend not only on the parameters of the tight-binding model itself, but also the real space geometry of the tight-binding model, the so-called "orbital embedding". One explicitly geometry-independent quantity is therefore the minimal value of the quantum geometric quantity out of all possible real space geometries. In this work, we demonstrate that, if the tight-binding model is compatible with certain spatial symmetries, then the real space geometry that minimizes the variance of the Berry curvature or the integral of the trace of the quantum metric tensor must obey all those spatial symmetries. We further show that the statement is applicable to systems with magnetic translation symmetries and other composite symmetries, with implications for the quantum geometry of the Hofstadter model.