Flat Bands

A flat band is a family of one-particle eigenstates whose energy is independent of crystal momentum over the Brillouin zone:

$$\varepsilon_f(\mathbf k) = E_f.$$

The equality may be exact in an ideal lattice Hamiltonian or approximate over a specified momentum and energy window. Exact flatness quenches the usual dispersion-driven group velocity and produces an extensive degeneracy. It does not erase the Bloch wavefunctions, their orbital texture, quantum metric, Berry curvature, topology, or interaction form factors.

We will build a flat band by canceling the hopping out of one cell. This solvable example separates two ideas that are often confused: all states can have the same energy while their wavefunctions still vary across momentum. Those wavefunctions determine how particles interact, whether a localized basis exists, and how a paired state can carry current.

Required background. Tight-Binding Models supplies the hopping matrices used to construct interference examples. Bloch’s Theorem supplies crystal-momentum sectors, while Density of States supplies normalized state counting for a narrow or degenerate band. Berry Curvature supplies the geometric information in eigenstates that an energy dispersion alone does not determine.

Narrow bandwidths and flat-band physics

A bandwidth measures the range of energies, not the distance between quantum states. First we will compare kinetic and interaction scales, then connect destructive interference to the geometry of the Bloch eigenstates. The resulting tools apply to both designed lattices and moiré materials.

Suppressed Kinetic Energy

Exact, nearly flat, and heavy are different

For an isolated Bloch band, the semiclassical band velocity is $\mathbf v_n(\mathbf k)=\hbar^{-1}\nabla_{\mathbf k}\varepsilon_n(\mathbf k)$.

An exactly flat band has $\mathbf v_f(\mathbf k)=\mathbf0$ everywhere that the band is differentiable. Its curvature also vanishes, so assigning an infinite curvature mass is a useful shorthand only when the band is isolated and smooth. At a touching with another band, a single-band effective-mass expansion is not defined.

For a nearly flat band, define the bandwidth

$$W = \max_{\mathbf k}\varepsilon_f(\mathbf k) - \min_{\mathbf k}\varepsilon_f(\mathbf k).$$

If it is separated from all other bands, useful direct isolation gaps are

$$\begin{aligned} \Delta_+ &= \min_{\mathbf k,m\in\mathrm{upper}} \left[ \varepsilon_m(\mathbf k) - \varepsilon_f(\mathbf k) \right], \\ \Delta_- &= \min_{\mathbf k,m\in\mathrm{lower}} \left[ \varepsilon_f(\mathbf k) - \varepsilon_m(\mathbf k) \right]. \end{aligned}$$

A common flatness ratio is

$$\mathcal F = \frac{\min(\Delta_+,\Delta_-)}{W}.$$

This definition assumes $W>0$ and both neighboring groups exist with positive direct gaps. For a lowest band, use its upper gap alone. Direct gaps compare states at the same momentum; an indirect gap compares extrema at different momenta and can be smaller or negative. The ratio says nothing about Berry-curvature uniformity, disorder broadening, or interaction-induced mixing.

A heavy quasiparticle band is conceptually different. Its small velocity may result from a frequency-dependent many-body self-energy and a small quasiparticle residue $Z$, not from an exactly degenerate one-electron Hamiltonian. A Landau level is different again: magnetic translations replace ordinary translations, and the degeneracy is tied to flux rather than a zero-field lattice interference pattern.

Density of states is not transport

For $N_f$ exactly degenerate one-particle states in volume $V$, the flat-band contribution is distributionally

$$D_f(E) = \frac{N_f}{V} \delta(E-E_f).$$

For a fixed exactly flat one-particle Hamiltonian, temperature changes occupations, not this spectral delta function. Residual dispersion, disorder, or a lifetime self-energy can broaden the spectrum; temperature-dependent changes of the Hamiltonian or self-energy can do so as well. Thermal probe windows and instrumental resolution broaden a measured line even when the underlying DOS is unchanged. A peak in tunneling or photoemission is therefore compatible with a flat band but does not establish one without momentum-resolved dispersion and a resolution budget.

Current response contains velocities and interband matrix elements as well as state counting. A large $D(E_F)$ can coexist with a small conventional Drude weight. Conversely, geometric interband matrix elements can support optical response and superfluid weight even when the intraband group velocity vanishes.

Destructive Interference and Compact States

An exactly solvable cross-stitch chain

Consider two orbitals $a_n,b_n$ in each one-dimensional cell. Let every orbital in cell $n$ hop with amplitude $-t$ to both orbitals in cells $n\pm1$, and let the intracell hopping be $-t_\perp$. Set the cell spacing to one, so $k$ is dimensionless, and define $c_k\equiv2t\cos k$. In the basis $(a_k,b_k)$,

$$H(k) = - \begin{pmatrix} c_k & t_\perp+c_k \\ t_\perp+c_k & c_k \end{pmatrix}.$$

The symmetric and antisymmetric cell combinations diagonalize the matrix:

$$\begin{aligned} \lvert +,k\rangle &= \frac{\lvert a,k\rangle+\lvert b,k\rangle}{\sqrt2}, \\ \lvert -,k\rangle &= \frac{\lvert a,k\rangle-\lvert b,k\rangle}{\sqrt2}. \end{aligned}$$

Their energies are

$$\begin{aligned} E_+(k) &= -t_\perp-4t\cos k, \\ E_-(k) &= t_\perp. \end{aligned}$$

The antisymmetric state is exactly flat. In real space,

$$\lvert\psi_n^{\mathrm{CLS}}\rangle = \frac{ \lvert n,a\rangle-\lvert n,b\rangle }{\sqrt2}$$

has support on one cell. The two amplitudes leak toward any neighboring orbital with equal magnitude and opposite sign, so the leakage cancels. This is a compact localized state (CLS): an exact eigenstate with strictly finite support.

Top: every orbital couples equally to both orbitals in the next cell. The amplitudes $+1/\sqrt2$ and $-1/\sqrt2$ therefore give zero net amplitude at either neighboring orbital. Bottom: exact bands for $t>0$ and $t_\perp=3t$. The flat band is isolated by $2t$ from the top of the dispersive band. Changing $t_\perp$ can close that gap without destroying the compact eigenstate: a flat band need not be an isolated band.

Compact states need not form a complete basis

Translating a CLS through every cell produces many flat-band eigenstates. They are not always linearly independent or complete. If the flat band touches a dispersive band, its normalized Bloch eigenvector can become singular at the touching. The missing states on a torus may be noncontractible loop states that extend around the system.

For finite-range lattice models, Rhim and Yang (2019) distinguish these cases by the singularity of the flat-band Bloch states:

• a nonsingular flat band has no immovable Bloch-state singularity and can be isolated without sacrificing exact flatness; nonsingularity alone does not assert an existing energy gap;
• a singular flat band touches another band in a way that obstructs a globally smooth one-band eigenvector, so translated CLSs alone do not span the degenerate subspace.

Kagome and Lieb nearest-neighbor models provide canonical singular examples. The band touching is not a minor plotting detail: a perturbation that gaps it can generate dispersion and, when time-reversal symmetry is broken appropriately, a Chern band.

The cross-stitch chain uses orbital antisymmetry. Other constructions use sublattice imbalance, line-graph constraints or magnetic interference. The shared question is whether amplitudes leaving a finite cluster cancel. In this example the translated one-cell states are orthonormal and complete for the flat branch, even when its energy touches the other branch; a touching alone therefore does not imply a singular flat band.

Spectral Flattening

Energy and eigenvectors can be varied independently. For isolated-band projectors $P_n(\mathbf k)$,

$$H(\mathbf k) = \sum_n \varepsilon_n(\mathbf k) P_n(\mathbf k).$$

A spectrally flattened Hamiltonian replaces the dispersions by constants:

$$H_{\mathrm{flat}}(\mathbf k) = \sum_n E_n^{\mathrm{flat}} P_n(\mathbf k).$$

The eigenvectors, Berry curvature, Chern numbers, symmetry representations, and the fact that adjacent manifolds remain separated are preserved when the constants preserve their ordering and remain distinct; the numerical gap sizes are deliberately changed. This construction is central in topological classification because it shows that topology lives in the projectors, not in the detailed dispersion.

The real-space cost matters. A smooth but nonpolynomial $P_n(\mathbf k)$ generally Fourier transforms into hopping over arbitrarily long distances, often with exponential decay when the band is analytic and isolated. Exact spectral flattening is therefore not the same as finding a short-range material Hamiltonian with $W=0$.

For a nonzero-Chern isolated band, there is no complete exponentially localized Wannier basis respecting translation. More precisely, for periodic tight-binding models with finitely many orbitals per cell, Chen, Mazaheri, Seidel, and Tang (2014) prove that an isolated exactly flat band with nonzero Chern number cannot have strictly finite-range hopping. Nearly flat short-range Chern models and exactly flat long-range parent models realize different compromises.

Enhanced Interactions

Projection removes dispersion, not matrix structure

Let $P$ now denote the many-particle projector onto states whose particles all lie in the chosen active bands. A two-body interaction becomes

$$H_{\mathrm{proj}} = P H_{\mathrm{int}} P.$$

For a translation-invariant density interaction,

$$H_{\mathrm{proj}} = \frac{1}{2A} \sum_{\mathbf q} V(\mathbf q) : \bar\rho(-\mathbf q) \bar\rho(\mathbf q) :,$$

where $A$ is sample area and normal ordering removes the self-interaction:

$$\bar\rho(\mathbf q) = \sum_{\mathbf k,m,n} \Lambda_{mn}(\mathbf k,\mathbf q) c^\dagger_{\mathbf k+\mathbf q,m} c_{\mathbf k,n}.$$

The form factor

$$\Lambda_{mn}(\mathbf k,\mathbf q) = \langle u_{m,\mathbf k+\mathbf q} \vert u_{n,\mathbf k} \rangle$$

is written here in a continuum Bloch convention with cell-averaged inner products; it remembers how the Bloch states vary across momentum. Transfers outside the first zone require the corresponding reciprocal-vector folding and Bloch phases. In an orbital tight-binding representation, include the physical orbital positions and density matrix elements consistently rather than assuming every coefficient overlap is a physical density overlap. Two bands with identical $\varepsilon(\mathbf k)$ can therefore have different projected interactions and different many-body phases.

The scale hierarchy controls the projection

A useful hierarchy compares

$$W,\quad U_{\mathrm{eff}},\quad \Delta_{\mathrm{iso}},\quad \Gamma,\quad k_BT.$$

The interaction-dominated but single-manifold regime is most controlled when

$$W,\Gamma,k_BT \ll U_{\mathrm{eff}} \ll \Delta_{\mathrm{iso}}.$$

Here $\Delta_{\mathrm{iso}}$ must bound the energy costs of the virtual processes being eliminated, including relevant indirect separations; a positive same-momentum gap alone need not do that. Off-diagonal interaction matrix elements, not only a nominal onsite $U$, determine the mixing. If $U_{\mathrm{eff}}\gtrsim\Delta_{\mathrm{iso}}$, interactions mix remote bands and a one-band projection may fail. If $\Gamma\gtrsim W$, experiment may be unable to distinguish intrinsic flatness from inhomogeneous broadening. If $W=0$, arbitrarily weak perturbations act at first order inside the degenerate subspace; “small” must then be defined relative to the next retained scale.

At partial filling, the projected interaction can favor polarization, charge order, pairing or fractional topology. Their competition depends on the form factors, filling and interaction range. A large interaction-to-bandwidth ratio strengthens the need to solve this problem; it does not specify its answer.

Band Geometry

The projector can vary while the energy does not

For an isolated band with normalized cell-periodic state $\lvert u_{\mathbf k}\rangle$, define

$$P(\mathbf k) = \lvert u_{\mathbf k}\rangle \langle u_{\mathbf k}\rvert.$$

Neighboring projectors differ even when $\varepsilon(\mathbf k)$ is constant. Their gauge-invariant distance begins as

$$\begin{aligned} d^2(\mathbf k,\mathbf k+d\mathbf k) &\equiv 1- \left| \langle u_{\mathbf k} \vert u_{\mathbf k+d\mathbf k} \rangle \right|^2 \\ &= g_{ij}(\mathbf k) dk_i\,dk_j + O(dk^3). \end{aligned}$$

The quantum geometric tensor is

$$\mathcal Q_{ij} = \langle \partial_i u \vert (1-P) \vert \partial_j u \rangle = g_{ij} - \frac{i}{2}\Omega_{ij},$$

using the convention

$$\Omega_{ij} = i\, \mathrm{Tr} \left( P[\partial_iP,\partial_jP] \right).$$

These derivatives compare states in the same physical Hilbert space. In a finite-orbital model, hold the orbital embedding fixed; a momentum-dependent change of the orbital basis must also transform the position operator. This is more than the band-phase gauge freedom $u_{\mathbf k}\mapsto e^{i\phi(\mathbf k)}u_{\mathbf k}$.

The real symmetric part $g_{ij}$ is the quantum metric. The imaginary antisymmetric part gives the Berry curvature. Both depend only on the projector and survive spectral flattening.

For a rank-one band,

$$g_{ij} = \frac{1}{2} \mathrm{Tr} \left( \partial_iP\,\partial_jP \right).$$

The Brillouin-zone integral of $\mathrm{tr}\,g$ controls the gauge-invariant part of Wannier spread. A band can therefore have zero dispersion but spatially extended or obstructed Wannier functions.

Geometry supports motion of pairs and collective states

In a conventional isolated dispersive band, superfluid weight contains a contribution related to band curvature. In a multiband flat system, virtual interband motion encoded in $g_{ij}$ can contribute even when that conventional term vanishes. Schematically,

$$D_{s,ij}^{\mathrm{geom}} \propto \sum_{\mathbf k} \frac{|\Delta_{\mathbf k}|^2} {E_{\mathbf k}} \tanh\!\left( \frac{E_{\mathbf k}}{2k_BT} \right) g_{ij}(\mathbf k),$$

with prefactors and matrix structure determined by the pairing model. Peotta and Törmä (2015) derive such a geometric contribution within multiband pairing mean-field theory; the schematic expression is not a universal stiffness formula for arbitrary interacting flat bands. A large density of states can enhance pair formation, while quantum geometry helps determine whether those pairs possess phase stiffness. Pairing and superfluidity are not the same criterion.

Using Cartesian momentum coordinates with a common length unit, the pointwise two-dimensional inequality

$$\mathrm{tr}\,g(\mathbf k) \ge \left| \Omega_{xy}(\mathbf k) \right|$$

implies an integrated metric bound for a Chern band. Topology therefore forbids the Bloch states from being geometrically trivial everywhere.

Chern Bands

Flatness imitates one Landau-level ingredient

An ideal Landau level combines exact kinetic degeneracy with a nonzero Chern number and highly constrained guiding-center geometry. A nearly flat lattice Chern band aims to reproduce enough of that structure without a net continuum magnetic field.

For an isolated two-dimensional band,

$$C = \frac{1}{2\pi} \int_{\mathrm{BZ}} d^2k\, \Omega_{xy}(\mathbf k).$$

Energy flatness does not determine $C$. A trivial CLS band can have $C=0$; a spectrally flattened Chern band retains $C\ne0$. What nonzero $C$ forbids is a globally smooth periodic eigenvector and a complete exponentially localized Wannier basis for that isolated band.

Fractional phases require more than a large flatness ratio

At fractional filling, repulsive interactions in a suitable Chern band can produce a fractional Chern insulator (FCI), a lattice analogue of a fractional quantum Hall phase. Small dispersion helps, but remote-band mixing, disorder and temperature must also be small compared with the relevant excitation gaps. The projected-density matrix elements and the distribution of Berry curvature and quantum metric control which interactions act within the band.

These are useful conditions, not a theorem of existence. Regnault and Bernevig (2011) give a concrete numerical fractional-Chern-insulator example, testing spectral flow and fractionalized excitations as well as its low-energy spectrum. Geometry that is nonuniform can favor competing charge order; longer-range interactions can stabilize or destabilize candidate fractions.

Evidence for an FCI should combine a fractional Hall response with a bulk charge gap, ground-state degeneracy or flux-insertion spectral flow in numerics, fractional quasiparticle signatures when available, and tests against conventional symmetry breaking. A fractional filling and resistance minimum are not enough.

Flat-Band Ferromagnetism

The Stoner shortcut is suggestive but uncontrolled

The weak-coupling Stoner criterion is often written

$$U \rho(E_F)>1.$$

Here $\rho(E_F)$ is the one-spin density of states per Hubbard site, so the product with onsite energy $U$ is dimensionless. It is not the DOS per physical volume defined earlier. An ideal flat band appears to make $\rho(E_F)$ divergent. But the same degeneracy invalidates ordinary nondegenerate perturbation theory, and the projected interaction depends on wavefunction overlaps. The criterion motivates a polarization tendency; it does not prove a ferromagnetic ground state.

Mielke–Tasaki ferromagnetism has explicit hypotheses

For a repulsive Hubbard model,

$$H = H_0 + U\sum_i n_{i\uparrow}n_{i\downarrow}, \qquad U>0,$$

take $H_0$ to have a finite real symmetric, spin-independent hopping matrix and $N_d$ degenerate lowest spatial one-particle states. Shift their common energy to zero and set the electron number to $N_e=N_d$. Let $\{u_a\}_{a=1}^{N_d}$ be any orthonormal basis of that lowest eigenspace and define its physical-site projector

$$P_{xy}=\sum_{a=1}^{N_d}u_a(x)u_a^*(y).$$

On the support $\Lambda_0=\{x:P_{xx}>0\}$, require $P$ to be irreducible: no partition into two nonempty site sets makes every matrix element between the sets vanish. Under these hypotheses, Mielke’s projector criterion states that the ground state is the saturated-spin $S=N_e/2$ multiplet, with no further degeneracy, if and only if this irreducibility condition holds. The fully polarized state avoids double occupancy; irreducibility excludes independent spin choices in disconnected parts of its physical support. This statement assumes the real spin-independent Hubbard model above, not arbitrary spin–orbit coupling or additional interactions.

The model constructions of Mielke and Tasaki (1993) express connectivity through specially constructed localized states. An overlap graph of an arbitrary flat-space basis is insufficient: for two disconnected atomic sites, the basis $(e_1\pm e_2)/\sqrt2$ overlaps on both sites, but $P=I$ is reducible and the singly occupied singlet and triplet remain degenerate. The projector condition is independent of such basis changes. Band touching, partial flatness, additional orbitals, longer-range interactions, and nonuniform hopping require the applicable theorem’s hypotheses rather than a slogan.

Lieb’s theorem (1989) is related but distinct. For a finite connected bipartite lattice with real spin-independent hopping only between opposite sublattices, onsite $U>0$ and one electron per site, the theorem gives

$$S_{\mathrm{gs}} = \frac{1}{2} \left| N_A-N_B \right|.$$

Sublattice imbalance often produces a flat band, but the theorem establishes a ferrimagnetic total spin for the full half-filled model. It is not simply the Stoner argument applied to a delta-function density of states.

Recognizing a flat band in a measurement

A nearly nondispersing spectral peak over one cut can be a saddle point, a localized impurity state, a matrix-element effect or an unresolved multiplet. Measure the dispersion across the relevant momentum region, compare its width with instrumental and intrinsic broadening, and check the number and orbital character of the states. A gap to neighboring bands is an additional property, not part of the definition of flatness.

Artificial Lattices and Designer Matter compares experimental implementations. Twisted Bilayer Graphene explains one continuum interference mechanism. Wannier Functions develops localized bases, while Moiré Superconductivity connects pairing and phase coherence to measurements.

Exercises

Exercise 1: diagonalize the cross-stitch chain

Diagonalize the cross-stitch Hamiltonian and verify directly that $(1,-1)^{\mathsf T}/\sqrt2$ has momentum-independent energy.

Solution

The matrix has equal diagonal entries and equal off-diagonal entries, so its eigenvectors are the symmetric and antisymmetric combinations. Acting on the antisymmetric vector gives

$$\begin{aligned} H(k) \frac{1}{\sqrt2} \begin{pmatrix} 1\\-1 \end{pmatrix} &= \frac{1}{\sqrt2} \begin{pmatrix} t_\perp\\-t_\perp \end{pmatrix} \\ &= t_\perp \frac{1}{\sqrt2} \begin{pmatrix} 1\\-1 \end{pmatrix}. \end{aligned}$$

All terms proportional to $\cos k$ cancel. The symmetric vector instead has energy $-t_\perp-4t\cos k$.

Exercise 2: flatness and isolation

A band has $W=2.0\,\mathrm{meV}$, $\Delta_+=18\,\mathrm{meV}$, $\Delta_-=12\,\mathrm{meV}$, interaction scale $U_{\mathrm{eff}}=8\,\mathrm{meV}$, and broadening $\Gamma=3\,\mathrm{meV}$. Compute $\mathcal F$ and assess the hierarchy.

Solution

The smaller direct gap is $12\,\mathrm{meV}$, so

$$\mathcal F = \frac{12}{2} = 6.$$

$U_{\mathrm{eff}}/W=4$ suggests interaction-dominated dispersion, while $U_{\mathrm{eff}}/\min(\Delta_+,\Delta_-)=2/3$ gives only a modest separation relative to the direct gaps. Before using this ratio to justify projection, we must also determine the relevant indirect separations and virtual excitation costs that define $\Delta_{\mathrm{iso}}$. Moreover $\Gamma>W$: the peaks are broader than the dispersion, although sufficiently precise fits can sometimes resolve shifts smaller than a linewidth. A measured bandwidth therefore needs an uncertainty estimate and a model of the broadening. Neither the one-band approximation nor a measured $2\,\mathrm{meV}$ bandwidth follows from the supplied scales alone.

Exercise 3: a flat band with nonzero quantum metric

In one dimension, set the cell spacing to one and fix two orthogonal pointlike internal orbitals at the same position in each cell. In that physical orbital basis, take a normalized flat-band eigenvector

$$\lvert u_k\rangle = \frac{1}{\sqrt2} \begin{pmatrix} 1\\e^{ik} \end{pmatrix}.$$

Compute $g_{kk}$.

Solution

The derivative is

$$\lvert\partial_k u_k\rangle = \frac{1}{\sqrt2} \begin{pmatrix} 0\\ie^{ik} \end{pmatrix}.$$

Therefore

$$\begin{aligned} \langle\partial_k u_k \vert \partial_k u_k\rangle &= \frac12, \\ \left| \langle u_k \vert \partial_k u_k\rangle \right|^2 &= \frac14. \end{aligned}$$

Hence

$$g_{kk} = \frac12-\frac14 = \frac14.$$

The dispersion can be exactly constant while the relative orbital phase winds with $k$. Indeed, $|\langle u_k|u_{k+q}\rangle|^2=\cos^2(q/2)=1-q^2/4+O(q^4)$, which gives the same metric directly. Restoring cell spacing $a$ gives $g_{kk}=a^2/4$ for dimensional momentum. Relabeling one orbital into a neighboring cell does not remove this physical spread if its position operator is transformed as well.

Exercise 4: why spectral flattening becomes long range

Explain why replacing $\varepsilon_n(\mathbf k)$ by a constant while retaining a nontrivial $P_n(\mathbf k)$ generally introduces long-range hopping.

Solution

Real-space hopping amplitudes are Fourier coefficients of $H(\mathbf k)$. A finite-range tight-binding Hamiltonian is a finite Laurent polynomial in $e^{i\mathbf k\cdot\mathbf R}$. The projector

$$P_n(\mathbf k) = \prod_{m\ne n} \frac{ H(\mathbf k)-\varepsilon_m(\mathbf k) }{ \varepsilon_n(\mathbf k)-\varepsilon_m(\mathbf k) }$$

contains momentum-dependent energy denominators. Even when it is smooth, it is generally not a finite Laurent polynomial. Its Fourier series therefore extends to arbitrarily distant cells, with decay set by analyticity and the complex-momentum gap.

Exercise 5: connectivity in flat-band ferromagnetism

Two compact orbitals are disjoint in real space and each hosts one electron. Does repulsive onsite $U$ select their spins to be parallel?

Solution

No. Each electron already avoids double occupancy, and disjoint orbitals have no shared site on which onsite $U$ can compare their spin configuration. Parallel and antiparallel states remain degenerate in this ideal projection. A connectivity condition, residual hopping, exchange, or another interaction is needed to couple the spins and select global order.

Exercise 6: the metric bound of a Chern band

For a two-dimensional isolated band with $C=1$, use $\mathrm{tr}\,g\ge|\Omega_{xy}|$ to show a lower bound on the integrated metric.

Solution

Integrating the pointwise inequality gives

$$\begin{aligned} \frac{1}{2\pi} \int_{\mathrm{BZ}} d^2k\, \mathrm{tr}\,g &\ge \frac{1}{2\pi} \int_{\mathrm{BZ}} d^2k\, \left| \Omega_{xy} \right| \\ &\ge \left| \frac{1}{2\pi} \int_{\mathrm{BZ}} d^2k\, \Omega_{xy} \right| \\ &= |C| = 1. \end{aligned}$$

The first inequality is geometric; the second is the triangle inequality. Nonzero Chern number therefore forces a nonzero Brillouin-zone-integrated quantum metric.

References

• L. Chen, T. Mazaheri, A. Seidel, and X. Tang, “The Impossibility of Exactly Flat Non-Trivial Chern Bands in Strictly Local Periodic Tight Binding Models,” Journal of Physics A: Mathematical and Theoretical 47, 152001 (2014), doi:10.1088/1751-8113/47/15/152001.
• E. H. Lieb, “Two Theorems on the Hubbard Model,” Physical Review Letters 62, 1201–1204 (1989), doi:10.1103/PhysRevLett.62.1201.
• A. Mielke, “Stability of Ferromagnetism in Hubbard Models with Degenerate Single-Particle Ground States,” Journal of Physics A: Mathematical and General 32, 8411–8418 (1999), arXiv:cond-mat/9910385.
• A. Mielke and H. Tasaki, “Ferromagnetism in the Hubbard Model: Examples from Models with Degenerate Single-Electron Ground States,” Communications in Mathematical Physics 158, 341–371 (1993), doi:10.1007/BF02108079.
• S. Peotta and P. Törmä, “Superfluidity in Topologically Nontrivial Flat Bands,” Nature Communications 6, 8944 (2015), doi:10.1038/ncomms9944.
• N. Regnault and B. A. Bernevig, “Fractional Chern Insulator,” Physical Review X 1, 021014 (2011), doi:10.1103/PhysRevX.1.021014.
• J.-W. Rhim and B.-J. Yang, “Classification of Flat Bands According to the Band-Crossing Singularity of Bloch Wave Functions,” Physical Review B 99, 045107 (2019), doi:10.1103/PhysRevB.99.045107.