Electrons can carry a geometry that has nothing to do with the shape of the material. That hidden geometry can control superconductivity, transport, optical response, and even how large a Cooper pair is allowed to become.
A crystal has an obvious geometry. Its atoms occupy positions in space, its lattice has lengths and angles, and its electronic energy bands form surfaces over momentum space. Yet quantum mechanics adds another geometry that is considerably less visible. Even if two electronic states have almost the same momentum and nearly the same energy, their wavefunctions can point in very different directions inside Hilbert space. The rate at which a quantum state changes as its parameters change defines a genuine metric, complete with distances, angles, curvature, and tensor structure.
This object is called the quantum metric. Together with the better-known Berry curvature, it forms the quantum geometric tensor. For decades Berry curvature became one of the central languages of topological matter, explaining anomalous velocities, Chern numbers, quantum Hall physics, and topological band structures. The real part of the same tensor, the quantum metric, received much less attention. That situation is changing rapidly. In 2025, experiments directly reconstructed the complete quantum metric tensor of Bloch electrons in a real solid, black phosphorus, while related spectroscopy established routes for extracting the broader quantum geometric tensor in crystalline materials. By 2026, quantum geometry was being connected experimentally and theoretically to tunable moiré superconductivity, Josephson transport, superconducting coherence lengths, nonlinear Hall responses, and new geometric descriptions of Cooper pairs. [1][2][8][9]
The central idea is strange but simple: energy tells us how expensive a quantum state is, while quantum geometry tells us how different that state is from its neighbors. In some of the most interesting modern materials, particularly nearly flat electronic bands, the second quantity can become just as important as the first.
To understand what it means for quantum states to possess a metric, begin with a normalized state $|\psi(\lambda)\rangle$ depending smoothly on parameters $\lambda^\mu$. These parameters could represent magnetic fields, coupling constants, atomic coordinates, or, in a crystal, the components of electron momentum.
Two nearby quantum states differ by
$$ |\psi(\lambda+d\lambda)\rangle-|\psi(\lambda)\rangle $$
but this difference cannot itself define a physical distance because multiplying a quantum state by an arbitrary phase does not change the physical state. The states
$$ |\psi\rangle $$
and
$$ e^{i\chi}|\psi\rangle $$
represent exactly the same physical configuration.
The correct metric must therefore ignore changes that correspond only to an unobservable phase. Provost and Vallée formalized this geometry in 1980 by constructing a Riemannian metric directly from Hilbert space. For a family of normalized quantum states, the infinitesimal physical distance can be written as [1]
$$ ds^2=g_{\mu\nu}d\lambda^\mu d\lambda^\nu $$
where $g_{\mu\nu}$ is the quantum metric tensor.
A useful way to interpret it comes from the overlap between nearby states. If two states are almost identical, their overlap is close to one. If they rotate rapidly away from one another in Hilbert space, the overlap decreases more quickly. To leading order, the metric measures precisely this loss of fidelity:
$$ 1-|\langle\psi(\lambda)|\psi(\lambda+d\lambda)\rangle|^2\approx g_{\mu\nu}d\lambda^\mu d\lambda^\nu $$
Large quantum metric therefore means that a tiny change in the underlying parameter produces a large change in the wavefunction. Small quantum metric means that neighboring states remain almost parallel in Hilbert space. [1]
This makes the quantum metric conceptually different from an energy derivative. A band can have almost no dispersion while its eigenvectors twist dramatically through Hilbert space. The energy landscape can be nearly featureless while the wavefunction landscape remains geometrically rich.
For electrons in a periodic crystal, the natural parameters are the components of crystal momentum $\mathbf{k}$. Bloch states take the form
$$ |\psi_{n\mathbf{k}}\rangle=e^{i\mathbf{k}\cdot\mathbf{r}}|u_{n\mathbf{k}}\rangle $$
where $n$ labels the band and $|u_{n\mathbf{k}}\rangle$ contains the periodic internal structure of the Bloch state.
The complete local geometry is encoded by the quantum geometric tensor
$$ Q_{ij}=\langle\partial_i u|(1-|u\rangle\langle u|)|\partial_j u\rangle $$
where $\partial_i$ means differentiation with respect to the momentum component $k_i$. The projection operator in the middle removes the physically meaningless component corresponding merely to a phase change.
This tensor contains two distinct geometrical objects. Using a common convention,
$$ g_{ij}=\mathrm{Re}(Q_{ij}) $$
is the quantum metric, while
$$ \Omega_{ij}=-2\mathrm{Im}(Q_{ij}) $$
is the Berry curvature.
They are not separate inventions. They are the symmetric and antisymmetric pieces of the same underlying quantum geometry. The Berry curvature tells us about geometric phase accumulated as a state moves around a loop, while the metric tells us the infinitesimal distance between neighboring states. Modern quantum-materials research increasingly treats them as complementary pieces of the same structure rather than topology and geometry as unrelated subjects. [3]
A two-band system makes this geometry particularly intuitive. Any normalized two-component quantum state can be represented by a pseudospin direction $\mathbf{n}(\mathbf{k})$ on a Bloch sphere. As momentum changes through the Brillouin zone, that pseudospin traces a texture over the sphere.
For such a system, the quantum metric takes the simple form
$$ g_{ij}=\frac{1}{4}\partial_i\mathbf{n}\cdot\partial_j\mathbf{n} $$
while the magnitude and sign of the Berry curvature are determined by how the pseudospin texture wraps through three-dimensional orientation space. Up to the band-sign convention,
$$ \Omega_{ij}=\frac{1}{2}\mathbf{n}\cdot(\partial_i\mathbf{n}\times\partial_j\mathbf{n}) $$
The metric asks how rapidly the pseudospin changes, whereas the Berry curvature asks how that change twists and encloses oriented area.
This distinction immediately explains why two materials can possess similar-looking energy dispersions but very different quantum geometry. Energy depends on the eigenvalues of the Hamiltonian. Quantum geometry depends on its eigenvectors. Changing orbital composition, sublattice character, spin texture, or layer hybridization can rotate the eigenstates dramatically without producing an equally dramatic change in the energy spectrum.
This becomes especially important in a flat band. The semiclassical velocity of an electron in an ordinary band is related to the energy dispersion by
$$ \mathbf{v}=\frac{1}{\hbar}\nabla_{\mathbf{k}}E(\mathbf{k}) $$
If the band becomes perfectly flat,
$$ E(\mathbf{k})=\mathrm{constant} $$
then
$$ \mathbf{v}=0 $$
Every state in the band has essentially the same energy, so the conventional kinetic mechanism for transport disappears.
That seems to create a paradox for superconductivity. A superconductor must possess phase stiffness: changing the phase of the condensate across space must cost energy so that a persistent supercurrent can exist. Conventional intuition ties that stiffness to electron mobility and band dispersion. If the Fermi velocity vanishes in a perfectly flat band, one might therefore expect the superfluid stiffness to vanish as well.
Yet flat bands can support a finite superfluid response because the wavefunctions themselves possess geometry. Peotta and Törmä showed in 2015 that the superfluid weight of certain topologically nontrivial flat bands contains a contribution controlled by the quantum metric. Later work generalized and sharpened this relationship, including symmetry-based lower bounds on superfluid weight. [4][5]
Schematically, the superfluid response can be separated into
$$ D_{ij}=D_{ij}^{\mathrm{conv}}+D_{ij}^{\mathrm{geom}} $$
The conventional contribution depends primarily on band dispersion. The geometric contribution instead has the structure
$$ D_{ij}^{\mathrm{geom}}\propto\sum_{\mathbf{k}}W(\mathbf{k})g_{ij}(\mathbf{k}) $$
where $W(\mathbf{k})$ depends on pairing and occupation. Even when the ordinary band velocity becomes extremely small, the geometric term need not vanish.
A flat band can therefore be kinetically frozen but geometrically alive.
This changes how superconductivity should be visualized. In ordinary BCS theory, electrons near the Fermi surface possess finite velocity and form overlapping Cooper pairs. The supercurrent arises from the collective phase coherence of these mobile paired electrons. A conventional estimate of the coherence length is
$$ \xi_{\mathrm{BCS}}\sim\frac{\hbar v_F}{\Delta} $$
where $v_F$ is the Fermi velocity and $\Delta$ is the superconducting gap.
If $v_F$ approaches zero, this expression appears to predict an extremely small coherence length. But work published in 2025 showed that the quantum metric introduces an additional intrinsic length scale. Under the conditions analyzed there, the superconducting coherence length takes the form [6]
$$ \xi=\sqrt{\xi_{\mathrm{BCS}}^2+\ell_{\mathrm{qm}}^2} $$
where $\ell_{\mathrm{qm}}$ is a length determined by the quantum metric.
This means that increasing the pairing strength cannot necessarily squeeze a Cooper pair indefinitely. Even when the conventional BCS contribution becomes tiny, the geometry of the underlying Bloch wavefunctions can impose a lower limit:
$$ \xi\geq\ell_{\mathrm{qm}} $$
The spatial size of a superconducting object can therefore be constrained by distance in Hilbert space. The result connects something physically measurable in real space, the coherence length governing vortices and superconducting correlations, to an abstract metric defined between momentum-space wavefunctions. [6]
The connection becomes even stranger when topology enters. Because the quantum metric and Berry curvature belong to the same positive quantum geometric tensor, they cannot always vary independently. In two dimensions one obtains an inequality of the form
$$ \sqrt{\det g}\geq\frac{|\Omega_{xy}|}{2} $$
with the precise notation depending on convention.
A band with sufficiently nontrivial Berry curvature therefore cannot make its quantum metric arbitrarily small everywhere. Integrating such relations over momentum space can connect global topological invariants to lower bounds on geometric quantities. In flat-band superconductors, this logic can translate topology into a lower bound on superfluid response or superconducting length scales. [5][6]
This provides a deeper interpretation of the relation between topology and superconductivity. Topology does not necessarily create the attractive interaction binding two electrons. Instead, topology can constrain the geometry of the band, and that geometry can determine whether the paired state possesses enough stiffness to sustain macroscopic superconducting coherence.
In two-dimensional superconductors, this stiffness can directly influence the Berezinskii-Kosterlitz-Thouless transition. In common conventions the transition satisfies a relation of the form
$$ k_BT_{\mathrm{BKT}}=\frac{\pi}{2}D_s(T_{\mathrm{BKT}}) $$
where $D_s$ is the superfluid stiffness. If quantum geometry contributes substantially to $D_s$, the geometry of Bloch wavefunctions can therefore influence an actual thermodynamic transition temperature.
For years this entire discussion faced an obvious experimental problem: the quantum metric is not something that can be photographed directly. Conventional angle-resolved photoemission spectroscopy, or ARPES, measures the energy and momentum of electrons emitted from a material and has become one of the main tools for mapping electronic band structure. The quantum metric, however, resides in the structure of the wavefunctions, not merely their energies.
That barrier has now begun to fall. Work appearing in Nature Physics developed a framework for extracting the quantum geometric tensor of electronic states in real crystalline solids using polarization-, spin-, and angle-resolved photoemission methods, with CoSn as a representative kagome system. [7]
A subsequent 2025 Science experiment went further and reported direct measurement of the complete quantum metric tensor in bulk black phosphorus. The researchers exploited polarization-dependent ARPES to reconstruct the momentum-space pseudospin texture of the valence-band states. Once that pseudospin field was known, the full tensor components of the quantum metric could be obtained. [2]
In two dimensions, a symmetric metric has three independent components:
$$ g_{xx},\qquad g_{yy},\qquad g_{xy} $$
so measuring the complete tensor is substantially richer than extracting a single scalar “quantum distance.” It reveals how quickly states change along different momentum directions and how those directions are geometrically coupled.
The experiment effectively transformed quantum geometry from something inferred through indirect consequences into something that can be mapped across momentum space.
Black phosphorus is especially useful for visualizing why the metric is genuinely a tensor. Its electronic structure is strongly anisotropic. Moving through momentum space in one crystallographic direction can change the electronic wavefunction very differently from moving through another. A scalar measure would erase that directional structure. A tensor preserves it.
The distance element becomes
$$ ds^2=g_{xx}dk_x^2+2g_{xy}dk_xdk_y+g_{yy}dk_y^2 $$
so an equal momentum displacement in two different directions need not correspond to an equal displacement in Hilbert space.
Mathematically this is exactly what a metric does in differential geometry. On a curved surface, one cannot determine physical distance merely by measuring coordinate changes. The metric tells us how coordinate increments translate into actual lengths. Quantum mechanics applies the same idea to the space of states.
The “surface” being measured is not embedded somewhere inside ordinary three-dimensional space. It is a manifold of normalized quantum states modulo their overall phase.
The metric is also beginning to show up in transport experiments where Berry curvature alone cannot explain the signal. In 2025, researchers reported a third-order nonlinear Hall response in non-centrosymmetric ferromagnetic Fe5GeTe2 and attributed the observed contribution to the quantum metric. Importantly, the effect was observed at room temperature, showing that quantum-geometric transport is not restricted to ultracold systems or millikelvin superconducting states. [10]
This is conceptually important because the Hall effect is normally associated with magnetic fields, Berry curvature, or scattering asymmetries. Quantum geometry creates additional nonlinear response channels. When the applied electric field becomes strong enough that higher-order terms matter, the current can contain contributions schematically of the form
$$ j_a=\sigma_{ab}E_b+\chi_{abcd}E_bE_cE_d+\cdots $$
and geometric tensors can enter the nonlinear coefficients.
The quantum metric is therefore not merely an abstract ruler between wavefunctions. It can alter how charge flows through real materials.
Perhaps the sharpest paradox appears in a Josephson junction. Consider two ordinary superconductors separated by a flat-band material. In a conventional normal metal, superconducting correlations penetrate across the junction because electronic states have a finite Fermi velocity. If the central material had an exactly flat band with
$$ v_F=0 $$
standard intuition suggests that the proximity effect should collapse.
A 2025 Physical Review Research study showed that this need not happen. For the flat-band Josephson junction considered by the authors, the penetration of superconducting correlations and the critical Josephson current are controlled by a quantum-metric length. Interface bound states can penetrate through the flat-band material and hybridize into Andreev bound states capable of carrying a sizable supercurrent even when the single-particle Fermi velocity vanishes. [9]
This is one of the clearest ways to express what quantum geometry does. An individual electron may have essentially no conventional group velocity in the flat band, yet a collective superconducting current can cross the material because the wavefunctions possess enough geometric overlap.
Transport survives without ordinary kinetic transport.
Moiré materials provide a natural laboratory for this physics because twisting atomically thin layers can produce extremely narrow electronic bands while simultaneously generating complicated layer, sublattice, valley, and orbital wavefunctions. The energy dispersion can become small exactly where the internal wavefunction texture becomes highly structured.
In July 2026, an experiment on alternating twisted quadralayer graphene reported superconductivity with an electrically tunable relationship between flat and Dirac-like bands. The authors found vanishingly small flat-band Fermi velocity alongside large superfluid stiffness and argued, using their theoretical analysis, that displacement-field-driven hybridization creates quantum-metric hot spots that strengthen the geometric contribution to superconductivity. [11]
This is an especially interesting form of control because an electric field does not simply shift the number of carriers. It changes how different electronic bands hybridize, thereby altering the wavefunctions themselves. The experiment supports the possibility of engineering superconductivity by manipulating Hilbert-space geometry rather than only density of states or interaction strength. [11]
The design principle is unusual:
$$ \mathrm{electric\ field}\rightarrow\mathrm{band\ hybridization}\rightarrow\mathrm{quantum\ metric}\rightarrow\mathrm{superfluid\ response} $$
The internal geometry of the eigenstates becomes something that can, in principle, be tuned experimentally.
The same geometric language is now being pushed beyond single electrons. A 2026 line of theoretical work asks whether Cooper pairs themselves possess an emergent quantum geometry rather than simply inheriting the geometry of the underlying bands. One recent preprint derives a “pair quantum geometry” contributing to the effective mass of two-body bound states and Cooper pairs, while another August 2026 work develops a composite quantum geometry of Bogoliubov-de Gennes superconducting states in which normal-state and pairing contributions can either separate cleanly or become intertwined. These are very recent theoretical results rather than established experimental facts, but they suggest that quantum geometry may become a many-body concept in its own right.
The distinction matters because superconductivity creates new quasiparticles. The Bogoliubov state is a coherent mixture of electron and hole degrees of freedom,
$$ |\gamma\rangle=u|e\rangle+v|h\rangle $$
so its geometry does not have to be identical to the geometry of the normal electron band from which it emerged.
Pairing can generate additional structure.
If this viewpoint continues to prove useful, the geometry relevant to superconductivity may eventually be understood as a hierarchy:
$$ \mathrm{orbital\ geometry}\rightarrow\mathrm{band\ geometry}\rightarrow\mathrm{pair\ geometry}\rightarrow\mathrm{collective\ geometry} $$
Each level describes distances not in ordinary space, but between increasingly complicated quantum states.
Recent work has even connected quantum geometry to how tightly two electrons can bind. A 2026 study of flat-band superconductors found that two-body bound-state size and many-body Cooper-pair size can remain finite and small in the weak-coupling limit because they are controlled by quantum geometry, while the macroscopic superconducting coherence length can behave differently. The result emphasizes that “Cooper-pair size” and “coherence length” need not be identical objects in flat-band systems. [12]
This separation is easy to miss in ordinary superconductors because several characteristic lengths are often of comparable scale. Flat bands expose the difference by suppressing kinetic energy so strongly that geometric contributions become visible.
The underlying lesson is broader. Quantum geometry can determine not only whether a collective state moves but also how large its constituent quantum objects are.
The quantum metric also provides a new way to think about localization. A Bloch state is extended across a crystal, but one can combine Bloch states into localized Wannier functions. How tightly those Wannier states can be localized depends on how smoothly the Bloch eigenstates vary through momentum space. Large quantum metric signals substantial variation and therefore limits localization. This relationship is one reason the same geometry appears repeatedly in flat-band physics: localization, superfluid stiffness, coherence length, and topology are all different manifestations of how the Bloch states are arranged relative to one another in Hilbert space. [3][6]
This gives the quantum metric a useful physical interpretation. It is not merely a measure of abstract distinguishability. It tells us how difficult it is to construct a state that is simultaneously simple in momentum space and tightly localized in real space.
Momentum-space geometry leaves a real-space footprint.
This entire subject also reveals why a band structure by itself is incomplete information. Two Hamiltonians can have exactly the same eigenvalues,
$$ E_n(\mathbf{k}) $$
while having different eigenvectors,
$$ |u_{n\mathbf{k}}\rangle $$
If one looks only at the energy dispersion, the systems may appear identical. Their quantum metric, Berry curvature, optical selection rules, interaction matrix elements, and collective responses can nevertheless be radically different.
In conventional condensed-matter intuition, one first draws the energy bands and then asks where the Fermi level lies. Quantum geometry suggests a richer workflow. One must ask not only
What are the energies?
but also
How do the eigenstates move through Hilbert space?
A band is therefore not just an energy surface. It is an energy surface carrying a quantum-state bundle with its own geometry.
The distinction between topology and geometry becomes particularly useful here. A topological invariant such as a Chern number compresses global information into an integer. That robustness is extraordinarily powerful, but it also discards local detail. Two bands may share the same Chern number while having very different distributions of Berry curvature and quantum metric across the Brillouin zone.
Quantum geometry restores that local information.
Instead of asking only whether a band is topologically nontrivial, one can ask where its metric becomes large, whether the geometry is isotropic, whether it satisfies special metric-curvature relations, how electric fields redistribute geometric hot spots, and which regions dominate superconductivity or optical response. A 2025 review of quantum geometry in materials emphasizes precisely this transition from global topological classification toward local geometric structure as an experimentally consequential quantity. [3]
The future of topological materials may therefore depend as much on engineering the shape of quantum states as on finding new topological invariants.
Quantum mechanics is often described as a theory of probabilities. The wavefunction determines amplitudes, measurements produce distributions, and interference reveals relative phases. Quantum geometry adds another layer: the set of possible wavefunctions forms a structured space in which physical states possess distances and curvature.
That statement might sound purely mathematical if the metric remained hidden inside formalism. It does not. The quantum metric can determine superfluid weight when ordinary band motion disappears, impose a minimum superconducting coherence length, enable Josephson transport through a flat band, contribute to nonlinear Hall currents, constrain localization, and now be reconstructed experimentally in actual electronic materials. Recent moiré experiments indicate that this geometry can even be modified through electric-field-controlled band hybridization.
The most surprising case remains the flat band. When
$$ \nabla_{\mathbf{k}}E=0 $$
the energy dispersion says that nothing should move. Quantum geometry answers that the energy spectrum is not the whole quantum system. Even if the eigenvalues remain fixed, the eigenvectors can twist, overlap, interfere, and carry the geometric structure required for collective motion.
A flat band can have zero ordinary velocity and still possess a nonzero geometric stiffness.
The quantum metric therefore changes the meaning of motion in a solid. Electrons do not respond only to the slopes of their energy bands. Their collective behavior can depend on the distances between their wavefunctions in an invisible state space. A current flowing through a material can carry information about geometry that exists nowhere in ordinary three-dimensional space.
The next generation of quantum materials may be engineered not only by arranging atoms, tuning interactions, or designing topology, but by deliberately shaping the geometry of Hilbert space itself.