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arXiv:2610.14048

Real-Space Interference from Quantum Metric

Bo-Qing Wu, Xing-Lei Ma, Jin-Xin Hu, Jinchao Zhao, K. T. Law

cond-mat.mes-hallcond-mat.supr-conquant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026
v1 · Oct 5, 2026v2 · Oct 26, 2026
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Abstract

Real-space interference of wave packets is governed by their central momentum separation and momentum width, which set the oscillation wavelength and spatial envelope, respectively. Here, we show that in crystalline systems, quantum metric distribution in an isolated band can similarly give rise to real-space interference through the evanescent tails of a localized bound state. The location and width of a quantum metric hot spot in the momentum space set a complex wavevector as one evanescent plane wave component of a bound state. Consequently, with multiple quantum metric hot spots, the density distribution of a bound state or the total density of two spatially overlapped bound states exhibits an interference pattern, where the oscillation period and long-range decay length are determined by the momentum-space separation of the hot spots and the width of the narrowest hot spots, respectively. This quantum-geometric interference is particularly pronounced in (nearly) flat bands, where competing scales from band dispersion are suppressed. Furthermore, we demonstrate that using a Josephson STM the real-space interference can be experimentally observed through the tip-position dependence of the Josephson critical current. Our results reveal a direct correspondence between momentum-space quantum geometry and real-space interference, showing how bound-state profiles encode the momentum-resolved quantum metric.

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Real-space interference of wave packets is governed by their central momentum separation and momentum width, which set the oscillation wavelength and spatial envelope, respectively. Here, we show that in crystalline systems, quantum metric distribution in an isolated band can similarly give rise to real-space interference through the evanescent tails of a localized bound state. The location and width of a quantum metric hot spot in the momentum space set a complex wavevector as one evanescent plane wave component of a bound state. Consequently, with multiple quantum metric hot spots, the density distribution of a bound state or the total density of two spatially overlapped bound states exhibits an interference pattern, where the oscillation period and long-range decay length are determined by the momentum-space separation of the hot spots and the width of the narrowest hot spots, respectively. This quantum-geometric interference is particularly pronounced in (nearly) flat bands, where competing scales from band dispersion are suppressed. Furthermore, we demonstrate that using a Josephson STM the real-space interference can be experimentally observed through the tip-position dependence of the Josephson critical current. Our results reveal a direct correspondence between momentum-space quantum geometry and real-space interference, showing how bound-state profiles encode the momentum-resolved quantum metric.
By acting the projector to the filled lattice eigenstates on a specific position, or applying local electron annihilation operator on the many-body ground state, one can construct a quantum state localized around a specific position in a solid. The overlap of two such local states at slightly different positions defines a quantum metric in real space, which manifests even in systems as simple as particles in a box. For continuous systems like electron gas, this metric weighted by the density gives the momentum variance of electrons, which is readily measurable by ARPES. The presence of disorder curves the real space manifold and gives rise to various differential geometrical quantities like Riemann tensor and Ricci scalar, indicating the possibility of engineering differential geometrical properties by disorder, as demonstrated by lattice models of 2D metals and topological insulators.
By applying the projector to the filled lattice eigenstates on a specific position, or applying the local electron annihilation operator on the many-body ground state, one can construct a quantum state localized around a specific position in a solid. The overlap of two such local states at slightly different positions defines a quantum metric in real space, which manifests even in systems as simple as particles in a box. For continuous systems like electron gas, this metric weighted by the density gives the momentum variance of electrons, which is readily measurable by ARPES. The presence of disorder curves the real space manifold and gives rise to various differential geometrical quantities like Riemann tensor and Ricci scalar, indicating the possibility of engineering differential geometrical properties by disorder, as demonstrated by lattice models of 2D metals and topological insulators.
A magnetic impurity embedded in a metal is collectively screened by Fermi-surface quasiparticles into a many-body spin-singlet ground state, forming a Kondo cloud of size ξK∼ℏvF/(kBTK)ξ_{\rm K}\sim\hbar v_F/(k_B T_{\rm K})ξK​∼ℏvF​/(kB​TK​). This kinematic picture collapses in flat bands, where vF=0v_F=0vF​=0 and the hierarchy of dispersive energy shells is absent. Here we show that the missing organizing principle is quantum geometry. A magnetic impurity coupled to an isolated flat band selects a single active bath mode: a coherent superposition of flat-band Bloch states weighted by the hybridization factor v(k)v(\mathbf{k})v(k), while all orthogonal flat-band modes remain dark. The resulting flat-band Kondo problem is a quantum geometric molecule, with an algebraic Kondo scale set by the total projected hybridization strength rather than a logarithmic-renormalization scale. In real space, the impurity-bath spin correlation defines a quantum geometric Kondo cloud. Its cloud-size tensor admits a gauge-invariant decomposition into a hybridization-weighted quantum metric, a dressed Berry-connection covariance, and a positive hybridization-gradient term, yielding the lower bound ξK2≥∑kρ(k)Tr g(k)ξ_{\rm K}^2\geq \sum_{\mathbf{k}}ρ(\mathbf{k}){\rm Tr}\,g(\mathbf{k})ξK2​≥∑k​ρ(k)Trg(k), where ρ(k)=∣v(k)∣2/∑k∣v(k)∣2ρ(\mathbf{k})=|v(\mathbf{k})|^2/\sum_{\mathbf{k}}|v(\mathbf{k})|^2ρ(k)=∣v(k)∣2/∑k​∣v(k)∣2. Our result reveals that, in flat bands, Kondo screening is governed by the quantum geometry and interference structure of the impurity-selected Bloch wave packet, rather than Fermi-surface kinematics.
We observe matter wave interference of a single cesium atom in free fall. The interferometer is an absolute sensor of acceleration and we show that this technique is sensitive to forces at the level of 3.2×10(-27) N with a spatial resolution at the micron scale. We observe the build up of the interference pattern one atom at a time in a free-space interferometer where the mean path separation extends far beyond the coherence length of the atom. Using the coherence length of the atom wave packet as a metric, we directly probe the velocity distribution and measure the temperature of a single atom in free fall.
Multimode squeezing is a key resource for continuous-variable quantum technologies, but its spatial range in bosonic lattices is usually tied to dispersive propagation. Here we show that parametric pairing can create an exactly flat Bogoliubov band spanned by compact Bogoliubov generators, while producing phase-sensitive anomalous correlations and sub-vacuum collective-mode squeezing that extend far beyond their finite support. Each compact generator mixes annihilation and creation operators, thereby encoding the Bogoliubov squeezing structure, and neighboring translated generators can have nonzero commutator overlap. Enforcing canonical bosonic commutation relations therefore requires spatially extended linear combinations of these translated compact generators, which define the canonical Bogoliubov modes. The squeezing transformation of these modes varies with momentum and is quantified by the squeezing-sector symplectic quantum metric. A complex-momentum singularity of the analytically continued canonical Bogoliubov modes sets both the anomalous-correlation decay length and the momentum-space width of this metric, defining an intrinsic quantum-geometric length scale. This singularity can be tuned continuously while preserving exact flatness, thereby controlling the spatial range of correlations and squeezing. Away from exact flatness, long-distance correlations persist through multiple decay channels, while modes localized by defects and dimerized boundaries provide complementary probes of the underlying quantum geometry. In the weak-damping limit, the same quantum-geometric length scale can be extracted from frequency-filtered two-port correlations. Our results establish the quantum geometry of canonical Bogoliubov modes as a mechanism for spatially extended quantum resources arising from compact Bogoliubov generators.
Quantum geometry underlies a wide range of correlated phenomena in electronic systems, yet accessing the geometry of individual bands remains experimentally challenging. Advances in quantum simulators, such as ultracold atoms in optical lattices, have enabled local state preparation and site-resolved measurements, offering direct access to quantum dynamics beyond conventional transport probes. These developments naturally raise the question of whether band-resolved quantum geometry can be extracted from local occupation readouts. Here, we propose an on-site quench protocol to probe the flat-band quantum geometry via local occupation measurements in an atomic lattice. Our scheme dynamically isolates the flat-band sector by coupling a one-dimensional flat-band lattice to a two-dimensional flat-band bath. The bath selectively removes dispersive excitations, while compact localized states remain trapped due to their overlap with the flat-band modes of the bath. We show that the late-time occupation pattern directly encodes the integrated overlaps between flat-band states separated by the wavevector of the initial density pattern, thereby enabling access to integrated quantum distances between the states separated by the commensurate momenta δk=2πs/rδk = 2 πs/rδk=2πs/r, s,r∈Zs,r \in \mathbb Zs,r∈Z and to the integrated flat-band quantum metric. Our results thereby establish a direct connection between quantum geometry and the long-time memory of an initial spatial pattern in flat-band systems.