An isolated group of $k$ bands in an $N$-level quantum system defines a rank-$k$ spectral projector and hence a map into the complex Grassmannian $\mathrm{Gr}(k,N)$. For a smooth gapped Hamiltonian, this projector is globally smooth and periodic even when band topology obstructs a globally smooth periodic, or symmetry-compatible, Bloch frame. We take the globally defined differential $\dd P$ as the central object: it is the tangent field of the Grassmannian map, removes unphysical rotations within the selected subspace, and retains the physical interband transition. The interband block of this tangent data simultaneously determines the quantum metric and Berry curvature; the associated horizontal generator produces finite Grassmannian motion, while wedge products of $\dd P$ enter topological forms. We construct a shortest path between two projectors in the ambient Grassmannian and show that the singular values of its horizontal generator block are the principal angles between the endpoint subspaces. This construction provides a piecewise-geodesic interpretation of discrete geometric phases. In a separate development, we derive a basis-independent expression for the determinant of a multiband Wilson loop from traces of powers of an ordered projector product, without decomposing a degenerate band multiplet into individual bands. Finally, the same tangent-vector calculus organizes Chern characters, chiral winding numbers, and the time-reversal $\mathbb Z_2$ index. The resulting framework unifies local quantum geometry, finite subspace distance, holonomy, and topology while avoiding global gauge fixing.
An isolated group of k bands in an N-level quantum system defines a rank-k spectral projector and hence a map into the complex Grassmannian Gr(k,N). For a smooth gapped Hamiltonian, this projector is globally smooth and periodic even when band topology obstructs a globally smooth periodic, or symmetry-compatible, Bloch frame. We take the globally defined differential \ddP as the central object: it is the tangent field of the Grassmannian map, removes unphysical rotations within the selected subspace, and retains the physical interband transition. The interband block of this tangent data simultaneously determines the quantum metric and Berry curvature; the associated horizontal generator produces finite Grassmannian motion, while wedge products of \ddP enter topological forms. We construct a shortest path between two projectors in the ambient Grassmannian and show that the singular values of its horizontal generator block are the principal angles between the endpoint subspaces. This construction provides a piecewise-geodesic interpretation of discrete geometric phases. In a separate development, we derive a basis-independent expression for the determinant of a multiband Wilson loop from traces of powers of an ordered projector product, without decomposing a degenerate band multiplet into individual bands. Finally, the same tangent-vector calculus organizes Chern characters, chiral winding numbers, and the time-reversal Z2 index. The resulting framework unifies local quantum geometry, finite subspace distance, holonomy, and topology while avoiding global gauge fixing.
Quantum geometry is universally bounded from below by Wilson-loop windings. In this work, we define an isolated set of bands to be Wilson-loop-ideal, if their quantum metric saturates the Wilson-loop lower bound. The definition naturally incorporates the known Chern-ideal and Euler-ideal bands, and allows us to define other types of ideal bands, such as Kane-Mele Z2-ideal and inversion-fragile-ideal bands. In particular, we find that in the case of zero total Chern number, an isolated WL-ideal set of two bands with non-singular nonabelian Berry curvature and nontrivial normal Wilson-loop winding always admits a Chern-ideal gauge, without the need of a global good quantum number (such as spin). This enables the direct construction of new topologically ordered states, such as fractional topological insulator wavefunctions. We further propose a general framework of constructing monotonic flows that achieve Wilson-loop-ideal states starting from non-ideal bands through band mixing, where Wilson-loop-ideal states are not energy eigenstates but have smooth projectors similar to isolated bands. We apply the constructed flows to the realistic model of 3.89∘ twisted bilayer MoTe2, a moiré Rashba model and another moiré time-reversal-breaking models, and numerically find Chern-ideal, Z2-ideal and inversion-fragile states, respectively, with relative error in the integrated quantum metric below 5×10−3. Our exact-diagonalization calculations on the numerically ideal states demonstrate the potential of our general definition of Wilson-loop-ideal bands and general procedure of constructing Wilson-loop-ideal states for future study of novel correlated physics.
Geometry of degenerate quantum states, configurations of m-planes and invariants on complex Grassmannians
Understanding the geometric information contained in quantum states is valuable in various branches of physics, particularly in solid-state physics when Bloch states play a crucial role. While the Fubini-Study metric and Berry curvature form offer comprehensive descriptions of non-degenerate quantum states, a similar description for degenerate states did not exist. In this work, we fill this gap by showing how to reduce the geometry of degenerate states to the non-abelian (Wilczek-Zee) connection A and a previously unexplored matrix-valued metric tensor G. Mathematically, this problem is equivalent to finding the U(N) invariants of a configuration of subspaces in Cn. For two subspaces, the configuration was known to be described by a set of m principal angles that generalize the notion of quantum distance. For more subspaces, we find 3m2−3m+1 additional independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces. We also develop a procedure for calculating these invariants as integrals of A and G over geodesics on the Grassmannain manifold. Finally, we briefly discuss possible application of these results to quantum state preparation and PT-symmetric band structures.
For full-rank density matrices the state space is contractible, the Uhlmann bundle is topologically trivial, and the holonomy admits no quantized invariants; although the Uhlmann phase is genuinely geometric, this topological poverty has kept it from serving as a robust physical diagnostic of mixed-state matter. Mixed states of fixed rank below the Hilbert-space dimension, however, are ubiquitous: reduced states of constrained subsystems, states confined to invariant sectors, and states supported on decoherence-free subspaces can have a support smaller than the Hilbert space, a support that can vary with parameters. Their geometry is far richer: the support carries genuine curvature, non-Abelian holonomy, and Chern topology. We formulate Uhlmann's theory directly on the manifold of rank-k density matrices: minimal purifications make the fixed-rank stratum the base of a principal U(k)-bundle whose Uhlmann connection is uniquely determined by a Sylvester equation, and in an eigenframe this connection takes a closed form reducing to the Berry, Wilczek--Zee, and faithful Uhlmann connections at k=1, at equal weights, and at k=N. The fixed-rank bundle inherits the topology of the Grassmannian, and non-factorizable higher Chern topology requires failure of the global eigenline splitting, which in the present spectral setting requires degeneracy, minimally realized by a rank-2 Yang monopole whose quantized second Chern number links the geometry to the four-dimensional quantum Hall response. Solvable models, from a genuinely non-Abelian holonomy to a dissipatively driven orbit whose Uhlmann holonomy is steered by the reservoir through an elliptic-integral spectral dressing, illustrate the content: the supporting subspace carries the topology, the spectral weights shape the transport, and Uhlmann holonomy provides a direct geometric probe of the resulting mixed-state structures.
Every object in this paper is a subspace held in place by a gap. The paper builds the quantum geometric tensor directly on the Grassmannian, the space of those subspaces, with no coordinates and no chosen basis, and follows what that one construction says about motion, cost, curvature, geodesics and measurement. The intrinsic tensor is Q_Π(V, W) = Tr(ΠVW). Its real part is the quantum metric, which measures how fast the subspace moves. Its imaginary part is the Berry curvature, which measures how it turns. A subspace can only move by mixing across its own gap, so both parts are exact sums over cross-gap pairs weighted by inverse gaps. The determinant inequality 4 det g ≥ Ω² of band geometry holds in operator form for any pair of tangent directions, with the extremal partner iX given explicitly. Every bound is sharp and rigid. Equality in the gap bound, the speed limit, the dissipation floor, the curvature bound and the trace condition each forces one recognisable configuration: one gap with equal couplings, a rigid rotation, motion at constant intrinsic speed, a complex line of the Grassmannian. The curvature deficit is an exact quadratic form whose kernel is the complex line of the given direction. What follows from it:- Speed and cost. The metric is the quantum Fisher information of the moving subspace, and the subspace can never move faster than the drive divided by the gap. Under an explicit bath model, to first order in the driving rate, the friction on the subspace is the metric reweighted by the spectrum, so the excess work of finite-time driving is at least the squared intrinsic length over the time allowed. The friction weight is checked against slow-driving simulations of two different baths.- Transport tensors. The quantum metric dipole, the nonlinear Hall response and the orbital moment are one cross-gap pair sum read with different powers of the gap.- Geodesic refraction. Across degenerate strata, geodesics of the pullback metric obey a refraction law whose index is the rate at which the subspace rotates. Where the subspace stops rotating they turn back by total internal reflection; where the gap narrows they are guided. A stratum on which the metric collapses along itself is a single projector; a fold is silent at first order and moves at second, and the bending of every geodesic is a quantum metric dipole. The shape sphere of the planar three-body problem is the worked instance.- Mixed states, through the symmetric logarithmic derivative.- Band geometry. Every two-level band meets the curvature bound exactly; the measured polariton tensor of Gianfrate et al. is a device realisation of that extremal case. The lowest band of every spin j meets it too, because it is a spin coherent state, the lowest Landau level on a sphere, while generic three-level bands fall strictly inside.- Data. On public single-shot data from a superconducting quantum heat engine and on the NIST dark-soliton Bose–Einstein condensate images, a three-level standard for extracting angles shows the same law in both: single eigenvectors trade identity where their gap is unresolved, while the subspace above a resolved gap keeps it. In the condensate a single image turns 40.7° while its subspace turns 5.2°, and every resolved cut lands within 0.93–1.12 of the gap prediction, 18 of 18. These data test the stability of the subspace, the reason for building the geometry on the projector, rather than the tensor itself. What ties it together is the Davis–Kahan bound, read as a rule for looking: one vector is held only by its nearest neighbour, the cluster it belongs to is held by the outer gap. Look at each part separately, name a part only where its own gap holds it, and treat everything above a resolved gap as one whole. Light travelling both ways round a ring is the same picture: the two directions are the tail, rotation and backscatter compete to set the tip. The same principal angles reach permutations: the Hamming distance between two permutations is a sum of squared sines of the angles between their graph subspaces, and the intrinsic metric adds the geodesic length, which tells apart permutations the Hamming count cannot. On 20 points, five swaps and one 10-cycle have the same Hamming count and geodesic lengths 3.512 and 2.896; swaps are the longest motion for a given count, and doubling on nine residues is the worked example. Files- iqgt_grassmannian_v7_3.pdf: the paper.- iqgt_grassmannian_v7_3.html: the paper as a web page.- grassmannian_mixer_v8.html: the Grassmannian Mixer, an interactive page that opens in any browser. One operator through eight live views: eigenvalues and the subspaces they fuse into, principal angles and the speed under its gap ceiling, the Bloch sphere, tip and tail, the corridor, the degenerate-strata surface with its geodesics and turning points, the curvature envelope, and permutation angles; plus light both ways round a ring, and two facing tip-and-tail mirrors, each tip landing on the other's tail.- iqgt_grassmannian_v7_3_code_and_proofs.zip: numerical checks for every section with their run logs, figure scripts, condensate data, the test vault, the mixer source, the scripts that build the paper, and the Lean 4 proofs. Machine-checked proofs: 90 theorems in Lean 4 with Mathlib, including the equality cases, no sorry and no project axioms, checked on every commit at https://github.com/dicipler-pixel/iqgt-grassmannian-lean Created with AI assistance and reviewed; see the reviewer advisory at the top of the abstract. This version: equality and rigidity cases for every bound, with the curvature deficit identity; the lowest spin-j bands as the lowest Landau level; strata sorted by the first order that moves, with the three-body shape sphere as the worked instance; light both ways round a ring; swaps as the longest permutation motion, with doubling on nine residues; the friction bound stated with its first-order qualifier; Lean proofs of the equality cases; the mixer with its ring and two-mirror panels.
This work develops an operator-first framework for directional transport in which the primary mathematical objects are spectral projector bundles rather than individual trajectories, eigenvectors, or prescribed metrics. The central idea is that many geometric structures traditionally treated as fundamental—including transport metrics, curvature, persistence, and directional organization—can instead be reconstructed from the continuous evolution of admissible transport subspaces. The framework combines non-normal operator theory, Grassmannian geometry, spectral projectors, quantum geometric tensors, and geometric transport into a unified mathematical architecture. A hierarchy of progressively more stable invariants is developed, replacing fragile local descriptors with structurally persistent objects whenever degeneracies, Exceptional Points, or strong non-normal effects occur. The manuscript introduces the Hierarchy Principle of Structural Persistence, which formalizes this progression. Under appropriate spectral-separation and analytic hypotheses, unstable descriptors such as individual eigenvectors are replaced by higher-level invariant structures including Riesz projector bundles, Grassmannian continuation, and global spectral invariants. This provides a systematic framework for analyzing transport across singularities while preserving the underlying admissible subspace. Within this framework, intrinsic transport metrics and Berry curvature emerge naturally from projector evolution through the quantum geometric tensor, while semiclassical spectral methods connect discrete transport operators to continuum geometric invariants under stated convergence assumptions. The theoretical framework is illustrated across several settings, including: Non-normal transport operators and pseudospectral dynamics. Grassmannian projector geometry and principal-angle continuation. Exceptional Point regularization using Riesz projectors and contour methods. Relational multi-body transport and collision-induced refraction. Covariance transport in quantum heat engine experiments. Directional support geometry motivated by Kakeya-type transport constraints. Frequency-dependent transport and geometric interpretations of psychophysical color spaces. Rather than presenting these as unrelated applications, the manuscript shows how each example arises from the same operator-theoretic hierarchy. The resulting framework suggests that the intrinsic geometry governing admissible transport can be reconstructed from progressively more persistent operator invariants, providing a common mathematical language for transport, geometry, and spectral organization across a broad range of physical systems. If you find value in my work please share it. I hope this helps, please enjoy. JB
Consider a set of quantum states $|\ensuremath{\psi}(x)\ensuremath{\rangle}$ parameterized by x taken from some parameter space M. We demonstrate how all geometric properties of this manifold of states are fully described by a scalar gauge-invariant Bargmann invariant P(3)(x1,x2,x3)=tr[P(x1)P(x2)P(x3)], where $P(x)=|\ensuremath{\psi}(x)\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\psi}(x)|$ are the projectors. Mathematically, P(x) defines a map from M to the complex projective space CPn and this map is uniquely determined by P(3)(x1,x2,x3) up to a symmetry transformation. The phase argP(3)(x1,x2,x3) can be used to compute the Berry phase for any closed loop in M, however, as we prove, it contains other information that cannot be extracted from any Berry phase. When the arguments xi of P(3)(x1,x2,x3) are taken close to each other, to the leading order, it reduces to the familiar Berry curvature $\ensuremath{\omega}$ and quantum metric g. We show that higher orders in this expansion are functionally independent of $\ensuremath{\omega}$ and g and are related to the extrinsic properties of the map of M into CPn giving rise to new local gauge-invariant objects, such as the fully symmetric 3-tensor T. Finally, we show how our results have immediate applications to the modern theory of polarization, calculation of the electric current response to a modulated field and physics of flat bands.