Intrinsic Quantum Geometric Tensor on the Grassmannian and Spectral Lower Bounds on Dissipation

Abstract

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.