Exactly solvable quantum many-body models are rare, and even when their spectra are algebraically organized, dynamical responses generally remain difficult to obtain because they probe an extensive number of excited states. Here we show that quantum geometric nesting (QGN) models admit an unusually strong form of solvability rooted in \emph{Fock-space fragmentation}: excitations on top of the exact frustration-free ground states decouple into Krylov subspaces with a fixed number of particle and hole operators, and hence remain dynamically invariant. Exploiting this structure, we prove that the stiffness of the spontaneously broken continuous symmetry in QGN models is exactly equal to its variational value in the Gaussian manifold, confirming a conjecture from quantum many-body bootstrap~\cite{GaoHanKhalaf2026}. The proof shows that an infinitesimal phase twist couples the ground state only to the one-particle, one-hole fragment, which coincides with the tangent space of the ground state within the variational manifold, thereby making the variational curvature exact. More generally, perturbations whose action remains within a fixed Fock-space fragment have response functions determined exactly by the corresponding few-body sector, enabling exact access to quantities including static susceptibility, optical conductivity, dynamical structure factors, and single-particle Green's functions.
Exactly solvable quantum many-body models are rare, and even when their spectra are algebraically organized, dynamical responses generally remain difficult to obtain because they probe an extensive number of excited states. Here we show that quantum geometric nesting (QGN) models admit an unusually strong form of solvability rooted in \emph{Fock-space fragmentation}: excitations on top of the exact frustration-free ground states decouple into Krylov subspaces with a fixed number of particle and hole operators, and hence remain dynamically invariant. Exploiting this structure, we prove that the stiffness of the spontaneously broken continuous symmetry in QGN models is exactly equal to its variational value in the Gaussian manifold, confirming a conjecture from quantum many-body bootstrap~\cite{GaoHanKhalaf2026}. The proof shows that an infinitesimal phase twist couples the ground state only to the one-particle, one-hole fragment, which coincides with the tangent space of the ground state within the variational manifold, thereby making the variational curvature exact. More generally, perturbations whose action remains within a fixed Fock-space fragment have response functions determined exactly by the corresponding few-body sector, enabling exact access to quantities including static susceptibility, optical conductivity, dynamical structure factors, and single-particle Green's functions.
We introduce the concept of “quantum geometric nesting” (QGN) to characterize the idealized ordering tendencies of certain flat-band systems implicit in the geometric structure of the flat-band subspace. Perfect QGN implies the existence of an infinite class of local interactions that can be explicitly constructed and give rise to solvable ground states with various forms of possible fermion bilinear order, including flavor ferromagnetism, density waves, and superconductivity. For the ideal Hamiltonians constructed in this way, we show that certain aspects of the low-energy spectrum can also be exactly computed including, in the superconducting case, the phase stiffness. Examples of perfect QGN include flat bands with certain symmetries (e.g., chiral or time reversal) and non-symmetry-related cases exemplified with an engineered model for pair-density wave. Extending this approach, we obtain exact superconducting ground states with nontrivial pairing symmetry. Published by the American Physical Society 2024
In this paper, we determine the complete ground-state spaces for a broad class of generalized stabilizer quantum models defined over CW-decompositions of non-compact manifolds of arbitrary dimensions and topological type. We introduce a C∗-algebraic framework in which the quantum spin algebra $\A$ is realized as the twisted group C∗-algebra of an infinite discrete Pauli module P endowed with a symplectic structure. We establish a correspondence between stabilizer models and isotropic subgroups of P: Every Lagrangian extension of such a subgroup determines a C∗-diagonal of $\A$, whose Weyl groupoid encodes the dynamics through a continuous 1-cocycle. This realization yields an explicit family of extremal ground-states for every admissible Lagrangian model, together with their decomposition into sectors. When the sector space is T0, this gives a complete characterization of the ground-state spaces. We also prove that every Lagrangian stabilizer model admits a unique frustration-free ground state with a spectral gap. Our framework applies in arbitrary dimension, including higher abelian quantum doubles, homological codes, and a broad class of stabilizer models.
We investigate a hierarchy of invariant Hilbert-space structures connecting Hilbert-space fragmentation, partial solvability, and quantum many-body scars. For spin chains with fragmented spin-1/2 subspaces and their local deformations, we show, using the spin-1XXC chain as a representative example, how successive local deformations reduce extensive irreducible-string fragmentation through an implicitly embedded spin-1/2 invariant space and ultimately to a single scar tower, connecting the XXC, perturbed XXC, and spin-1XY scar models. Through the intermediate embedded space, local dynamics couples states from distinct frozen irreducible-string sectors and selects a new decomposition into representations, from which smaller invariant spaces can be retained. We also establish a rigidity constraint on isolating nonsymmetric towers and a constructive correspondence that transfers a reduction between locally realizable representations of the embedded dynamics. The latter demonstrates that the reduction hierarchy can be transferred through a unitary transformation within the embedded subspace, yielding an exact scar tower built on a Néel reference state without requiring unitary equivalence of the full Hamiltonians.
Preprint and reproducibility archive; not peer reviewed. WHAT IS THE QUESTION? Quantum phases are often characterized through properties of their many-body states: order parameters, correlations, reduced density matrices, entanglement spectra, and local-unitary equivalence classes. Transport, however, is a response to a specified physical probe. This paper asks a sharp version of that distinction: Can two exact condensates be indistinguishable by every spatial reduced-density spectrum, yet exhibit different charge stiffness when the same physical gauge field is applied to both? The answer is yes. The paper constructs an exactly solvable finite-range flat-band pairing model in which two number-projected condensates are related by onsite product unitaries and therefore have identical spatial reduced-density spectra for every bipartition, while their response to the same fixed physical U(1) Peierls twist is qualitatively different. MAIN EXACT RESULT The model contains an exactly degenerate manifold of number-projected condensates. Two branches in this manifold are related by onsite product unitaries. For every spatial region R, their reduced density matrices are therefore unitarily conjugate and have exactly identical spectra. Consequently, all spatial entanglement spectra and all Rényi entropies coincide between the two branches. Nevertheless, under the same fixed physical U(1) twist, one branch is exactly soft while the other has positive static charge curvature. The difference is exact rather than a small perturbative correction. The soft tower has zero curvature, while the rigid branch satisfies an exact all-filling response law K_N^r(epsilon) = [N(M-N)/(M-1)] K_1(epsilon), with M = 2L, and retains a nonzero response density in the fixed-filling density limit evaluated in the paper. An arbitrarily weak symmetry-allowed selector epsilon > 0 uniquely chooses the rigid texture while the odd-charge gap remains open. The odd-charge gap is obtained exactly: Delta_odd(epsilon) = 3/4 + epsilon/2. Thus the switch between zero and finite static curvature is not produced by closing the protected odd-charge gap. WHAT INFORMATION IS MISSING? The two condensates are equivalent as static states under onsite local unitary transformations. However, the external gauge coupling is held physically fixed rather than being transformed together with the state. The paper formulates this distinction as an equivariance problem for the twist-defect map. Static onsite-local-unitary orbit invariants determine the charge curvature only if the physical twist-defect map transforms equivariantly along the orbit. In the constructed model, it does not. The conceptual conclusion is therefore more precise than the familiar observation that condensate fraction and superfluid density are different quantities. Here the two states agree on an exceptionally strong collection of static state information: every spatial reduced-density spectrum. They nevertheless respond differently to the same physical probe. The missing datum is the relative orientation between the state orbit and the physical gauge embedding. In other words: state equivalence does not automatically imply response equivalence when the probe is specified independently of the equivalence transformation. The result does not deny that a full microscopic wavefunction written in a fixed physical basis, together with the complete physical coupling to the external field, determines the response. Rather, it identifies exactly which information is lost when one reduces the system to probe-independent state-orbit data. TWIST-INDUCED UNSELECTION AND NONCOMMUTING LIMITS The work also studies competition between the infinitesimal selector and a finite gauge twist. In the joint scaling limit epsilon -> 0+ with A_x^2/epsilon fixed, the leading texture energies cross at lim_(epsilon -> 0+) A_(x,c)^2(epsilon) / epsilon = 24/7. The twist can therefore drive the selected rigid texture back toward the soft texture while the states remain within the same static onsite-local-unitary orbit. This provides an explicit example of noncommuting limits: selecting the state with epsilon > 0 and then taking epsilon -> 0+ does not reproduce every feature of setting epsilon = 0 from the outset. WHY THIS MATTERS For readers outside flat-band superconductivity, the broader issue is: What information about a quantum state is sufficient to determine its response to an external probe? Entanglement spectra, reduced-density spectra, and local-unitary orbit data are powerful state descriptors. This construction shows that even the complete collection of spatial reduced-density spectra does not, by itself, determine static charge response when the physical gauge coupling supplies additional orientation information. The distinction is conceptually relevant to: - quantum many-body response; - quantum geometry; - transport inferred from wavefunction information; - exact paired states; - entanglement-based classification; - local-unitary invariants; - gauge coupling and probe dependence. The value of the construction is that this separation is realized exactly, without disorder, uncontrolled effective theory, or approximate numerical diagonalization. SCOPE AND NONCLAIMS Throughout this work, “stiffness” means finite-volume static charge curvature K_ij^(N) = [d^2 E_(0,N)(A) / dA_i dA_j]_(A=0) and the explicitly evaluated selected density limit. The result is not identified with an unspecified dynamical order of limits, finite-frequency optical conductivity, or a universal dynamical definition of superfluid weight. The thermodynamic sequence used in the manuscript is an L x 1 torus, periodic in both directions. It is not claimed to be a fully two-dimensional thermodynamic family in which both linear dimensions diverge. Neutral collective excitations can soften with increasing size, so finite-twist uniformity and dynamical response limits remain separate problems. VERSION CLARIFICATION The current release should be quoted using the asymptotic scaling statement A_(x,c)^2(epsilon) / epsilon -> 24/7 as epsilon -> 0+. Earlier wording could be read as claiming the stronger finite-epsilon identity A_x^2 = (24/7) epsilon. That stronger interpretation is not supported. The current manuscript explicitly shows finite-epsilon crossing values displaced from 24/7 and states the scaling limit correctly. This clarification does not alter the paper’s main exact result: identical spatial reduced-density spectra can coexist with zero versus finite static charge curvature under the same physical twist. It also does not alter the exact odd-charge-gap formula. REPRODUCIBILITY The deposit contains the Letter, Supplemental Material, exact and numerical certificates, figure generators, manuscript-build scripts, SHA-256 manifests, and a claim ledger. The archive retains refuted or quarantined claims instead of silently removing them and separates exact identities from finite-precision numerical stress tests. The assumptions behind the selected-state, curvature, gap, and scaling-limit statements are explicitly recorded. SEARCH AND INDEXING CONTEXT Relevant concepts include: flat-band superconductivity; static charge stiffness; Kohn curvature; Peierls twist; exact condensates; eta pairing; quantum geometric nesting; reduced density matrix; entanglement spectrum; Rényi entropy; local-unitary orbit; onsite product unitary; response from wavefunction data; gauge embedding; physical probe orientation; twist-defect map; transport switching; noncommuting limits; quantum many-body response.
Exactly solvable models are often identified from structural features of their interactions. For quantum spin Hamiltonians, these are captured by the frustration graph of anticommuting Pauli terms. The frustration graph characterizes broad families of spin models that map to free fermions for arbitrary, or generic, values of their coupling strengths. Nevertheless, there exist non-generically solvable models that admit free-fermion solutions only at finely tuned couplings. In this work, we characterize a family of such models in terms of graph structures. Given an arbitrary instance of Fendley's free-fermions-in-disguise (FFD) chain, we construct a set of operators, called Krylov currents, that are exactly quadratic in its effective fermionic modes and thus extend it to a linear family of free-fermion solvable Hamiltonians. As the Krylov currents are nonlinear in the FFD Hamiltonian terms, the elements of this generated family are non-generically solvable, and they accordingly fail the known conditions for generic solvability. We prove that every Krylov current admits an expansion in terms of induced paths of the frustration graph and that they generate the full special-orthogonal algebra of Gaussian transformations. Our results identify path support as a signature of non-generic solvability, and they provide a systematic route to finding exactly solvable many-body Hamiltonians.
We adopt a geometric perspective on Fock space to provide two complementary insights into the eigenstates in many-body localized fermionic systems. On the one hand, individual many-body-localized eigenstates are well approximated by a Slater determinant of single-particle orbitals. On the other hand, the orbitals of different eigenstates in a given system display a varying, and generally imperfect, degree of compatibility, as we quantify by a measure based on the projectors onto the corresponding single-particle subspaces. We study this incompatibility between states of fixed and differing particle number, as well as inside and outside the many-body localized regime. This gives detailed insights into the emergence and strongly correlated nature of quasiparticlelike excitations in many-body localized systems, revealing intricate correlations between states of different particle numbers down to the level of individual realizations.