This manuscript develops a geometric theory of quantum phase transitions that does not begin from a preassigned order parameter, symmetry-breaking pattern, or topological invariant. The basic object is a phase-sensitive state kernel extracted from a ground state, thermal state, band projector, correlation matrix, or quantum geometric tensor. A kernel-resolved phase is defined operationally as a maximal parameter region on which the principal phase-sensitive direction of this kernel can be transported continuously with finite accumulated projective variation. The central diagnostic is the rupture index\kappa=\alpha(1-\beta)=\alpha-\gamma,where \alpha is the local closing exponent of the relevant spectral or kernel separation, \gamma is the independently measured exponent of the transverse carrier amplitude, and \beta=\gamma/\alpha is the relative carrier-amplitude exponent. The theorem-bearing class treated in this paper is the finite-dimensional free-fermion / two-level Dirac-mass class. The transverse-field Ising chain, the SSH chain, and the Kitaev chain are developed as the first model realizations. In all three cases the local channel is a nondegenerate Dirac mass closure with\alpha=1,\qquad \gamma=0,\qquad \beta=0,\qquad \kappa=1.The numerical section is presented as an implementation check of the common local theorem, not as an independent proof of three unrelated mechanisms. A next-harmonic deformation of the SSH model verifies stability under nonuniversal velocity and curvature corrections, while a gapped noncritical control confirms saturation rather than an artificial rupture peak. The paper explicitly delimits BKT, quadratic-touching, multicritical, infinite-dimensional-kernel, interacting-kernel, and non-gap-controlled cases as frontier problems rather than classifying them under the present theorem.
This manuscript develops a geometric theory of quantum phase transitions that does not begin from a preassigned order parameter, symmetry-breaking pattern, or topological invariant. The basic object is a phase-sensitive state kernel extracted from a ground state, thermal state, band projector, correlation matrix, or quantum geometric tensor. A kernel-resolved phase is defined operationally as a maximal parameter region on which the principal phase-sensitive direction of this kernel can be transported continuously with finite accumulated projective variation. The central diagnostic is the rupture index\kappa=\alpha(1-\beta)=\alpha-\gamma,where \alpha is the local closing exponent of the relevant spectral or kernel separation, \gamma is the independently measured exponent of the transverse carrier amplitude, and \beta=\gamma/\alpha is the relative carrier-amplitude exponent. The theorem-bearing class treated in this paper is the finite-dimensional free-fermion / two-level Dirac-mass class. The transverse-field Ising chain, the SSH chain, and the Kitaev chain are developed as the first model realizations. In all three cases the local channel is a nondegenerate Dirac mass closure with\alpha=1,\qquad \gamma=0,\qquad \beta=0,\qquad \kappa=1.The numerical section is presented as an implementation check of the common local theorem, not as an independent proof of three unrelated mechanisms. A next-harmonic deformation of the SSH model verifies stability under nonuniversal velocity and curvature corrections, while a gapped noncritical control confirms saturation rather than an artificial rupture peak. The paper explicitly delimits BKT, quadratic-touching, multicritical, infinite-dimensional-kernel, interacting-kernel, and non-gap-controlled cases as frontier problems rather than classifying them under the present theorem.
We resolve the nature of the quantum phase transition between antiferromagnetic and valence-bond solid ground states in square-lattice spin-1/2 magnets - the putative deconfined quantum-critical point. Using quantum Monte Carlo simulations of J-Q models, in which Heisenberg exchange J competes with multi-spin interactions Qn formed by n singlet projectors, we demonstrate first-order discontinuities increasing with n. For n=2 and n=3, the first-order signatures are very weak and we extract universal scaling dimensions of a proximate multicritical point. Their values coincide closely with known bounds for an SO(5) symmetric conformal field theory (CFT) containing a relevant SO(5) singlet operator that destabilizes the generic deconfined critical point. We characterize the emergent SO(5) symmetry by its scaling dimensions of the leading irrelevant perturbations. Using the J-Q2-Q6 model, we tune the transition from weak to stronger first-order and find scaling of the coexisting order parameters with a large exponent beta ~ 0.85, in agreement with the relevant scaling dimensions of the multicritical point. The exponent that had previously been identified with the correlation length does not have a corresponding CFT level, and we explain it as pseudo-critical length scale of bubbles of one phase inside the other that are compressed in size by an SO(5)-violating irrelevant repulsive interaction between the two order parameters. The bubble scale reflects a hierarchy of excitation gaps generated by the SO(5) perturbation in the neighborhood of the first-order transition, akin to a Higgs mechanism. We argue that the multicritical point is the tip of a gapless spin liquid phase in frustrated Heisenberg models into which the J-Q models can be deformed, but the spin liquid, and perhaps the multicritical point itself, exist outside the regime amenable to sign-free simulations.
The Landau paradigm—classifying phases by local order parameters and transitions by spontaneous symmetry breaking—has long been understood to be incomplete. Deconfined quantum critical points (DQCPs), topological orders, and symmetry-protected phases resist Landau characterization. A candidate structural pattern now emerging across the cond-mat.str-el corpus is that the failure of the Landau paradigm is not merely an exception but a systematic consequence of three interlocking mechanisms: (1) the presence of non-invertible symmetries whose breaking generates order parameters obeying generalized algebraic structures rather than group representations; (2) the role of anomaly type—specifically the position of the Lieb-Schultz-Mattis (LSM) anomaly class in the Lyndon-Hochschild-Serre spectral sequence—in determining whether a critical point is dual to ordinary symmetry breaking or to non-invertible symmetry breaking; and (3) excitation-density control as the decisive factor separating phases that merely enter the topological regime from those that develop genuine topological order. This synthesis draws on six papers from cond-mat.str-el and one from cond-mat.mes-hall (as a mechanistic comparator), all posted within the last 30 days. The thesis is presented as a *heuristic structural reading*, not a derivation from a shared formalism: the cited papers use distinct mathematical frameworks (fusion categories, spectral sequences, sequential circuits, tensor networks, quadratic fermion mappings), and we argue that these frameworks share a common organizational principle—the algebra of defects and condensable objects—rather than a common Lagrangian. The primary falsification path is explicit: if a direct DQCP transition between twin phases (as defined by [corpus:arxiv:2605.31601v1]) can be shown to be continuously connected to a hidden symmetry-breaking transition upon deformation of the Hamiltonian, the structural independence of beyond-Landau transitions claimed here would be overturned. Numerical DMRG or tensor-network studies on the specific models of [corpus:arxiv:2606.06343v1] and [corpus:arxiv:2605.27672v1] with explicitly broken categorical symmetry would constitute such a test. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166v1, 2605.27193v1, 2605.27672v1, 2605.31601v1, 2605.31602v1, 2606.05856v1, 2606.06343v1, 2606.11303v1, 2606.11527v1
We investigate the nature of the topological quantum phase transition between the gapless and gapped Kitaev quantum spin liquid phases away from the exactly solvable point. The transition is driven by anisotropy of the Kitaev couplings. At the critical point, the two Dirac points of the gapless Majorana modes merge, resulting in the formation of a semi-Dirac point with quadratic and linear band touching directions. We derive an effective Gross-Neveu-Yukawa-type field theory that describes the topological phase transition in the presence of additional magnetic interactions. We obtain the infrared scaling form of the propagator of the dynamical Ising order parameter field and perform a renormalization-group analysis. The universality of the transition is found to be different from that of symmetry-breaking phase transitions of semi-Dirac electrons. However, as in the electronic case, the Majorana fermions acquire an anomalous dimension, indicative of the breakdown of the fractionalized quasiparticle description.
We study quantum phases and phase transitions in a one-dimensional interacting fermion system with a Lieb-Schultz-Mattis (LSM) type anomaly. Specifically, the inversion symmetry enforces any symmetry-preserving gapped ground state of the system to be a Kitaev chain, following an LSM type theorem that we prove. Alternatively, via the Jordan-Wigner transformation, this system describes a spin system whose gapped ground states must break either the inversion or the Ising symmetry associated with fermion parity. We obtain a phase diagram using analytical methods and variational matrix product state simulations and study the critical behaviors of the quantum phase transitions therein using entanglement entropy, energy variance, and finite-size scaling of order parameters. In particular, we observe a continuous phase transition between two ordered phases that are beyond the Ginzburg-Landau-Wilson paradigm, in analogy to the deconfined quantum critical points in two spatial dimensions. We show this type of a 1D deconfined quantum critical point is described by the Tomonaga-Luttinger liquid theory and extract its Luttinger parameter and critical exponents. We also identify a gapless phase that emerges from a deconfined quantum critical point between two ordered phases, which cannot be described by a U(1) Luttinger liquid.
The exact multi-resolvent hierarchy of dense many-body spectra encodes higher-order spectral-weight correlations, but closing it predictively for interacting Hamiltonians remains a challenge. We introduce the multi-resolvent self-consistency (MRSC) equations, which separate an exact algebraic hierarchy from an approximate finite-parameter closure. The exact Feshbach geometric identity organizes r-resolvent cumulants into a ladder over irreducible vertices Gamma^(r), and the projected hierarchy expresses those vertices as functionals of the cumulants. The closure works in finite parameters: a connectivity basis organizes a general Hamiltonian's seed into class amplitudes, and a causal r-fold Stieltjes representation with a moment-field identity reduce moment matching to a finite-dimensional problem closed at a predictive fixed point. On the free-fermion benchmark the tested exact identities are reproduced and the Faddeeva closure is recovered at r=1. In a blind test on the interacting Ising chain, with the exact two-resolvent data withheld, the Hamiltonian-derived closure predicts the tested probes to 0.35%-0.97%. At the observable level the exact two-resolvent sector supplies a sign-correct, energy-resolved correction to the single-resolvent baseline for the off-diagonal eigenstate-thermalization-hypothesis (ETH) variance, of order one relative to it. Closure errors organize hierarchically: a level-by-level deficit decomposition, anchor measurements of the isolatable terms, and an explicit propagation bound. The same moments further yield a diagnostic chain for the window conditions of local rotational invariance -- an exact drift-kernel identity, a proven single-state identifiability boundary, and a transferability census over couplings and realizations -- a finite-order framework, not a proof of universal emergence.
A recurring obstacle in quantum many-body physics is the identification of phase boundaries from microscopic, experimentally accessible quantities rather than from effective field-theoretic arguments applied post hoc. This synthesis examines five specific findings from recent cond-mat.str-el, cond-mat.dis-nn, cond-mat.mes-hall, and physics.atom-ph preprints and argues — as a heuristic reading, not a derivation — that they converge on a shared structural pattern: phase boundaries are most sharply located when excitation production, spectral topology, or symmetry-defect statistics can be directly controlled or measured at the microscopic level. The sources span (i) non-Hermitian delocalization realizing random Dirac criticality in 1D via spectral winding [corpus:arxiv:2606.12089], (ii) protocol-dependent SPT order emergence governed by Kibble-Zurek defect suppression [corpus:arxiv:2606.11303], (iii) categorical symmetry structure encoding deconfined quantum critical points beyond the Landau paradigm [corpus:arxiv:2606.05856], (iv) a constrained Rydberg spin chain whose phase diagram is fully resolved by DMRG and exact factorization lines [corpus:arxiv:2605.27166], and (v) strain-tunable bandgap and doping in carbon nanotube quantum dots as a direct mechanical handle on transport phase boundaries [corpus:arxiv:2606.12180]. Two additional sources serve supporting roles: the non-Hermitian interacting SSH model demonstrating that exceptional-point proximity amplifies charge-density-wave instabilities [corpus:arxiv:2606.06466], and the Localization Landscape Theory analysis on the Bethe lattice distinguishing percolation from Anderson-transition criticality [corpus:arxiv:2605.29745]. The central falsification path is concrete: if SPT order can be produced by sudden quench protocols in systems where excitation density is independently controlled (e.g., via post-selection or feedback cooling), the excitation-suppression mechanism claimed in [corpus:arxiv:2606.11303] would require revision. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166, 2605.29745, 2606.04958, 2606.05856, 2606.06343, 2606.06432, 2606.06466, 2606.09759, 2606.11048, 2606.11060, 2606.11303, 2606.12089, 2606.12180