We introduce a universal criterion for criticality in mean-field rotor Hamiltonians based on the geometric structure of the constant-energy shell. Rather than characterizing the onset of a phase transition through the conventional thermodynamic singularities alone, we show that the relevant information is already encoded in the way the geometry of the shell reorganizes along distinguished collective directions. For a broad class of finite-dimensional trigonometric mean-field interactions, the trace of the Weingarten operator (representing the principal curvatures) admits a universal collective expansion in terms of the order-parameter amplitudes. This expansion defines an energy-dependent quadratic form whose eigenmodes identify the geometrically unstable channels of the system. Criticality is then associated with the vanishing of the corresponding curvature coefficients, yielding a direct geometric selection principle for the modes that become unstable at the transition. In this way, the phase transition in mean-field systems (usually of first- or second-order) is reformulated as a geometric instability phenomenon intrinsic to the microcanonical energy shell. The resulting framework is geometrically universal within the class considered, independent of model-specific details except for a finite set of collective couplings. Moreover, our approach recovers the known critical channels in standard mean-field rotor models while extending naturally to multimode and spectrally coupled cases. These results support a view in which critical behavior can be understood as reorganizations of energy-shell geometry triggered by a collective restructuring of the underlying energy-shell geometry.
We introduce a universal criterion for criticality in mean-field rotor Hamiltonians based on the geometric structure of the constant-energy shell. Rather than characterizing the onset of a phase transition through the conventional thermodynamic singularities alone, we show that the relevant information is already encoded in the way the geometry of the shell reorganizes along distinguished collective directions. For a broad class of finite-dimensional trigonometric mean-field interactions, the trace of the Weingarten operator (representing the principal curvatures) admits a universal collective expansion in terms of the order-parameter amplitudes. This expansion defines an energy-dependent quadratic form whose eigenmodes identify the geometrically unstable channels of the system. Criticality is then associated with the vanishing of the corresponding curvature coefficients, yielding a direct geometric selection principle for the modes that become unstable at the transition. In this way, the phase transition in mean-field systems (usually of first- or second-order) is reformulated as a geometric instability phenomenon intrinsic to the microcanonical energy shell. The resulting framework is geometrically universal within the class considered, independent of model-specific details except for a finite set of collective couplings. Moreover, our approach recovers the known critical channels in standard mean-field rotor models while extending naturally to multimode and spectrally coupled cases. These results support a view in which critical behavior can be understood as reorganizations of energy-shell geometry triggered by a collective restructuring of the underlying energy-shell geometry.
A microcanonical approach to criticality in the mean-field ϕ4 model: evidence of intrinsic microcanonical structure before the thermodynamic limit
Collective critical behavior is often identified with thermodynamic nonanalyticities and divergences emerging only in the infinite-size limit. Here we adopt a complementary viewpoint: criticality is a structural property due to the rearrangement of the interactions among system's constituents that already exists at finite size and becomes singular only asymptotically. We show that the microcanonical entropy derivatives provide a natural finite-N arena where such structure is encoded in intrinsic extremal/inflection morphologies, and that microcanonical inflection-point analysis (MIPA) turns these morphologies into a unique finite-size critical marker and a well-defined critical trajectory. Using the mean-field ϕ4 model as a stringent benchmark, we reconstruct βN(ε) and γN(ε) from microcanonical simulations, validate them against analytic results, and demonstrate that the MIPA trajectory converges to the exact thermodynamic critical point while simultaneously organizing the approach of other observables to their asymptotic behavior. Our results elevate finite-size criticality from a rounded remnant of the thermodynamic limit to a measurable and predictive object in its own right, with direct relevance to modern finite-system platforms and numerical studies.
We develop a unified geometric framework for local non-adiabaticity in driven quantum systems. We show that the previously introduced AMT non adiabaticity parameter arises as a special realization of a more general geometric instability criterion governed by the normalized Fubini Study distinguishability speed. The local geometric evolution speed is identified as the physically relevant quantity controlling the onset of non-adiabatic instability.
We introduce a universal dimensionless instability parameter measuring the competition between quantum-state evolution speed and spectral-gap protection. This quantity provides a local, gauge-invariant, and basis-independent criterion for arbitrary driven Hamiltonians. Near quantum critical points, the instability parameter diverges through inverse gap amplification, recovering the Kibble Zurek freeze-out condition directly from local geometric data.
We prove that monotonic occupation-dependent nonlinear regulators geometrically compress the quantum metric, establishing a self-limitation theorem in which nonlinear spectral deformation confines the accessible region of projective Hilbert space under strong driving. The multimode extension yields a matrix-valued instability criterion that identifies collective instability channels invisible to scalar descriptions.
The framework naturally extends to open quantum systems through the Bures metric and quantum Fisher geometry, where thermal mixing and Lindblad decay increase the instability threshold through geometric suppression of state distinguishability. The instability threshold further implies a universal geometric lower bound on coherent control time and quantum gate duration.
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 paper introduces the Negative-Curvature Energy Phase (NCEP), a novel theoretical framework that reformulates quantum many-body phase transitions and state classification through the intrinsic geometry of the energy density landscape. By defining a Riemannian metric tensor on the parameter space via the second derivatives of the ground state energy, we construct the associated scalar curvature R_E. We establish that when the scalar curvature satisfies the condition R_E < 0, the quantum system enters a fundamentally distinct energy organization phase. This negative-curvature regime signals an inherent geometric instability, leading to striking physical phenomena such as anomalous energy aggregation, pronounced non-Gaussian fluctuations, and novel critical behaviors. The proposed framework transcends traditional symmetry-breaking paradigms, offering a powerful, model-independent geometric taxonomy for complex quantum states of matter.
For a many-body system in equilibrium, described by a thermodynamically stable Hamiltonian, quantum criticality is associated with structural changes of the many-body ground state. However, there exist physically relevant models, notably, certain quadratic bosonic Hamiltonians (QBHs), which fail to have a ground state. QBHs can be dynamically stable or unstable. We show the notion of criticality is meaningful for the entire class of QBHs that are dynamically stable or at the boundary of instability, regardless of thermodynamic stability, and that the key state for such QBHs is a naturally and unambiguously defined quasiparticle vacuum (QPV). This state is Gaussian, and coincides with the ground state if the QBH is thermodynamically stable. We identify a relevant spectral gap, the Krein gap, associated to the minimal spectral separation between creation and annihilation operators, and show that the QPV is unique when the Krein gap is positive. We prove that, for dynamically stable QBHs with finite-range couplings, correlations are exponentially bounded unless the Krein gap closes, which is associated with one of two spectral degeneracies: an exceptional point or a Krein collision. Consequently, long-range QPV correlations can ensue. Thus, the Krein gap takes the role of the spectral gap for dynamically stable QBHs, and the boundary of dynamical stability and criticality (associated to exceptional points) or multicriticality (associated to Krein collisions) are the same. We also find that bosonic critical behavior beyond thermodynamic stability is witnessed by the scaling of the entanglement entropy and other indicators of equilibrium criticality from information geometry. Our framework opens the door to investigating all dynamically stable QBHs through the lens of critical phenomena, including thermodynamically unstable ones from photonics, cavity-QED, and magnonics.
We develop a mathematically rigorous and experimentally testable framework for the classification of physical regimes based on the modular structure of quantum states. The central object of the theory is the modular generator K = −log ρ, which provides a unified language for spectral, geometric, and information-theoretic diagnostics. We introduce a universal instrument panel (MSRO: Modular–Spectral RG Observables) that integrates spectral quantiles, commutator-based response functions, partition-based locality diagnostics, and information-geometric structures. A key requirement of the framework is strong portability: the same diagnostic protocol applies across multiple domains—including open quantum systems, information geometry, entanglement-based models, and RG-like flows—without domain-specific retuning. The main result of the work is the derivation of a universal and experimentally testable scaling law for the modular response signal: ν(λ) ∼ 1 log λ, which characterizes critical regimes in a wide class of quantum systems. We prove that this scaling emerges from spectral asymptotics of the modular operator and establish a strict equivalence between spectral, geometric, and information-theoretic descriptions of criticality: ν ↔ k(q) ↔ I(λ) ↔ K ↔ R ↔ Δ. The framework is supported by functional analytic results (including Fr´echet differentiability of log ρ), operator inequalities (Golden–Thompson, monotonicity of relative entropy), and stability estimates. We further provide a concrete experimental protocol for quantum simulators, including statistical error bounds and Fisher information constraints, making the prediction directly testable. Beyond regime classification, the results suggest a new perspective on criticality as a geometric phenomenon associated with vanishing modular curvature. The approach opens pathways toward large-N limits, quantum field extensions, and the discovery of new classes of non-standard critical behavior.