Quantum criticality often lies beyond the scope of the conventional Landau paradigm, and a unifying framework has yet to emerge, due in part to the wide variety of quantum orders. We propose a geometric approach to quantum phase transitions that shifts focus from microscopic order to the competition between noncommuting operators. This competition is encoded in the boundary geometry of their expectation values, defining a quantum observable space. We show that quantum phase transitions occur at zero-curvature points on the quantum observable space boundary, signaling maximal commutativity and suggesting an underlying integrable structure at criticality.
Quantum criticality often lies beyond the scope of the conventional Landau paradigm, and a unifying framework has yet to emerge, due in part to the wide variety of quantum orders. We propose a geometric approach to quantum phase transitions that shifts focus from microscopic order to the competition between noncommuting operators. This competition is encoded in the boundary geometry of their expectation values, defining a quantum observable space. We show that quantum phase transitions occur at zero-curvature points on the quantum observable space boundary, signaling maximal commutativity and suggesting an underlying integrable structure at criticality.
Quantum phase transitions are sudden changes in the ground-state wavefunction of a many-body system that can occur as a control parameter such as a concentration or a field strength is varied. They are driven purely by the competition between quantum fluctuations and mutual interactions among constituents of the system, not by thermal fluctuations; hence they can occur even at zero temperature. Examples of quantum phase transitions in many-body physics may be found in systems ranging from high-temperature superconductors to topological insulators. A quantum phase transition usually can be characterized by nonanalyticity/discontinuity in certain order parameters or divergence of the ground state energy eigenvalue and/or its derivatives with respect to certain physical quantities. Here in a circular one-dimensional spin model with Heisenberg XY interaction and no magnetic field, we observe critical phenomena for the n0=1/N→0 Mott insulator caused by a qualitative change of the boundary condition. We demonstrate in the vicinity of the transition point a sudden change in ground-state properties accompanied by an avoided level-crossing between the ground and the first excited states. Notably, our result links conventional quantum phase transitions to microscopic boundary conditions, with significant implications for quantum information, quantum control, and quantum computing.
The geometric properties of quantum states are fully encoded by the quantum geometric tensor. The real and imaginary parts of the quantum geometric tensor are the quantum metric and Berry curvature, which characterize the distance and phase difference between two nearby quantum states in Hilbert space, respectively. For conventional Hermitian quantum systems, the quantum metric corresponds to the fidelity susceptibility and has already been used to specify quantum phase transitions from the geometric perspective. In this paper, we extend this wisdom to the non-Hermitian systems for revealing non-Hermitian critical points. To be concrete, by employing numerical exact diagonalization and analytical methods, we calculate the quantum metric and corresponding order parameters in various non-Hermitian models, which include two non-Hermitian generalized Aubry-Andr\'e models and non-Hermitian cluster and mixed-field Ising models. We demonstrate that the quantum metric of eigenstates in these non-Hermitian models exactly identifies the localization transitions, mobility edges, and many-body quantum phase transitions with gap closings, respectively. We further show that this strategy is robust against the finite-size effect and different boundary conditions.
While phases and phase transitions are conventionally described by local order parameters in real space, we present a unified framework characterizing the phase transition through the geometry of configuration space defined by the statistics of pairwise distances rH between configurations. Focusing on the concrete example of Ising spins, we establish crucial analytical links between this geometry and fundamental real-space observables, i.e., the magnetization and two-point spin correlation functions. This link unveils the universal scaling law in the configuration space: the standard deviation of the normalized distances exhibits universal criticality as Var(rH)∼L−2β/ν, provided that the system possesses zero magnetization and satisfies 4β/ν<d. We validate this scaling with stochastic series expansion quantum Monte Carlo simulations of the transverse-field Ising model(TFIM). Furthermore, we propose configuration-space diagnostics that go beyond local real-space observables. First, the distribution probability P(rH) parameterized by the transverse field h forms a one-dimensional manifold. Information-geometric analyses, particularly the Fisher information defined on this manifold, successfully pinpoint the TFIM phase transition, regardless of the measurement basis. Second, for the Su-Schrieffer-Heeger Heisenberg model, a parity index derived from P(rH) successfully characterizes the symmetry-protected topological phase and its transition. Our work establishes configuration space geometry as a novel perspective on quantum criticality, revealing how macroscopic universal phenomena are encoded within its global statistical features.
The mathematical operation involved in quantum--classical limits constitutes an active research area. The limit is usually associated with semiclassical expansions or environment-induced decoherence. In this work we argue that a complementary and conceptually sharp characterization is available: the quantum--classical transition can be understood as a flattening of the information geometry underlying quantum state space. From this perspective, \textbf{quantumness is associated with curvature-induced rigidity of probability assignments}, while classical behavior emerges when this geometric rigidity becomes operationally irrelevant. This geometric viewpoint provides a structural criterion for classicality based on the scalar curvature of the information metric and unifies semiclassical limits, decoherence, and thermodynamic coarse-graining within a single framework.
This paper introduces the concept of Quantum State Boundary Phase Transition (QSBT), a novel theoretical framework proposing that phase transitions can occur not only at fixed points of Hamiltonian systems, but also at boundaries within the Hilbert space of a quantum system. Specifically, we hypothesize that the partial derivative of the wavefunction, [∂ψ], at a boundary represents a critical point where a phase transition can initiate or be amplified. The core of this theory rests on the observation that quantum systems, particularly in topologically complex environments, exhibit emergent behaviors that defy traditional phase transition models. We explore the potential of QSBT within topological quantum systems, suggesting it could lead to the discovery of entirely new quantum phases. This work lays the foundation for a paradigm shift in our understanding of phase transitions, with the potential to unlock significant advancements in quantum materials research and fundamental physics. The presented model offers a mechanism for understanding transitions driven by boundary conditions, offering a pathway to predict and control novel quantum phenomena. ---
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.