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Negative-Curvature Energy Phase (NCEP): A Geometric Approach to Quantum Many-Body States

Jincheng Zhang

Published Aug 11, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.21885329

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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.
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From the Aharonov-Bohm effect to general relativity, geometry plays a central role in modern physics. In quantum mechanics, many physical processes depend on the Berry curvature. However, recent advances in quantum information theory have highlighted the role of its symmetric counterpart, the quantum metric tensor. In this paper, we perform a detailed analysis of the ground state Riemannian geometry induced by the metric tensor, using the quantum XYXYXY chain in a transverse field as our primary example. We focus on a particular geometric invariant, the Gaussian curvature, and show how both integrals of the curvature within a given phase and singularities of the curvature near phase transitions are protected by critical scaling theory. For cases where the curvature is integrable, we show that the integrated curvature provides a new geometric invariant, which like the Chern number characterizes individual phases of matter. For cases where the curvature is singular, we classify three types (integrable, conical, and curvature singularities) and detail situations where each type of singularity should arise. Finally, to connect this abstract geometry to experiment, we discuss three different methods for measuring the metric tensor: via integrating a properly weighted noise spectral function or by using leading-order responses of the work distribution to ramps and quenches in quantum many-body systems.
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We propose a generalized quantum geometric tenor to understand topological quantum phase transitions, which can be defined on the parameter space with the adiabatic evolution of a quantum many-body system. The generalized quantum geometric tenor contains two different local measurements, the non-Abelian Riemannian metric and the non-Abelian Berry curvature, which are recognized as two natural geometric characterizations for the change in the ground-state properties when the parameter of the Hamiltonian varies. Our results show the symmetry-breaking and topological quantum phase transitions can be understood as the singular behavior of the local and topological properties of the quantum geometric tenor in the thermodynamic limit.
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