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arXiv:2110.08703 · Physics Letters B

Emergence of kinematic space from quantum modular geometric tensor

Xing Huang, Chen‐Te Ma

Published Jan 11, 2022• 4 citations
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Abstract

We generalize the Quantum Geometric Tensor by replacing a Hamiltonian with a modular Hamiltonian. The symmetric part of the Quantum Geometric Tensor provides a Fubini-Study metric, and its anti-symmetric sector gives a Berry curvature. Our generalization dubbed Quantum Modular Geometric Tensor gives the metric and curvature of a Kinematic Space. We also use the result of the identity Virasoro block to relate the connected correlator of two Wilson lines to the two-point function of a modular Hamiltonian. This relation realizes a novel holographic entanglement formula for two intervals of a general separation. This result also provides a new interpretation to the connected correlators as the mutual information. Our study remains valid at the quantum level. Hence it provides an opportunity to explore the Kinematic Space in a generic case without conformal symmetry.

DOI
10.1016/j.physletb.2022.136893

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We generalize the Quantum Geometric Tensor by replacing a Hamiltonian with a modular Hamiltonian. The symmetric part of the Quantum Geometric Tensor provides a Fubini-Study metric, and its anti-symmetric sector gives a Berry curvature. Our generalization dubbed Quantum Modular Geometric Tensor gives the metric and curvature of a Kinematic Space. We also use the result of the identity Virasoro block to relate the connected correlator of two Wilson lines to the two-point function of a modular Hamiltonian. This relation realizes a novel holographic entanglement formula for two intervals of a general separation. This result also provides a new interpretation to the connected correlators as the mutual information. Our study remains valid at the quantum level. Hence it provides an opportunity to explore the Kinematic Space in a generic case without conformal symmetry.
Abstract We discover the connection between the Berry curvature and the Riemann curvature tensor in any kinematic space of minimal surfaces anchored on spherical entangling surfaces. This new holographic principle establishes the Riemann geometry in kinematic space of arbitrary dimensions from the holonomy of modular Hamiltonian, which in the higher dimensions is specified by a pair of time‐like separated points as in CFT1 and CFT2. The Berry curvature that we constructed also shares the same property of the Riemann curvature for all geometry: internal symmetry; skew symmetry; first Bianchi identity. We derive the algebra of the modular Hamiltonian and its deformation, the latter of which can provide the maximal modular chaos to the modular scrambling modes. The algebra also dictates the parallel transport, which leads to the Berry curvature exactly matching to the Riemann curvature tensor. Finally, we compare CFT1 to higher dimensional CFTs and show the difference from the OPE block.
We introduce a quantum geometric tensor in a curved space with a parameter-dependent metric, which contains the quantum metric tensor as the symmetric part and the Berry curvature corresponding to the antisymmetric part. This parameter-dependent metric modifies the usual inner product, which induces modifications in the quantum metric tensor and Berry curvature by adding terms proportional to the derivatives with respect to the parameters of the determinant of the metric. The quantum metric tensor is obtained in two ways: By using the definition of the infinitesimal distance between two states in the parameter-dependent curved space and via the fidelity susceptibility approach. The usual Berry connection acquires an additional term with which the curved inner product converts the Berry connection into an object that transforms as a connection and density of weight one. Finally, we provide three examples in one dimension with a nontrivial metric: an anharmonic oscillator, a Morse-like potential, and a generalized anharmonic oscillator; and one in two dimensions: the coupled anharmonic oscillator in a curved space.
Geometric Quantum Mechanics is a novel and prospecting approach motivated by the belief that our world is ultimately geometrical. At the heart of that is a quantity called Quantum Geometric Tensor (or Fubini-Study metric), which is a complex tensor with the real part serving as the Riemannian metric that measures the `quantum distance', and the imaginary part being the Berry curvature. Following a physical introduction of the basic formalism, we illustrate its physical significance in both the adiabatic and non-adiabatic systems.
Abstract The geometry of the parameter space is encoded by the quantum geometric tensor, which captures fundamental information about quantum states and contains both the quantum metric tensor and the curvature of the Berry connection. We present a formulation of the Berry connection and the quantum geometric tensor in the framework of the phase space or Wigner function formalism. This formulation is obtained through the direct application of the Weyl correspondence to the geometric structure under consideration. In particular, we show that the quantum metric tensor can be computed using only the Wigner functions, which opens an alternative way to experimentally measure the components of this tensor. We also address the non-Abelian generalization and obtain the phase space formulation of the Wilczek–Zee connection and the non-Abelian quantum geometric tensor. In this case, the non-Abelian quantum metric tensor involves only the non-diagonal Wigner functions. Then, we verify our approach with examples and apply it to a system of N coupled harmonic oscillators, showing that the associated Berry connection vanishes and obtaining the analytic expression for the quantum metric tensor. Our results indicate that the developed approach is well adapted to study the parameter space associated with quantum many-body systems.
A series of geometric concepts are formulated for PT\mathcal{PT}PT-symmetric quantum mechanics and they are further unified into one entity, i.e., an extended quantum geometric tensor (QGT). The imaginary part of the extended QGT gives a Berry curvature whereas the real part induces a metric tensor on the system's parameter manifold. This results in a unified conceptual framework to understand and explore physical properties of PT\mathcal{PT}PT-symmetric systems from a geometric perspective. To illustrate the usefulness of the extended QGT, we show how its real part, the metric tensor, can be exploited as a tool to detect quantum phase transitions as well as spontaneous PT\mathcal{PT}PT symmetry breaking in PT\mathcal{PT}PT-symmetric systems.
We develop a unified geometric framework in which the quantum geometric tensor (QGT) simultaneously governs both the metrology of gauge-invariant curvature and the symmetry classification of random-matrix ensembles. The symmetric Fubini–Study sector, generated by the Jordan product of symmetric logarithmic derivatives (SLDs), determines the quantum Fisher information and therefore the information gain associated with closed-loop curvature acquisition. The antisymmetric Berry sector, classified by the de Rham cohomology class of the SLD connection, determines whether the accumulated holonomy phase is linearly observable. We show that time-reversal symmetry acts as a topological gate on the readable Berry channel: the holonomy survives only in the unitary Wigner–Dyson class (GUE), where the cohomology class is nontrivial. This converts the previously phenomenological time-reversal “gate factor” into a cohomological indicator. In the symplectic class (GSE), the non-Abelian obstruction is identified with the Jordan associator curvature and appears operationally as a Holevo incompatibility penalty. The resulting cost-efficiency functional is shown to be genuinely U-shaped across the Dyson classes, with a strict minimum at GUE arising from the non-monotonic topology of the readable holonomy sector. We further establish a bridge between the QGT and thermodynamic fluctuation geometry, derive curvature-statistics scaling laws from Wigner level repulsion, and analyze the deformation of the Kähler structure under decoherence through the Bures metric and mean Uhlmann curvature. The framework unifies Berry geometry, quantum metrology, random-matrix universality, and information geometry into a single cohomological description of curvature acquisition.