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arXiv:2404.03234

Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians

Alexander Avdoshkin

Published Apr 4, 2024• 1 citation
arXivHTMLPDF

Abstract

Understanding the geometric information contained in quantum states is valuable in various branches of physics, particularly in solid-state physics when Bloch states play a crucial role. While the Fubini-Study metric and Berry curvature form offer comprehensive descriptions of non-degenerate quantum states, a similar description for degenerate states did not exist. In this work, we fill this gap by showing how to reduce the geometry of degenerate states to the non-abelian (Wilczek-Zee) connection $A$ and a previously unexplored matrix-valued metric tensor $G$. Mathematically, this problem is equivalent to finding the $U(N)$ invariants of a configuration of subspaces in $\mathbb{C}^n$. For two subspaces, the configuration was known to be described by a set of $m$ principal angles that generalize the notion of quantum distance. For more subspaces, we find $3 m^2 - 3 m + 1$ additional independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces. We also develop a procedure for calculating these invariants as integrals of $A$ and $G$ over geodesics on the Grassmannain manifold. Finally, we briefly discuss possible application of these results to quantum state preparation and $PT$-symmetric band structures.

DOI
10.48550/arxiv.2404.03234

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Geometry of degenerate quantum states, configurations of mmm-planes and invariants on complex Grassmannians
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Consider a set of quantum states $|\ensuremath{\psi}(x)\ensuremath{\rangle}$ parameterized by xxx taken from some parameter space MMM. We demonstrate how all geometric properties of this manifold of states are fully described by a scalar gauge-invariant Bargmann invariant P(3)(x1,x2,x3)=tr[P(x1)P(x2)P(x3)]{P}^{(3)}({x}_{1},{x}_{2},{x}_{3})=tr[P({x}_{1})P({x}_{2})P({x}_{3})]P(3)(x1​,x2​,x3​)=tr[P(x1​)P(x2​)P(x3​)], where $P(x)=|\ensuremath{\psi}(x)\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\psi}(x)|$ are the projectors. Mathematically, P(x)P(x)P(x) defines a map from MMM to the complex projective space CPn\mathbb{C}{P}^{n}CPn and this map is uniquely determined by P(3)(x1,x2,x3){P}^{(3)}({x}_{1},{x}_{2},{x}_{3})P(3)(x1​,x2​,x3​) up to a symmetry transformation. The phase argP(3)(x1,x2,x3)arg{P}^{(3)}({x}_{1},{x}_{2},{x}_{3})argP(3)(x1​,x2​,x3​) can be used to compute the Berry phase for any closed loop in MMM, however, as we prove, it contains other information that cannot be extracted from any Berry phase. When the arguments xi{x}_{i}xi​ of P(3)(x1,x2,x3){P}^{(3)}({x}_{1},{x}_{2},{x}_{3})P(3)(x1​,x2​,x3​) are taken close to each other, to the leading order, it reduces to the familiar Berry curvature $\ensuremath{\omega}$ and quantum metric ggg. We show that higher orders in this expansion are functionally independent of $\ensuremath{\omega}$ and ggg and are related to the extrinsic properties of the map of MMM into CPn\mathbb{C}{P}^{n}CPn giving rise to new local gauge-invariant objects, such as the fully symmetric 3-tensor TTT. Finally, we show how our results have immediate applications to the modern theory of polarization, calculation of the electric current response to a modulated field and physics of flat bands.
N\mathcal{N}N-bein formalism for degenerate states in the parameter space of quantum geometry
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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.
The complete geometry of quantum states in parameter space is characterized by the quantum geometric tensor, which contains the quantum metric and Berry curvature as the real and imaginary parts, respectively. When the quantum states are degenerate, the quantum metric and Berry curvature take non-Abelian forms. The non-Abelian (Abelian) Berry curvature and Abelian quantum metric have been experimentally measured. However, an experimentally feasible scheme to extract all the components of the non-Abelian quantum metric tensor is still lacking. Here we propose a generic protocol to directly extract the non-Abelian quantum metric tensor in arbitrary degenerate quantum states in any dimensional parameter space based on measuring the transition probabilities after parameter quenches. Furthermore, we show that the non-Abelian quantum metric can be measured to obtain the real Chern number of a generalized Dirac monopole and the second Chern number of a Yang monopole, which can be simulated in three- and five-dimensional parameter space of artificial quantum systems, respectively. We also demonstrate the feasibility of our quench scheme for these two applications with numerical simulations.
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.