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arXiv:2205.15353 · Physical review. B./Physical review. B

Extrinsic geometry of quantum states

Alexander Avdoshkin, Fedor K. Popov

Published Jun 26, 2023• 24 citations
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Abstract

Consider a set of quantum states $|\ensuremath{\psi}(x)\ensuremath{\rangle}$ parameterized by $x$ taken from some parameter space $M$. We demonstrate how all geometric properties of this manifold of states are fully described by a scalar gauge-invariant Bargmann invariant ${P}^{(3)}({x}_{1},{x}_{2},{x}_{3})=tr[P({x}_{1})P({x}_{2})P({x}_{3})]$, where $P(x)=|\ensuremath{\psi}(x)\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\psi}(x)|$ are the projectors. Mathematically, $P(x)$ defines a map from $M$ to the complex projective space $\mathbb{C}{P}^{n}$ and this map is uniquely determined by ${P}^{(3)}({x}_{1},{x}_{2},{x}_{3})$ up to a symmetry transformation. The phase $arg{P}^{(3)}({x}_{1},{x}_{2},{x}_{3})$ can be used to compute the Berry phase for any closed loop in $M$, however, as we prove, it contains other information that cannot be extracted from any Berry phase. When the arguments ${x}_{i}$ of ${P}^{(3)}({x}_{1},{x}_{2},{x}_{3})$ are taken close to each other, to the leading order, it reduces to the familiar Berry curvature $\ensuremath{\omega}$ and quantum metric $g$. We show that higher orders in this expansion are functionally independent of $\ensuremath{\omega}$ and $g$ and are related to the extrinsic properties of the map of $M$ into $\mathbb{C}{P}^{n}$ giving rise to new local gauge-invariant objects, such as the fully symmetric 3-tensor $T$. 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.

DOI
10.1103/physrevb.107.245136

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Consider a set of quantum states ∣\ensuremathψ(x)\ensuremath⟩|\ensuremath{\psi}(x)\ensuremath{\rangle}∣\ensuremathψ(x)\ensuremath⟩ 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)=∣\ensuremathψ(x)\ensuremath⟩\ensuremath⟨\ensuremathψ(x)∣P(x)=|\ensuremath{\psi}(x)\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\psi}(x)|P(x)=∣\ensuremathψ(x)\ensuremath⟩\ensuremath⟨\ensuremathψ(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 \ensuremathω\ensuremath{\omega}\ensuremathω and quantum metric ggg. We show that higher orders in this expansion are functionally independent of \ensuremathω\ensuremath{\omega}\ensuremathω 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.
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The geometric properties of quantum states are crucial for understanding many physical phenomena in quantum mechanics, condensed matter physics, and optics. The central object describing these properties is the quantum geometric tensor, which unifies the Berry curvature and the quantum metric. In this work, we use the differential-geometric framework of vector bundles to analyze the properties of parameter-dependent quantum states and generalize the quantum geometric tensor to this setting. This construction is based on a general connection on a Hermitian vector bundle, which defines a notion of quantum state transport in parameter space, and a sub-bundle projector, which constrains the set of accessible quantum states. We show that the sub-bundle geometry is similar to that of submanifolds in Riemannian geometry and is described by generalized Gauss-Codazzi-Mainardi equations. This leads to a novel definition of the quantum geometric tensor that contains an additional curvature contribution. To illustrate our results, we describe the sub-bundle geometry arising in the semiclassical treatment of Dirac fields propagating in curved spacetime and show how the quantum geometric tensor, with its additional curvature contributions, is obtained in this case. As a concrete example, we consider Dirac fermions confined to a hyperbolic plane and demonstrate how spatial curvature influences the quantum geometry. This work sets the stage for further exploration of quantum systems in curved geometries, with applications in both high-energy physics and condensed matter systems.
The complete quantum metric of a parametrized quantum system has a real part (usually known as the Provost-Vallee metric) and a symplectic imaginary part (known as the Berry curvature). In this paper, we first investigate the relation between the Riemann curvature tensor of the space described by the metric, and the Berry curvature, by explicit parallel transport of a vector in Hilbert space. Subsequently, we write a generating function from which the complex metric, as well as higher-order geometric tensors (affine connection, Riemann curvature tensor), can be obtained in terms of gauge-invariant cumulants. The generating function explicitly relates the quantities which characterize the geometry of the parameter space to quantum fluctuations. We also show that for a mixed quantum-classical system both real and imaginary parts of the quantum metric contribute to the dynamics, if the mass tensor is Hermitian. A many-operator generalization of the uncertainty principle results from taking the determinant of the complex quantum metric. We also calculate the quantum metric for a number of Lie group coherent states, including several representations of the SU(1,1)\text{SU}(1,1)SU(1,1) group. In our examples nontrivial complex geometry results for generalized coherent states. A pair of oscillator states corresponding to the SU(1,1)\text{SU}(1,1)SU(1,1) group gives a double series for its spectrum. The two minimal uncertainty coherent states show trivial geometry, but, again, for generalized coherent states nontrivial geometry results.
An isolated group of kkk bands in an NNN-level quantum system defines a rank-kkk spectral projector and hence a map into the complex Grassmannian Gr(k,N)\mathrm{Gr}(k,N)Gr(k,N). For a smooth gapped Hamiltonian, this projector is globally smooth and periodic even when band topology obstructs a globally smooth periodic, or symmetry-compatible, Bloch frame. We take the globally defined differential $\dd P$ as the central object: it is the tangent field of the Grassmannian map, removes unphysical rotations within the selected subspace, and retains the physical interband transition. The interband block of this tangent data simultaneously determines the quantum metric and Berry curvature; the associated horizontal generator produces finite Grassmannian motion, while wedge products of $\dd P$ enter topological forms. We construct a shortest path between two projectors in the ambient Grassmannian and show that the singular values of its horizontal generator block are the principal angles between the endpoint subspaces. This construction provides a piecewise-geodesic interpretation of discrete geometric phases. In a separate development, we derive a basis-independent expression for the determinant of a multiband Wilson loop from traces of powers of an ordered projector product, without decomposing a degenerate band multiplet into individual bands. Finally, the same tangent-vector calculus organizes Chern characters, chiral winding numbers, and the time-reversal Z2\mathbb Z_2Z2​ index. The resulting framework unifies local quantum geometry, finite subspace distance, holonomy, and topology while avoiding global gauge fixing.
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.
Abstract The quantum geometric tensor, composed of the quantum metric tensor and Berry curvature, fully encodes the parameter space geometry of a physical system. We first provide a formulation of the quantum geometrical tensor in the path integral formalism that can handle both the ground and excited states, making it useful to characterize excited state quantum phase transitions (ESQPT). In this setting, we also generalize the quantum geometric tensor to incorporate variations of the system parameters and the phase-space coordinates. This gives rise to an alternative approach to the quantum covariance matrix, from which we can get information about the quantum entanglement of Gaussian states through tools such as purity and von Neumann entropy. Second, we demonstrate the equivalence between the formulation of the quantum geometric tensor in the path integral formalism and other existing methods. Furthermore, we explore the geometric properties of the generalized quantum metric tensor in depth by calculating the Ricci tensor and scalar curvature for several quantum systems, providing insight into this geometric information.