The quantum geometric tensor - the Berry curvature together with the quantum metric - now underlies a long list of observables, from the anomalous Hall effect to the superfluid weight of a flat band. We ask which of these observables actually require quantum mechanics. To answer this question, we study a purely classical system: a point particle carrying a classical magnetic moment $\boldsymbol{\ell}$ that precesses in a momentum-dependent magnetic field $\mathbf{B}(\mathbf{p})$. Within Hamiltonian classical mechanics, the component of $\boldsymbol{\ell}$ along the field reproduces the Berry-curvature phenomena, while its precessing transverse component reproduces the quantum-metric phenomena. The particle acquires a position spread whose second moment is the metric, an orbital magnetic moment, and - most strikingly - an inertial mass generated by a position-dependent force, and with it a nonzero Drude weight in a system that is nominally dispersionless.
Quantum geometric quantities have featured heavily in the discussion of the properties of quantum systems in recent years. Among quantities most commonly discussed is the variance of Berry curvature and the integral of the trace of the quantum metric tensor. Despite their usefulness, it is known that they suffer from one significant complication: for a tight-binding model, the quantum geometric quantities depend not only on the parameters of the tight-binding model itself, but also the real space geometry of the tight-binding model, the so-called"orbital embedding". One explicitly geometry-independent quantity is therefore the minimal value of the quantum geometric quantity out of all possible real space geometries. In this work, we demonstrate that, if the tight-binding model is compatible with certain spatial symmetries, then the real space geometry that minimizes the variance of the Berry curvature or the integral of the trace of the quantum metric tensor must obey all those spatial symmetries. We further show that the statement is applicable to systems with magnetic translation symmetries and other composite symmetries, with implications for the quantum geometry of the Hofstadter model.
Ziwei Wang, Charlie Raca, S. H. Simon· 2 citations
We study the quantum geometry of doubly degenerate energy levels in a four-dimensional parameter space. We find that the scalar quantum metric $g$ and the Berry curvature $F$ obey $(\operatorname{tr} g)^2/16\geq\sqrt{\det g}\geq |2\Tr(F\wedge F)-\Tr F\wedge \Tr F|/24$. The first inequality characterizes the anisotropy in the metric. The second determinant inequality measures the self-duality of the traceless $SU(2)$ part of the curvature under Hodge star operation and the algebraic closedness of the inter-level polarization amplitudes under $SU(2)$ rotations in the doubly degenerate levels. The saturation of the determinant bound imposes a quaternionic Cauchy-Riemann equation, analogous to the complex analyticity imposed by ideal-band conditions in two-dimensional Chern insulators. As examples, four-band Dirac Hamiltonians automatically saturate the determinant bound and possess a topological zero in $\operatorname{tr}(F\wedge F)$. We compare the differences between Kramers degeneracy and ordinary $U(2)$ degeneracy. In addition to the non-Abelian geometric bound, the latter also obeys an independent first-Chern bound.
Junwen Zhao, Zhiming Pan, Kang Yang et al.· 1 citation· ⚡1
The Quantum Geometric Tensor, encompassing the quantum metric and Berry curvature, is a central concept in modern condensed matter physics. However, its standard calculation via $k$-derivatives of the Bloch projector conceals a fundamental ambiguity regarding the choice of unit-cell convention, specifically in the treatment of intra-cell orbital positions (i.e., with or without the orbital position $e^{ikx_\alpha}$). We resolve this inconsistency by introducing a convention-independent physical QGT defined via a covariant derivative that explicitly incorporates the full position operator. We demonstrate that this formulation is uniquely mandated by the microscopic derivation of the physical current via the Peierls substitution. Notably, we uncover a leading-order failure in standard $k \cdot p$ effective theories for systems with bond-ordered gaps, identifying a need for caution in their application. Finally, we propose geometric engineering as a new design paradigm, enabling the independent tuning of geometric responses without altering the energy dispersion.
Starting from a formal comparison between the Schr\"{o}dinger equation and the Jacobi field equation, this paper proposes the hypothesis that, within the framework of non-relativistic quantum mechanics, the sectional curvature of microscopic physical space can be expressed as \(K(x) = 2m(E-V(x))/\hbar^2\), and attempts to offer a possible unified geometric understanding of quantum mechanics and relativity. The paper is divided into three parts. The first part identifies the Schr\"{o}dinger equation, in its mathematical form, as a Jacobi field equation. The wave function is made to correspond to a Jacobi field, giving it the geometric meaning of the deviation of geodesics in physical space. Phenomena such as quantum tunneling, energy quantization, and the path integral are then interpreted, from this perspective, as geometric manifestations of different spatial curvatures. The second part proposes a geometric projection scheme based on semi-geodesic coordinates. The transverse metric factor of a curved space, which satisfies a Jacobi field equation, is taken as a unified geometric framework. It is demonstrated that the metric correction terms in classical mechanics, special relativity, general relativity, and quantum mechanics can all be derived from the same equation, with the curvature \(K\) arising from different physical sources. The third part extends this framework to statistical mechanics.
We propose a gradient-flow method for quantum states in lattice models, generated by an action consisting of the quantum-metric and the square of the Berry curvature. These two terms drive the spectral projector toward Bogomolny saturation and uniform Berry curvature, respectively. We show that, due to a no-go theorem for finite-dimensional projectors, the two conditions cannot in general be satisfied simultaneously in lattice models. Hence the flow is expected to approach a nontrivial fixed point that balances the two geometric requirements. For the Wilson-Dirac model, we demonstrate that the flowed projector exhibits almost uniform Berry curvature while remaining close to the Bogomolny bound. We further construct a short-range truncated flattened Hamiltonian from the flowed projector and obtain a lattice model with nearly flat bands and nearly uniform Berry curvature. We also apply the method to the Hofstadter model and confirm the roles of the metric and Berry-curvature terms in a Chern band with higher Chern number.
The geometry of quantum states is a fundamental research area with applications ranging from band theory in condensed matter to variational algorithms in quantum information. Due to their relative simplicity, pure states are usually studied, while mixed ones are needed in general, for instance to allow for finite temperatures. The geometry of mixed states, however, is much more challenging, with important aspects yet to be explored. Here, we provide one missing piece by identifying the structure that relates distance measure and curvature for general (mixed or pure) quantum states. This structure is shown to carry the physical meaning of fluctuation-dissipation---by directly relating distance measure with fluctuations and curvature with linear response---and it turns into the corresponding well-known structure for pure states, where it is K\"ahler, and only then. We thus obtain a general, simple fluctuation-dissipation picture, which, among its consequences, implies that response functions inherently probe the mixed-state quantum geometric tensor. We end by using this picture to give geometric characterizations of generic transport behaviors.
Per Moosavi· 0 citations
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