The geometry of the Hermitian matrix space and the Schrieffer--Wolff transformation

• URL: http://arxiv.org/abs/2407.10478v2
• Date: Tue, 20 Aug 2024 07:20:29 GMT
• Title: The geometry of the Hermitian matrix space and the Schrieffer--Wolff transformation
• Authors: Gergő Pintér, György Frank, Dániel Varjas, András Pályi,
• Abstract summary: In quantum mechanics, the Schrieffer--Wolff (SW) transformation is known as an approximative method to reduce the perturbation dimension of Hamiltonian. We prove that it induces a local coordinate in the space of Hermitian matrices near a $k$-fold degeneracy submanifold.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In quantum mechanics, the Schrieffer--Wolff (SW) transformation (also called quasi-degenerate perturbation theory) is known as an approximative method to reduce the dimension of the Hamiltonian. We present a geometric interpretation of the SW transformation: We prove that it induces a local coordinate chart in the space of Hermitian matrices near a $k$-fold degeneracy submanifold. Inspired by this result, we establish a `distance theorem': we show that the standard deviation of $k$ neighboring eigenvalues of a Hamiltonian equals the distance of this Hamiltonian from the corresponding $k$-fold degeneracy submanifold, divided by $\sqrt{k}$. Furthermore, we investigate one-parameter perturbations of a degenerate Hamiltonian, and prove that the standard deviation and the pairwise differences of the eigenvalues lead to the same order of splitting of the energy eigenvalues, which in turn is the same as the order of distancing from the degeneracy submanifold. As applications, we prove the `protection' of Weyl points using the transversality theorem, and infer geometrical properties of certain degeneracy submanifolds based on results from quantum error correction and topological order.

Related papers

• Equivalence of dynamics of disordered quantum ensembles and semi-infinite lattices [44.99833362998488]
  We develop a formalism for mapping the exact dynamics of an ensemble of disordered quantum systems onto the dynamics of a single particle propagating along a semi-infinite lattice. This mapping provides a geometric interpretation on the loss of coherence when averaging over the ensemble and allows computation of the exact dynamics of the entire disordered ensemble in a single simulation.
  arXiv  Detail & Related papers  (2024-06-25T18:13:38Z)
• Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space [45.9982965995401]
  We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators. We find conditions for their strong continuity and establish properties of their generators.
  arXiv  Detail & Related papers  (2024-06-15T17:39:32Z)
• Spiral flow of quantum quartic oscillator with energy cutoff [0.0]
  The cutoff dependence of the corrected matrices is found to be described by a spiral motion of a three-dimensional vector. This foundational combination of a limit-cycle and a floating fixed-point behaviors warrants further study.
  arXiv  Detail & Related papers  (2024-04-26T14:34:05Z)
• Curvature of Gaussian quantum states [0.0]
  The space of quantum states can be endowed with a metric structure using the second order derivatives of the relative entropy, giving rise to the so-called Kubo-Mori-Bogoliubov inner product.
  arXiv  Detail & Related papers  (2024-04-15T09:14:10Z)
• The Tempered Hilbert Simplex Distance and Its Application To Non-linear Embeddings of TEMs [36.135201624191026]
  We introduce three different parameterizations of finite discrete TEMs via Legendre functions of the negative tempered entropy function. Similar to the Hilbert geometry, the tempered Hilbert distance is characterized as a $t$-symmetrization of the oriented tempered Funk distance.
  arXiv  Detail & Related papers  (2023-11-22T15:24:29Z)
• Digitizing lattice gauge theories in the magnetic basis: reducing the breaking of the fundamental commutation relations [0.0]
  We present a digitization scheme for the lattice $mathrmSU(2)$ gauge theory Hamiltonian in the $mathitmagnetic$ $mathitbasis$, where the gauge links are unitary and diagonal. The digitization is obtained from a particular partitioning of the $mathrmSU(2)$ group manifold, with the canonical momenta constructed by an approximation of the Lie derivatives on this partitioning.
  arXiv  Detail & Related papers  (2023-11-20T17:11:20Z)
• Intrinsic Bayesian Cramér-Rao Bound with an Application to Covariance Matrix Estimation [49.67011673289242]
  This paper presents a new performance bound for estimation problems where the parameter to estimate lies in a smooth manifold. It induces a geometry for the parameter manifold, as well as an intrinsic notion of the estimation error measure.
  arXiv  Detail & Related papers  (2023-11-08T15:17:13Z)
• Quantum quenches in a pseudo-Hermitian Chern insulator [1.0093662416275693]
  We show the bulk-surface duality of the pseudo-Hermitian phases, then build a concrete relation between the static band topology and quench dynamics. We propose a possible scheme to realize the seemingly challenging model in a bilayer lattice and detect the dynamics with a double-quench protocol.
  arXiv  Detail & Related papers  (2022-09-30T03:20:45Z)
• Annihilating Entanglement Between Cones [77.34726150561087]
  We show that Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation.
  arXiv  Detail & Related papers  (2021-10-22T15:02:39Z)
• The semiring of dichotomies and asymptotic relative submajorization [0.0]
  We study quantum dichotomies and the resource theory of asymmetric distinguishability using a generalization of Strassen's theorem on preordered semirings. We find that an variant of relative submajorization, defined on unnormalized dichotomies, is characterized by real-valued monotones that are multiplicative under the tensor product and additive under the direct sum.
  arXiv  Detail & Related papers  (2020-04-22T14:13:26Z)
• Bulk detection of time-dependent topological transitions in quenched chiral models [48.7576911714538]
  We show that the winding number of the Hamiltonian eigenstates can be read-out by measuring the mean chiral displacement of a single-particle wavefunction. This implies that the mean chiral displacement can detect the winding number even when the underlying Hamiltonian is quenched between different topological phases.
  arXiv  Detail & Related papers  (2020-01-16T17:44:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.