Beyond the Berry Phase: Extrinsic Geometry of Quantum States

• URL: http://arxiv.org/abs/2205.15353v1
• Date: Mon, 30 May 2022 18:01:34 GMT
• Title: Beyond the Berry Phase: Extrinsic Geometry of Quantum States
• Authors: Alexander Avdoshkin, Fedor K. Popov
• Abstract summary: We show how all properties of a quantum manifold of states are fully described by a gauge-invariant Bargmann. We show how our results have immediate applications to the modern theory of polarization.
• Score: 77.34726150561087
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Consider a set of quantum states $| \psi(x) \rangle$ 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 $P^{(3)}(x_1, x_2, x_3)=\operatorname{tr}[P(x_1) P(x_2)P(x_3)]$, where $P(x) = |\psi(x)\rangle \langle\psi(x)|$. 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 determined 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 $\omega$ and quantum metric $g$. We show that higher orders in this expansion are functionally independent of $\omega$ 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 electrical response to a modulated field and physics of flat bands.

Related papers

• Measuring quantum relative entropy with finite-size effect [53.64687146666141]
  We study the estimation of relative entropy $D(rho|sigma)$ when $sigma$ is known. Our estimator attains the Cram'er-Rao type bound when the dimension $d$ is fixed.
  arXiv  Detail & Related papers  (2024-06-25T06:07:20Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find 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.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• The SUSY partners of the QES sextic potential revisited [0.0]
  We study the SUSY partner Hamiltonians of the quasi- solvable (QES) sextic potential $Vrm qes(x) = nu, x6 + 2, nu, mu,x4 + left[mu2-(4N+3)nu right], x2$, $N in mathbbZ+$.
  arXiv  Detail & Related papers  (2023-11-10T18:38:02Z)
• Quantum connection, charges and virtual particles [65.268245109828]
  A quantum bundle $L_hbar$ is endowed with a connection $A_hbar$ and its sections are standard wave functions $psi$ obeying the Schr"odinger equation. We will lift the bundles $L_Cpm$ and connection $A_hbar$ on them to the relativistic phase space $T*R3,1$ and couple them to the Dirac spinor bundle describing both particles and antiparticles.
  arXiv  Detail & Related papers  (2023-10-10T10:27:09Z)
• Space-time-symmetric quantum mechanics in 3+1 dimensions [0.0]
  In conventional quantum mechanics, time is treated as a parameter, $t$, and the evolution of the quantum state with respect to time is described by $hat H|psi(t)rangle=ihbar fracddt|psi(t)rangle$. In a recently proposed space-time-symmetric (STS) extension of QM, position becomes the parameter and a new quantum state, $|phi(x)rangle$, is introduced.
  arXiv  Detail & Related papers  (2023-08-08T16:27:43Z)
• Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies [14.084478426185266]
  We show how to define a quantized many-body charge polarization $vecmathscrP$ for (2+1)D topological phases of matter, even in the presence of non-zero Chern number and magnetic field. We derive colored Hofstadter butterflies, corresponding to the quantized value of $vecmathscrP$, which further refine the colored butterflies from the Chern number and discrete shift.
  arXiv  Detail & Related papers  (2022-11-16T19:00:00Z)
• Time Evolution of Typical Pure States from a Macroscopic Hilbert Subspace [0.0]
  We consider a macroscopic quantum system with unitarily evolving pure state $psi_tin mathcalH$. We prove two facts about the evolution of the superposition weights $|P_nupsi_t|2$.
  arXiv  Detail & Related papers  (2022-10-18T17:37:42Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• Completing the quantum formalism in a contextually objective framework [0.0]
  In standard quantum mechanics, a state vector $| psi rangle$ may belong to infinitely many different orthogonal bases. In an idealized case, measuring $A$ again and again will give repeatedly the same result, with the same eigenvalue. The answer is obviously no, since $| psi rangle$ does not specify the full observable $A$ that allowed us to obtain $mu$.
  arXiv  Detail & Related papers  (2020-03-06T10:27:10Z)

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.