Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum walk with a local spin interaction

Manami Yamagishi, Naomichi Hatano, Kohei Kawabata, Chusei Kiumi, Akinori Nishino, Franco Nori, Hideaki Obuse

Abstract

We introduce a model of quantum walkers interacting with a magnetic impurity localized at the origin. First, we study a model of a single quantum walker interacting with a localized magnetic impurity. For a simple case of parameter values, we analytically obtain the eigenvalues and the eigenvectors of bound states, in which the quantum walker is bound to the magnetic impurity. Second, we study a model with two quantum walkers and one magnetic impurity, in which the two quantum walkers indirectly interact with each other via the magnetic impurity, as in the Kondo model. We numerically simulate the collision dynamics when the spin-spin interaction at the origin is of the XX type and the SU(2) Heisenberg type. In the case of the XX interaction, we calculate the entanglement negativity to quantify how much the two quantum walkers are entangled with each other, and find that the negativity increases drastically upon the collision of the two walkers. We compare the time dependence for different statistics, namely, fermionic, bosonic, and distinguishable walkers. In the case of the SU(2) interaction, we simulate the dynamics starting from the initial state in which one fermionic walker is in a bound eigenstate around the origin and the other fermionic walker is a delta function colliding with the first walker. We find that a bound eigenstate closest to the singlet state of the first walker and the magnetic impurity is least perturbed by the collision of the second walker. We speculate that this is a manifestation of Kondo physics at the lowest level of the real-space renormalization-group procedure.

Quantum walk with a local spin interaction

Abstract

We introduce a model of quantum walkers interacting with a magnetic impurity localized at the origin. First, we study a model of a single quantum walker interacting with a localized magnetic impurity. For a simple case of parameter values, we analytically obtain the eigenvalues and the eigenvectors of bound states, in which the quantum walker is bound to the magnetic impurity. Second, we study a model with two quantum walkers and one magnetic impurity, in which the two quantum walkers indirectly interact with each other via the magnetic impurity, as in the Kondo model. We numerically simulate the collision dynamics when the spin-spin interaction at the origin is of the XX type and the SU(2) Heisenberg type. In the case of the XX interaction, we calculate the entanglement negativity to quantify how much the two quantum walkers are entangled with each other, and find that the negativity increases drastically upon the collision of the two walkers. We compare the time dependence for different statistics, namely, fermionic, bosonic, and distinguishable walkers. In the case of the SU(2) interaction, we simulate the dynamics starting from the initial state in which one fermionic walker is in a bound eigenstate around the origin and the other fermionic walker is a delta function colliding with the first walker. We find that a bound eigenstate closest to the singlet state of the first walker and the magnetic impurity is least perturbed by the collision of the second walker. We speculate that this is a manifestation of Kondo physics at the lowest level of the real-space renormalization-group procedure.

Paper Structure

This paper contains 22 sections, 1 theorem, 157 equations, 16 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Model of impurity interaction
  3. Identification of a quantum walk as scattering of a Dirac particle due to a series of delta potentials
  4. Hamiltonian and the scattering operator for a magnetic impurity
  5. Time evolution with chiral symmetry
  6. Analytical solution of a bound state
  7. Reformulation of the time-evolution operator
  8. Transfer-matrix analysis of eigenvalues
  9. Bound-state eigenfunctions
  10. Two walkers with a magnetic impurity
  11. Definition
  12. Numerical simulation
  13. Collision of delta functions in the XX case
  14. Collision of a bound state and a delta function in the XX case
  15. Collision of a bound state and a delta function in the SU(2) case
  16. ...and 7 more sections

Key Result

Theorem 1

There exists a bound-state eigenvalue $\mathrm{e}^{\mathrm{i}\lambda_\mu}$ of the time-evolution operator $\tilde{\hat{U}}_{\mathrm{K}}^{\mathrm{1w}}=\hat{S}^{\mathrm{1w}}\tilde{\hat{C}}^{\mathrm{1w}}$ if and only if

Figures (16)

  • Figure 1: Scattering of a Dirac particle due to a delta potential at $x=0$.
  • Figure 2: Scattering of a Dirac particle in a series of delta potentials.
  • Figure 3: Eigenvalues of the time-evolution operators on the unit circle (green circle) with chiral symmetry [(a)--(d)] and with SU(2) symmetry [(e)--(i)]. For (a)--(d), we set the XX interaction $J_x=J_y=J, J_z=0$ with (a) $J=0$, (b) $J=1$, (c) $J=3$, and (d) $J=20$. For (e)--(i), we set the Heisenberg interaction $J_x=J_y=J_z=J$ with (e) $J=1$, (f) $J=2$, (g) $J=3$, (h) $J=7$, and (i) $J=20$.
  • Figure 4: Transfer matrix $T$ that relates $\{\psi_{\mathrm{L}\uparrow,\downarrow}(x-1), \psi_{\mathrm{R}\uparrow,\downarrow}(x)\}$ and $\{\psi_{\mathrm{L}\uparrow,\downarrow}(x), \psi_{\mathrm{R}\uparrow,\downarrow}(x+1)\}$ for $x\neq0$.
  • Figure 5: Schematic figure of the eigenstate \ref{['eq3023']}.
  • ...and 11 more figures

Theorems & Definitions (1)

  • Theorem 1