Flux-threaded ring: level flow and persistent current#

A magnetic flux through a mesoscopic ring changes the boundary phase acquired by a charged particle around the loop. The resulting level flow produces an equilibrium persistent current, a standard manifestation of phase coherence in multiply connected conductors.[1][2]

For a spinless ring of \(L\) sites,

\[ H(\phi)=-t\sum_{j=0}^{L-1} \left( e^{i\phi/L}c_j^\dagger c_{j+1} +e^{-i\phi/L}c_{j+1}^\dagger c_j \right), \]

where

\[ \phi=2\pi\frac{\Phi}{\Phi_0}, \qquad \Phi_0=\frac{h}{e}. \]

The example uses \(L=4\), \(N=2\), and \(t=1\).

Complex hopping#

import numpy as np
from edinpy import fermion as edf

L = 4
N = 2
t = 1.0
phi = np.pi / 2

theta = phi / L
phase = np.exp(1j * theta)

modes = edf.FermionModes(edf.DoF(L, name="site"))
sector = edf.NParticleSector(modes, N=N).build()
c = edf.set_notation(edf.Annihilation, modes)
cd = edf.set_notation(edf.Creation, modes)

H = 0
for i in range(L):
    j = (i + 1) % L
    hop = cd(i) * c(j)
    H += -t * (phase * hop + phase.conjugate() * hop.dag)

The conjugate hopping amplitudes preserve Hermiticity while making the matrix genuinely complex.

Single-particle level flow#

For momenta

\[ k_m=\frac{2\pi m}{L}, \]

the one-particle energies are

\[ \varepsilon_m(\phi)=-2t\cos\left(k_m-\frac{\phi}{L}\right). \]

Single-particle energy levels of a four-site ring versus magnetic flux — Single-particle level flow of the four-site ring. Crossings change which orbitals are occupied in the zero-temperature many-body ground state.#

Many-body ground-state energy#

For noninteracting fermions,

\[ E_0(\phi)=\sum_{m\in\mathrm{occ}}\varepsilon_m(\phi). \]

Ground-state energy of the half-filled four-site ring versus magnetic flux — Many-body ground-state energy at half filling. Exact diagonalization of the complex Fock-space Hamiltonian reproduces the occupied single-particle result, including the cusps produced by level crossings.#

Persistent current as a custom operator#

The equilibrium current is

\[ I(\phi)=-\frac{\partial E_0}{\partial\phi} =\left\langle-\frac{\partial H}{\partial\phi}\right\rangle. \]

Differentiating the Hamiltonian term by term gives

\[ \hat I=\frac{t}{L}\sum_j \left[ i e^{i\phi/L}c_j^\dagger c_{j+1} -i e^{-i\phi/L}c_{j+1}^\dagger c_j \right]. \]

The literal operator is:

current = 0
for i in range(L):
    j = (i + 1) % L
    hop = cd(i) * c(j)
    current += (t / L) * (
        1j * phase * hop
        - 1j * phase.conjugate() * hop.dag
    )

hamiltonian = edf.Hamiltonian(H, sector)
hamiltonian.eigsolve(k=None)
psi0 = hamiltonian.eigenstate(0)
I = np.real(psi0.dag * current * psi0)

Persistent current of the half-filled four-site ring versus magnetic flux — Persistent current of the half-filled ring. The discontinuous changes at zero temperature coincide with changes in the occupied single-particle levels.#