Peierls Phase Preview

The Peierls phase is the lattice version of electromagnetic phase holonomy. In a tight-binding model, a charged particle hops between localized orbitals. A vector potential is incorporated, to leading approximation, by multiplying each hopping amplitude by a link phase:

$$t_{ij} \quad\longmapsto\quad t_{ij} \exp(i\theta_{ij}),$$

where, for hopping from site $j$ to site $i$,

$$\theta_{ij} = \frac{q}{\hbar} \int_{\mathbf r_j}^{\mathbf r_i} \mathbf A(\mathbf r)\cdot d\mathbf r.$$

This is called the Peierls substitution or Peierls phase prescription. It is the bridge from continuum minimal coupling to lattice magnetic flux, Hofstadter-type spectra, and lattice quantum Hall models.

Helpful background. The tight-binding model explains hopping amplitudes and boundaries. Here the essential distinction is between a phase on one link and the invariant product around a loop. We use static electromagnetic fields; time-dependent gauge changes also require the scalar-potential terms.

From Continuum Phase to Link Phase

In continuum wave mechanics, a charged particle accumulates a phase

$$\exp \left( \frac{iq}{\hbar} \int_C \mathbf A\cdot d\mathbf r \right)$$

along a path $C$. In a tight-binding model, the microscopic path between two localized orbitals is compressed into a link. The hopping term

$$H_{\mathrm{hop}} = \sum_{ij} t_{ij}c_i^\dagger c_j$$

is modified to

$$H_{\mathrm{hop}}(\mathbf A) = \sum_{ij} t_{ij}e^{i\theta_{ij}} c_i^\dagger c_j.$$

Hermiticity requires the reverse link to carry the opposite phase when $t_{ji}=t_{ij}^*$:

$$\theta_{ji} = -\theta_{ij}.$$

The line integral is usually taken along the bond or along a chosen path connecting the localized orbital centers. The approximation is controlled by the physical field varying slowly across an orbital and by weak field-induced orbital distortion or interband mixing. The size or gradient of a gauge-dependent vector potential alone cannot establish validity: even a large pure-gauge potential can describe zero field.

Gauge Transformation on Sites and Links

Under an electromagnetic gauge transformation,

$$\mathbf A' = \mathbf A+\nabla\chi,$$

the link phase changes by endpoint phases:

$$\theta_{ij}' = \theta_{ij} + \frac{q}{\hbar} \left( \chi_i-\chi_j \right), \qquad \chi_i=\chi(\mathbf r_i).$$

With the convention

$$c_j' = e^{iq\chi_j/\hbar}c_j,$$

the hopping operator transforms as

$$c_i^{\prime\dagger}c_j' = e^{-iq\chi_i/\hbar} e^{iq\chi_j/\hbar} c_i^\dagger c_j.$$

The phase change of the hopping coefficient cancels the phase change of the site operators, so the Hamiltonian describes the same physics. This is the lattice version of gauge covariance.

Plaquette Flux Is Invariant

An individual link phase is gauge dependent. The gauge-invariant object is the phase around a closed loop. For a plaquette $p$ with oriented boundary $\partial p$,

$$\prod_{\langle ij\rangle\in\partial p} e^{i\theta_{ij}} = \exp \left( \frac{iq}{\hbar} \oint_{\partial p} \mathbf A\cdot d\mathbf r \right).$$

By Stokes’ theorem, for a small plaquette in a smooth field,

$$\oint_{\partial p} \mathbf A\cdot d\mathbf r = \int_p \mathbf B\cdot d\mathbf S = \Phi_p.$$

Therefore the plaquette phase is

$$\exp \left( \frac{iq\Phi_p}{\hbar} \right).$$

This is the lattice Aharonov–Bohm phase. Link phases are like gauge potentials; plaquette phases are like magnetic flux.

One-Dimensional Ring

On a static open nearest-neighbor chain, all link phases can be removed recursively by site rephasing: there is no closed loop on which a residual phase can remain. The same statement holds on a tree. It does not remove electric fields or magnetic changes to hopping magnitudes from a more complete model. On a ring, the total phase around the ring cannot generally be removed:

$$\Theta = \sum_{j=1}^N\theta_{j+1,j} = \frac{q\Phi}{\hbar}, \qquad c_{N+1}=c_1.$$

One may distribute this phase uniformly over all links, concentrate it on one boundary link, or choose another gauge. The spectrum depends on $\Theta$ modulo $2\pi$, not on the particular distribution of link phases.

For $N\ge3$ sites with hopping magnitude $J>0$, choose

$$H=-J\sum_{j=1}^N \left(e^{i\Theta/N}c_{j+1}^\dagger c_j+\mathrm{h.c.}\right).$$

The single-particle modes $\psi_j=e^{2\pi i mj/N}/\sqrt N$ have energies

$$E_m(\Theta)=-2J\cos\!\left(\frac{2\pi m-\Theta}{N}\right), \qquad m=0,\ldots,N-1.$$

Increasing $\Theta$ by $2\pi$ relabels $m$ by $m-1$ modulo $N$. The spectrum as a set is periodic, although an individual branch with its label held fixed need not be. This is the tight-binding Aharonov–Bohm effect.

Magnetic Unit Cells

For a two-dimensional lattice with primitive vectors $\mathbf a_1,\mathbf a_2$, the magnetic flux through a unit cell is

$$\Phi_{\mathrm{cell}} = \mathbf B\cdot \left( \mathbf a_1\times\mathbf a_2 \right).$$

The dimensionless flux is

$$\alpha = \frac{q\Phi_{\mathrm{cell}}}{h}.$$

If $\alpha$ is an integer, magnetic translations along the primitive lattice directions commute. If $\alpha=P/Q$ is rational in lowest terms, an enlarged magnetic unit cell with $Q$ ordinary cells can restore a Bloch-like description. This is the symmetry seed of Hofstadter spectra.

Magnetic translations derives the operator algebra behind this enlarged cell.

Relation to Minimal Coupling

In continuum mechanics, minimal coupling uses

$$\hat{\mathbf p} \mapsto \hat{\mathbf p}-q\mathbf A.$$

On a lattice, there is no infinitesimal position derivative inside a finite hopping term. The Peierls phase is the finite-link version of the same gauge principle. The covariant derivative has become a covariant parallel transport between neighboring localized orbitals.

This perspective also explains why the phase belongs on links, not on sites alone. Sites carry local phase conventions; links compare neighboring phase conventions.

Validity and Limitations

The Peierls substitution is powerful, but it is an approximation unless derived within a controlled tight-binding construction. It assumes that localized orbitals and hopping matrix elements remain meaningful in the field and that the vector potential is sampled primarily through phase transport between orbital centers.

Important limitations include:

• strong fields can distort orbitals and change hopping magnitudes;
• multiband models can acquire additional matrix-valued Berry or orbital effects;
• Zeeman coupling and spin–orbit terms are not generated by scalar hopping phases alone;
• nonuniform fields can require path and orbital-shape information beyond a bond-center approximation;
• interactions and lattice relaxation can modify the effective Hamiltonian.

The gauge covariance of the chosen lattice model is exact. Its accuracy as a description of a continuum particle or real material remains an approximation.

Common Mistakes

• Treating a single link phase as observable rather than the product around a closed loop.
• Forgetting the sign of the charge or the orientation of the hopping path.
• Applying the same phase to both directions of a link instead of using opposite phases.
• Gauging away a vector potential on a ring and forgetting the boundary phase.
• Assuming the Peierls substitution is exact for arbitrarily strong or rapidly varying fields.
• Confusing magnetic flux through a plaquette with Berry curvature in momentum space; both use curvature language, but they live in different spaces.

Exercises

1. Show that the Peierls hopping term is gauge covariant.

Solution

The transformed phase is

$$\theta_{ij}' = \theta_{ij} + \frac{q}{\hbar} (\chi_i-\chi_j),$$

and the transformed operators obey

$$c_i^{\prime\dagger}c_j' = e^{-iq\chi_i/\hbar} e^{iq\chi_j/\hbar} c_i^\dagger c_j.$$

Therefore

$$e^{i\theta_{ij}'} c_i^{\prime\dagger}c_j' = e^{i\theta_{ij}} c_i^\dagger c_j.$$

The hopping term is unchanged as a physical operator.

2. Compute the phase around a square plaquette of area $a^2$ in a uniform field $B\hat{\mathbf z}$.

Solution

The flux through the plaquette is

$$\Phi_p = Ba^2$$

with the sign determined by the chosen orientation. The plaquette phase is

$$\exp \left( \frac{iqBa^2}{\hbar} \right).$$

Reversing the orientation complex-conjugates this phase.

3. A one-dimensional ring has $N$ identical hoppings and total flux phase $\Theta=q\Phi/\hbar$. Show that a gauge with uniform link phase $\Theta/N$ is equivalent to a gauge with all phase placed on one link.

Solution

Choose site phases recursively so that each ordinary link phase is removed:

$$\chi_{j+1}-\chi_j = -\frac{\hbar}{q}\frac{\Theta}{N}$$

for $j=1,\ldots,N-1$. This moves the phase from those links into the site basis. Around the final link, the accumulated mismatch is the total phase $\Theta$, so one boundary hopping carries $e^{i\Theta}$. The distribution changed, but the loop product did not.

4. Apply the ring Hamiltonian to $\psi_j=e^{2\pi i mj/N}/\sqrt N$ and verify the sign of the flux shift.

Solution

The amplitude arriving at site $j$ is

$$(H\psi)_j=-J\left(e^{i\Theta/N}\psi_{j-1} +e^{-i\Theta/N}\psi_{j+1}\right) =-2J\cos\!\left(\frac{2\pi m-\Theta}{N}\right)\psi_j.$$

Reversing the bond orientation or charge changes the sign of $\Theta$. A gauge redistribution of the same oriented total phase does not change the energies.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976.
• D. R. Hofstadter, “Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,” Physical Review B 14, 2239–2249, 1976.
• C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley, 2005.
• R. Peierls, “Zur Theorie des Diamagnetismus von Leitungselektronen,” Zeitschrift für Physik 80, 763–791, 1933.
• S. H. Simon, The Oxford Solid State Basics, Oxford University Press, 2013.