Zak Phase Preview

The Zak phase is the Berry phase acquired by a one-dimensional Bloch band as crystal momentum goes once around the Brillouin zone. It is the first band-theory example where the phrase “Berry phase over momentum space” becomes concrete.

For an isolated band with cell-periodic Bloch state $\lvert u_{nk}\rangle$, define the one-dimensional Berry connection

$$\mathcal A_n(k) = i\langle u_{nk}|\partial_k u_{nk}\rangle.$$

The Zak phase is

$$\gamma_n = \oint_{\mathrm{BZ}} \mathcal A_n(k)\,dk \quad \operatorname{mod}2\pi,$$

when a smooth periodic gauge is used. More generally, the endpoint comparison must be included. Berry-Phase Polarization and Charge Pumping owns the full material treatment of polarization classes, charge pumping, and ionic-plus-electronic bookkeeping; this page is the geometric preview.

Bloch-Band Setup

For a one-dimensional crystal with lattice spacing $a$, Bloch states have the form

$$\psi_{nk}(x) = e^{ikx}u_{nk}(x), \qquad u_{nk}(x+a)=u_{nk}(x).$$

Crystal momentum is defined modulo a reciprocal lattice vector

$$G=\frac{2\pi}{a}, \qquad k\sim k+G.$$

Thus the one-dimensional Brillouin zone is a circle:

$$\mathrm{BZ}\simeq S^1.$$

The Zak phase is the holonomy of the band eigenstate line as $k$ goes around that circle.

Gauge Dependence and the Endpoint Term

The cell-periodic state has a phase freedom:

$$\lvert u_{nk}\rangle \mapsto e^{i\chi(k)} \lvert u_{nk}\rangle.$$

The Berry connection transforms as

$$\mathcal A_n(k) \mapsto \mathcal A_n(k)-\partial_k\chi(k).$$

If the gauge is smooth and periodic over the Brillouin-zone loop, the integral is gauge invariant modulo $2\pi$. If the chosen gauge is not periodic, the gauge-invariant closed-loop phase includes an endpoint comparison:

$$\gamma_n = \int_{k_0}^{k_0+G} \mathcal A_n(k)\,dk + \arg \langle u_{n k_0}|u_{n,k_0+G}\rangle \quad \operatorname{mod}2\pi.$$

This formula is the one-dimensional version of a general lesson: open-path Berry integrals are not gauge invariant until endpoint data are supplied.

What the Zak Phase Measures

The Zak phase measures how the phase of a Bloch eigenvector twists as $k$ goes around the Brillouin-zone circle. It is not local curvature; in one dimension there is no two-form Berry curvature on the Brillouin zone. The invariant is the loop holonomy itself.

This makes the Zak phase closer in spirit to an Aharonov–Bohm phase around a circle than to a two-dimensional Chern number. There is a connection, a closed loop, and a phase modulo $2\pi$.

Because it is a phase, $\gamma_n$ is only defined modulo $2\pi$. Because it is a band quantity, it also depends on the convention for the unit cell and orbital positions unless the physical observable is formulated carefully. This is one reason the modern theory of polarization requires its own treatment.

SSH Model Intuition

The SSH model is the standard two-band example. In a common convention,

$$H(k) = d_x(k)\sigma_x+d_y(k)\sigma_y,$$

with

$$d_x(k)=v+w\cos(ka), \qquad d_y(k)=w\sin(ka).$$

As $k$ traverses the Brillouin zone, the vector

$$(d_x(k),d_y(k))$$

traces a closed curve in the plane. If the curve winds around the origin, the band eigenstate has a nontrivial Berry phase. With chiral symmetry and the usual convention, the lower-band Zak phase is

$$\gamma_- = \pi W \quad \operatorname{mod}2\pi,$$

where $W$ is the winding number of the vector around the origin.

For the SSH convention above, the gap closes when $\lvert v\rvert=\lvert w\rvert$. The winding can change only at this gap closing. This is the one-dimensional analogue of the more general rule that topological invariants change only when the relevant gap or isolation assumption fails.

Symmetry Quantization

The Zak phase is not automatically $0$ or $\pi$. In a generic one-dimensional band it can take any value modulo $2\pi$, and it can shift under changes of origin or unit-cell convention.

Symmetries can quantize it. Examples include:

• inversion symmetry, which can force the Zak phase to be $0$ or $\pi$ modulo $2\pi$ for an isolated band or occupied subspace under suitable conventions;
• chiral symmetry in the SSH model, where the winding number controls the phase;
• combined symmetry constraints in multi-band settings, where the invariant belongs to an occupied subspace rather than to one labeled band.

The quantized value is meaningful only after the protecting symmetry, gauge convention, occupied subspace, and boundary convention are stated.

Relation to Polarization

In one-dimensional band insulators, the electronic polarization is related to the Berry phase of occupied Bloch states. Schematic single-band formulas look like

$$P \sim -\frac{e}{2\pi} \gamma \quad \operatorname{mod} \frac{e}{a},$$

with important convention, spin, occupancy, and unit-cell factors suppressed.

This is not merely a formula for the center of a localized orbital. It is a bulk Berry phase defined modulo a polarization quantum. The dedicated material owner supplies the convention, filling, ionic-charge, branch, and observable ledger suppressed here.

Relation to Chern Numbers

The Zak phase is a one-dimensional Berry holonomy:

$$\gamma = \oint_{S^1}A.$$

A first Chern number is a two-dimensional curvature integral:

$$C = \frac{1}{2\pi} \int_{M^2}F.$$

Both are global Berry-geometry quantities, but they are not the same kind of invariant. A Zak phase is a phase modulo $2\pi$ and is not necessarily quantized. A Chern number is an integer when the relevant bundle is defined over a closed two-dimensional surface.

One useful bridge is the Thouless pump: a one-dimensional Hamiltonian with an additional cyclic parameter $(k,\lambda)$ can define a two-dimensional parameter torus. The pumped charge over a cycle is then governed by a Chern number on that torus. The material owner cited above supplies the physical charge and adiabatic-cycle audit, while the geometry begins here.

Common Mistakes

• Calling every one-dimensional Berry phase a topological invariant without checking symmetry or convention.
• Forgetting that the Brillouin zone is a circle and that endpoint matching matters.
• Ignoring the unit-cell and origin dependence of the raw Zak phase.
• Confusing the SSH winding number with a two-dimensional Chern number.
• Assigning a Zak phase to a band that is not isolated over the Brillouin zone.
• Treating edge states as guaranteed without specifying boundary termination and protecting symmetry.
• Using a discontinuous numerical eigenvector gauge and trusting the raw integral of $\mathcal A(k)$.

Cross-Links

• Berry Phase
• Berry Connection
• Holonomy
• Parallel Transport
• Chern Numbers
• SSH Model
• Bloch Theorem
• Berry Curvature Formula Card
• Chern Number Formula Card
• Homotopy and Winding
• Topological Invariants

References

• J. Zak, “Berry’s phase for energy bands in solids,” Physical Review Letters 62, 2747-2750, 1989.
• W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Physical Review Letters 42, 1698-1701, 1979.
• R. D. King-Smith and D. Vanderbilt, “Theory of polarization of crystalline solids,” Physical Review B 47, 1651-1654, 1993.
• R. Resta, “Macroscopic polarization in crystalline dielectrics: the geometric phase approach,” Reviews of Modern Physics 66, 899-915, 1994.
• D. Xiao, M.-C. Chang, and Q. Niu, “Berry phase effects on electronic properties,” Reviews of Modern Physics 82, 1959-2007, 2010.
• J. K. Asbóth, L. Oroszlány, and A. Pályi, A Short Course on Topological Insulators, Springer, 2016.

Exercises

1. Show how the Zak integral changes under a gauge transformation.

Let $\lvert u_k'\rangle=e^{i\chi(k)}\lvert u_k\rangle$. Show that

$$\int_{k_0}^{k_0+G} \mathcal A'(k)\,dk = \int_{k_0}^{k_0+G} \mathcal A(k)\,dk - \chi(k_0+G)+\chi(k_0).$$

Solution

The Berry connection transforms as

$$\mathcal A'(k) = \mathcal A(k)-\partial_k\chi(k).$$

Integrating gives

$$\int_{k_0}^{k_0+G} \mathcal A'(k)\,dk = \int_{k_0}^{k_0+G} \mathcal A(k)\,dk - \int_{k_0}^{k_0+G} \partial_k\chi(k)\,dk.$$

The last integral is $\chi(k_0+G)-\chi(k_0)$, so the stated result follows.

2. Why does the endpoint overlap restore gauge invariance for a nonperiodic gauge?

Solution

The overlap transforms as

$$\langle u'_{k_0}|u'_{k_0+G}\rangle = e^{-i\chi(k_0)} e^{i\chi(k_0+G)} \langle u_{k_0}|u_{k_0+G}\rangle.$$

Therefore its argument changes by

$$\chi(k_0+G)-\chi(k_0).$$

This cancels the change in the Berry-connection integral, so

$$\int \mathcal A\,dk + \arg\langle u_{k_0}|u_{k_0+G}\rangle$$

is gauge invariant modulo $2\pi$.

3. For the SSH Bloch vector $(d_x,d_y)=(v+w\cos ka,w\sin ka)$, when does the curve enclose the origin?

Solution

As $k$ varies, the curve is a circle of radius $\lvert w\rvert$ centered at $(v,0)$ in the $(d_x,d_y)$ plane. It encloses the origin when the distance from the center to the origin is smaller than the radius:

$$\lvert v\rvert<\lvert w\rvert.$$

At $\lvert v\rvert=\lvert w\rvert$, the circle passes through the origin and the band gap closes.

4. Explain why a Zak phase is not generally a Chern number.

Solution

The Zak phase is a one-dimensional loop integral of a connection:

$$\gamma=\oint_{S^1}A.$$

A Chern number is a normalized integral of curvature over a closed two-dimensional surface:

$$C=\frac{1}{2\pi}\int_{M^2}F.$$

The Zak phase is a phase modulo $2\pi$ and need not be quantized. The Chern number is an integer under its closed-surface and gap assumptions.

5. Why should edge-state claims in the SSH model mention boundary termination?

Solution

The bulk winding diagnoses a gapped phase under symmetry assumptions, but an actual finite chain also has a choice of where the chain is cut. Different terminations expose different bonds at the boundary. The familiar zero-energy edge modes occur for the boundary convention compatible with the nontrivial winding and chiral symmetry. Changing termination can move or remove the simple edge-mode statement without changing the bulk Hamiltonian convention.