Topology¶

Pysic-rs provides the topological invariants central to modern condensed-matter and field theory: Berry phases and curvatures, Chern and TKNN numbers, and winding numbers — including discrete (lattice) evaluations.

Python binding status: berry_phase, berry_phase_bloch, chern_number, winding_number, and skyrmion_number are bound. berry_curvature, the discrete variants, and the TKNN helpers are Rust-only. See the API page.

1. Berry Phase¶

1.1 Adiabatic Evolution & Geometric Phase¶

Consider a parameter-dependent Hamiltonian \(H(\mathbf{R})\) with instantaneous eigenstates \(|n(\mathbf{R})\rangle\). As \(\mathbf{R}\) is slowly varied around a closed loop \(\mathcal{C}\) in parameter space, the adiabatic theorem guarantees that a system starting in \(|n(\mathbf{R}_0)\rangle\) returns to that eigenstate — but acquires a phase:

\[ |\psi(T)\rangle = e^{i\gamma_n}\,e^{-\frac{i}{\hbar}\int_0^T E_n(t)\,dt}\,|n(\mathbf{R}_0)\rangle \]

The first exponential is the dynamical phase; the second is the geometric (Berry) phase.

1.2 Berry Phase Formula¶

Theorem (Berry, 1984). The geometric phase is

\[ \boxed{\gamma_n = i\oint_{\mathcal{C}}\langle n(\mathbf{R})|\nabla_{\mathbf{R}}|n(\mathbf{R})\rangle\cdot d\mathbf{R} = \oint_{\mathcal{C}}\mathbf{A}_n\cdot d\mathbf{R}} \]

where \(\mathbf{A}_n = i\langle n|\nabla_{\mathbf{R}}|n\rangle\) is the Berry connection.

Proof. Under adiabatic evolution, the state picks up a phase from the overlap of neighbouring eigenstates:

\[ \langle n(\mathbf{R})|n(\mathbf{R}+d\mathbf{R})\rangle \approx 1 + \langle n|\nabla_{\mathbf{R}}|n\rangle\cdot d\mathbf{R} \]

The total phase after a closed loop is the path-ordered product (Wilson loop) of these overlaps. In the adiabatic limit, this reduces to the integral of the Berry connection along the loop. The gauge-invariance of \(\gamma_n\) follows from the fact that changing the phase convention \(|n\rangle\to e^{i\phi(\mathbf{R})}|n\rangle\) shifts \(\mathbf{A}_n \to \mathbf{A}_n-\nabla\phi\), which does not change the closed-loop integral. \(\square\)

1.3 Berry Curvature¶

The Berry connection \(\mathbf{A}_n\) is a gauge field on parameter space. Its “field strength” is the Berry curvature:

\[ \boxed{\Omega_n^{ab} = \partial_a A_n^b - \partial_b A_n^a = -2\,\mathrm{Im}\sum_{m\neq n}\frac{\langle n|\partial_a H|m\rangle\langle m|\partial_b H|n\rangle}{(E_n-E_m)^2}} \]

Proof. Differentiate \(H|n\rangle = E_n|n\rangle\) with respect to \(R^a\):

\[ (\partial_a H)|n\rangle + H|\partial_a n\rangle = (\partial_a E_n)|n\rangle + E_n|\partial_a n\rangle \]

Projecting onto \(\langle m|\) for \(m\neq n\):

\[ \langle m|\partial_a n\rangle = \frac{\langle m|\partial_a H|n\rangle}{E_n-E_m} \]

Substituting into \(\Omega_n^{ab} = i(\partial_a A_n^b - \partial_b A_n^a)\) and using \(\mathbf{A}_n = i\langle n|\nabla|n\rangle\) gives the result. \(\square\)

Gauge invariance: \(\Omega_n^{ab}\) is invariant under \(|n\rangle\to e^{i\phi}|n\rangle\) (the connection transforms, but the curvature does not — just as in electromagnetism).

1.4 Flux Quantisation (Stokes)¶

For a closed surface \(\Sigma\) bounding \(\mathcal{C}\):

\[ \gamma_n = \iint_\Sigma \Omega_n^{ab}\,dS_{ab} \]

This relates the Berry phase to the flux of Berry curvature through a surface bounded by the loop.

2. Chern Number (TKNN Invariant)¶

2.1 Definition¶

For a 2D parameter space (e.g. the Brillouin zone), the first Chern number of band \(n\) is the total Berry flux:

\[ \boxed{C_n = \frac{1}{2\pi}\int_{\mathrm{BZ}}\Omega_n^{xy}\,d^2k \in \mathbb{Z}} \]

Theorem (Thouless–Kohmoto–Nightingale–den Nijs, 1982). \(C_n\) is an integer.

Proof. The Berry curvature \(\Omega_n^{xy}\) is a closed 2-form (\(d\Omega = 0\) follows from the Bianchi identity of the Berry connection). By Chern–Weil theory, the integral of a closed 2-form over a compact 2-manifold (the BZ torus) is a topological invariant — it cannot change under smooth deformations of the Hamiltonian. The integer is the first Chern class of the Berry connection bundle. \(\square\)

2.2 Quantum Hall Effect¶

TKNN formula (1982): the Hall conductance of a 2D electron gas in a magnetic field is

\[ \sigma_{xy} = \frac{e^2}{h}\sum_{n\,\mathrm{filled}}C_n \]

Each filled Landau level contributes \(C_n = 1\) (for the lowest level), giving the quantised Hall conductance \(\sigma_{xy} = ne^2/h\) — the integer quantum Hall effect.

2.3 Lattice (Discrete) Evaluation¶

On a discretised BZ with plaquette \((k_x, k_y)\to(k_x+\Delta k, k_y)\to(k_x+\Delta k, k_y+\Delta k)\to(k_x, k_y+\Delta k)\), the Berry phase around each plaquette is

\[ \gamma_{\square} = \mathrm{Im}\ln\bigl[U_1(k)\,U_2(k+\hat{1})\,U_1^*(k+\hat{2})\,U_2^*(k)\bigr] \]

where \(U_\mu(k) = \langle n(k)|n(k+\hat{\mu})\rangle/|\langle n(k)|n(k+\hat{\mu})\rangle|\) is the link variable (Wilson loop element). The Chern number is

\[ C = \frac{1}{2\pi}\sum_{\square}\gamma_{\square} \in \mathbb{Z} \]

This is the lattice Berry flux method implemented by chern_number_discrete.

3. Winding Number¶

3.1 Definition¶

For a map \(g: S^1\to S^1\) (a closed curve in the complex plane avoiding the origin), the winding number counts the net number of counterclockwise encirclements of the origin:

\[ \boxed{w(g) = \frac{1}{2\pi i}\oint\frac{g'(z)}{g(z)}\,dz \in \mathbb{Z}} \]

Proof. Write \(g(z) = r(z)e^{i\theta(z)}\). Then \(g'/g = r'/r + i\theta'\), and the contour integral of \(r'/r\) vanishes (single-valued), while \(\frac{1}{2\pi}\oint d\theta\) counts the net winding of \(\theta\) around the origin — an integer. \(\square\)

3.2 Discrete Version¶

For a polygon with vertices \(z_0, z_1, \ldots, z_N = z_0\), the winding number is

\[ w = \frac{1}{2\pi}\sum_{k=0}^{N-1}\arg\!\left(\frac{z_{k+1}}{z_k}\right) \]

where the argument is taken in \((-\pi,\pi]\) and the sum of branch cuts gives the total winding.

3.3 Physical Applications¶

  • Topological insulators: the winding number classifies 1D chiral symmetry-protected topological phases (SSH model).

  • Vortices in superfluids/superconductors: the winding number of the order parameter around a vortex core.

  • Anyons: the braiding of anyonic quasiparticles in 2D is described by the winding number of their worldlines.

4. Skyrmion Number¶

4.1 Definition¶

For a map \(\mathbf{n}: \mathbb{R}^2\to S^2\) (a spin configuration), the skyrmion number is the degree of the map:

\[ \boxed{N_{\rm sky} = \frac{1}{4\pi}\iint\mathbf{n}\cdot\left(\frac{\partial\mathbf{n}}{\partial x}\times\frac{\partial\mathbf{n}}{\partial y}\right)dx\,dy \in \mathbb{Z}} \]

Proof. The integrand is the pullback of the area form on \(S^2\) under \(\mathbf{n}\). Integrating over all space counts how many times \(\mathbf{n}\) wraps around \(S^2\) — the degree of the map. For a compact manifold without boundary, this must be an integer (degree theorem). \(\square\)

4.2 Physical Applications¶

  • Chiral magnets: skyrmions are topologically stabilised magnetic vortices.

  • Skyrmion tubes in QCD: baryons can be viewed as skyrmions in the pion field.

  • Spintronics: skyrmion racetrack memory proposals.

5. Routines¶

Routine

Description

berry_phase(psi, dparam, ...)

\(\gamma_n\) around a loop

berry_curvature(H_params, psi, ...)

\(\Omega_n^{ab}\) at a point

berry_phase_bloch(...)

Bloch-state Berry phase

chern_number(...)

\(C_n\) integral over BZ

chern_number_discrete(lattice, ...)

lattice flux version

winding_number(path_real, path_imag)

\(w(g) = \frac{1}{2\pi i}\oint g'/g\)

winding_number_discrete(points)

discrete encirclement

skyrmion_number(spin_config)

degree of the map \(\mathbb{R}^2\to S^2\)

6. Usage Examples¶

Discrete Chern number of a Landau-level band¶

from pysicrs import chern_number_discrete

# Lattice Berry flux per plaquette on a 12Ă—12 BZ sample
C = chern_number_discrete(plaq_flux)
print(int(round(C)))   # 1 for lowest Landau level

Winding number of a closed curve¶

from pysicrs import winding_number_discrete
import math, cmath

# z(t) = e^{i·2t}: winds twice around the origin
points = [cmath.exp(2j*t) for t in (i/100*2*math.pi for i in range(101))]
w = winding_number_discrete([p.real for p in points], [p.imag for p in points])
print(w)   # 2

Berry phase around a loop¶

from pysicrs import berry_phase
# Spin-1/2 in a varying magnetic field: loop that encloses a monopole
# gives Îł = 2Ď€ Ă— (solid angle)/4Ď€

7. Advantages & Limitations¶

âś… Direct access to geometric/topological invariants (no ad-hoc derivations)

âś… Both continuum and discrete evaluation for lattice-style problems

✅ Connects naturally to the Gauge module (winding → instanton quantisation)

❌ Requires pre-computed eigenstates / Berry connections from outside

❌ No coverage of higher-genus (genus \(g\)) index theorems besides the winding/skyrmion counts

❌ Reflection of a generically non-abelian (Wilson-loop) connection is out of scope

8. References¶

  1. Berry, M.V. (1984). “Quantal phase factors accompanying adiabatic changes.” Proc. R. Soc. A 392:45.

  2. Thouless, D.J., Kohmoto, M., Nightingale, M.P. & den Nijs, M. (1982). Phys. Rev. Lett. 49:405.

  3. Simon, B. (1983). “Holonomy, the quantum adiabatic theorem, and Berry’s phase.” Phys. Rev. Lett. 51:2167.

  4. Göckeler, M. & Schücker, T. (1990). Differential Geometry, Gauge Theories, and Gravity. Cambridge.

  5. Hasan, M.Z. & Kane, C.L. (2010). “Colloquium: topological insulators.” Rev. Mod. Phys. 82:3045.