Spin Coherent States and Berry Phases

A spin coherent state replaces a quantum spin by a unit vector while retaining the phase accumulated as that vector moves on the sphere. The resulting Berry term is first order in time, fixes spin precession, and carries quantized global information that no ordinary classical energy functional contains.

Required background. Exchange Interactions and Effective Spin Hamiltonians fixes the microscopic spin representation and couplings; Coherent-State Path Integrals supplies time slicing and endpoint prescriptions. Helpful background. Wess–Zumino and WZW Terms supplies the extension and quantization logic for geometric terms.

Coherent states on the sphere

For spin $S$, a normalized coherent state $|\mathbf n\rangle$ points along $\mathbf n=(\sin\vartheta\cos\varphi,\sin\vartheta\sin\varphi,\cos\vartheta)$ and satisfies

$$\langle\mathbf n|\mathbf S|\mathbf n\rangle=S\mathbf n, \qquad \frac{2S+1}{4\pi}\int d\Omega\, \lvert\mathbf n\rangle\langle\mathbf n\rvert=1.$$

Inserting this resolution between short imaginary-time steps gives

$$S_E[\mathbf n]=iS\,\Omega[\mathbf n] +\int_0^\beta d\tau\,H(S\mathbf n)+O(S^0),$$

where $\Omega$ is the oriented solid angle enclosed by the path and a closing geodesic. In a north-patch gauge we choose

$$iS\,\Omega=iS\int d\tau\,(1-\cos\vartheta)\,\dot\varphi.$$

Changing the gauge patch moves a Dirac string and changes the action for a closed path by $i4\pi S$ times an integer. Because $2S\in\mathbb Z$, $e^{-S_E}$ is unchanged. The local one-form is gauge dependent; the solid-angle phase is physical. Coherent-state quantization and the geometric phase are developed in Klauder 1960, pp. 123–168.

Precession from the Berry curvature

Varying the solid angle gives

$$\delta\Omega=\int d\tau\, \delta\mathbf n\cdot(\dot{\mathbf n}\times\mathbf n).$$

After returning to real time and enforcing $\mathbf n^2=1$, stationarity yields

$$S\,\dot{\mathbf n}=\frac{\partial H}{\partial\mathbf n}\times\mathbf n =-\mathbf n\times\frac{\partial H}{\partial\mathbf n}.$$

For $H=-S\mathbf B\cdot\mathbf n$, this gives $\dot{\mathbf n}=\mathbf n\times\mathbf B$ in the convention where $\mathbf B$ includes the gyromagnetic sign. The first-order term supplies the symplectic form $S\sin\vartheta\,d\vartheta\wedge d\varphi$ and hence the spin Poisson brackets. Omitting it would produce second-order rotor dynamics rather than Landau–Lifshitz precession.

From a lattice of spins to continuum fields

For a ferromagnet, neighboring coherent-state vectors vary slowly and their Berry phases add. The continuum action remains first order in time, which leads to quadratic magnon dispersion. For a bipartite antiferromagnet, write

$$\frac{\mathbf S_j}{S}\simeq(-1)^j\mathbf n(x) \sqrt{1-\frac{a^2\mathbf L^2(x)}{S^2}}+\frac aS\mathbf L(x), \qquad \mathbf n^2=1,\quad \mathbf n\cdot\mathbf L=0.$$

Here $\mathbf L$ is the smooth spin density, with the same normalization used in the sigma-model derivation. The leading staggered Berry phases cancel pairwise, while the smooth part makes $\mathbf L$ conjugate to $\mathbf n$. Integrating out $\mathbf L$ produces a second-order nonlinear sigma model plus a residual topological contribution. In one spatial dimension that contribution becomes $i\theta Q$ with $\theta=2\pi S$; in higher dimensions singular hedgehog events retain sublattice-dependent Berry phases. The cancellation must be performed on the lattice before replacing the sum by an integral Haldane 1983, pp. 464–468.

Assumptions and failure modes

The classical symbol $H(S\mathbf n)$ receives ordering corrections of relative order $1/S$. A smooth-field expansion additionally requires gradients small compared with the inverse lattice spacing. Berry phases can remain decisive even when their local contribution cancels: discarding lattice-scale instantons too early erases the distinction between integer and half-integer chains and between competing paramagnets.

For open time paths, endpoint wave functions and the chosen closure matter. For closed traces the ambiguity reduces to the quantized patch change above. A continuum calculation should state the coherent-state gauge, Euclidean sign, spin normalization, and orientation convention for the solid angle.

Exercises

1. Show that the Berry curvature integrates to $4\pi S$ over the sphere.

Solution

From $A=S(1-\cos\vartheta)d\varphi$, $F=dA=S\sin\vartheta\,d\vartheta\wedge d\varphi$. Therefore $\int_{S^2}F=S\int_0^\pi\sin\vartheta\,d\vartheta\int_0^{2\pi}d\varphi=4\pi S$. Dividing by $2\pi$ gives Chern number $2S\in\mathbb Z$.

2. Derive precession for $H=-S B n_z$.

Solution

$\partial H/\partial\mathbf n=-SB\hat{\mathbf z}$. The equation gives $S\dot{\mathbf n}=(-SB\hat{\mathbf z})\times\mathbf n=SB\mathbf n\times\hat{\mathbf z}$, hence $\dot{\mathbf n}=B\mathbf n\times\hat{\mathbf z}$. The polar angle is constant and the azimuth advances with the convention-dependent signed frequency.

References

• Haldane, F. D. M. “Continuum Dynamics of the 1-D Heisenberg Antiferromagnet: Identification with the O(3) Nonlinear Sigma Model.” Physics Letters A 93 (1983): 464–468. DOI.
• Klauder, J. R. “The Action Option and a Feynman Quantization of Spinor Fields in Terms of Ordinary $c$-Numbers.” Annals of Physics 11 (1960): 123–168. DOI.