Following Haldane’s seminal work, we show how to map the spin Heisenberg model to a field theory of non-linear sigma model. The additional geometrical phase term gives rise to the famous “Haldane conjecture”, which states that the integer-spin antiferromagnetic Heisenberg model shows a energy gap, while the half-integer spin system is described by the Wess-Zumino-Witten model.
copyright © jxu@ustb.edu.cn
We start from the antiferromagnetic Heisenberg (AFH) model
\begin{eqnarray}
H=J\sum_{\left\langle ij\right\rangle}\vec{S}_i\cdot \vec{S}_j,
\end{eqnarray}
with $J>0$, and the spin operator $\vec{S}_i=(S_i^x, S_i^y, S_i^z)$ includes the three generators of SU(2) symmetry.
To get a low-energy field theory of this model, we need to sum over all the “classical” paths (i.e., the path-integral approach), where each space-time point on these paths can be described by a “classical” single-spin state $\vert\hat{\Omega}\rangle$, with $\hat{\Omega}$ a unit vector lives on a Bloch sphere. This is exactly the spin coherent state, and is defined as
\begin{eqnarray}
\vec{S}\cdot\hat{\Omega}\vert\hat{\Omega}\rangle=s\vert\hat{\Omega}\rangle.
\end{eqnarray}
In other words, the spin coherent state is just the maximum spin polarized state along the direction that specified by $\hat{\Omega}$.
To get a path-integral form, we start from the partition function of a single spin
\begin{eqnarray}
Z&=&\rm{Tr}\left[e^{-\beta H}\right]\nonumber\\
&=&\int D[\hat{\Omega}] \langle\hat{\Omega}_1\vert e^{-\frac{\beta H}{N}}\vert\hat{\Omega}_2\rangle\langle\hat{\Omega}_2\vert e^{-\frac{\beta H}{N}}\vert\hat{\Omega}_3\rangle\langle\hat{\Omega}_3\vert\cdots\cdots\vert\hat{\Omega}_N\rangle\langle\hat{\Omega}_N\vert e^{-\frac{\beta H}{N}}\vert\hat{\Omega}_1\rangle\nonumber\\
&=&\int D[\hat{\Omega}] \langle\hat{\Omega}_1\vert (1-H\tau) \vert\hat{\Omega}_2\rangle\langle\hat{\Omega}_2\vert (1-H\tau)\vert\hat{\Omega}_3\rangle\langle\hat{\Omega}_3\vert\cdots\cdots\vert\hat{\Omega}_N\rangle\langle\hat{\Omega}_N\vert (1-H\tau)\vert\hat{\Omega}_1\rangle\nonumber\\
&=&\int D[\hat{\Omega}] \prod_{i=1}^N \langle\hat{\Omega}_i\vert (1-H\tau) \vert\hat{\Omega}_{i+1}\rangle\nonumber\\
&=&\int D[\hat{\Omega}]\left(\prod_{i=1}^N \langle\hat{\Omega}_i\vert\hat{\Omega}_{i+1}\rangle-\prod_{i=1}^N \langle\hat{\Omega}_i\vert\hat{\Omega}_{i+1}\rangle\cdot\sum_{i=1}^N\frac{\langle\hat{\Omega}_i\vert H\vert\hat{\Omega}_{i+1}\rangle}{\langle\hat{\Omega}_i\vert\hat{\Omega}_{i+1}\rangle}\tau\right)\nonumber\\
&=&\int D[\hat{\Omega}]\prod_{i=1}^N \langle\hat{\Omega}_i\vert\hat{\Omega}_{i+1}\rangle\cdot\left(1-\sum_{i=1}^N\frac{\langle\hat{\Omega}_i\vert H\vert\hat{\Omega}_{i+1}\rangle}{\langle\hat{\Omega}_i\vert\hat{\Omega}_{i+1}\rangle}\tau\right)\nonumber\\
&=&\int D[\hat{\Omega}]\left[\prod_{i=1}^N \left(1+\tau\langle\hat{\Omega}_i\vert \partial_\tau\vert\hat{\Omega}_{i}\rangle\right)\right]\left(1-\sum_{i=1}^N\langle\hat{\Omega}_i\vert H\vert\hat{\Omega}_{i}\rangle\tau\right)\nonumber\\
&=&\int D[\hat{\Omega}] e^{-\tau\sum_{i=1}^N-\langle\hat{\Omega}_i\vert \partial_\tau\vert\hat{\Omega}_{i}\rangle+\langle\hat{\Omega}_i\vert H\vert\hat{\Omega}_{i}\rangle}\nonumber\\
&=&\int D[\hat{\Omega}] e^{-\int\left(-\langle\hat{\Omega}\vert \partial_\tau\vert\hat{\Omega}\rangle+H(\hat{\Omega})\right)d\tau}.
\end{eqnarray}
So we get the Lagrangian of a single-spin system
\begin{eqnarray}
\mathcal{L}=-\langle\hat{\Omega}\vert \partial_\tau\vert\hat{\Omega}\rangle+H(\hat{\Omega}),
\label{lag}
\end{eqnarray}
where $H(\hat{\Omega})=\langle\hat{\Omega}\vert H\vert\hat{\Omega}\rangle$. Now considering the unit vector on the Bloch sphere with angular coordinate $\theta$ and $\phi$, the coherent state can be viewed as a rotation from the $S_z$-polarized state
\begin{eqnarray}
\vert\hat{\Omega}(\theta,\phi)\rangle=e^{-i\phi S_z}e^{-i\theta S_y}\vert s,s\rangle,
\end{eqnarray}
and the first term in the Lagrangian can be written as
\begin{eqnarray}
\langle\hat{\Omega}\vert \partial_\tau\vert\hat{\Omega}\rangle&=&-i\dot{\phi}\langle s,s\vert e^{i\theta S_y}e^{i\phi Sz}\cdot S_z\cdot e^{-i\phi S_z}e^{-i\theta S_y}\vert s,s\rangle-i\dot{\theta}\langle s,s\vert e^{i\theta S_y}e^{i\phi Sz} e^{-i\phi S_z}\cdot S_y\cdot e^{-i\theta S_y}\vert s,s\rangle\nonumber\\
&=&-i\dot{\phi}\langle s,s\vert e^{i\theta S_y}\cdot S_z\cdot e^{-i\theta S_y}\vert s,s\rangle\nonumber\\
&=&-is\cos\theta\dot{\phi},
\end{eqnarray}
where in the last line we have calculated the rotation of spin operator
\begin{eqnarray}
e^{-i\theta S_y} S_z e^{i\theta S_y}=\sin\theta S_x+\cos\theta S_z.
\end{eqnarray}
Now we can get the action of our single-spin system
\begin{eqnarray}
S=\int\mathcal{L}d\tau=-is\cdot\omega[\hat{\Omega}]+\int H(\hat{\Omega})d\tau,
\end{eqnarray}
where $\omega[\hat{\Omega}]=-\int\cos\theta\dot{\phi}d\tau$ corresponds to the Berry phase that measures the area enclosed by the path of the local spin $\hat{\Omega}$ on the Bloch sphere, and it is easy to show that $\omega[-\hat{\Omega}]=-\omega[\hat{\Omega}]$.
Now let us consider a small perturbation around the spin direction $\hat{\Omega}=\hat{n}\sqrt{1-\vec{L}^2}+\vec{L}$, where $\vec{L}$ is small and perpendicular to $\hat{n}$, that is, $\hat{n}\cdot\vec{L}=0$. Physically, this is based on the assumption that the ground state of the system is the ordered state $\hat{n}$ that disturbed by small spin fluctuations governed by $\vec{L}$. Here we consider a low-energy description of the system, so that both fields vary slowly as a function of time. By Taylor expansion around this ordered state, up to the second order we have
\begin{eqnarray}
\omega[\hat{\Omega}]&\approx&\omega[\hat{n}]+\int \left(\frac{\delta \omega}{\delta \hat{n}}\cdot\vec{L}\right)d\tau\nonumber\\
&=&\omega[\hat{n}]+\int \left(\frac{\delta \omega}{\delta \theta}\cdot L_\theta+\frac{1}{\sin\theta}\frac{\delta \omega}{\delta \phi}\cdot L_\phi\right)d\tau\nonumber\\
&=&\omega[\hat{n}]+\int \left(\sin\theta\dot{\phi}\cdot L_\theta-\dot{\theta}\cdot L_\phi\right)d\tau\nonumber\\
&=&\omega[\hat{n}]+\int \hat{n}\cdot\left(\vec{L}\times\frac{\partial \hat{n}}{\partial \tau}\right)d\tau\nonumber\\
&=&\omega[\hat{n}]+\int \vec{L}\cdot\left(\frac{\partial \hat{n}}{\partial \tau}\times\hat{n}\right)d\tau.
\label{phase}
\end{eqnarray}
It would be straightforward to generalize to multiple spins on a AFH chain. In such case, we sum over product states of local coherent states $\vert \{\hat{\Omega}\}\rangle=\vert\hat{\Omega}(x_1)\rangle\cdot\vert\hat{\Omega}(x_2)\rangle\cdot\vert\hat{\Omega}(x_3)\rangle\cdots\vert\hat{\Omega}(x_N)\rangle$, where $\hat{\Omega}(x_i)=(-1)^i\hat{n}(x_i)\sqrt{1-\vec{L}(x_i)^2}+\vec{L}(x_i)$. Here, it is crucial that both $\hat{n}(x)$ and $\vec{L}(x)$ are slowly varying fields as a function of spatial coordinate. This can be gained from the observation that the classical spin wave modes have two minimum excitations at $k=0$ and $\pi$, which corresponds to the fields $\vec{L}(x)$ and $\hat{n}(x)$, respectively. If we assume that the actual ground state to be around $k=\pi$, which is obvious from a classical point of view, then the most important part contributes to the effective low-energy description would be the excitation from $k=0$, which is the slowly varying field $\vec{L}(x)$. Thus under a continuum limit we can generalize the local phase to the whole system as
\begin{eqnarray}
-is\cdot\omega[\{\hat{\Omega}\}]&=&-\sum_i\langle\hat{\Omega}(x_i)\vert \partial_\tau\vert\hat{\Omega}(x_i)\rangle d\tau\nonumber\\
&=&-is\sum_i\omega[\hat{\Omega}(x_i)]\nonumber\\
&\approx&-is\sum_i\omega[\hat{n}(x_i)]-is\sum_i \vec{L}\cdot\left(\frac{\partial \hat{n}}{\partial \tau}\times\hat{n}\right)d\tau\nonumber\\
&=&-is\sum_i\frac{1}{2}\bigg(\omega[\hat{n}(x_i)]-\omega[\hat{n}(x_{i+1})]\bigg)-is\sum_i \vec{L}\cdot\left(\frac{\partial \hat{n}}{\partial \tau}\times\hat{n}\right)d\tau\nonumber\\
&\approx&-is\sum_i\frac{1}{2}\int\left(\frac{\delta \omega}{\delta n}\cdot \frac{\partial \hat{n}}{\partial x}\cdot a\right) d\tau-is\sum_i \vec{L}\cdot\left(\frac{\partial \hat{n}}{\partial \tau}\times\hat{n}\right)d\tau\nonumber\\
&\approx&-\frac{is}{2}\int\left(\frac{\delta \omega}{\delta n}\cdot \frac{\partial \hat{n}}{\partial x}\right) dxd\tau-\frac{is}{a}\int \vec{L}\cdot\left(\frac{\partial \hat{n}}{\partial \tau}\times\hat{n}\right)dxd\tau\nonumber\\
&=&-\frac{is}{2}\int\hat{n}\cdot\left(\frac{\partial \hat{n}}{\partial x}\times \frac{\partial \hat{n}}{\partial \tau}\right) dxd\tau-\frac{is}{a}\int \vec{L}\cdot\left(\frac{\partial \hat{n}}{\partial \tau}\times\hat{n}\right)dxd\tau\nonumber\\
&=&-2\pi is\Theta-\frac{is}{a}\int \vec{L}\cdot\left(\frac{\partial \hat{n}}{\partial \tau}\times\hat{n}\right)dxd\tau,
\end{eqnarray}
where the topological winding number
\begin{eqnarray}
\Theta=\frac{1}{4\pi}\int\hat{n}\cdot\left(\frac{\partial \hat{n}}{\partial x}\times \frac{\partial \hat{n}}{\partial \tau}\right) dxd\tau=\frac{1}{4\pi}\int\hat{n}\cdot d\vec{S}
\end{eqnarray}
maps the spacetime to a 2-sphere which counts the numbers of wraps that cover this sphere (the area divided by $4\pi$), and thus takes integer numbers.
Keeping the same order in the Hamiltonian term, we arrive at
\begin{eqnarray}
H(\{\hat{\Omega}\})&=&Js^2\sum_{\left\langle ij\right\rangle}\hat{\Omega}(x_i)\cdot \hat{\Omega}(x_j)\nonumber\\
&\approx&Js^2\sum_i\left((-1)^i\hat{n}(x_i)+\vec{L}(x_i)-\frac{(-1)^i}{2}\vec{L}(x_i)^2\hat{n}(x_i)\right)\cdot \left((-1)^{i+1}\hat{n}(x_{i+1})+\vec{L}(x_{i+1})-\frac{(-1)^{i+1}}{2}\vec{L}(x_{i+1})^2\hat{n}(x_{i+1})\right)\nonumber\\
&\approx&Js^2\sum_i\left(-1-a\hat{n}(x_i)\cdot\frac{\partial \hat{n}(x_i)}{\partial x}-\frac{a^2}{2}\hat{n}(x_i)\cdot\frac{\partial^2 \hat{n}(x_i)}{\partial x^2}+2\vec{L}(x_i)^2\right)\nonumber\\
&=&Js^2\sum_i\left(-1-\frac{a^2}{2}\hat{n}(x_i)\cdot\frac{\partial^2 \hat{n}(x_i)}{\partial x^2}+2\vec{L}(x_i)^2\right)\nonumber\\
&=&\frac{Js^2}{a}\int \left(-1+\frac{a^2}{2}\left\vert\frac{\partial \hat{n}}{\partial x}\right\vert^2+2\vec{L}^2\right)dx,
\end{eqnarray}
with the assumption that both the N\’eel field $\hat{n}$ and spin fluctuation $\vec{L}$ are slowly varying fields.
We are now ready to sum over the “classical” paths of the multiple spins
\begin{eqnarray}
Z&=&\int D[\{\hat{\Omega}\}] e^{-S}\nonumber\\
&=&\int D[\{\hat{\Omega}\}] \exp\left\{is\cdot\omega[\{\hat{\Omega}\}]-\int H(\{\hat{\Omega}\})d\tau\right\}\nonumber\\
&\approx&\int D[\hat{n}]D[\vec{L}] \exp\left\{2\pi is\Theta+\int \left[\frac{is}{a}\vec{L}\cdot\left(\frac{\partial \hat{n}}{\partial \tau}\times\hat{n}\right)-\frac{Js^2}{a}\left(-1+\frac{a^2}{2}\left\vert\frac{\partial \hat{n}}{\partial x}\right\vert^2+2\vec{L}^2\right)\right]dxd\tau\right\}.
\end{eqnarray}
The field $\vec{L}$ is readily to calculate and we arrive at
\begin{eqnarray}
Z&\propto&\int D[\hat{n}] \exp\left\{2\pi is\Theta-\int \left[\frac{1}{8Ja}\left(\frac{\partial \hat{n}}{\partial \tau}\times\hat{n}\right)^2+\frac{Jas^2}{2}\left\vert\frac{\partial \hat{n}}{\partial x}\right\vert^2\right]dxd\tau\right\}\nonumber\\
&=&\int D[\hat{n}] \exp\left\{2\pi is\Theta-\frac{Jas^2}{2}\int \left[\frac{1}{(2Jas)^2}\hat{n}_\tau^2+\hat{n}_x^2\right]dxd\tau\right\}\nonumber\\
&=&\int D[\hat{n}] \exp\left\{2\pi is\Theta-\frac{\rho_s}{2}\int \left(\frac{1}{c^2}\hat{n}_\tau^2+\hat{n}_x^2\right)dxd\tau\right\},
\end{eqnarray}
where we have defined the spin wave velocity $c=2Jas$ and spin stiffness $\rho_s=Jas^2$.
Now we are ready to scale the time scale $c\tau\to\tau$ to reach the effective action of our AFH model
\begin{eqnarray}
S&=&-2\pi is\Theta+\frac{1}{2g}\int \left(\hat{n}_\tau^2+\hat{n}_x^2\right)dxd\tau\nonumber\\
&=&-2\pi is\Theta+\frac{1}{2g}\int \left(\partial_\mu\hat{n}\cdot\partial_\mu\hat{n}\right)d^2x,
\end{eqnarray}
where the dimensionless coupling constant $g=c/\rho_s=2/s$. This action is the Wess-Zumino-Witten (WZW) model, where the first term corresponds to a topological phase called the Wess-Zumino (WZ) term.
It is now obvious that the topological phase depends only on the spin quantum number $s$. If $s$ is half-integer, then WZ term contributes a plus or minus sign, depending on the winding number. This results a coherent effect between different integral paths, and leads to a gapless state. On the other hand, if $s$ is integer, then we can see all the paths now share the same trivial phase, and the system is described solely by the non-linear sigma model (NLSM) in 1+1 dimensions, with the Lagrangian density
\begin{eqnarray}
\mathcal{L_{NLSM}}=\frac{1}{2g}\partial_\mu\hat{n}\cdot\partial_\mu\hat{n},
\end{eqnarray}
where $\hat{n}$ is a unit vector that lives on a curved manifold (that’s why it’s called “non-linear”), i.e. a 2-sphere. It has already been shown by a large-$N$ expansion that the system is gapped.
Now we are ready to state the Haldane conjecture: all AFH chains with integer spin are gapped, whereas the half-integer spin are gapless.