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I am learning nonlinear sigma model from Assa Auerbach's book "Interacting Electrons and Quantum Magnetism" and getting some difficulties in taking continuum limit. I am following chapter 12: The continuum approximation.

Let me formulate the problem first. Consider the Heisenberg antiferromagnet on $d$-dimensional cubic lattice:

\begin{equation} H[\hat{\Omega}]=\frac{1}{2}S^2 \sum_{ij}^{}J_{ij} \hat{\Omega}_i \cdot \hat{\Omega}_j. \end{equation}

where summation runs over first, second, third nearest neighbours. The prefactor $1/2$ is introduced because of over counting. Using Haldane's mapping and taking some approximations (discussed in the book), we get

\begin{eqnarray} H \approx \frac{1}{2}S^2 \sum_{ij}^{}J_{ij} \eta_i\eta_j - \frac{1}{4}S^2 \sum_{ij}^{}J_{ij} \eta_i\eta_j (x_{ij}^\mu \partial_\mu \hat{n})(x_{ij}^\nu \partial_\nu \hat{n}) + \frac{1}{4}\sum_{ij}^{}J_{ij} \left[2\mathbf{L}_i \cdot \mathbf{L}_j - \eta_i \eta_j \left(\mathbf{L}_i^2 +\mathbf{L}_j^2 \right)\right] \end{eqnarray} or \begin{eqnarray} H \approx \frac{1}{2}S^2 \sum_{ij}^{}J_{ij} \eta_i\eta_j - \frac{1}{4}S^2 \sum_{ij}^{}J_{ij} \eta_i\eta_j (x_{ij}^\mu \partial_\mu \hat{n})(x_{ij}^\nu \partial_\nu \hat{n}) + \frac{1}{2}\sum_{ij}^{}J_{ij} \left[\mathbf{L}_i \cdot \mathbf{L}_j - \eta_i \eta_j \mathbf{L}_i^2 \right]. \end{eqnarray} where repeated indices $\mu,\nu=1,\dots,d$ are summed over.

The author takes the continuum limit using Equation (12.17) [Page 132 in the book]: \begin{equation} \sum_{i}^{} F_i \; \rightarrow \; a^{-d}\int d^dx \sum_{i}^{}\delta(\mathbf{x}-\mathbf{x}_i)F(\mathbf{x}) = a^{-d}\int d^dx \; F(\mathbf{x}) \quad (12.17) \end{equation}

The continuum expression is given in Equation (12.18) [Page 132 in the book].

\begin{equation} H\approx E_0^{cl} + \frac{1}{2}\int d^dx\; \rho_s \sum_{\mu}^{} |\partial_\mu \hat{n}|^2 + \frac{1}{2}\int d^dx \int d^dx^\prime \left( \mathbf{L}_x\chi_{xx^\prime}^{-1}\mathbf{L}_{x^\prime} \right) \quad (12.18) \end{equation}

The classical energy \begin{equation} E_0^{cl}=\frac{1}{2} S^2 \sum_{ij}^{}J_{ij}\eta_i\eta_j \quad (12.19) \end{equation} and the "stiffness constant" is \begin{equation} \rho_s = -\frac{S^2}{2dNa^d} \sum_{ij}^{}J_{ij} \eta_i\eta_j |\mathbf{x}_i-\mathbf{x}_j|^2 \quad \quad (12.20) \end{equation} where $N$ is the number of spins (or lattice sites).

The inverse uniform susceptibility is \begin{equation} \chi_{\mathbf{x},\mathbf{x}^\prime}^{-1}= \frac{1}{Na^d} \sum_{ij}^{} J_{ij} \left[ \delta(\mathbf{x}-\mathbf{x}_i)\delta(\mathbf{x}^\prime-\mathbf{x}_j)-\delta(\mathbf{x}^\prime-\mathbf{x})\delta(\mathbf{x}-\mathbf{x}_i)\eta_i\eta_j \right]\quad \quad (12.21) \end{equation}

I am facing problem in understanding how the continuum limit has been performed to get Equation (12.18). I have following technical and conceptual difficulties:

  • In the stiffness and susceptibility, they still have summation over $i$ and $j$. Which lattice sums they have replaced? Do they introduced some other lattice sums using Kronecker delta function.

  • How the prefactor in stiffness and susceptibility expressions comes?

  • In n-field part, we have one integral so there is one $a^d$ in denominator of stiffness. In L-field part, we have two integrals but we still have one $a^d$ in denominator of susceptibility. How?

  • I think the volume of unit cell, $a^d$, is same either we consider first, second or third nearest neighbours. Is it right?

I stuck somehow to derive the Equations (12.18), (12.20), and (12.21). I would appreciate very much if someone illustrates by giving intermediate steps.

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  • $\begingroup$ You have a better chance of getting answers if you put the equations you're asking about in your question. $\endgroup$ – user1504 Aug 23 '13 at 18:55
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    $\begingroup$ @user1504: I have edited my post by putting those equations where I am getting problem. $\endgroup$ – maxr Aug 23 '13 at 22:22

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