What is the Single Mode Approximation? When Girvin and co-workers solved the excited collective modes called magneto-rotons in Fractional Quantum Hall liquids, they used something called the Single Mode Approximation (SMA). My question is: what is the SMA? Can you explain it easily?
 A: The Single Mode Approximation for the FQHE was formulated in analogy to Feynman's "single mode approximation" for roton excitations in superfluid $\text{He}^4$. 
Feynman's general idea was that since the superfluid ground state is a Bose condensate, low-lying excitations just above it cannot be single-particle excitations, but are necessarily collective excitations that should manifest as density waves of the superfluid, similar to elastic waves in a continuum. Then if $\Phi_0$ denotes the translationally invariant condensate ground state of N particles, the simplest density wave variational ansatz reads
$$
\psi_{\vec k} = \frac{1}{\sqrt{N}}\sum_{j=1}^N{e^{-i{\vec k}\cdot{\vec r}_j}} \Phi_0
$$
It can be easily verified that each of these collective excitations is orthogonal to the ground state $\Phi_0$, as expected of well-defined excited states, $\langle \psi_{\vec k} | \Phi_0\rangle = 0$. In addition, they conveniently include both the main condensate trends through the presence of $\Phi_0$ and genuine density wave features through the plane-wave terms.  Also note that for each $\vec k$ there is only a single density mode, wherefrom the name of Single Mode Approximation. 
Unlike superfluid $\text{He}^4$, an FQHE system involves fermions, not bosons. Yet its behavior is very much boson-like, since the current flow is dissipationless and in the "composite boson" description the FQHE ground state may be regarded as a Bose condensate. By the same reasoning as above, Girvin surmised that the SMA might work for neutral FHQE magneto-roton excitations and found indeed that it gives excellent results. 
See his own excellent "Introduction to the Fractional Quantum Hall Effect", especially Sec.3, "Neutral Collective Excitations".  
