RGApproachToInteractingFermions

Introduction

The renormalization group (RG) is a very powerful tool in physics. Essentially, it is a way to continuously map a given theory onto other theories possessing the same low-energy physics by successively integrating out "fast", or high-energy, modes. This is expressed in terms of differential equations giving the "flows" of different coupling constants that appear in the theory. We are often interested in determining the fixed points of these flows, or points which these flows may end at, since these give us some important information about the system. For example, each fixed point corresponds to a certain phase of the system. Since there are many different sets of initial parameters that all flow to the same fixed point, we have an explanation of universality, or the observation that many different systems all possess similar physical properties, such as critical exponents [1]. Here, we will discuss how to perform RG for fermions, which differs in some respects from the case with bosonic fields. This discussion will follow that in Shankar's paper [1]. The most important difference, one which is true of any fermionic system, is that we have a Fermi sea in the ground state of our system. This means that, unlike in the bosonic case, we cannot simply impose a cutoff on the momentum, restricting it to the inside of a sphere of radius Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} . We will illustrate the correct procedure as we go along for different cases.

System at half-filling in one dimension

We will begin with a simple system - Fermions on a one-dimensional tight-binding lattice at half filling. The Hamiltonian for such a system without interactions is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0=-t\sum_{\langle ij\rangle}\left (c^{\dagger}_{i+1} c_i+c^{\dagger}_i c_{i+1}\right ),}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle ij\rangle} means that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} are nearest neighbors, with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j>i} . We can diagonalize this Hamiltonian by performing a Fourier transform, upon which we find that the energies are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon(k)=-2t\cos{K}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\pi<K<\pi} . If we write the partition function for this Hamiltonian as a path integral, with the action written in momentum and frequency space, we get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_0=\int{D[\psi^{\ast},\psi] e^{-S_0}},}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_0=\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\int_{-\pi}^{\pi}\frac{dK}{2\pi}\,\psi^{\ast}(K,\omega)(-i\omega+\cos{K})\psi(K,\omega).}

Note that we write an integral over the frequency, rather than a Matsubara sum; we are working at zero temperature, and will do so throughout this article. We have absorbed the constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} into the field and frequency integral to bring our notation into contact with that of Shankar [1].

The RG transformation

We will now discuss how to perform RG for this system. First, we will find an RG transformation that will leave the above "bare" action invariant. We first impose a cutoff on the momentum integral. As noted before, we cannot simply restrict its range to a small region Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\Lambda<K<\Lambda} due to the fact that the system's ground state is a filled Fermi sea. In this case, all states with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\tfrac{\pi}{2}<K<\tfrac{\pi}{2}} are occupied. This means that our low-energy excitations are not excitations with small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} , but with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} close to the Fermi surface. Therefore, our cutoff should restrict Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} to small regions around Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\tfrac{\pi}{2}} . Let us define a new momentum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=|K|-\tfrac{\pi}{2}} . Since there are two regions that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is restricted to, we will introduce a new label to our fields, specifying whether the momentum is near the left (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\tfrac{\pi}{2}} ) or the right (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\tfrac{\pi}{2}} ) Fermi point. Since we are interested only in small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} , we will expand the energy to leading (in this case, linear) order. The action becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_0=\sum_{i=L,R}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}\,\psi^{\ast}_i(k,\omega)(-i\omega+k)\psi_i(k,\omega),}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} labels the "branch" of the Fermi surface, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} is our cutoff.

We are now ready to begin finding the appropriate RG transformation. This is done in three steps [1]. First, we split the fields into "slow" (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \psi_{<}} ) and "fast" modes (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \psi_{>}} ) and integrate out the fast modes. The slow modes are defined over a range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq k\leq\tfrac{\Lambda}{s}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s>1} , and the fast modes are defined over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{\Lambda}{s}\leq k\leq\Lambda} . We can then write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle \psi_i(k,\omega)=\psi_{i,<}(k,\omega)+\psi_{i,>}(k,\omega)} . We will split the action into two parts - one containing only slow modes, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{0,<}} and one containing only fast modes, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{0,>}} . There are no terms mixing the two types of modes since they are defined on two disjoint intervals. We can easily integrate out the fast modes since the integral is over a Gaussian, leaving just the slow modes. The result of the integration is just a constant, which we will drop. The action for the slow modes is thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_0=\sum_{i=L,R}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\int_{-\Lambda/s}^{\Lambda/s}\frac{dk}{2\pi}\,\psi^{\ast}_{i,<}(k,\omega)(-i\omega+k)\psi_{i,<}(k,\omega).}

In order to compare this theory to other theories with the same cutoff, we must now rescale the momenta and frequencies to restore the momentum integral back to its original range [1]. To this end, let us define a new momentum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k'=sk} and new frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega'=s\omega} . Upon introducing these variables into the action, we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_0=\frac{1}{s^3}\sum_{i=L,R}\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\int_{-\Lambda}^{\Lambda}\frac{dk'}{2\pi}\,\psi^{\ast}_{i,<}(\tfrac{k'}{s},\tfrac{\omega'}{s})(-i\omega'+k')\psi_{i,<}(\tfrac{k'}{s},\tfrac{\omega'}{s}).}

Finally, we rescale the fields so that the coefficient of some quadratic term in the action remains constant [1]. We define a new field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi'_i(k',\omega')=\tfrac{1}{\zeta}\psi_{i,<}(\tfrac{k'}{s},\tfrac{\omega'}{s})} . We only have one quadratic term in our action, so we will leave its coefficient invariant. In terms of this new field, the action becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_0=\frac{\zeta^2}{s^3}\sum_{i=L,R}\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\int_{-\Lambda}^{\Lambda}\frac{dk'}{2\pi}\,{\psi'_i}^{\ast}(k',\omega')(-i\omega'+k')\psi'_i(k',\omega').}

If we define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta=\tfrac{1}{s^{3/2}}} , then we recover the original action in terms of rescaled variables.

We see that the three steps of the RG transformation we seek are: integrate out the fast modes, introduce rescaled momenta Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k'=sk} and frequencies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega'=s\omega} , and introduce rescaled fields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi'_i(k',\omega')=s^{3/2}\psi_{i,<}(\tfrac{k'}{s},\tfrac{\omega'}{s})} [1].

Quadratic perturbations

Now that we know the appropriate transformation, we may now look at how different perturbations scale under this transformation. Let us start with a quadratic perturbation,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta S_2=\sum_{i=L,R}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\int_{-\Lambda}^{\Lambda}\frac{dk}{2\pi}\,\mu(k,\omega)\psi^{\ast}_i(k,\omega)\psi_i(k,\omega).}

Note that this perturbation preserves the symmetry between the two Fermi points. We now perform our RG transformation on this system. This separates directly into a piece depending only on the slow modes and one depending only on the fast modes, just as in the bare action. Therefore, when we integrate out the fast modes, we only generate a constant term in the action, which we drop. Performing the remaining two steps of the transformation, we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta S_2=s\sum_{i=L,R}\int_{-\infty}^{\infty}\frac{d\omega'}{2\pi}\int_{-\Lambda}^{\Lambda}\frac{dk'}{2\pi}\,\mu(\tfrac{k'}{s},\tfrac{\omega'}{s}){\psi'_i}^{\ast}(k',\omega')\psi'_i(k',\omega'),}

We now expand the coupling function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} in a Taylor series [1],

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(k,\omega)=\mu_{00}+\mu_{10}k+\mu_{01}i\omega+\cdots+\mu_{mn}k^m(i\omega)^n+\cdots.}

We may now determine how each of these terms scales with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} in the transformed action. In general, defining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu'(k',\omega')=s\mu(\tfrac{k'}{s},\tfrac{\omega'}{s})} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu'_{mn}=\frac{1}{s^{m+n-1}}\mu_{mn}.}

We see that $\displaystyle \mu _{00}$ scales with a positive power of $s$, so that it grows as we increase $s$, and therefore integrate out more modes. Such a perturbation that grows under the RG transformation is called relevant [1]. As was guaranteed by our definition of the transformation, we find that $\displaystyle \mu _{10}$ and $\displaystyle \mu _{01}$ remain constant. Such perturbations are referred to as marginal [1]. Finally, if $m+n>1$, then $\displaystyle \mu _{mn}$ decreases under the RG transformation; these types of perturbations are referred to as irrelevant [1].

References

[1] R. Shankar, Rev. Mod. Phys. 66, 129 (1994).