Renormalization group for non-relativistic fermions

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Ramamurti Shankar (2010), Scholarpedia, 5(9):9575. doi:10.4249/scholarpedia.9575 revision #121948 [link to/cite this article]
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Curator: Ramamurti Shankar

Figure 1: The low energy region for nonrelativistic fermions is not near the origin but near the Fermi surface, shown by the thick line

The renormalization group (RG) is a method for trading a given problem for an equivalent one that gives the same answers to a limited subset of questions. Often one begins with a theory that is good for all energies and trades it for a different one that has the same low energy behavior by systematically eliminating high energy degree of freedom. The case of non-relativistic fermions at finite density is atypical in that the low energy region is not centered around a point (the origin) in momentum space, but a surface, namely the Fermi surface. We see that in this case the RG analysis naturally leads to Landau's Fermi Liquid (which emerges as a fixed point) and its instabilities (which are associated with its relevant perturbations).


The Renormalization Group Philosophy

The renormalization group (RG) is useful in problems where the variables of interest fall into two sets, say \( x \) and \( y\) and we want to focus on just \(x\). The RG provides a way to start from the original theory containing both \( x\) and \( y\ ,\) eliminate \( y\) and obtain a theory involving just \( x \) which yields exactly the same results for all questions involving just \(x \) Kadanoff (1977), Wilson (1975), Fisher (1974,1998).

As a concrete example consider a partition function \[\tag{1} Z = \int dx \int dy\; e^{-S(x,y;\; a, b, c..)} \ .\]

The action (or Boltzmann weight) \(S(x,y;\; a, b,..)\) is a function of the variable x and y and a host of parameters \( (a,b,c, ...)\) that define it. For example one could have \[\tag{2} S(x,y;\; a, b, c..)= a (x^2+y^2) + b (x^2 + y^2 )^2 \ .\]

The average of any function \( F(x,y)\) is given by \[\tag{3} <F(x,y)> = \frac {1 }{Z}\int dx \int dy\; F(x,y)\;e^{-S(x,y;\; a, b, c..)} \ .\]

Suppose we want the average of functions of just \(x\ .\) We then define an effective action \(S'(x; a', b'...)\) by \[\tag{4} e^{-S'(x;\; a', b'...)}= \ \int dy\; e^{-S(x,y;\; a, b, c..)} \ .\]

It is clear that for any function of just \( x \) we could use \( S'\) as the action \[\tag{5} <F(x)> = \frac {1 }{Z'}\int dx\; F(x)\;e^{-S'(x;\; a', b', c'..)}\ \ \ \ Z'= \int dx\; e^{-S'(x;\; a', b', c'..)} \ .\]

Usually the elimination of \(y\) is followed by rescaling of some kinematical variables (like length) and the integration variable x. The parameters \( (a', b',..)\ ,\) also called coupling constants, are the ones appearing in the final action \(S'\) after all these transformations have been made.

The change of parameters from \((a,b,c,...)\) to \((a',b',c',...)\) is called renormalization. Since \( y \) in general stands for many variables \(y_1, y_2 ... \ ,\) the couplings change after each \( y_i\) is eliminated. Parameters that grow (shrink) under such renormalization are called relevant (irrelevant) while those that remain fixed are called marginal.

Typically the variables \(x\) refer to low energy degrees of freedom and \(y\) to high energy degrees of freedom. In a relativistic field theory (in Euclidean space) low energy would correspond to long wavelength. In this case modes with wave number outside a ball of some size \( \Lambda\) would be integrated out to define the low energy theory. The RG would then focus on the flow of parameters as \(\Lambda\) is reduced to zero. The situation is very different for nonrelativistic fermions. Whereas in \(d=1\ ,\) the Fermi surface consists of just two points and the situation resembles quantum field theory Solyom (1979), Bourbonnais and Caron (1991), in higher dimensions novel features stem from the extended Fermi surface.

Nonrelativistic Fermions at finite density

Figure 2: The low energy region is an annulus of width \( 2 \Lambda \) symmetrically enclosing the Fermi circle.

Let us consider such fermions in two spatial dimensions, ignore interactions and spin and set \(\hbar =1\ .\) The energy of the fermion of momentum \( K\) and mass \(m\) is \[\tag{6} E(K) =\frac{K^2}{2m} \ .\]

If we demand that there be some nonzero density of fermions, we subtract a chemical potential \[\tag{7} E(K) =\frac{K^2}{2m}- \frac{K_{F}^{2}}{2m}. \]

In the ground state all momentum states within a circle of radius equal to the Fermi momentum \(K_F\) are occupied by one fermion. One chooses \(K_F\) to ensure the desired density. Excitations on top of this ground state are obtained by knocking electrons out of the filled Fermi sea out to empty states. Since such "particle-hole" states can have arbitrarily low energy cost (if they are arbitrarily close to the Fermi energy) we say the system has no gap.

What happens to the ground state if we add interactions? It will be an admixture of filled Fermi sea and "particle-hole" states. The particles and the holes will be near the Fermi surface if the interactions are weak. So the ground state and its low energy excitations are determined by a band of single-particle states with \( |k|<\Lambda \) where \( k = K-K_F \ ,\) as depicted in Figure 2. The effective theory is obtained by integrating states outside the cut-off \( \Lambda \ .\) As we send \(\Lambda \to 0\) the flow will tell us if any given interaction is relevant, irrelevant or marginal Benfatto and Gallavotti (1990), Feldman and Trubowitz (1990), Shankar (1991), Polchinski (1992), Shankar (1994).

A preview of main results

Figure 3: Kinematics in d=2. Typically incoming and outgoing momenta are equal up to a permutation (top), except in the Cooper channel (bottom).

Before describing the process in detail, let consider some major results that can be anticipated easily, provided we ignore some subtleties. Consider an interaction \( u(K_1,..K_4)\) describing two incoming fermions of momenta \( (K_1, K_2) \) scattering into two particles with momenta \( (K_3, K_4) \ .\) In the limit of vanishing bandwidth \( \Lambda \ ,\) all four momenta lie on the Fermi circle \( |K_i|=K_F \ .\) Together with momentum conservation, this implies that the incoming momenta and outgoing momenta are identical up to a permutation, as depicted in the upper half of Figure 3. Thus we find (upon invoking rotational invariance) \[\tag{8} u=u(\theta_1, \theta_2)= u(\theta_1- \theta_2)\equiv F(\theta ) \ .\]

Decades ago Landau discovered the need for the function \(F(\theta )\ ,\) named after him, in his description of the low energy physics of fermions, without explicitly using the renormalization group.

The RG also yields another function \(V(\theta )\ .\) If \(K_1=-K_2\ ,\) or \( \theta_1=-\theta_2\ ,\) we can see from the lower half of Figure 3 that \(K_3=-K_4\) can point in any direction, unrelated to \(\theta_1=-\theta_2\ .\) In this case \[\tag{9} u = u(\theta_1,\theta_3)= u(\theta_1-\theta_3)=V(\theta )\]

whose Fourier coefficients \(V_\ell\ ,\) defined by \[\tag{10} V(\theta )=\sum_\ell V_\ell e^{i\ell\theta} \ .\]

define the interaction between Cooper pairs with relative angular momentum \( \ell \!\ .\)

While \(F \) and \(V \) are the only kinematically allowed functions, are they relevant, irrelevant or marginal? Could the couplings have dependence on the frequency \(\omega\) and \( k = K-K_F \) of each particle? For this we must turn to the calculation.

The Details

We will proceed in two steps: Write a path integral for free fermions and device an RG transformation that renders the action \(S_0 \) a fixed point and then test all possible perturbations for relevance or irrelevance.

The path integral for free fermions in terms of Grassmann fields is \[\tag{11} Z_0= \int \left[d\psi d\overline{\psi}\right]\; e^{S_0} \ .\]

where \[\tag{12} S_0= \int_{0}^{ 2\pi}d\theta \int_{-\infty}^{\infty} d\omega \int_{-\Lambda}^{\Lambda} dk\; \overline{\psi} (\theta, \omega,k)\left[i\omega - k\right]\psi (\theta, \omega , k)\ .\]

We have approximated \( {K^2-K_{F}^{2}\over 2m}\simeq {K_F\over m}\cdot k=v_F\ k\) and set \(v_F=K_F/m\ ,\) the Fermi velocity, to unity.

Let us now perform mode elimination and reduce the cut-off by a factor \( s\ .\) Since this is a Gaussian integral, mode elimination just leads to a multiplicative constant in front of \(Z_0\) that we are not interested in. So the result is just the same action as above, but with \(|k| \le \Lambda /s \ .\) If we make the following additional transformations

\[\tag{13} \begin{align} (\omega ', k') &= s(\omega , k) \\ \left((\psi ' (\omega ', k'), \overline{\psi}' (\omega ', k')\right) &= s^{-3/2} \left(\psi ({\omega '\over s}, {k'\over s}) , \overline{\psi} ({\omega '\over s}, {k'\over s})\right). \end{align} \]

we find the action is invariant, that is, it is a fixed point of the RG transformation. We are now ready to find the response of various interactions to our RG. Benfatto and Gallavotti (1990), Shankar (1991), Shankar (1994), Feldman and Trubowitz (1990)

Consider generic four-Fermi interaction in path-integral form: \[\tag{14} \begin{array}{lcl} S_4 &=& \int\; \overline{\psi}(4)\, \overline{\psi}(3)\, \psi (2)\,\psi(1)\, u(4,3,2,1)\\ \ \int & \equiv& \prod_{i=1}^{3} \int{d \theta_{i}}\int_{- \Lambda}^{\Lambda} dk_{i} \int_{-\infty}^{\infty} d\omega_{i} \end{array} \]

where the labels \(1,2,3,4\) stand for the frequencies and momenta of each particle. Mode elimination is now nontrivial since the quartic interaction couples low and high energies. So we shall proceed in steps of increasing accuracy.

Tree Level Approximation

In the tree approximation, we simply ignore the high energy modes, rescale the remaining fields, frequencies and momenta, and read off the new coupling. We find, upon dropping unwanted labels,

\[\tag{15} u'(k',\omega' , \theta ) = u(k'/s, \omega' /s, \theta) \]

This is the evolution of the coupling function. To deal with coupling constants with which we are more familiar, we expand the functions in a Taylor series (schematic)

\[\tag{16} u = u_o + k u_1 + k^2 u_2 ... \]

where \( k \) stands for all the \( k\)'s and \(\omega \)'s. An expansion of this kind is possible since couplings in the Lagrangian are nonsingular in this problem assumed to have only short range interactions. Expanding both sides of Eq. (15) and comparing coefficients we find that \(u_0 \) is marginal and the rest are irrelevant, as are all coupling of more than four fields. However \(u_0 \) depends on the angles on the Fermi surface\[u_0 = u(\theta_1 , \theta_2 , \theta _3 , \theta_4 ) \]

As we have seen before \( u(\theta_1-\theta_2)=F(\theta)\ ,\) the Landau function. The function \( V(\theta) \) likewise arises as a marginal coupling in the Cooper channel.

Beyond Tree Level

Figure 4: One loop flow

In interacting theories, mode elimination is done perturbatively. After the tree approximation, the corrections to the flow are just as in \(\phi^4\) theory, with the difference that all propagators are restricted to the modes being eliminated. The perturbation series is ordered in the number of loops.

In our problem the flow of the quartic interaction to one loop is given by the diagrams in Figure 4. All external legs are on the Fermi surface and carry \(\omega =0\) since dependence on \( k, \omega\) has been shown to be irrelevant. Only the spatial momenta of the lines are indicated and all internal lines have to lie in shells of thickness \(dk = d \Lambda\) near the cut-off, while \(\omega\) and \(\theta\) are integrated over their full range.

Consider the first diagram labeled ZS. The diagram has no momentum or frequency transfer at the left vertex. Both propagators in the loop are identical, their poles in \(\omega\) lie on the same half-plane for any \(k\ ,\) and the contour can be closed the other way to give zero. (The name ZS comes from Fermi liquid theory in which summing bubbles of this types leads to a collective excitation called zero sound for small values of external momentum and frequency. In the diagrammatic approach the loop momenta take all values within the cut-off and the diagram is very sensitive to the ratio of external momentum transfer and frequency transfer. In the RG context, the loop momenta are at the cut-off and the diagram has a unique value when we set external momentum and frequency transfers to zero. Next we will consider the ZS' diagram which has a large momentum transfer. It does not play a major role in the theory and the name ZS' is arbitrary.)

The ZS' and BCS diagrams give no contribution for a different reason. In ZS' there is a large momentum transfer \(Q'=K_4-K_1 = K_2-K_1\) which is of order \(K_F\ .\) The two momenta \(K,K+Q'\) must both lie on the shells being eliminated and also be related by a difference \(Q'\ .\) The reader may easily verify with a sketch that this restricts the range of the angle of the loop momentum to order \( d \Lambda\ .\) Since the \(dk\) integral is also of order \( d \Lambda\ ,\) the flow is quadratic in \( d \Lambda\) and the logarithmic derivative with respect to the cut-off vanishes. Likewise the angle of the internal Cooper pair in the BCS diagram is limited to a width \( d \Lambda\ ,\) in addition to \(dk\ ,\) again leading to a flow of order \((d \Lambda )^2\ .\)

Thus at one loop \(u\) or \(F\) is marginal.

When we consider the flow of \(V\ ,\) we find \(ZS\) and \(ZS'\) do not contribute to order \(d\Lambda\) for the same reasons as in the case of \(F\ ,\) while the BCS diagram does. The reason is that with zero incoming momentum \(P\ ,\) if one of the loop momenta \(K\) is in the shell being eliminated, so is the other \(-K\ ,\) for the kinetic energy is an even function of momentum. Thus the loop angle may be integrated with no restriction, leading to a flow with respect to \(dt = -d\Lambda /\Lambda \ :\)

\[\tag{17} {dV(\theta_{1} - \theta_{3} ) \over dt} = - \int \ d\theta\ V(\theta_{1}- \theta )\ V(\theta - \theta_{3} ) \]

By expanding in terms of angular momentum eigenfunctions we get an infinite number of flow equations

\[\tag{18} \frac{dV_l}{dt} = - V_{l}^2. \]

one for each coefficient. These equations tell us that if the potential in angular momentum channel \(l\) is repulsive, it will get renormalized down to zero ( a result derived many years ago by Morel and Anderson (1962) while if it is attractive, it will run off, causing the BCS instability. The couplings \(V\)'s are not a part of Landau theory which assumes there are no phase transitions.

Solving the Fixed Point Theory

The RG not only gives us a fixed point theory, it facilitates a solution nonperturbative in \(F\) by giving us a small parameter \(\frac{\Lambda}{K_F}=\frac{1}{N}\ .\) We call the small parameter \(1/N\) because the annulus with cut off \(\Lambda\) can be divided into roughly \(N\) patches of extent \(\Lambda \times \Lambda \) in both radial and angular directions and each of these patches behaves like one of \(N\) isospin components of a large \(N\) theory Feldman 'et al' (1993), Shankar(1994). Skipping details, suffice it to say here that in the end only RPA diagrams survive. Thus the susceptibility for small \((q,\omega)\) is a sum over bubbles in the ZS channel which may be summed geometrically to yield well known results of Landau's theory Nozieres and Pines (1966).

As in all the large \(N\) theories, the one loop flow (or \(\beta\)-function) is exact.

Extensions, uses and generalizations

The theory, when extended to non-circular Fermi surfaces predicts the charge density wave instability at half filling if the Fermi surface is nested: besides \(F\) and \(V\ ,\) the CDW instability being driven by a third coupling \(W\ ,\) that is marginally at tree level and flows to strong coupling at one loop in the repulsive case Shankar (1994).

Besides giving us a way to understand Fermi liquid theory and its instabilities, the RG allows other applications. One can ask how a non-Fermi liquid of the type Anderson (1990) envisaged may emerge by asking how the above derivation could fail. One way is at strong coupling where the perturbative RG does not converge, say due to singular Fermi liquid parameters Metzner and DiCastro (1993). Another is to consider a Fermi surface coupled to gauge fields either in strongly correlated electrons Polchinski (1992) or in the Fractional Quantum Hall problem at half-filling Halperin, Lee and Read (1993). One can ask if a repulsive interaction can spawn superconductivity by generating attractive interactions upon integrating out high energy states diagrammatically Shankar (1994). This idea has been implemented many times, generally in two stages: integrate high energy modes perturbatively (to one loop say) to get an effective set of interactions and then run the flow as above starting with these as initial couplings Zanchi and Schulz (2000), Schulz (1995), Honerkamp 'et al' (2001). One can also try to derive the ideas espoused here more rigorously, Salmhofer (1998), though as of now, no small parameters besides \( \Lambda/K_F= 1/N\) has been found. Rigorous versions of the ideas discussed in this paper have been used to prove theorems about the convergence of the expansions for the correlation functions at temperatures larger than an exponentially small critical temperature Disertori and Rivasseau (2000) and the Fermi (non-Fermi) liquid behavior of a system of non-relativistic 2D fermions with convex ( flat and nested) Fermi surface at temperatures larger than an exponentially small one Benfatto, A. Giuliani and V. Mastropietro (2006); Rivasseau (2002) ; Afchain, Magnen and Rivasseau (2005).

The theory may be applied to nuclear matter or stars Schwenk, Friman and Brown, (2003), as well as quarks at finite density Rajagopal and Wilczek (2000).

One can also vary the power law of the force between fermions continuously to a controllable fixed point. Nayak and Wilczek (1993)

The theory may also be extended to quantum dots to address a problem with finite size, disorder and interactions Murthy and Shankar, 2008.

There is a natural way to extend the theory to higher spatial dimensions and to include spin. If the present approach is applied to fermions in \( d \) spatial dimensions, the scaling is done only for \( \omega \)and \( k=K-K_F \ ,\) since the other angles ( like \(\theta \)) on the Fermi surface behave like internal or isospin degrees of freedom. Thus the Fermi liquid behaves like a \( 1+1 \) dimensional field theory for all \( d \ge 2\ .\) A new generalization has been proposed Senthil and Shankar (2009) to Fermi surfaces with more co-dimensions on which the energy depends linearly. For example in three dimensions a line Fermi surface has co-dimension 2 and behaves like a \( 2+1\) dimensional theory. It is also possible to vary these extra dimensions in an \( \varepsilon \) expansion and obtain new weak coupling fixed points.


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See Also

Bardeen-Cooper-Schrieffer theory, Path integral, Renormalization group

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