Ann. Henri Poincaré 4, Suppl. 2 (2003) S921 – S931
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/02S921-11
DOI 10.1007/s00023-003-0972-4
Annales Henri Poincaré
Functional Renormalization Group
for Interacting Fermi Systems
Walter Metzner
Abstract. We describe a Wick ordered functional renormalization group method
for interacting Fermi systems, where the effective low-energy action is generated
from the bare action of the microscopic model by a differential flow equation. We
apply this renormalization group approach to a prototypical two-dimensional lattice electron system, the Hubbard model on a square lattice. The flow equation
for the effective interactions is evaluated numerically on 1-loop level. The effective
interactions diverge at a finite energy scale which is exponentially small for small
bare interactions. To analyze the nature of the instabilities signaled by the diverging interactions, the flow of the singlet superconducting susceptibilities for various
pairing symmetries and also charge and spin density susceptibilities are computed.
This and similar RG calculations by others conclusively establish the existence of
d-wave superconductivity in the Hubbard model.
1 Introduction
Renormalization group (RG) methods provide the most reliable and efficient approach to low dimensional Fermi systems with competing singularities at weak
coupling. Such methods have been developed long ago for one-dimensional systems
where, combined with exact solutions of fixed point models, they have been an important source of physical insight [1, 2]. A major complication in two-dimensional
systems compared to 1D is that the effective interactions cannot be parameterized
accurately by a small number of running couplings, even if irrelevant momentum
and energy dependences are neglected, since the tangential momentum dependence
of effective interactions along the Fermi “surface” (a line in 2D) is strong and important in the low-energy limit. This has been demonstrated in particular in a
1-loop RG study for a model system with two parallel flat Fermi surface pieces [3].
In a series of papers Zanchi and Schulz [4] have developed a new renormalization
group approach for interacting Fermi systems, which is based on a flow equation
derived by Polchinski [5] in the context of local quantum field theory. In this RG
version the complete flow from the bare action of an arbitrary microscopic model
to the effective low-energy action, as a function of a continuously decreasing infrared energy cutoff, is given by an exact differential flow equation. Zanchi and
Schulz have applied this approach to the 2D Hubbard model (with nearest neighbor hopping) in a 1-loop approximation, with a discretized tangential momentum
dependence of the effective 2-particle interaction. The expected [6] presence of an-
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Walter Metzner
Ann. Henri Poincaré
tiferromagnetism and d-wave superconductivity as major instabilities of the model
close to half-filling was thereby confirmed.
The development of continuous renormalization group methods for interacting Fermi systems has made further progress with a work by Salmhofer [7]. By expanding the effective action in Wick-ordered monomials instead of bare monomials
he obtained an exact flow equation for the effective interactions with a particularly convenient structure: The β-function is bilinear in the effective interactions
and local in the flow parameter, i.e. it does not depend on effective interactions at
higher energy scales.
Here we briefly review the Wick ordered functional renormalization group and
discuss its application it to a prototypical two-dimensional electron system, the
Hubbard model [9] with nearest and also next-nearest neighbor hopping amplitudes
on a square lattice. We will focus on the electron density regime near half-filling,
where the Hubbard model serves as a model for the copper-oxygen planes of hightemperature superconductors [10]. The flow of effective 2-particle interactions is
evaluated on 1-loop level, neglecting the irrelevant energy dependence and also the
irrelevant normal momentum dependence, but keeping the important tangential
momentum dependence. The effective interactions diverge at a finite energy scale,
which is exponentially small for a small bare interaction. To analyze the physical nature of the instabilities signaled by the diverging interactions, Salmhofer’s
flow equations are extended for the calculation of susceptibilities. Charge and spin
susceptibilities as well as singlet superconducting susceptibilities for various pairing symmetries are computed. Depending on the choice of the model parameters,
hopping amplitudes, interaction strength and band-filling, commensurate or incommensurate antiferromagnetism or d-wave superconductivity turn out to be the
leading instability. The results agree qualitatively with those by Zanchi and Schulz
[4] (for t = 0), and also with results obtained from a one-particle irreducible version of the functional RG by Honerkamp et al. [8].
2 Two-dimensional Hubbard model
The Hubbard model [9]
H=
i,j
σ
tij c†iσ cjσ + U
nj↑ nj↓ ,
(1)
j
is a lattice electron model with quantum mechanical hopping amplitudes tij and
a local repulsion U > 0. Here c†iσ and ciσ are creation and annihilation operators
for fermions with spin projection σ ∈ {↑, ↓} on a lattice site i, and njσ = c†jσ cjσ . A
hopping amplitude −t between nearest neighbors and an amplitude −t between
next-nearest neighbors on a square lattice leads to the dispersion relation
k = −2t(cos kx + cos ky ) − 4t (cos kx cos ky )
(2)
Vol. 4, 2003
Functional Renormalization Group for Interacting Fermi Systems
π
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π
ky
ky
0
−π
−π
0
π
0
kx
−π
−π
π
0
kx
Figure 1: The Fermi surfaces of the non-interacting 2D Hubbard model with t = 0
(a) and t = −0.16t (b) for various densities.
for single-particle states. This dispersion relation has saddle points at k = (0, π)
and (π, 0), which lead to logarithmic van Hove singularities in the non-interacting
density of states at the energy vH = 4t .
In Fig. 1 we show the Fermi surfaces of the non-interacting system for various
electron densities n for the case without next-nearest neighbor hopping (t = 0) and
a case with t < 0. For a chemical potential µ = vH the Fermi surface contains the
van Hove points. In this case a perturbative calculation of the two-particle vertex
function leads to several infrared divergencies already at second order in U , i.e. at
1-loop level [12, 13, 14]. In particular, the particle-particle channel diverges as log2
for vanishing total momentum k1 +k2 , and logarithmically for k1 +k2 = (π, π). The
particle-hole channel diverges logarithmically for vanishing momentum transfer; for
momentum transfer (π, π) it diverges logarithmically if t = 0 and as log2 in the
special case t = 0. Note that µ = 0 for t = 0 corresponds to half-filling (n = 1). In
this case there are also logarithmic divergences for all momentum transfers parallel
to (π, π) or (π, −π) due to the strong nesting of the square shaped Fermi surface.
For µ = vH only the usual logarithmic Cooper singularity at zero total momentum
in the particle-particle channel remains. However, the additional singularities at
µ = vH lead clearly to largely enhanced contributions for small |µ−vH |, especially
if t is also small.
Hence, for small |µ − vH | one has to deal with competing divergencies in
different channels. This problem can be treated systematically by the RG method.
3 Renormalization group approach
We now briefly review the Wick ordered functional renormalization group approach. For details, see Salmhofer [7] and Ref. [11].
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Walter Metzner
Ann. Henri Poincaré
K1
=
ΣΣ
V Λn
n, j Perm.
K2
...
V Λm
...
...
∂
∂Λ
V Λm-n+j
Kj
Figure 2: Diagrammatic representation of the flow equation for VmΛ . The line with
a slash corresponds to ∂DΛ /∂Λ, the others to DΛ ; all possible pairings leaving m
ingoing and m outgoing external legs have to be summed.
A) Wick ordered flow equations
The infrared singularities are regularized by introducing an infrared cutoff
Λ > 0 into the bare propagator such that contributions from momenta with
|k − µ| < Λ are suppressed. All Green functions of the interacting system will
then depend on Λ, and the true theory is recovered only in the limit Λ → 0.
The RG equations are most conveniently obtained from the effective potential
V Λ , which is the generating functional for connected Green functions with bare
propagators amputated from the external legs. Taking a Λ-derivative one obtains
an exact functional flow equation for this quantity. Expanding V Λ on both sides
of the flow equation in powers of the fermionic fields (i.e. Grassmann variables),
and comparing coefficients, one obtains the so-called Polchinski equations [5] for
the effective m-body interactions used by Zanchi and Schulz [4]. Salmhofer [7]
has pointed out that an alternative expansion in terms of Wick (normal) ordered
monomials of fermion fields yields flow equations for the corresponding m-body
interactions VmΛ with a particularly simple structure (see Fig. 2). The flow of VmΛ is
given as a bilinear form of other n-body interactions (at the same scale Λ), which
are connected by lines corresponding to the propagator
DΛ (k) =
Θ(Λ − |ξk |)
,
ik0 − ξk
(3)
where ξk = k − µ, and one line corresponding to ∂DΛ (k)/∂Λ. For small Λ, the
momentum integrals on the right hand side of the flow equation are thus restricted
to momenta close to the Fermi surface (see Fig. 3). With the initial condition
V Λ0 = bare interaction, where Λ0 = max |ξk |, the above flow equations determine
the exact flow of the effective interactions as Λ sweeps over the entire energy range
from the band edges down to the Fermi surface. The effective low-energy theory
can thus be computed directly from the microscopic theory without introducing
any ad hoc parameters. The structure of these flow equations is very convenient
for a power counting analysis to arbitrary loop order [7], and also for a concrete
numerical solution.
Vol. 4, 2003
Functional Renormalization Group for Interacting Fermi Systems
π
ky
0
−π
−π
111111111
000000000
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
0
kx
S925
π
Figure 3: The support of the propagator DΛ in momentum space
B) One-loop flow
To detect instabilities of the system in the weak-coupling limit, it is sufficient
to truncate the infinite hierarchy of flow equations described by Fig. 2 at one-loop
level, and neglect all components of the effective interaction except the two-particle
interaction V2Λ . The effective 2-particle interaction V2Λ then reduces to the oneparticle irreducible 2-particle vertex ΓΛ , and its flow is determined by ΓΛ itself
(no other m-body interactions enter). Putting arrows on the lines to distinguish
creation and annihilation operators one thus obtains the flow equation for ΓΛ
shown graphically in Fig. 4. Flow equations for susceptibilities are obtained by
considering the exact RG equations in the presence of suitable external fields,
which leads to an additional 1-body term in the bare interaction, and expanding
everything in powers of the external fields to sufficiently high order [11]. On 1-loop
level one obtains the flow equations shown in Fig. 5. The flow of a susceptibility
χΛ is determined by the corresponding response vertex RΛ , the flow of which is in
turn given by ΓΛ and RΛ itself. The initial conditions are given by χΛ0 = 0 for the
susceptibilities and by the bare response vertices for RΛ . For pairing susceptibilities
only the particle-particle channel contributes to the propagator pair in Fig. 5, for
charge and spin density susceptibilities only the particle-hole channel.
It is clearly impossible to solve the flow equations with the full energy and
momentum dependence of the vertex function, since ΓΛ has three independent
energy and momentum variables. The problem can however be much simplified
by ignoring dependences which are irrelevant (in the RG sense) in the low energy limit, namely the energy dependence and the momentum dependence normal
to the Fermi surface. Hence, we compute the flow of the vertex function at zero
energy and with at least three momenta on the Fermi surface (the forth being determined by momentum conservation). On the right hand side of the flow equation
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Walter Metzner
2
∂
∂Λ
1
1011
0001
0001
0111
2′
2
=
1′
1
10
000
111
000
111
01
1011
00
11
0100
1′
+
1
PH
PP
10
00
11
0001
11
11
00
11
00
00
11
11
00
2′
2
Ann. Henri Poincaré
2′
+
1′
0011
11
11
00
00
000
111
+
00 111
000
1111
00
1110
00
PH’
2′
1′
1
2
Figure 4: Flow equation for the vertex function ΓΛ in 1-loop approximation with
the particle-particle channel (PP) and the two particle-hole channels (PH and
PH’).
(a)
∂
∂Λ
(b)
∂
∂Λ
1111
0000
0000
1111
0000
1111
0000
1111
1
0
00
11
000
11
1
=
=
11
00
11
00
00
11
00
00 11
11
00
11
1
0
00
11
00
11
00 11
11
000
1
Figure 5: Flow equations for (a) the susceptibilities χΛ and (b) the response vertices RΛ in 1-loop approximation.
we approximate the vertex function by its zero energy value with three momenta
projected on the Fermi surface (if not already there), as indicated in Fig. 6. This
projection procedure is exact for the bare Hubbard vertex, and asymptotically exact in the low-energy regime, since only irrelevant dependences are neglected. The
remaining tangential momentum dependence is discretized. Exploiting all symmetries (time reversal, spin-rotation, lattice point group) one has to deal with 880
”running couplings” for a discretization with 16 points on the Fermi surface, and
about 5000 for 32 points. Most of the results where obtained for 16 points, but it
has been checked that increasing the number of points does not change our results
too much [11].
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Functional Renormalization Group for Interacting Fermi Systems
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π
ky
5
6
4
7
0
3
8
2
10
16
9
1
11
15
12
14
13
−π
−π
0
kx
π
Figure 6: Projection of momenta on the Fermi surface; discretization and labeling
of angle variables.
4 Results
The flow of the vertex function has been computed for many different model parameters t and U (t just fixes the absolute energy scale) and densities close to
half-filling [11, 15]. In all cases the vertex function develops a strong momentum
dependence for small Λ with divergencies for several momenta at some critical scale
Λc > 0, which vanishes exponentially for U → 0. To see which physical instability
is associated with the diverging vertex function, the following susceptibilities have
been computed: commensurate and incommensurate spin susceptibilities χS (q)
with q = (π, π), q = (π − δ, π) and q = (1 − δ)(π, π), where δ is a function of
density [16], the commensurate charge susceptibility χC ((π, π)), and singlet pair
susceptibilities with form factors

1
(s-wave)


 √1 (cos kx + cos ky ) (extended s-wave)
2
d(k) =
(4)
√1 (cos kx − cos ky ) (d-wave dx2 −y 2 )


 2
(d-wave dxy ).
sin kx sin ky
Some of these susceptibilities diverge together with the vertex function at the scale
Λc . Depending on the choice of U , t and µ the strongest divergence is found for the
commensurate or incommensurate spin susceptibility or for the pair susceptibility
with dx2 −y2 symmetry [11, 15]. In Fig. 7 we show a typical result for the flow of the
vertex function and susceptibilities as a function of Λ. Only the singlet part (from
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Walter Metzner
50
Γs(1,1;5)
Γs(1,5;1)
Γs(1,1;9)
Γs(1,9;1)
Γs(1,9;5)
Γs(3,3;11)
Γs(3,11;3)
Γs(2,10;2)
Γs(2,10;4)
40
30
Γs(i1,i2;i3)/t
Ann. Henri Poincaré
20
10
0
-10
-20
0.01
(a)
100
0.1
Λ/t
1
sdw (π,π)
sdw (π−δ,π−δ)
sdw (π,π−δ)
cdw (π,π)
sc dx2-y2
sc dxy
sc s, sc xs
χ/χ
0
10
1
0.1
0.01
(b)
0.1
Λ/t
1
Figure 7: (a) The flow of the singlet vertex function ΓΛ
s as a function of Λ for
several choices of the momenta kF 1 , kF 2 and kF 1 , which are labelled according to
the numbers in Fig. 6. The model parameters are U = t and t = 0, the chemical
potential is µ = −0.02t; (b) the flow of the ratio of interacting and non-interacting
susceptibilities, χΛ /χΛ
0 , for the same system.
Vol. 4, 2003
Functional Renormalization Group for Interacting Fermi Systems
S929
a spin singlet-triplet decomposition) of the vertex function is plotted, for various
choices of the three independent external momenta. The triplet part of the vertex
function flows generally more weakly than the singlet part. Note the threshold at
Λ = 2|µ| below which the amplitudes for various scattering processes, especially
umklapp scattering, renormalize only very slowly. The flow of the antiferromagnetic spin susceptibility is cut off at the same scale. The infinite slope singularity
in some of the flow curves at scale Λ = |µ| is due to the van Hove singularity being
crossed at that scale. The pairing susceptibility with dx2 −y2 -symmetry is obviously
dominant here (note the logarithmic scale).
Following the flow of the vertex function and susceptibilities, one can see
that those interaction processes which enhance the antiferromagnetic spin susceptibility (especially umklapp scattering) also build up an attractive interaction
in the dx2 −y2 pairing channel. This confirms the spin-fluctuation route to d-wave
superconductivity [6].
5 Strong coupling problem
Within the one-loop calculation described above, the renormalized interaction ΓΛ
always diverges in some momentum channels at a finite energy scale Λc , even if
the bare interaction U was very small. Hence one is always running into a strong
coupling problem in the low-energy limit.
If the vertex function diverges only in the Cooper channel, driven only by the
particle-particle contribution to the flow, the strong coupling problem emerging
in the low energy region can be controlled by exploiting Λc as a small parameter
(Λc is small for a small bare interaction) [17]. The formation of a superconducting
ground state can then be described essentially by a BCS theory with renormalized
input parameters. One could also continue the RG flow below Λc down to Λ → 0
by including pairing counterterms into the effective action. This would remove
the divergence of the vertex function at finite Λ. In the Hubbard model the pure
Cooper channel instability is realized for µ = vH at sufficiently small U . In that
regime one can safely infer superconductivity with a d-wave order parameter from
the divergence of the one-loop pairing susceptibility! At finite temperature the
off-diagonal long-range order will of course turn into the quasi long-range order of
a Kosterlitz-Thouless phase.
In general the one-loop calculation can produce divergencies of the vertex
function in various momentum channels, with big contributions from both particleparticle and particle-hole diagrams. In the weak coupling limit U → 0 this happens
for µ → vH , that is when the Fermi surface touches the van Hove points. In that
case different possible instabilities can compete in a complicated way. Besides spin
density wave and pairing instabilities one has to deal with ferromagnetism (at
moderate |t /t| in the Hubbard model) [18, 19] and Pomeranchuk instabilities of
the Fermi surface [15, 20] as alternative or coexisting candidates. Honerkamp et
al. [8] have pointed out that for a sizable t the behavior of the one-loop flow may
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Walter Metzner
Ann. Henri Poincaré
also signal the formation of an insulating spin liquid for certain densities close to
half-filling.
A complete theory of the effective strong coupling problem characterized by
strong particle-particle and particle-hole fluctuations has still to be found. For
weak bare coupling, one may again try to exploit the smallness of the scale Λc
where strong fluctuations appear, to construct a tractable effective low-energy
theory.
What will change, if the bare coupling is big? Since strong particle-particle
and particle-hole fluctuations appear already at weak coupling close to half-filling
with a small t , it is not unplausible to conjecture that the Hubbard model might
exhibit most (if not all) of its low-energy features qualitatively already at weak
bare coupling. This is in fact the case in one dimension [2]. Of course bigger U
will lead to larger energy scales for binding phenomena, and also the scales for
pairing and pair condensation will separate, as the pairs get more tightly bound
at stronger coupling.
6 Conclusion
In summary, the pairing instability obtained from a one-loop RG analysis of the
two-dimensional Hubbard model conclusively establishes the existence of dx2 −y2
superconductivity in this model, at least at weak coupling.
A complete analysis of the weak coupling behavior of the Hubbard model,
especially for a Fermi surface near the van Hove points, can be expected to yield
further important clues for a better understanding of its low-energy behavior at
moderate and possibly even at strong coupling.
Acknowledgments
The renormalization group calculations reviewed in Secs. 3 and 4 have been performed in collaboration with Christoph Halboth. I would also like to thank Maurice
Rice, Daniel Rohe, and Manfred Salmhofer for valuable discussions.
References
[1] For an early review on 1D Fermi systems, see J. Solyom, Adv. Phys. 28, 201
(1979).
[2] For a comprehensive modern review on 1D Fermi systems, see J. Voit, Rep.
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[3] A. T. Zheleznyak, V. M. Yakovenko, and I. E. Dzyaloshinskii, Phys. Rev. B
55, 3200 (1997).
Vol. 4, 2003
Functional Renormalization Group for Interacting Fermi Systems
S931
[4] D. Zanchi and H. J. Schulz, Z. Phys. B 103, 339 (1997); Europhys. Lett. 44,
235 (1998); Phys. Rev. B 61, 13609 (2000).
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035109 (2001).
[9] See, for example, The Hubbard Model, ed. A. Montorsi (World Scientific 1992).
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[16] See, for example, H. J. Schulz, Phys. Rev. Lett. 64, 1445 (1990), where also
the choice of δ as a function of doping is discussed.
[17] J. Feldman, J. Magnen, V. Rivasseau, and E. Trubowitz, Europhys. Lett. 24,
437 (1993).
[18] R. Hlubina, S. Sorella, and F. Guinea, Phys. Rev. Lett. 78, 1343 (1997).
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[20] I. Grote, E. Körding, and F. Wegner, J. Low Temp. Phys. 126, 1385 (2002).
Walter Metzner
Max Planck Institute for Solid State Research
Heisenbergstr. 1
D-70569 Stuttgart
Germany