VOLUME 84, NUMBER 15

PHYSICAL REVIEW LETTERS

10 APRIL 2000

Topological Approach to Luttinger’s Theorem and the Fermi Surface of a Kondo Lattice
Masaki Oshikawa
Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan(Received 22 September 1999)
A nonperturbative proof of Luttinger’s theorem, based on a topological argument, is given for Fermi
liquids in arbitrary dimensions. Application to the Kondo lattice shows that even completely localized
spins contribute to the Fermi sea volume as electrons, whenever the system can be described as a Fermi
liquid.
PACS numbers: 71.10.Ay, 05.30.Fk, 75.30.Mb

Landau’s Fermi liquid theory is among the most important theories in the quantum many-body problem. At zerotemperature, a Fermi liquid has a Fermi surface, similar to
the noninteracting fermions. One of the most fundamental
results on the Fermi liquid is Luttinger’s theorem, whichstates that the volume inside the Fermi surface is invariant
by the interaction, if the number of particles is held fixed.
Luttinger argued, in his 1960 paper [1], the correction to
the volume vanishes order by order in the perturbation
expansion.
Recently, there has been renewed interest in Luttinger’s
theorem. Since Luttinger’s original proof was based on theperturbation theory, Luttinger’s theorem could be violated
by nonperturbative effects. In fact, several claims of possible breakdown of Luttinger’s theorem have been reported
recently [2–6]. On the other hand, such nonperturbative
effects can violate the Fermi liquid theory itself. In fact,
the Fermi liquid theory is known to be invalid generally in
one dimension, where Tomonaga-Luttinger (TL) liquid isthe generic behavior. Although not being a Fermi liquid, a
TL liquid in one dimension has a well-defined Fermi surface (actually Fermi points in one dimension). Thus thequestion of the validity of Luttinger’s “theorem” still exists in this case, where Luttinger’s original proof certainly
does not apply. This question was answered recently bya perturbative proof [7] and a more general nonperturbative proof [8], which can be applied to one-dimensional TL
liquids. However, the question on higher (especially two)
dimensions remains unanswered. In fact, it is not clear
whether a Fermi liquid which violates Luttinger’s theorem
can exist.
Another interesting problem, which is not answered by
the Luttinger’s perturbative proof, is the Fermi surface ofthe Kondo lattice. The Kondo lattice contains a periodic
array of localized spins which are coupled to conduction
electrons. The Kondo lattice is believed to belong to the
Fermi liquid (or TL liquid, in one dimension) in some region of the phase diagram. Even if we assume Luttinger’s
theorem to be valid, there is a problem in how to countthe number of particles. It is rather difficult, by conventional methods, to clarify whether a localized spin should
be counted as an electron (“large Fermi surface” picture) ornot (“small Fermi surface” picture). In one dimension, thenonperturbative proof of the Luttinger’s theorem was also
applied to the Kondo lattice [8], to show that the localized
spins do participate in the Fermi sea. (See also [9,10] fornumerical evidence.) On the other hand, there has been no
definite answer for higher dimensions, although there areseveral results [11,12] supporting the large Fermi surface
picture.
The argument in Ref. [8] is a generalization of the LiebSchultz-Mattis (LSM) theorem [13], which was given at
about the same time as the apparently unrelated Luttinger’s
theorem [1]. Since the LSM argument itself cannot be applied to higher dimensions, the discussion in Ref. [8] wasrestricted to one dimension. However, very recently the
LSM argument was combined with Laughlin’s gauge invariance argument [14] on the quantum Hall effect (QHE)
and extended to higher dimensions [15]. Inspired by thisobservation, we will extend the nonperturbative proof of
the Luttinger’s theorem to arbitrary dimensions in thepresent Letter.
We consider an interacting fermion system on a Ddimensional lattice with periodic boundary conditions. We
start from a finite system of size Lx 3 Ly 3 · · · 3 LD ,where the length is defined so that the unit cell has the size1 3 1 3 · · · 3 1. The number of fermions is assumed tobe conserved. If the system satisfies a commensurability condition, it can have a finite excitation gap [15]. In
this Letter, we will rather focus on the gapless case, whichis expected for general incommensurate particle density.
For simplicity, let us first start with the case of spinlessfermion of single species. We introduce a fictitious electric charge e for each particle, and a coupling to an externally controlled fictitious electromagnetic field. Becauseof the periodic boundary condition, the system is topologically equivalent to torus. Following Refs. [14,15], we
consider an adiabatic increase of a (fictitious) magnetic
flux F piercing through the “hole” of the torus so that theuniform electric field is induced, say, in the x direction.
While in general the Hamiltonian of the system H共F兲depends on the flux F reflecting the Aharonov-Bohm (AB)effect, the AB effect is absent when the flux reaches theunit flux quantum F0 苷 hc兾e. We consider the adiabatic
increase of the flux from F 苷 0 to F 苷 F0 . In the following, we will consider how the total momentum of the3370© 2000 The American Physical Society

0031-9007兾00兾84(15)兾3370(4)$15.00

VOLUME 84, NUMBER 15

PHYSICAL REVIEW LETTERS

system is changed during the adiabatic process in two different ways, and compare those results. In the remainder of this Letter, we take the units in which h̄ 苷 1, forsimplicity.
First, we analyze the momentum change in a system ofinteracting fermions for general. We remind the reader
that the momentum itself is a gauge dependent quantity
in the presence of the gauge field; a meaningful comparison between momenta can be made only under the samegauge choice. In a simplest gauge choice, the AB flux F
is represented by the uniform vector potential Ax 苷 F兾Lx
in the x direction. In this gauge, the Hamiltonian always
commutes with the translation operator Tx to the x direction. We further assume that the translation symmetry is
not spontaneously broken, as it should not be in a Fermi
liquid. Thus the ground state is an eigenstate of the total
momentum Px : Px jC0 典 苷 Px0 jC0 典 with the eigenvalue Px0 .
The x component of the total momentum Px is related toTx as Tx 苷 eiPx . After the adiabatic process, the original
ground state jC0 典 evolves into some state jC00 典. While thestate jC00 典 could be different from jC0 典, it belongs to thesame eigenvalue Px0 of Px , because the Hamiltonian always
commutes with Tx (and thus Px ) in the uniform gauge [15]during the adiabatic process. Although it naively meansthat the momentum is unchanged after the adiabatic process, it is not true. The Hamiltonian H共F0 兲 with the unit
flux quantum in the uniform gauge is different from theoriginal one H共0兲, although the spectrum should be identical. Namely, they correspond to different choices of thegauge for the same physics. In order to get back to theoriginal gauge, we must perform a large gauge transformation [15]
"
#
2pi X
U 苷 exp
xnr៬ ,(1)
Lx r៬
where nr៬ is the particle number operator at site r៬ , and x is
the x coordinate of r៬ . This transforms the Hamiltonian
H共F0 兲 back to the original one: UH共F0 兲U 21 苷 H共0兲.
After this gauge transformation, the adiabatic evolution of
the ground state becomes UjC00 典.
Now we can examine the total momentum Px of thisstate, and compare it with the original one Px0 . Here we
can employ the arguments used in the LSM theorem andits generalizations [8,13]. By using the identity
"
#
X nr៬
21
U Tx U 苷 Tx exp 2pi
,
(2)
Lx
r៬
we see that UjC00 典 is an eigenstate of Px withPx 苷 Px0 1 2pnLy Lz · · · LD ,(3)

where n is the particle density (number of particles perunit cell). This result is valid regardless of the interaction
strength.
Next, we analyze the momentum change assuming thatthe system is a Fermi liquid. The Fermi liquid is describedin terms of quasiparticles, which are almost noninteracting.

10 APRIL 2000

More precisely, the low-energy effective Hamiltonian of a
Fermi liquid is given byXX៬ ñ ៬ 1៬ k៬ 0 兲ñ ៬ ñ ៬ ,
H ⬃
(4)
e共k兲
f共k,
k k0
k
k៬

៬ k៬ 0k,where ñk៬ is the quasiparticle number operator of momen៬ Namely, there is an interaction energy due to thetum k.
second term but no scattering between the quasiparticles.
Thus the eigenstates of ñk៬ are also eigenstates of Hamiltonian. In the ground state, the Fermi sea (region inside
the Fermi surface) is completely filled with quasiparticles,
while the outside is empty in terms of quasiparticles. Excitations on the ground state are given by quasiparticles outside the Fermi sea and/or quasiholes inside the Fermi sea.
In fact, the quasiparticle (or quasiholes) are free fromscattering only in the vicinity of the Fermi surface; thevery notion of quasiparticle and/or quasihole is useful onlyin this case. The Fermi liquid theory is valid for the lowenergy phenomena, in which the relevant excitations
consist only of quasiparticles (quasiholes) near the Fermi
surface.
共L兲
Let us define the Fermi sea volume VF in the finite
size system Lx 3 Ly 3 · · · 3 LD . The quasiparticles arescattering free, and their momenta are discretized as in
the case of free particles. Thus we can define the Fermi
共L兲
sea volume VF by an integer “occupation number” of the
共L兲
quasiparticles NF :共L兲共2p兲D NF.
(5)
Lx Ly · · · L D
Although the quasiparticles are not free from scattering(and thus are not meaningful) away from the Fermi surface,
this expression is still valid because the Fermi sea volume
共L兲
is uniquely determined by its surface. The VF should
approach the true volume of the Fermi sea VF , in thethermodynamic limit Lj ! `.
The adiabatic evolution is determined by the low-energy
effective Hamiltonian (4). In the Fermi liquid theory, thecharge of the quasiparticle is identical to that of the original
particle e. The coupling of the quasiparticles to the uniform vector potential Ax is thus given by the substitutionof the momentum kx ! kx 1 eAx 兾c in the Hamiltonian.
After the adiabatic insertion of the unit flux quantum, andgetting back to the original Hamiltonian by the gauge transformation, each quasiparticle gets a momentum shift: kx isincreased by 2p兾Lx . This produces quasiparticles on one
side of the Fermi surface, and quasiholes on the opposite
side.
Since the result of the adiabatic process is equivalent tothe shift of the whole Fermi sea by 2p兾Lx , the changeof the x component of total momentum Px of the system
during the adiabatic process is given by
共L兲

VF 苷

DPx 苷

2p 共L兲
NF .
Lx

(6)

3371

VOLUME 84, NUMBER 15

PHYSICAL REVIEW LETTERS

We note that the only changes after the adiabatic process
involve the quasiparticles and quasiholes near the Fermi
surface, so that the Fermi liquid theory is still valid. To
violate Eq. (6), the system must break some of the properties of Fermi liquid used in the present argument. For
example, if a quasiparticle had a charge e0 which is different from the charge e of the original particle, we would
obtain a different result.
Now, comparing the two results Eqs. (3) and (6) ob共L兲
tained with different arguments, we obtain NF 兾Lx 2
nLy Lz · · · LD 苷 integer, where we have used the fact thateach component of momenta is defined modulo 2p. Let us
choose the system size so that Lx , Ly , . . . and LD are mutually prime with the others. We also assume Lx 苷 qlx
where lx is an integer. (It should be recalled that system size should be an integral multiple of q, to allow thefilling factor n 苷 p兾q.) Then, from Eq. (5) we obtain
共L兲
NF 2 plx Ly Lz · · · LD 苷 Lx 3 共integer兲.
Furthermore, we can consider other adiabatic processes,
in which the gauge field is induced in one of the otherdirections y, z, . . . , instead of x. Similar calculations for
共L兲these cases lead to NF 苷 La 3 共integer兲, where a 苷y, z, . . . , D. Because we have chosen the lengths Lj ’s mu共L兲
tually prime, we conclude that NF 2 plx Ly Lz · · · LD 苷
nLx Ly Lz · · · LD , where n is an integer. Writing this interms of Fermi sea volume, we arrive atVF
2 n 苷 n,
(7)
共2p兲D
共L兲

where we have replaced the Fermi sea volume VF for thefinite size system by its thermodynamic limit VF , because
this relation is exact already for the finite system. The thermodynamic limit VF should be independent of our special(mutually prime) choice of Lj ’s, if VF is well defined.
The relation (7) is nothing but the statement of Luttinger’s theorem. The integer n corresponds to the numberof completely filled bands. It is valid also when the Fermi
sea consists of several disjoint regions, if VF is understoodas the sum of volumes of all regions. Our proof is much
simpler than the original one [1]. Moreover, in contrast toRef. [1], our argument is nonperturbative and relies only
on some of the basic properties of Fermi liquid.
It is straightforward to extend our argument to spinful
electrons. When the numbers of up-spin electrons anddown-spin electrons are conserved separately, we consider
the fictitious electromagnetic field coupled to only up-spin(or down-spin) electrons. Assuming the spinful Fermi liquids, the volume of the Fermi sea VFs for spin s is given byVFs 苷 共2p兲D ns , where ns is the number of particles withspin s per unit cell. For the spin-symmetric case n" 苷 n# ,"#it reads VF 苷 VF 苷 VF 苷 共2p兲D n兾2 where n is the total
particle density n" 1 n# .
As a nontrivial application, let us consider the Kondo
lattice. Luttinger’s original perturbative proof does not apply to this case, and the question on the volume of the3372

10 APRIL 2000

Fermi sea has remained. For the sake of clarity, we consider the Kondo lattice model given by the Hamiltonian
X
X
y
y
y
H 苷2
tjk cjs cks 1 H.c. 1
Uj cj" cj" cj# cj#
j

j,k

1

X

Jl s៬ l ? S៬ l ,

(8)

l
y

where cjs and cjs are standard fermion creation or annihiylation operators at site j with spin s, s៬ l 苷 cla s៬ ab clb 兾2
is the spin operator of the conduction electron, and S៬ l
is the localized spin at site l. As in the previous case,we couple the fictitious electromagnetic field only to the
up-spin electrons. After the adiabatic insertion of the AB
flux of unit flux quantum, we make the gauge transformation as in theprevious cases. However, the naive one
2pi P
eU" 苷 exp关 Lx r៬ xnr៬ s 兴, does not bring the Hamiltonian
back to the original one, because it changes the Kondo
coupling. In order to recover the original Hamiltonian, wemust also twist the localized spins. The transformation
#
"
2pi X
z
(9)
x共nr៬ s 1 Sr៬ 兲U" 苷 exp
Lx r៬
does the required job. We obtain the total momentum after
the adiabatic process as
Px 苷 Px0 1 2p关n" 1 Ns 共S 1 m兲兴Ly Lz · · · LD ,(10)

where Ns is the number of localized spins per unit cell andm is the magnetization per single localized spin. The special contribution proportional to S comes from the boundary term exp共2piNs S1z 兲 appearing in U"21 Tx U" , similar tothe one-dimensional case [8,13,16].
Thus, provided that the system belongs to a spinful
Fermi liquid, the volume of the Fermi sea is given byVFs 苷 共2p兲D 关ns 1 Ns 共S 6 m兲兴, where 6 takes 1 fors 苷 " and 2 for s 苷 #. For the spin-symmetric casen" 苷 n# and m 苷 0, we obtain
共2p兲D关n 1 2Ns S兴 ,(11)
2for the total particle density n 苷 n" 1 n# 苷 2n" . This isexactly what we obtain if we apply the Luttinger’s theorem
to the Anderson-type model in which the localized spins
are represented by electrons. It means that the localized
spin S does contribute to the Fermi sea volume as 2S
electrons, even though it is completely immobile. This is
the picture conventionally called the large Fermi surface.
It should be noted that we did not answer the nontrivial
question whether (or when) the Kondo lattice belongs to
the Fermi liquid. We have proved only that, if the Kondo
lattice is a Fermi liquid (as it is believed to be true in someregion of the phase diagram), the localized spins participate
in the Fermi sea.
Finally, let us comment on claims [2–6] of the violationof the Luttinger’s theorem. There are several possibilities"#

VF 苷 V F 苷 V F 苷

VOLUME 84, NUMBER 15

PHYSICAL REVIEW LETTERS

regarding the apparent contradiction to our nonperturbative
proof. Of course, it should be checked whether our argument applies to the model under consideration. However,
our argument does apply to a very wide range of latticemodels, including the Hubbard and t-J models, for which
the violation of the Luttinger’s theorem has also been proposed. A possibility is that the system is not a Fermi liquidin these cases. In other words, a violation of the Luttinger’s
theorem requires the system to be a non-Fermi liquid. We
note that, however, merely not being a Fermi liquid is insufficient, as the TL liquid in one dimension does satisfy
the Luttinger’s theorem [7,8]. Our approach could be extended to a non-Fermi liquid which has an appropriately
defined Fermi surface, if such a liquid does exist. Our argument reveals a rigid relationship between the structureof low-energy excitations and the Fermi sea volume.
Another possibility is that the claimed violation of Luttinger’s theorem is actually incorrect. In particular, numerical results are available only for restricted system size
and/or temperature, and can miss the possibly small singularity at the true Fermi surface. On the other hand, even
if they are incorrect in identifying the true Fermi surface,
they might still be of physical relevance because the actual experiments are done also at finite energy scale; theexperimentally measured “Fermi surface” could be different from the true Fermi surface defined in the low-energy
limit, to which our argument applies. In any case, our definite result on the Fermi surface of the Fermi liquid in the
low-energy limit would be useful as a guideline. Claimsof the violation of the Luttinger’s theorem should be examined in the light of the present result.
During the 40 years after the Luttinger’s paper [1],several examples of “quantization” of a physical quantity have been found in many-body physics. Namely, despite the complexity of the interacting many-body states,
some physical quantity takes a special value which is stableagainst various perturbations such as interaction strength.
Presumably the most natural understanding of such a quantization is given by a topological argument. Indeed, typical examples of the quantization—QHE and the quantizedmagnetization plateaus—have been related to topologicalmechanisms [14–16].
Luttinger’s theorem perhaps does not look like a quantization, because the volume of the Fermi sea takes continuous values depending on the particle density. However, theinsensitivity to the interaction resembles other quantization
phenomena, and may well be regarded as a certain kind ofquantization, especially when written as in Eq. (7). In fact,

10 APRIL 2000

we have revealed a close theoretical relationship among
Luttinger’s theorem, QHE, and magnetization plateaus. In
addition, our argument can be related also to the chiralanomaly in quantum field theory [17]. Luttinger’s theorem might be actually the first example of the topologicalquantization discovered in quantum many-body problem,
although the topological understanding has been missing
for a long time.
I thank Ian Affleck, Hal Tasaki, and Masanori Yamanaka
for stimulating discussions which were essential for thepresent work. This work is supported by Grant-in-Aid
from Ministry of Education, Science, Sports, and Cultureof Japan.

[1] J. M. Luttinger, Phys. Rev. 119, 1153 (1960).
[2] M. Langer, J. Schmalian, S. Grabowski, and K. H. Bennemann, cond-mat/9505063.
[3] A. V. Chubukov, D. K. Morr, and K. A. Shakhnovich,cond-mat/9511138; B. L. Altshuler, A. V. Chubukov,
A. Dashevskii, A. M. Finkel’stein, and D K. Morr,cond-mat/9703120.
[4] W. O. Putikka, M. U. Luchini, and R. R. P. Singh, Phys.
Rev. Lett. 81, 2966 (1998).
[5] A. Sherman and M. Schreiber, cond-mat/9808087.
[6] C. Groeber, M. G. Zacher, and R. Eder, cond-mat/9902015.
[7] K. B. Blagoev and K. S. Bedell, Phys. Rev. Lett. 79, 1106
(1997).
[8] M. Yamanaka, M. Oshikawa, and I. Affleck, Phys. Rev.
Lett. 79, 1110 (1997).
[9] S. Moukouri and L. G. Caron, Phys. Rev. B 54, 12 212(1996).
[10] N. Shibata, K. Ueda, T. Nishino, and C. Ishii, Phys. Rev.
B 54, 13 495 (1996).
[11] L. Taillefer et al., J. Magn. Magn. Mater. 63-64, 372(1987).
[12] R. Eder, O. Stoica, and G. A. Sawatzky, Phys. Rev. B 55,
6109 (1997); R. Eder, O. Rogojanu, and G. A. Sawatzky,
Phys. Rev. B 58, 7599 (1998).
[13] E. H. Lieb, T. Schultz, and D. J. Mattis, Ann. Phys. (N.Y.)
16, 407 (1961).
[14] R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).
[15] M. Oshikawa, Phys. Rev. Lett. 84, 1535 (2000).
[16] M. Oshikawa, M. Yamanaka, and I. Affleck, Phys. Rev.
Lett. 78, 1984 (1997).
[17] R. Jackiw, Topological Investigations of Quantized Gauge
Theories, in Current Algebra and Anomalies (World Scientific, Singapore, 1985).

3373