Quantum fluctuations beyond the Gutzwiller approximation

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by  Michele Fabrizio

Quantum fluctuations beyond the Gutzwiller approximation
Michele Fabrizio1
1International School for Advanced Studies (SISSA), Via Bonomea 265, I-34136 Trieste, Italy
(Dated: January 17, 2017)
We present a simple scheme to evaluate linear response functions including quantum fluctuation
corrections on top of the Gutzwiller approximation. The method is derived for a generic multi-
bandlatticeHamiltonianwithoutanyassumptionaboutthedynamicsofthevariationalcorrelation
parameters that define the Gutzwiller wavefunction, and which thus behave as genuine dynamical
degrees of freedom that add on those of thevariational uncorrelated Slater determinant.
7
Weapplythemethodtothestandardhalf-filledsingle-bandHubbardmodel. Weareabletorecover
1
known results, but, as by-product, we also obtain few novel ones. In particular, we show that
0
quantum fluctuations can reproduce almost quantitatively the behaviour of the uniform magnetic
2
susceptibility uncovered by dynamical mean field theory, which, though enhanced by correlations,

n isfoundtobesmooth across theparamagnetic Mott transition. Bycontrast,thesimpleGutzwiller
a approximation predicts that susceptibility to divergeat the transition.
J

6 PACSnumbers: 71.10.-w,71.30.+h,71.10.Fd
1

] I. INTRODUCTION when the coordination number tends to infinity, rather
l
e than in Gutzwiller-only wavefunctions. In other words,
-
r The Gutzwiller approximation1,2 is likely the simplest the Gutzwiller approximation applied on a lattice with
st tool to deal with strong correlations in lattice models of finite z is just the variational counterpart of dynamical
t. interacting electrons. It consists in a recipe for approx- meanfieldtheory(DMFT)8 appliedonthatsamelattice.
a imate analytical expressions of expectation values in a Recently, several attempts to include the Gutzwiller ap-
m class of wavefunctions,named Gutzwiller wavefunctions, proximation inside DFT electronic structure codes have
- of the form beenperformedwithquiteencouragingoutcomes9–20. In
d
this perspective, it might be useful to have at disposal a
n
o |Ψi= P(i)|Ψ0i, (1) simple and flexible method to calculate linear response
c Yi functions within the Gutzwiller approximation, in
[ view of an extension of the so-called linear response
where |Ψ0i is a variational Slater determinant, and P(i) TDDFT21,22 to the case when DFT is combined with
1 a linear operator that acts on the local Hilbert space at
the Gutzwiller approximation.
v site i and depends on a set of variational parameters.
There are already several works dealing with linear
6 Curiously, the Gutzwiller approximation often pro-
0 responseintheGutzwiller approximation,mostofwhich
vides physically more sound results than a direct
3 limited to the single-band Hubbard model23–28. Exten-
evaluation of expectation values in wavefunctions like
4 sions to multi-band models have been attempted29,30,
Eq. (1). For instance, the numerical optimisation on
0 though under an assumption about the dynamics of
. a finite-dimensional lattice of a variational Gutzwiller
1 the variational parameters that determine the linear
wavefunctionforasingle-bandhalf-filledHubbardmodel
0 operators (i) in Eq. (1).
7 never stabilises a genuine Mott insulating phase3,4, i.e. Here we shPall instead present a very simple and general
an insulator that does not break any symmetry, which
1
method to evaluate linear response functions within
: intuitively is to be expected beyond a critical strength
v the Gutzwiller approximation without any preliminary
of the on-site repulsion. By contrast, the Gutzwiller
i assumption. The method is essentially an extension
X approximationis instead able to describe such a genuine
of the time-dependent Gutzwiller approximation of
r Mott transition5. The explanation of this strange
a outcome relies onthe followingobservations. The first is Ref. 31 to a generic multi-band Hamiltonian, where the
dynamics of the linear operators (i) and of the Slater
that, in order to describe a genuine Mott insulator, one P
determinant Ψ , see Eq. (1), are treated on equal
needs to add to the Gutzwiller wavefunction, Eq. (1), | 0i
footing. Linearisationof the equations ofmotion around
long range density-density Jastrow factors4. However,
the stationary solution, which is the equilibrium state,
the effect of such Jastrow factors disappears in lattices
thus allows calculating linear response functions.
with coordination number z , therefore, only in
→ ∞ We note that the results of the Gutzwiller approxi-
that limit, wavefunctions like Eq. (1) can faithfully
mation at equilibrium coincide with the saddle point
describe Mott insulators. Moreover, right in that limit
solution of the slave-boson theory in the path-integral
of z , the Gutzwiller approximation provides the
exact→exp∞ression of expectation values6,7. Therefore the formulation32,which, inmulti-bandmodels,corresponds
to the so-called rotationally invariant slave boson
Gutzwiller approximation should better be regarded as
formalism (RISB)33. Our present results in the linear
a recipe to evaluate approximate expectation values in
response regime can therefore be considered equivalent
Gutzwiller-Jastrow wavefunctions, which becomes exact
2
to the quantum fluctuations corrections above the with K (i,t) = K (i,t)∗, which maps c into a new
αβ βα iα
RISB saddle-point solution. We preferred here to basis set d of single particle operators
iα
derive such corrections to the action directly from the
time-dependent Gutzwiller approximation rather than d = (i,t)†c (i,t)= U (i,t)c . (4)
iα W iαW αβ iβ
from the RISB theory, since the former is at least a well β
X
controlled variational scheme in lattices with infinite
Evidently, if we consider the gauge transformation
coordination number. However, both the notations as
well as the language we shall use are actually closely (i,t) (i,t) (i,t)†, (5)
related to RISB theory. P →P W
Ψ (t) (i,t) Ψ (t) , (6)
0 0
| i→ W | i
The paper is organisedasfollows. InSec. II we briefly i
Y
present the time-dependent Gutzwiller approximation,
the Gutzwiller wavefunction Ψ(t) in (2) stays invariant
with some additional technical details postponed to the | i
and the transformed Ψ (t) remains a Slater determi-
0
Appendix. InSec.IIIwelinearisetheequationsofmotion | i
nant. Such gauge invariance, analogous to that of the
aroundthestationarysolutionandderiveaneffectiveac- RISB theory33, repeatedly appears in the calculations
tion for the fluctuations in the harmonic approximation.
that follow.
In Sec. IV we apply the method to the single-band half-
filled Hubbard model, which allows a comparison with The most general (i,t) can be written34,35 as
alreadyexistingresults. SectionVisdevotedtoconclud- P
ing remarks. (i,t)= λ (i,t) n;i m¯;i , (7)
nm¯
P | ih |
nm¯
X
wherenandm¯ canbechosentobelongtothelocalbasis
II. THE GUTZWILLER APPROXIMATION IN
BRIEF ofFockstatesbuiltwiththeoperatorsciα. Alternatively,
one can use a mixed-basis representation where n labels
Fockstatesintheoriginalbasisc ,andm¯ Fockstatesin
Besides the original works1,2 where M. Gutzwiller in- iα
troduced a novel class of variational wavefunctions as a different basis36, e.g. the basis of the operators diα in
well as an approximate scheme to compute expectation Eq.(4),whichisalsousedtobuilttheSlaterdeterminant
values, after him called Gutzwiller wavefunctions and Ψ0(t) . Wedefinetheuncorrelatedlocalprobabilitydis-
| i
approximation, and the subsequent demonstration that tributionPˆ0(i,t),whichispositivedefinite,byitsmatrix
such an approximation becomes exact in the limit of elements
infinite-coordination lattices6,7, there are by now many
articles where the Gutzwiller approximationis described P0n¯m¯(i,t)= Ψ0(t) m¯;i n¯;i Ψ0(t) , (8)
| ih |
in detail. Here we shall follow Ref. 34 and use its same (cid:12) (cid:12)
notations. as well as the Gutzw(cid:10)iller va(cid:12)(cid:12)riational mat(cid:12)(cid:12)rix (cid:11)
The time-dependent Gutzwiller wavefunction is defined
through24,31,34 Φˆ(i,t) λˆ(i,t) Pˆ (i,t) , (9)
0
≡
q
Ψ(t) = (i,t) Ψ0(t) , (2) with matrix elements Φnm¯(i,t). Expectation values of
| i P | i localandnon-localoperatorsintheGutzwillerwavefunc-
i
Y
tion (2) can be calculated explicitly in infinite coordina-
which is the analogous of Eq. (1) where now Ψ (t) tion lattices if one imposes the following two constraints
0
| i
is a time-dependent variational Slater determinant, and at any time7,34:
(i,t) linear operators on the local Hilbert space that
P
depend on time-dependent variational parameters. For Tr Φˆ(i,t)†Φˆ(i,t) =1, (10)
sake of simplicity, we shall not include in our analysis
(cid:16) (cid:17)
BCS wavefunctions nor operators (i,t) that are charge Tr Φˆ(i,t)†Φˆ(i,t)cˆ† cˆ n (i,t)
P iα iβ ≡ αβ
non-conserving. The extension to those cases is simple,
though notations get more involved. (cid:16) =(cid:17)hΨ0(t)|c†iαciβ |Ψ0(t)i, (11)
Suppose that the Hamiltonian is written in terms of
where the fermionic operators within the spur must be
fermionic operators c and c† , α = 1,...,2M, that
iα iα regardedas their matrix representationinthe localFock
correspond to annihilating or creating a fermion at site
space. The secondconstraintEq.(11) playsthe roleof a
i in a chosenbasis of Wannier functions φ (x,t), where
iα gauge-fixing condition, exactly as in the RISB model33.
αindicatesbothspinandorbitalindices. Letusimagine
Another importantingredient is the wavefunctionrenor-
a U(2M) unitary transformation malisationmatrixRˆ(i,t)withelementsR (i,t),defined
αβ
by solving the set of equations
(i,t)=exp i K (i,t)c† c , (3)
W (cid:18) Xαβ αβ iα iβ(cid:19) hΨ0(t)|c†iγP(i,t)†ciαP(i,t)|Ψ0(t)i
3
= n (i,t)R (i,t) , (12) A. The model
γβ αβ
β
X
We shall assume the generic Hamiltonian
where the left hand side can be straightforwardly evalu-
ated by the Wick’s theorem. As shown in the Appendix = c†tˆ c† + , (21)
A, the solution of the above equation reads H i ij j Hi
i6=j i
X X
Rˆ(i,t)=Qˆ(i,t)Sˆ(i,t), (13)
where includes all on-site terms. If the constraints
i
H
where Qˆ(i,t) has matrix elements Eq.(10)andEq.(11)aresatisfiedatanytimet,then, in
infinite coordination lattices, it holds that7,34
Q (i,t)=Tr Φˆ(i,t)†cˆ Φˆ(i,t)cˆ† , (14)
αβ iα iβ E(t)= Ψ(t) Ψ(t) = Ψ0(t) ∗(t) Ψ0(t)
h |H| i h |H | i
(cid:16) (cid:17)
and the hermitian matrix Sˆ(i,t) is defined through + Tr Φˆ(i,t)†HˆiΦˆ(i,t)
4Sˆ(i,t)−2 =1 ∆ˆ(i,t)2 (15) Xi (cid:16) (cid:17)
− E (t)+ Tr Φˆ(i,t)†Hˆ Φˆ(i,t) , (22)
∗ i
where the matrix elements of ∆ˆ(i,t) are ≡ Xi (cid:16) (cid:17)
where
∆ (i,t)=Tr Φˆ(i,t)†Φˆ(i,t) cˆ , cˆ† . (16)
αβ iα iβ (t)= c† Rˆ(i,t)†tˆ Rˆ(j,t)c , (23)
(cid:18) h i(cid:19) H∗ i ij i
i6=j
The meaning of Rˆ(i,t) is that the action of the anni- X
may be interpreted as the Hamiltonian of the quasipar-
hilation operator c on the Gutzwiller wavefunction is
iα ticles. Evidently, all expectation values can be straight-
equivalent to the action of the operator
forwardly evaluated since the uncorrelated wavefunction
P(i,t)†ci P(i,t)→Rˆ(i,t)ci (17) |Ψ0(t)i allows using Wick’s theorem.
on the Slater determinant Ψ (t) , where c is a spinor
0 i
| i
with components ciα. One can readily show that under B. The action
the gauge transformation Eq. (5),
Rˆ(i,t) Rˆ(i,t)W =Rˆ(i,t)Uˆ(i,t)†, (18) In the time-domain the variational principle corre-
→ sponds to searching for the saddle point of the action31
whereUˆ(i,t)hasthematrixelementsU (i,t)ofEq.(4),
αβ
so that Eq. (17) transforms into = dt i Ψ(t) Ψ˙(t) E(t)
S h | i−
Z (cid:20) (cid:21)
W(i,t)P(i,t)†ci P(i,t)W(i,t)† → Rˆ(i,t)W di. dt i Tr Φˆ(i,t)† ∂Φˆ(i,t)
Since we have complete freedom in choosing (i,t), a ≡ ( ∂t !
convenientchoiceistheunitarytransformationtWhatdiag- Z Xi
onalisesthe localsingle-particledensity matrix,inwhich +i Ψ (t) Ψ˙ (t) E(t) , (24)
0 0
case the operators d are associated to the natural or- h | i − )
iα
bitals and satisfy
where the equivalence holds on provision that the con-
Tr Φˆ(i,t)†Φˆ(i,t)dˆ† dˆ =δ n (i,t), (19) straints (10) and (11) are fulfilled at any time. The sad-
iα iβ αβ α dle point equations are readily obtained:
(cid:16) (cid:17)
while the matrix elements of Rˆ(i,t)W acquire the simple ∂Φˆ(i,t) ∂E (t)
expression i ∂t =HˆiΦˆ(i,t) + ∂Φˆ(∗i,t)† , (25)
Tr Φˆ(i,t)†cˆ Φˆ(i,t)dˆ† i Ψ˙ (t) = (t) Ψ (t) , (26)
iα iβ 0 ∗ 0
R (i,t)W = . (20) | i H | i
αβ (cid:16) (cid:17) where
n (i,t) 1 n (i,t)
β β
− ∂E (t) ∂ (t)
The matrix Φˆ(i,t) is irn this ca(cid:16)se convenien(cid:17)tly defined ∂Φˆ(∗i,t)† = Ψ0(t)(cid:12) ∂ΦˆH(i∗,t)† (cid:12)Ψ0(t)
D (cid:12) (cid:12) E
in the mixed-basis representation, where n in Φnm¯(i,t) Tˆ(i,t(cid:12))Φˆ(i,t). (cid:12) (27)
refers to a Fock state in the original basis, and m¯ to a ≡ (cid:12) (cid:12)
Fock state in the natural one. Such a mixed-basis repre- Tˆ(i,t) is a tensor with components Tnm;n′m′(i,t), which
sentation is useful since,throughout all calculations, one is still functional of the matrices Φˆ and Φˆ† at site i as
does not actually need to know what the natural basis well as at all sites connected to i by the hopping. One
is in terms of the original one36. Such a nice property is canshowthat this tensor is hermitean, Tˆ(i,t)=Tˆ(i,t)†,
linked to the gauge-invariance, equations (5) and (6), of which implies that the normalisation Eq. (10) is con-
the theory33. served by the time evolution.
4
C. Fate of the constraint
Ψ (t) + δΨ (t) = 1+i (i,t) Ψ (t) ,
0 0 0
| i | i K !| i
i
X
Concerning the second constraint, Eq. (11), we now
where the operator
prove that, if it is satisfied at the initial time, it will
remain so at the saddle point solutions of Eq. (25) and
(i,t)= K (i,t)c† c , (28)
Eq.(26). Supposewehaveindeedfoundthesaddlepoint K αβ iα iβ
Φˆ(i,t) and Ψ0(t) . By definition, any small variation Xαβ
| i
with respect to that solution must lead to a vanishing
has infinitesimal matrix elements K (i,t)=K (i,t)∗,
variation of the action. Let us consider the infinitesimal αβ βα
andKˆ(i,t)isitsmatrixrepresentationintheFockspace.
gauge transformation
We already mentioned that the energy E(t) is gauge in-
variantsothatthevariationoftheaction, δ = W ,
Φˆ(i,t)+δΦˆ(i,t)=Φˆ(i,t) 1 iKˆ(i,t) , simply reads S S −S
−
(cid:16) (cid:17)
∂Wˆ(i,t)†
δ = dt i Tr Φˆ(i,t)†Φˆ(i,t) Wˆ(i,t) +i Ψ (t) (i,t)† ˙ (i,t) Ψ (t)
0 0
S Z ( i (cid:18) ∂t (cid:19) i h |W W | i)
X X
dt Tr Φˆ(i,t)†Φˆ(i,t) K˙(i,t) Ψ (t) ˙(i,t) Ψ (t)
0 0
≃ i Z ( (cid:18) (cid:19) − h |K | i)
X
= dtK˙ (i,t) Tr Φˆ(i,t)†Φˆ(i,t) cˆ† cˆ† Ψ (t) c† c Ψ (t)
i αβ Z αβ ( (cid:18) iα iβ(cid:19)−h 0 | iα iβ | 0 i)
X X
∂
= dtK (i,t) Tr Φˆ(i,t)†Φˆ(i,t) cˆ† cˆ† Ψ (t) c† c Ψ (t) .
− i αβ Z αβ ∂t ( (cid:18) iα iβ(cid:19)−h 0 | iα iβ | 0 i)
X X
Since Φˆ(i,t)and Ψ (t) aresolutionsofthesaddlepoint + µ (i)Φˆ(i)dˆ† dˆ , (29)
| 0 i αβ iα iβ
equations, it follows that δS must strictly vanish for
αβ
X
any choice of the infinitesimally small matrix elements
E Ψ = µ (i)d† d Ψ , (30)
Kαβ(t), which implies ∗ | 0i H∗− αβ iα iβ | 0i
(cid:16) Xi (cid:17)
∂ Tr Φˆ(i,t)†Φˆ(i,t) cˆ† cˆ† where Λ(i) enforces Eq. (10), and the hermitean matrix
∂t ( (cid:18) iα iβ(cid:19)) µˆ(i)withcomponents µαβ(i)enforcesEq.(11). Inwhat-
ever follows we shall assume to work in a mixed-basis
−hΨ0(t)|c†iαciβ |Ψ0(t)i)=0, trhepernesaetnutraatlioonrbwithaelsr,estohtehoaptewraetomrussdtiαalsaoreenasssuorceiathteadtto
thus just the desired result. It actually means that Tr Φˆ†(i)Φˆ(i)dˆ† dˆ = Ψ d† d Ψ =δ n (i).
the term in parenthesis is conserved in the evolution. iα iβ h 0 | iα iβ | 0i αβ α
(cid:16) (cid:17)
Therefore, if it is initially vanishing, it will remain so at
The quasiparticle Hamiltonian in the natural basis, in-
anytime,whichthusimpliesthattheconstraintEq.(11)
cluding explicitly the Lagrange multipliers, is therefore
is fulfilled during the whole time evolution.
d† Rˆ(i)†tˆ Rˆ(j)d d†µˆ(i)d , (31)
H∗ → i ij i − i i
i6=j i
X X
D. Stationary problem
with Rˆ defined in Eq. (20). Working in the mixed-basis
representationwiththenaturalorbitalsconsiderablysim-
At equilibrium one needs to find the minimum of the
plifies all calculations.
energywiththetwoconstraintsEqs.(10)and(11),which RecallingthatTˆ(i)isstillfunctionalofΦˆ,Eq.(29)looks
can be enforced e.g. by Lagrange multipliers, leading to like a stationary non-linear Schrœdinger equation20,37.
the set of equations
One can for instance solve it as in any Hartree-Fock cal-
culation. Namely,onecanfindtheeigenstatesandeigen-
Λ(i)Φˆ(i)= Hˆi+Tˆ(i) Φˆ(i) values of Eq. (29) assuming Tˆ(i) fixed, and impose that,
(cid:16) (cid:17)
5
when Tˆ(i) is calculated substituting the actual expres- where Ψ (t) is properly normalised and includes the
0
| i
sion of the lowest energy solution Φˆ0(i), the two values zerothorder|Ψ(00)i,solutionofthesaddlepoint,aswellas
coincide. The Lagrangemultiplierµˆ isfixedbyimposing a fluctuation correction δΨ (t) . The unitary operator
0
| i
Eq. (11) and Eq. (19). In this way one finally gets the
self-consistent Tˆ(i), which we shall hereafter denote as (i,t)=exp itd†µˆ(i)d , (39)
W − i i
Tˆ(0)(i) Tˆ Φˆ ,Φˆ† . (32) (cid:18) (cid:19)
≡ 0 0
where µˆ(i) is the equilibrium Lagrange multiplier, and
h i
Once the latter is known, as well as the value of µˆ, one Wˆ(i,t) is the matrix representation of (i,t).
can also solve (29) for all eigenvectors, Φˆn(i) and corre- Through the above definitions, the actiWon becomes
sponding eigenvalues E (i), with E (i)=Λ(i). We shall
n 0
denote , Rˆ, Qˆ, nˆ and Sˆ calculated with Φˆ as (0),
Rˆ(0),Qˆ(H0)∗,nˆ(0) andSˆ(0),respectively,withthe0latterHt∗wo S= dt(i φn(i,t)∗φ˙n(i,t)+ihΨ0(t)|Ψ˙0(t)i
matrices diagonal in the natural basis, Z Xi nX>0
φ (i,t)∗V (i)φ (i,t)
n(0) =δ n(0), (33) − n nm m
αβ αβ α i nm
X X
−1/2
S(0) =δ S(0) =δ n(0) 1 n(0) . (34)
αβ αβ α αβ α − α +E0+E∗ E∗(t) , (40)
(cid:16) (cid:16) (cid:17)(cid:17) − )
where E (t)= Ψ (t) (t) Ψ (t) , being now
We conclude by noting that the saddle point Hamil- ∗ h 0 |H∗ | 0 i
tonian Eq. (31) with the inclusion of the Lagrange mul-
(t)= d† Rˆ(i,t)†tˆ Rˆ(j,t)d d†µˆ(i)d ,(41)
tipliers is not anymore invariant under the most general H∗ i ij i− i i
U(2M)gaugetransformation,butonlyunderasubgroup Xi6=j Xi
GwithgeneratorsTˆa thatcommutewithµˆ. Thisiscom- and
mon in theories where the gauge invariance implements
constraints about physical states. In the natural basis V (i)=Tr Φˆ (i)†Hˆ Φˆ (i) (42)
nm n i m
representation, µ = δ µ is diagonal, so that the
matrix elements oi,fαTβˆa muαsβt saiαtisfy (cid:16) +Tr Φˆ(cid:17)n(i)†Φˆm(i)dˆ†i µˆ(i)dˆi .
Ta µ µ =0, (35) (cid:16) (cid:17)
i,αβ iα− iβ We expand ∗(t) up to second order in the fluctu-
H
whose solution is st(cid:0)raightforw(cid:1)ard. For any non- ations. The zeroth order is just ∗(0). Since the sta-
H
degenerate α, i.e. such that µ = µ , β = α, we tionary solution is the saddle point of the action, the
iα iβ
associate the generators Tα = δ 6 δ of∀U(1)6 abelian expectation value of the first order expansion (1)(t)
i,γβ αβ γβ H
groups. On the contrary, for any set of αi, i = 1,...,k, over the saddle point Slater determinant |Ψ(00)i can-
such that µiαi = µiαj 6= µiβ, ∀β 6= α1,...,αk, we can cels with the first order expansion of the local energy
associate generators of a U(k) Lie algebra. φ (i,t)∗V (i)φ (i,t). Therefore (1)(t)con-
i nm n nm m H
tributestoE (t)withasecondordertermthat,bylinear
∗
P P
response theory, reads
III. FLUCTUATIONS ABOVE THE SADDLE δ E (t)= δΨ (t) (1)(t) Ψ(0) +c.c.
POINT SOLUTION 1 ∗ h 0 |H | 0 i
t
= i dτ (1)(t), (1)(τ) , (43)
− H H 0
Our goalis to determine the actionof the fluctuations Z (cid:10)h i(cid:11)
beyondthesaddlepointwithintheharmonicapproxima-
where, hereafter, ... will denote average over Ψ(0) ,
tion. To that purpose we assume that h i0 | 0 i
andtheoperatorsinEq.(43)haveanadditionaltimede-
Φˆ(i,t)=e−iE0t φ (i,t)Φˆ (i)Wˆ(i,t)†, (36) pendence since are evolved with the saddle point Hamil-
n n
tonian (0). The explicit expression of (1)(t) is
n ∗
X H H
where φ (i,t) for n > 0 is regarded as a first order fluc-
n
tuation, while, to enforce normalisation, (1)(t)= d†Rˆ(0)(j)†tˆ Rˆ(1)(i,t)d +H.c. ,(44)
H " j ji i #
φ (i,t)=1 1 φ (i,t)2 . (37) Xi6=j
0 n
− 2 n>0 | | where Rˆ(0)(i) is the stationary value, while the explicit
X
expression of the first order Taylor expansion Rˆ(1)(i,t)
In addition, the Slater determinant is defined through
is given in Appendix A1, see Eq. (A17).
Ψ (t) e−iE∗t (t) Ψ (t) , (38)
0 0
| i→ W | i
6
There are several second order terms upon expanding fluctuation correctionsto linear response functions of lo-
(t), which we shall consider separately. The first is caloperators. ForanylocalobservableOˆ(i),letusdefine
∗
H
simply the matrix element
H1(2)(t)= d†i Rˆ(1)(i,t)†tˆijRˆ(1)(j,t)dj, (45) On(i)≡Tr Φˆn(i)†Oˆ(i)Φˆ0(i) . (54)
Xi6=j (cid:16) (cid:17)
Suppose we add a perturbation that couples to the local
whose expectation value over |Ψ(00)i is an additional sec- density matrix
ond order contribution
δ (t)= c†Vˆ(i,t)c , (55)
δ2E∗(t)= H1(2)(t) 0, (46) H i i i
X
which, together with δ1E∗((cid:10)t) in Eq.(cid:11)(43), endow the ac- where the matrix Vˆ(i,t) with elements V (i,t) repre-
αβ
tion with spatial correlations among the φ (i,t)’s at dif-
n sentstheexternalfield. Withoutlossofgeneralitywecan
ferent sites.
assume that the expectation value of δ (t) in Eq. (55)
The next second order corrections to (t) derive from H
H∗ vanishes atthe stationarysolution. Since by assumption
the second order expansion of Rˆ(i,t) the external field is first order, the perturbation adds a
second order correction to the action (53) that is
Rˆ(2)(i,t)=Rˆ(2)(i,t)+Rˆ(2)(i,t), (47)
1 2
where we distinguish two different contributions, see V(t)= φ (i,t)∗Tr Φˆ (i)†ˆc†Vˆ(i,t)cˆ Φˆ (i)
n n i i 0
equations (A19) and (A20) in Appendix A1. The rea- "
son of this distinction is that Xi Xn (cid:16) (cid:17)
δ E (t)= φ (i,t)∗V (i)φ (i,t) +φn(i,t)Tr Φˆ0(i)†cˆ†iVˆ(i,t)cˆiΦˆn(i) (56)
3 ∗ n nm m #
i nm (cid:16) (cid:17)
X X
+hXi6=j (cid:16)d†jRˆ(0)(j)tˆjiRˆ1(2)(i,t)di +H.c.(cid:17)i0 ≡Xi Xn (cid:16)φn(i,t)∗Vn(i,t)+φn(i,t)Vn(i,t)∗(cid:17)
= E E φ (i,t)∗φ (i,t), (48) =√2 eVn(i,t)xn(i,t)+ mVn(i,t)pn(i,t) .
n 0 n n ℜ ℑ
− in (cid:18) (cid:19)
nX>0 (cid:16) (cid:17) X
InthepresenceofV(t)theactiontransformsintothatof
reproducesthebareexcitationenergyofthefluctuations.
forced harmonic oscillators,whose solution allows calcu-
The last contributionto the energy of the fluctuations is
lating the expectation value of any local operator Oˆ(i),
therefore
see Eq. (54),
δ E (t)= d†Rˆ(0)(j)tˆ Rˆ(2)(i,t)d +H.c. . (49)
4 ∗ hXi6=j(cid:16) j ji 2 i (cid:17)i0 O(i,t)=Tr Φˆ(i,t)†Oˆ(i)Φˆ(i,t)
(cid:16) (cid:17)
If we define new variables √2 eO (i)x (i,t)+ mO (i)p (i,t) ,
n n n n
≃ ℜ ℑ
x (i,t)= 1 φ (i,t)+φ (i,t)∗ , (50) Xn (cid:18) (cid:19)
n n n
√2 at linear order in the external field.
(cid:16) (cid:17)
i
p (i,t)= φ (i,t) φ (i,t)∗ , (51)
n n n
− √2 −
(cid:16) (cid:17) A. Residual gauge invariance and would-be
and the quadratic potential
Goldstone modes
U t, x,p =δ E (t)+δ E (t)+δ E (t), (52)
1 ∗ 2 ∗ 4 ∗
{ } Aswementioned,theactionEq.(40),withthetimede-
which(cid:0)has a re(cid:1)tarded component δ E (t), see Eq. (43), pendentquasiparticleHamiltoniandefinedinEq.(41),is
1 ∗
the action of the fluctuations reads, upon defining ω = invariantunderasubgroupGoftheinitialU(2M)gauge
n
E E , symmetry. This implies the existence of massless modes
n 0
− withsingularpropagatorsthatdivergeas1/ω2atlowfre-
quency,whicharethewould-beGoldstonemodesrelated
δ = dt p (i,t)x˙ (i,t) (53)
S Z ( i n>0 (cid:20) n n to the fact that the saddle-point Φˆ0(i) is not invariant
X X under G. Let us consider for instance a U(1) subgroup
ωn x (i,t)2+p (i,t)2 U t, x,p , ofGrelatedtothe non-degeneratestateαinthe natural
− 2 n n − { } ) basis. The associated adjoint charge is
(cid:16) (cid:17)(cid:21) (cid:0) (cid:1)
which is just the action of coupled harmonic oscillators. n (i,t) φ (i,t)∗Tr Φˆ (i)†Φˆ (i)dˆ† dˆ +c.c. ,
δS in Eq. (53) can be for instance used to evaluate the α ≃ n n 0 iα iα
nX>0(cid:16) (cid:16) (cid:17) (cid:17)
7
and its conjugate variable is readily found to be calculation to be really variational.
The localbasis comprises four states which we choose to
ϕ (i,t) i φ (i,t)∗Tr Φˆ (i)†Φˆ (i)dˆ† dˆ be, in order, the empty configuration, 0 , the doubly
α ≃ 2n(0) n n 0 iα iα occupied one, 2 , the singly occupied by|aispin up elec-
α nX>0(cid:18) (cid:16) (cid:17) tron, ,andt|haitoccupiedbyaspindownone, . The
φ (i,t)Tr Φˆ (i)†Φˆ (i)dˆ† dˆ . mostg|↑eineralcharge-conservingΦˆ hasthefollowin|↓giform,
− n 0 n iα iα
(cid:16) (cid:17)(cid:19) dropping for the meanwhile the site index,
The role of ϕ (i,t) is just to enforce the constraint
α 1 Φˆ 0
Eq. (11), i.e. Φˆ = c , (58)
√2 (cid:18) 0 Φˆs(cid:19)
n (i,t)= Ψ (t) c† c Ψ (t) c† c .
α h 0 | iα iα | 0 i≡h iα iα it where the chargecomponent, i.e. the matrix elements in
Indeedwecanalwaysperformagaugetransformationon the subspace 0 , 2 , is
| i | i
the fermions
(cid:0) (cid:1)
φ +φ 0
Φˆ = c0 c3 =φ σ +φ σ , (59)
c e−iϕα(i,t) c , c 0 φ φ c0 0 c3 3
iα → iα (cid:18) c0− c3(cid:19)
whichmakesϕα(i,t)todisappearfromtheenergyleaving withσ0 the2 2identitymatrix,andσi,i=1,...,3the
×
just the time derivative term in the action, Paulimatrices, whereasthe spin component, namely the
matrix elements in the subspace , , is instead
δ = dtϕ˙ (i,t) n (i,t) c† c . |↑i |↓i
S − α α −h iα iα it (cid:0) (cid:1)
Z (cid:16) (cid:17) Φˆ = 3 φ σ =φ σ +φ σ, (60)
The condition of vanishing derivative with respect to s si i s0 0 s·
ϕα(i,t)isthereforejusttheconditionthattheconstraint Xi=0
is conserved.
which allows a full spin-SU(2) invariant analysis26,41.
It follows that we can always drop from the action all
Normalisation implies that
termsthatcontainthe variablesconjugateto the adjoint
charges associated with the gauge symmetry G, on pro-
1= φ 2+ φ 2+ φ 2+φ∗ φ .
visionthat, wherevernα(i,t) appears,we replace it with c0 c3 s0 s · s
hc†iαciα it. One can read(cid:12)(cid:12)ily v(cid:12)(cid:12)erify(cid:12)(cid:12) tha(cid:12)(cid:12)t th(cid:12)(cid:12)e m(cid:12)(cid:12)atrix Qˆ with compo-
However,theaboveproceduredoesnotinvolvealltheco-
nents
efficients φ (i,t); some of their linear combinations are
n
untouched by gauge-fixing and remain genuine indepen-
dent dynamical degrees of freedom38. This fact, rather Qσσ′ =Tr Φˆ†cσΦˆc†σ′ , (61)
thanbeingalimitation,itendowsthetheorywitharicher (cid:16) (cid:17)
can be written as
dynamics.
Qˆ =Q σ +Q σ, (62)
0 0
·
IV. APPLICATION TO THE HALF-FILLED
where
HUBBARD MODEL
2Q = φ∗ φ +φ∗ φ + φ∗ φ φ∗ φ , (63)
We now apply the above formalism to the simple case 0 c0 s0 s0 c0 c3 s0− s0 c3
of a single band Hubbard model at half-filling, where 2Q =(cid:16)φ∗ φ φ∗ φ (cid:17)+ (cid:16)φ∗ φ +φ∗ φ (cid:17), (64)
all calculations can be worked out analytically and i c0 si− si c0 c3 si si c3
which also allows for a direct comparison with previous (cid:16) (cid:17) (cid:16) (cid:17)
with i=1,...,3. Seemingly,
works23–28,38–40. We will show that we can indeed
recover known results, but also find few novel ones.
∆ˆ φ∗ φ +φ∗ φ σ
≡ c0 c3 c3 c0 0
The Hamiltonian is in this case (cid:16) φ∗ φ +φ φ∗(cid:17)+iφ∗ φ σ
− s0 s s0 s s∧ s ·
t
= c† c +H.c. (cid:16)∆ σ +∆ σ. (cid:17) (65)
H − √z iσ jσ ≡ 0 0 ·
<Xij>σ (cid:16) (cid:17)
U 2
+ 2 ni 1 1 , (57) A. Stationary solution
4 − −
Xi (cid:20) (cid:16) (cid:17) (cid:21)
where < ij > means nearest neighbour bonds on a d- AscommonwhendiscussingtheMotttransitioninthe
dimensionalhyper cubic lattice, andz =2dis the lattice singlebandHubbardmodel,weshallbeinterestedinthe
coordination number that must be sent to + for the stationary solution within the paramagnetic sector, i.e.
∞
8
neglecting spontaneous breakdown of spin SU(2) sym- U isthe wellknownvalue ofthe Brinkman-Rice5 metal-
c
metry. Such solution at half-filling is characterised by a insulatortransitionwithintheGutzwillerapproximation.
site independent In conclusion, the lowest energy eigenstate is
θ
Φˆ0 = √12 φ(c000)σ0 φ(s000)σ0! , Φˆ0 = √12 sin 02 σ0 cos 0θ2 σ0! , (71)
where cosθ =min(1,U/U ), and has eigenvalue
with c
U Max(U,U )
1= φ(0) 2+ φ(0) 2. E0 = = c . (72)
c0 s0 − 4cosθ − 4
(cid:12) (cid:12) (cid:12) (cid:12)
Under this assumption(cid:12) (cid:12) (cid:12) (cid:12)
We can now find all other eigenvalues and eigenvectors.
The highest energy one is
Rˆ(i)= φ(0)∗φ(0)+φ(0)∗φ(0) σ =R(0)σ , i, (66)
c0 s0 s0 c0 0 0 ∀ θ
(cid:16) (cid:17) Φˆ = 1 cos 2 σ0 0 , (73)
sboindthinagt mthoedeqluwaistihparretnicolremHalaismeidlthoonpiapningis, ij.ues.t a tight- 3 √2 0 −sin 2θ σ0!
with eigenvalue
t
(0) = R(0)2 c†c +H.c. , (67)
H∗ − √z i j E3 = E0. (74)
<Xij> (cid:16) (cid:17) −
This eigenstate actually corresponds to the high energy
andnaturalandoriginalorbitalscoincide. Itfollowsthat Hubbard bands.
the stationary Slater determinant is the non-interacting Thelowestexcitedeigenstateisthreefolddegenerate(i=
Fermi sea. We define 1,2,3)
t
−Xi T0 ≡− √z <Xij>σhc†iσcjσ +H.c.i0, Φˆ1i = √12 (cid:18)00 σ0i(cid:19) , (75)
where ... is the average over the Fermi sea. There- with eigenvalue
0
h i
fore T is the hopping energy per site, and 2T /z the
− 0 − 0 U
hopping energy per bond of the Fermi sea. E = , (76)
1
The saddle point equations for Φˆ can be readily found − 4
0
and describes spin fluctuations. We note that above the
U
Eφ(0) = 2T R(0)φ + φ Brinkmann-Rice transition, U > Uc, this magnetic state
c0 − 0 s0 4 c0 becomes degenerate with the ground state. In what fol-
Eφ(0) = 2T R(0)φ U φ . lows we shall anyway expand always around Φˆ0, and, to
s0 − 0 c0− 4 s0 avoidproblems, we will mostly consider the metal phase
at U U .
The lowest energy eigenvalue is ≤ c
Finally, the last eigenstate is
1 2 1 σ 0
E = U2+ 8T R(0) , (68) Φˆ = 3 , (77)
0 − 2 r 0 2 √2 (cid:18)0 0(cid:19)
(cid:16) (cid:17)
and is characterisedby with eigenvalue
U
φ(c00) =sin θ2 , φ(s00) =cos 2θ , E2 =+ 4 , (78)
and describes instead charge fluctuations. This mode
with tanθ =8T0R(0)/U. Since through Eq. (66) R(0) = becomes degenerate with Φˆ3 above the transition.
sinθ, the self-consistency condition implies
8T R(0) 8T
0 0
tanθ = = sinθ, (69)
U U B. Action of the fluctuations
namely
Following section III we write
3
U/U U U =8T ,
cosθ = c ≤ c 0 (70) Φˆ(i,t)=φ0(i,t)Φˆ0+ φ1i(i,t)Φˆ1i
(1 U >Uc. i=1
X
9
3 θ −1
+ φn(i,t)Φˆn, (79) −√2 2sin 2 ∇·Jc(i)p2(i,t)
n=2 (cid:18) (cid:19)
X
with φ0(i,t) fixed by normalisation. Through equations +2√2 cotθh∗(i)x3(i,t) , (81)
(62), (63) and (64) we find that )
where ∇ is the lattice divergence, J (i) and J (i) the
θ s c
Rˆ(1)(i,t)=sin φ (i,t) φ (i,t)∗ σ spin and charge currents, respectively, defined through
2 1 − 1 ·
the continuity equations
θ(cid:16) (cid:17)
cos φ (i,t) φ (i,t)∗
− 2 2 − 2 i ∂ c†σ c = c†σ c , (0) i∇ J (i), (82)
+cosθ φ(cid:16)(i,t)+φ (i,t)∗ (cid:17) ∂t i 0 i i 0 i H∗ ≡− · c
3 3 ∂ (cid:16) (cid:17) h i
i√2 sin(cid:16) θ p (i,t) σ i(cid:17)√2 cos θ p (i,t) i ∂t c†i σci = c†iσci, H∗(0) ≡−i∇·Js(i). (83)
≡ 2 1 · − 2 2 (cid:16) (cid:17) h i
and finally h (i) the Hamiltonian density
+√2 cosθx (i,t), (80) ∗
3
t
where we have introduced the conjugate variables asso- h∗(i)=− 2√z R(0)2 c†icj +H.c. . (84)
ciated with φn and φ∗n. Eq. (44) reads explicitly jXn.n.i (cid:16) (cid:17)
θ −1 Therefore δ1E∗(t) defined in Eq. (43) becomes, due to
(1) = √2 2cos ∇ J (i) p (i,t) particle-hole and spin SU(2) symmetry
∗ s 1
H i ( (cid:18) 2 (cid:19) · ·
X
1 1
δ E (t)= dτ χ (i j,t τ)p (i,t) p (j,τ)+ χ (i j,t τ)p (i,t)p (j,τ)
1 ∗ ( 1+cosθ ∇J∇J − − 1 · 1 1 cosθ ∇J∇J − − 2 2
i,j Z −
X
+8 cot2θχ (i j,t τ)x (i,t)x (j,τ) , (85)
h∗h∗ − − 3 3 )
where χ is the linear response function of ∇J with and by definition
∇J∇J
the Hamiltonian (0), which is actually the same for
∗
H
chargeandspincurrents,andχ theresponsefunction
h∗h∗
of h∗. We observe that, because of charge and spin con- 2 d
tinuity equations, in Fourier space the following equiva- γq = cosqi [ 1,+1]. (87)
z ∈ −
lence holds i=1
X
2T sin2θ γ γ +χ (q,ω)=ω2χ(q,ω), (86)
0 0− q ∇J∇J
(cid:16) (cid:17) Without going into further details, we find that the
where χ(q,ω) is the density-density response function, following expressions for the remaining contributions
which is the same both in the charge and spin channels, δ E (t) in Eq. (46), and δ E (t) in Eq. (49):
2 ∗ 4 ∗
4T θ θ
δ E (t)= 0 sin2 p (i,t) p (j,t)+cos2 p (i,t)p (j,t)+cos2θx (i,t)x (j,t) , (88)
2 ∗ 1 1 2 2 3 3
− z ( 2 · 2 )
<ij>
X
θ θ
δ E (t)= 2T sin2θ cos2 x (i,t) x (i,t)+sin2 x (i,t)2 . (89)
4 ∗ 0 1 1 2
− ( 2 · 2 )
i
X
We have now all ingredients required to evaluate linear C. Hubbard-band dispersion mode
responsefunctionsoflocaloperatorswithintheharmonic
approximation for the fluctuations. As we mentioned, the Hubbard bands may be associ-
ated with the excited state Φˆ , hence with the operators
3
10
θ
x3 and p3. Their equations of motion in Fourier space iωp (q,ω)=√2 sin v(q,ω)
2
are − 2
2T 1+u u 2 u x(q,ω),
iωx (q,ω)=ω p (q,ω), (90) − 0 −
3 3 3
−
from which it follows that(cid:0)the dy(cid:1)na(cid:0)mical (cid:1)charge suscep-
iωp (q,ω)= ω 4T γ
− 3 − 3− 0 q tibility is
h
(1 u)χ(q,ω)
+8cot2θχh∗h∗(q,ω) x3(q,ω).(91) χc(q,ω)= −
(1 u) 2T 1+u u 2 u χ(q,ω)
0
i − − −
χ(q,ω)
Withinthemetalphase,U <Uc,ω3 =E3 E0 =4T0,so (cid:0), (cid:1) (cid:0) (cid:1) (94)
that, upon defining cosθ = U/Uc u, an−d noting that, ≡ 1+Γcχ(q,ω)
≡
for small q, χ (q,ω) = O(q4), the eigenmode energy
| | hh where it is evident the analogy with conventional RPA,
is solution of the equation
though with a renormalised coupling constant
ω2 =4T 4T 1 u2 +4T u2 γ γ U 1+u u
3q 0 0 − 0 0− q Γc = 1 <0. (95)
(cid:20) − 2 1 u − 2
(cid:0) (cid:1) u2 (cid:0) (cid:1) − (cid:16) (cid:17)
+8 χ (q,ω ) (92) We note that
1 u2 h∗h∗ 3q
− (cid:21)
χ(q 0,ω =0)= ,
∗
→ −N
16T2 1 u2 +u2 γ γ ,
≃ 0" − 0− q # where
(cid:0) (cid:1) (cid:0) (cid:1)
thus describes an optical mode that softens at the metal 0
= N , (96)
insulator transition, ω = 4T √1 u2 0 when N∗ 1 u2
30 0
− → −
u 1. We observe that the continuum of quasiparticle-
→ is the quasiparticle density of states (DOS) at the chem-
quasihole excitations extends up to an energy of order
T0 1 u2 , so that, upon approaching the transition, ical potential, as opposed to the bare DOS N0, and
− diverges approaching the Mott transition. Therefore,
ω must detach from the continuum and become a
3q
(cid:0) (cid:1) through Eq. (94), the charge compressibility is readily
genuine coherent excitation.
obtained
This coherent mode actually corresponds to the spin-
waveexcitationsoftheIsingfieldwithintheZ2slave-spin κ= N∗ N∗ ,
representation of the Hubbard model27,42,43. This is not 1−ΓcN∗ ≡ 1+F0S
surprising since, as shown in Ref. 27, the Gutzwiller
wavefunction is just the mean-field variational state of and defines the Landau FS parameter
0
the Z slave-spin theory. At the mean-field level, the
2 FS = Γ . (97)
Mott transition in this representation translates into 0 −N∗ c
the order-disorder transition of a quantum Ising model.
Since approachingthe transition, u 1, FS (1 u)−2
Therefore the mode x3 seems to be the real fingerprint diverges faster than (1 u)−→1, we0fin∼d th−at the
of the Mott transition. N∗ ∼ −
chargecompressibilitycorrectlyvanishesattheMIT.The
expression of FS coincides with that originally obtained
0
by Vollhardt23.
In the opposite limit of small q with respect to fre-
D. Dynamical charge susceptibility | |
quency,
We assume to perturb the system in the metal phase, 2T 1 u2 γ γ
0 0 q
u 1,byanexternalpotentialthatcouplestothecharge χ(q,ω) − − ,
de≤viation from half-filling, namely ≃ (cid:0) ω2(cid:1)(cid:0) (cid:1)
which,insertedintoEq.(94),allowscalculatingthepoles
δ (t)= v(i,t) ni 1 of the dynamical charge susceptibility, which are
H −
i
X√2 sin θ(cid:0) v(cid:1)(i,t)x (i,t). (93) ωc2q =4T02(1+u)2u(2−u) γ0−γq . (98)
2
≃− 2
i This acoustic mode is above the qua(cid:0)siparticle(cid:1)-quasihole
X
continuumandactuallycorrespondstotheLandau’szero
Since ω = E E = 2T 1 + u and by means of
2 2 − 0 0 sound. Once again this result is compatible with Voll-
Eq.(86),wefindinthepresenceofthefieldthefollowing
(cid:0) (cid:1) hardt’s description of the correlated metal within the
equationsofmotionfortheconjugatevariablesx andp
2 2 Gutzwiller approximation in the framework of Landau-
ω2 Fermiliquidtheory23. Indeedthezerosoundvelocityhas
iωx (q,ω)= χ(q,ω)p (q,ω),
2 2 the expected Landau’s expression, once one realises that
− 1 u
−