Table Of ContentReply to the Reply to the Comment on “Z -slave-spin theory for strongly correlated
2
fermions”
Alvaro Ferraz1, Evgeny Kochetov1,2
1International Institute of Physics - UFRN, Department of Experimental and Theoretical Physics - UFRN, Natal, Brazil and
2Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
The authors of the Reply [1] to our Comment[2] claim mutes with H , the constraint Q =0 must now be
approx i
that the equation enforced explicitly at any given time instant. This can
3
be done byaddingto the HamiltonianaglobalLagrange
01 A˙i|physi=i[Ai,H]|physi=DiAi|physi=0 (1) multiplier in the form λPiQi. Since the eigenvalues
2 of the on-site projection operator Qi are either 0 or 1,
insures that the on-site operator constraint A = 0 se-
i the limit λ → +∞ singles out the physical subspace.[3]
n
lects the physical subspace consistently with the time
a Only after this is done, can one safely use the equation
evolution generated by the Hamiltonian H. Here D is
J i A = 0. Within a mean-field (MF) approximation, one
i
some nonlocal operator constructed out of the spin and
8 may however replace the operator equation Qi =0 with
1 fermion operators. This is indeed the case since an ex- its ground-state expectation value, hQ i= 0. In spite of
i
plicit solution of Eq.(1) reads Ai(t) = exp(tDi)Ai. As thatit stilleliminates the unphysicalstates. In contrast,
] A |physi=0,onegetsA (t)|physi=0atanygiventime
l i i the constraint expectation value hAii can be nullified by
e instant.
the unphysical subspace as well. [2] Consequently, a MF
-
r Nocalculationsareinfactneededtoseethis,sincethe theory based solely on the condition hA i = 0 becomes
st equation Ai(t)|physi = e−itHAieitH|physi = 0 follows inconsistentsinceuncontrolledunphysicalidegreesoffree-
. simplyfromtheobservationthattheHamiltonianH does
t dom contribute to the theory as well.
a not couple the physical and unphysical subspaces. This
m
occursbecauseofthefactthatitcommuteswiththepro-
- jection operator Q :=A2 which has eigenvalues 0 and 1 To conclude, Reply [1] does not refute the criticism
d i i
n corresponding to the physical and unphysical subspaces, exposed in our Comment [2].
o respectively. In case the dynamics never takes the state
c outside the physical subspace, the requirements A = 0
i
[
and Qi =0 are truly equivalent. [1] A. Rüegg, S.D. Huber, and M. Sigrist, Phys. Rev. B 87,
1 However, as soon as any approximate treatment that
1037102 (2013).
v couples the physical and unphysical spaces is applied, [2] A. Ferraz and E. Kochetov, Phys. Rev. B 87, 1037101
8
those requirements are no longer identical. In this case (2013).
0
5 Qi =0remainstheonlyconstraintavailabletoselectthe [3] ThistrickcannotbeusedtoimposetherequirementAi=
4 physicalsubspace. Since the operatorQi no longer com- 0, since the operator Ai has an eigenvalue −1.
.
1
0
3
1
:
v
i
X
r
a
Comment on "Z -slave-spin theory for strongly correlated fermions"
2
Alvaro Ferraz1, Evgeny Kochetov1,2
1International Institute of Physics - UFRN, Department of Experimental and Theoretical Physics - UFRN, Natal, Brazil and
2Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
PACSnumbers: 71.27.+a,71.10.-w,71.30.+h
The authors of ref.[1] introduce a new slave-spin (SS) which appears as a necessary step to recover the trans-
3 mean-field (MF) approach to the Hubbard model of verse field Ising model representation.
1 strongly correlated electrons in terms of Ising spins and The projection of the Hubbard model
0 standardfermion operators,by constructing an enlarged
2 U 1
n HThilebeprutrpspoaseceofwtihtihscboontshtrpuhcytisoicnailsacnledaru:ntphhisysfoicramlusltaattieosn. H′ =−XtijIixIjxfi†σfjσ + 2 X(Iiz + 2) (6)
ijσ i
a
involvesamuchsmallernumberofslavefields,compared
J
with more standard slave particle frameworks. In this onto the physical subspace is equivalent to the original
8
way the complexity of the originalmodel is reduced to a model (1). Such a projection can be achieved explicitly
1
more readily tractable system of Ising spins in a trans- by imposing the requirement Qi := A2i = 1/2+Iiz[1−
] versefieldcoupledtofreefermions. InthisComment,we 2(ni−1)2] = 0, for each lattice site. The projection op-
l
e showthattheunphysicalstatesarenotproperlyexcluded eratorQi commutes with Hamiltonian(1) and generates
- and despite the claimed good agreement with previous a U(1) local gauge symmetry which takes care of the
r
t results,the consistencyoftheirMFtheoryisunjustified. redundancy of the decomposition (2). However, notice
s
. The paper [1] may be considered as an extension and that the operator constraints Ai =0 and Qi =0 are not
t
a animprovementoftheformalismdevelopedearlierin[2]. equivalent to each other.[7]
m It can also be thought of as a minimal formulation of The,sofar,exactSSformulationisthentreatedatMF
- previous slave-spin representations. [3] levelbyfullydecouplingtheSSandtheauxiliaryfermion
nd Thesingle-bandHubbardmodeliswritteninthe form operators in the form H′ →HMF =Hf +HI, where
o U
c H =−Xtijc†iσcjσ + 2 X(ni−1)2. (1) Hf = −Xgijtijfi†σfjσ,
[
ijσ i ijσ
v1 Tonhseitehorpeppuinlsgioanm. pTlhiteudoepeirsatdoernco(t†e)dcrbeyatteisjaanndeleUctriosnthaet HI = −XχijtijIixIjx+ U2 XIiz. (7)
8 iσ ij i
0 site i with spin σ, and ni =Pσc†iσciσ.
Here the hopping amplitude andthe Isingexchangecou-
5 The authorsintroduce auxiliarylocalpseudospinvari-
4 ables I~ =(Ix,Iy,Iz) with eigenstates Iz|±i =±1|±i . pling are renormalized by the factors gij and χij, re-
. i i i i i i 2 i spectively. These are to be determined through the self-
1 The physical electron operator is then represented as
0 consistency equations
3 c(†) =2Ixf(†), (2)
iσ i iσ
:1 wheref(†) isanauxiliaryfermionoperatordefinedonthe gij =4hIixIjxiHI, χij =4X(hfi†σfjσiHf +c.c.). (8)
v iσ σ
space spanned by the vectors |0i , |↑i , |↓i , |↑↓i . [4]
i i i i i
X The SS representation (2) expands the Hilbert space. To proceed, the authors replace the constraint Ai =0
r Theauthorsidentifytheon-sitephysicalHilbertspaceto by its ground state expectation value, ignoring com-
a be spanned by the states pletely the condition Qi = 0. In the enlarged
Hilbert space, the Q operator is a projection matrix,
i
H ={|+i|0i, |−i|↑i, |−i|↓i, |+i|↑↓i}. (3)
phys Q |H i = 0, Q |H i = |H i . Because of
i phys i i unphys i unphys i
The unphysical states are then this,thelocalMFconstrainthQii=0stilleliminatesthe
unphysical states. In contrast, the same condition does
H ={|−i|0i, |+i|↑i, |+i|↓i, |−i|↑↓i}. (4)
unphys not hold for the constraint expectation value hA i that
i
To obtain a faithful representation of the original prob- can be nullified by the unphysical subspace as well. To
lem,theauthorsclaimtosingleoutthephysicalsubspace see this, let us pick up a state
by the local constraint A := Iz +1/2−(n −1)2 = 0.
This equation justifies thei substiitution in (1i) |Φi=c(i1)|−ii|0ii+c(i2)|−ii|↑↓ii
U U
2 X(ni−1)2 → 2 X(Iiz +1/2), (5) +c(3)|+i |↑i +c(4)|+i |↓i ∈H ,
i i i i i i i i unphys
2
where |c(1)|2 +|c(2)|2 = |c(3)|2 +|c(4)|2. It then follows [2] S.D. Huber and A. Rüegg, Phys. Rev. Lett. 102, 065301
i i i i
thathΦ|A |Φi=0. Consequently,theconditionhA i=0 (2009).
i i
no longerdiscriminates between physicaland unphysical [3] L.de’Medici, A.Georges, andS.Biermann,Phys.Rev.B
72, 205124 (2005); S.R. Hassan and L. de’Medici, Phys.
states (as opposed to the requirement hQ i = 0), and
i Rev.B81,035106 (2010); R.YuandQ.Si,Phys.Rev.B
the resulting SS MF theory becomes inconsistent. The 84, 235115 (2011).
substitution (5) now implies that the uncontrolled un- [4] In the limit U ≫ t, the physical electron operators ad-
physical degrees of freedom contribute to the theory as mitrepresentationsintermsofthesu(2)spinsandspinful
well. fermions[5].Withinthatapproach,theunderdopedHub-
The authorsrefer to their MF approachas a Z gauge bardmodelreducestotheKondo-Heisenbergproblem[6].
2
theory. They claim that the local Z transformations, [5] T.C.RibeiroandX.-G.Wen,Phys.Rev.Lett.95,057001
2
f → u f u = −f , Ix → u Ixu = −Ix, u = 1−2Q , (2005);A.Ferraz,E.Kochetov,andB.Uchoa,Phys.Rev.
i i i i i i i i i i i i Lett. 98, 069701 (2007).
respecttheMFdecomposition(7)and"inthissense,the
[6] R. T. Pepino, A. Ferraz, and E. Kochetov, Phys. Rev. B
mean-field ansatz breaks the U(1) gauge theory down to 77, 035130 (2008).
Z2".[1] This statement may cause some confusion. [8] It [7] TheoperatorAi doesnotcommutewith theHamiltonian
is of course legitimate to restrict the space of variational (1). The consistency of the constraint Ai = 0 with time
states to a subspace generated by the states related by evolution requires that [Ai,H]H =0. The consistency
phys
the u transformations. However, this does not result of this secondary constraint implies in turn that Qi =
i
in a Z invariant MF theory: the gauge-related varia- 0. Therefore the constraint Ai = 0 can be imposed self-
2 consistently only if therequirement Qi=0 holds.
tional states still belong to the different Hamiltonians,
[8] Tostartwith,thereisaconfusionwiththeverydefinition
HMF and uiHMFui 6=HMF. In fact, a Z2 gauge invari- ofgaugeinvariance.TheauthorsclaimthataZ2gaugein-
antMFtheoryonlyarisesifoneconsidersbothfunctions variant operator O obeysthecondition uiOui =O.Since
χij and gij as Hubbard-Stratanovich dynamical fields, u2i =1, this implies that [O,Qi]=0. In other words, the
χij = χ¯ijσij, gij = g¯ijσij. Here χ¯ij and g¯ij are precisely discrete Z2 symmetry is automatically promoted back to
the solutions to the saddle point Eqs.(8), and σ = ±1 the continuous U(1) local gauge symmetry generated by
ij
is an Ising dynamical gauge field that takes care of the Qi. In contrast, the conservation of a genuine Z2 parity
Z gauge fluctuations beyond the saddle point approxi- operator implies the conservation of a Z2 charge modulo
2 2.
mation.
[1] A. Rüegg, S.D. Huber, and M. Sigrist, Phys. Rev. B 81,
155118 (2010).
