Limitations of spin-fermion models in studying underdoped cuprates

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by  Alvaro Ferraz

Limitations of spin-fermion models in studying underdoped cuprates
Alvaro Ferraz1, Evgenii Kochetov1,2
1International Institute of Physics - UFRN, Department of Experimental and Theoretical Physics - UFRN, Natal, Brazil and
2Bogoliubov Theoretical Laboratory, Joint Institute for Nuclear Research, 141980 Dubna, Russia
A microscopic basis is provided for the spin-fermion model used to describe the physics of the
underdoped cuprates. In this way, a spin-fermion coupling is shown to take care of the local no
double occupancy constraint, which is ignored in the weakly coupled regime. This ingredient is
proved to be essential to capture the physics of strong correlations, however. We elaborate further
onourrecent proposal for thestrong-coupling version of thespin-fermion model thatmay proveto
7
bethecorrectstartingpointtodealwiththestrongcorrelationphysicsdisplayedbytheunderdoped
1
cuprates.
0
2

n I. INTRODUCTION: SPIN-FERMION MODEL nature of the low-energy AF paramagnons singles out
a the so-called hotspots on the FS connected by Q~. Those
J
The purpose of this work is to establish a reliable mi- points are precisely the intersects of the large hole-like
3
1 croscopic basis for the spin-fermion (SF) model which FS with the magnetic zone boundary.
mightbeappropriatetodescribethephysicsofthelightly Due to the enormous enhancement of the AF fluctua-

] dopedcuprates. Sincethereareseveraldifferentversions tions at or near the quantum critical point (QCP), the
l
e of the SF model in use to describe the high-T supercon- fermions in the vicinity of the hotspots interact strongly
- c
r ductorsitisimportanttodiscussinitiallyhowtheydiffer with each other via singular AF paramagnons. This re-
st from each other. sults in strong damping of the propagating spin modes.
t. More than a decade ago the SF model was put for- The spin modes become dissipative and the fermions
a ward in an attempt to account for the observed non- quasiparticles acquire a short lifetime. As a result, the
m
Fermi liquid anomalies in the doped cuprates near op- electron system displays a singular non-FL infrared be-
d- timaldoping.[1]Thisoriginalmodelessentiallydescribes haviourdeclaredto be observedin the cuprates near op-
n low-energy fermions with a large Fermi surface (FS) in- timaldoping.[1]Inprinciple,theSFmodelisaFLtheory
o teracting with each other via soft collective spin excita- plagued with infrared singularities in an attempt to ac-
c tions, count for a non-FL low-energy physics of the optimally
[
doped cuprates.

1 sf = c+ λ+ φ, This SF model can be formally derived from the Hub-
L L L L
v bard model [1] which describes fermions hopping on a
8 where lattice with a tunneling amplitude t and are subjected
ij
0
to a short-range Coulomb repulsion U,
8 = c† (∂ µ)c t (c† c +c† c ),
3 Lc iα τ − iα− ij iα jα jα iα
0 Xi Xi<j HU =− tijα(c†iαcjα−µ)+U ni↑ni↓, (3)
1. Lλ = λ φaic†iασαaβciβ, Xi,j Xi
0 Xi
7 = χ−1φ~ φ~ . (1) where niα =c†iαciα is the on-site electron number opera-
1 Lφ 0 q~ −q~ tor. Followingstandardprocedurethequarticfermionin-
X
v: q teractioncanbedecoupledusingaHubbard-Stratonovich
Xi Here c†iα is the creation operator for electrons with spin field φ~ that physically corresponds to the collective spin
excitations which are "made out" of the fermion degrees
projection α = , on site i, the amplitudes t are hop-
r ↑ ↓ ij of freedom. This procedure can be justified only if the
a ping matrix elements describing the "large" Fermi sur-
face, µ is a chemical potential, and σa, a = x,y,z, are SF coupling λ ∼ U is smaller that the fermionic band-
width W t, where t is a nearest-neighbour hopping
the Pauli matrices. ∼
amplitude. This automatically brings the theory into a
Thebarespinpropagatoremergesfromtheintegration
weak-coupling regime controlled by a small parameter,
of the high-energy fermionic modes,
U/t.
ω2 The requirement U/t 1 implies that strong electron
χ0(q =~q,ωn)=(v2n +(~q−Q~)2+ξA−F2)−1, (2) correlationsdue a large≪on-site Coulombrepulsion U are
s
totally ignored. This may correspond to an overdoped
where ω = 2πn/β are Matsubara bosonic frequencies. regionof the cuprate phase diagramthat displays a con-
n
Theantiferromagnetic(AF)spincorrelationlength,ξ , ventional FL physics. With some modifications (the hot
AF
and the spin velocity v are high-energy phenomenolog- spots theory, etc.) being imposed, it exhibits a non-FL
s
ical parameters. At criticality (ξ = ), the singular behaviour that might be thought of as being relevant
AF
∞
2
to the optimally doped cuprates, provided the strange λ/t 1. Only under this condition, can the SF model
≪
metal phase has nothing to do with strong correlations. be justified as a low-energy approximation to the Hub-
However, this approach is inapplicable to lower dopings bard model. The physical meaning of the coupling λ in
in which strong correlations are known to be at work. theconventionalapproachisthatitrepresentsaspinden-
In particular, the underdoped pseudogap (PG) region is sity wave (SDW) gap. In contrast, in Eq.(4), λ should
characterized by strong electron correlations which keep be consideredas a coupling of ordert and it has nothing
the majority of the electrons welllocalized. Accordingly, to do with the SDW gap.
theFSconsistsofsmallseparatedpatches(electronpock- ii) The spin sector in the conventional SF model
ets) as observedin the quantum-oscillationexperiments. is described by a soft spin mode φ~ which appears as
The observation of quantum oscillations in the lightly a Hubbard-Stratonovich auxiliary field to linearize the
hole-doped cuprates is an important breakthrough since quartic fermion coupling. This mode is "made out" of
it indicates that coherent electronic quasiparticles may the fermionic degrees of freedom which necessarily im-
exist even in the PG regime. The PG state does not ex- plies that the spin velocity v is of order the relevant
s
hibitalargeFSenclosingatotalnumberofchargedcarri- Fermi velocity, v v ta, with a being the lattice
s F
∼ ∼
ers. Insteadthe FSconsistsofsmallpocketswithatotal spacing. In contrast, in the model (4), the spin sector is
areaproportionaltoadopantdensityx,ratherthan1+x described instead by the "hard-spin" field na that obeys
which is expected for conventional FL’s. There are two the local constraint (na)2 = 1. The spin-1 order pa-
a
routestoaFermisurfacereconstructionthatcouldapply rameter ~n can be spPlit into two spin-1/2 charge neutral
to the cuprates. A possible theoretical justification for spinons z :
α
this phenomenon might be the occurrence of a simulta-
neoussettingofanewlong-rangeordertogetherwiththe na =z¯iασαaβziβ, z¯iαziα =1.
PGphaseatacriticalholeconcentration,x 0.19. The
c
≈ The emergent gauge field, z z eiθi, glues them to-
resultingbreakdownoftranslationalsymmetrywouldcut iα iα
→
gether.
the large FS into small pieces but the Luttinger’s theo-
The compact U(1) gauge theory is at strong coupling
rem (LT) would still hold. This is exactly what the con-
(there is no Maxwell term in the action) and it is pre-
ventional SF model predicts. However, so far there are
sumably in a confining phase. If one assumes however
noobservationsofthe exhistanceofanyorderparameter
that there exists an energy window in which the de-
that breaks translation symmetry near x=x .
c
confinement does indeed take place, one can derive the
small-pocketFS within an effective FL∗ theory based on
II. FRACTIONALIZED FERMI-LIQUID MODEL the spinon decomposition of the order parameter and on
symmetry considerations.[2] It is not totally clear what
controls such a theory, however. For instance, the key
The fractionalized Fermi-liquid (FL∗) was proposed
equation (3.1) in [2] rewrites the conventional electron
phenomenologically as an alternative to describe lightly
on-siteoperatorc intermsofanew (unitary equivalent
doped antiferromagnets. It goes beyond the weak cou- α
) SU(2) spinor ψ :
pling regime and as a result it is able to produce small ±
Fermi pockets without translationalsymmetry breaking.
cα =zαψ+ ǫαβz¯βψ− Z(Fα+Gα), (5)
The Lagrangiandensity of this new model is [2] − ≈
whereZ issomeconstantfactor,ǫ isatotallyantisym-
αβ
L′sf =L′c+L′λ+L′n, metric tensor, ǫ↑↓ =1, and zα a set of complex numbers
subject to the condition z¯ z = 1. Although the
α α α
electron operator c by dePfinition transforms itself as an
′ = c† (∂ µ)c t (c† c +c† c ), α
Lc iα τ − iα− ij iα jα jα iα SU(2) spinor, the zαψ+ and ǫαβz¯βψ− operators taken
X X
i i<j apartfromeachotherdo nottransforminthe sameway.
′ = λ ( 1)xi+yinac† σa c , Nevertheless, a true single fermion operator splits into a
Lλ X − i iα αβ iβ sum of two independent fermion modes, F and G . It
i α α
1 is not obvious what controls the approximation made in
′ = d2r[(∂ na)2+v2(∂ na)2]. (4)
Ln 2g Z τ s r Eq.(5).
On the other hand, when an emergent gauge field is
Three phenomenological parameters enter into Eq.(4): indeed in the deconfined phase (e.g., 2 gauge field in
Z
the parameter λ represents a spin-fermion coupling, g 2+1) the spinons can appear in the physical spectrum.
controls the strength of the antiferromagnetic(AF) fluc- To get the gauge field deconfined one needs first to have
tuations, and v stands for the spin velocity. it gapped. This is a necessary condition to arrive at the
s
There are two basic distinctions between the conven- FL* phase. One route to achieve this is to apply a MF
tional SF model [1] and that given by (4). i) The con- theory that breaks U(1) down to 2 [3]. However, such
Z
ventional SF model is controlled by a small parameter a theory can be controlled only in the limit where the
3
spinexchangetermismuchlargerthanthehoppingterm Proceeding with Eq.(8) we end up with the lattice
which is not the case for the cuprates, however. Kondo-Heisenbergmodelwhich,atstrongcoupling(λ
≫
An alternative approach was recently proposed in [4]. t), is equivalent to the original t J model:
−
Roughlyspeakingitinvolvesadirectcouplingofaspinon
fifieelldd.inAthcerupchiaelnoremqeuniroelmogeincatlinSFthmisodapelp(r4o)actohaisZt2hagtauthgee Ht−J = Xijσ 2tijd†iσdjσ +JXij S~i(1−ndi)·S~j(1−ndj)
coupling λ is to be consideredas the largestenergyscale 3
+ λ (S~ s~ + nd), λ + . (9)
in the problem. The topological order and fractionaliza- i· i 4 i → ∞
X
i
tion appear in a regime inaccessible in a small λ expan-
sion. In spite of the global character of the parameter λ, it
enforces the NDO constraint locally due to the fact that
theon-sitephysicalHilbertsubspacecorrespondstozero
III. ITINERANT-LOCALIZED MODEL OF eigenvalues of the constraint, whereas the nonphysical
STRONG CORRELATIONS subspaceisspannedbytheeigenvectorswithstrictlypos-
itiveeigenvalues. Asweshowedelsewhere,in1d,Eq. (9)
In the present paper we show that the starting point reproduces the well-known exact results for the t J
−
employed in [2] – the phenomenological SF model (4) – model.[7]
can in fact be derived microscopically up to a certain The unphysical doubly occupied electron states are
λ-dependent extra term by employing the earlier estab- separatedfromthephysicalsectorbyanenergygap λ.
∼
lished mapping of the t J model of strongly correlated Intheλ + limit,i.e. inthelimitinwhichλismuch
− → ∞
electrons onto the Kondo-Heisenberg model at a domi- larger than any other existing energy scale in the prob-
nantly large Kondo coupling [5]. This enable us to clar- lem, those states are automatically excluded from the
ify the physical meaning of the coupling λ in (4) and, Hilbert space. On the other hand, in this limit, the high
in this way, to understand the possible physical limita- and low energy itinerant fermions cannot be separated
tions of the SF models in describing the physics of the out and this is another manifestation of the local Mott
underdoped cuprates. physics. In view of that it does not seem appropriate to
Let us start with the observation that, in the under- integrateoutthe high-energyfermions. We assumehow-
doped cuprates, one striking feature is the simultaneous ever that this separation still holds in the spin-dopon
existence of both localized and itinerant nature of the representation for well localized particles, i.e., the local-
lattice electrons. To include both aspects into consid- ized spin degrees of freedom. In this way we arrive [8]
eration on equal footing, Ribeiro and Wen proposed a ataneffectiveLagrangiantodescribestronglycorrelated
slave-particlespin-dopon representationof the projected electrons in the underdoped phase:
electron operators in the enlarged Hilbert space [6],
˜= ˜ + ˜ + ˜ ,
d λ n
L L L L
1 1
c˜† =c†(1 n )= ( 2S~ ~σ)d˜. (6)
i i − i−σ √2 2 − i· i 3λ
˜ = d† (∂ + µ)d
Ld iα τ 4 − iα
In this framework, the localized electron is represented Xi
by the lattice spin S~ su(2) whereas the doped hole + 2 t (d† d +d† d ),
∈ ij iα jα jα iα
(dopon) is described by the projected hole operator, X
i<j
d˜ =d (1 nd ). Herec˜† =(c˜†,c˜†)tandd˜=(d˜,d˜ )t.
iα iσ − i−α ↑ ↓ ↑ ↓ ˜ = λ ( 1)xi+yinad† σa d ,
In terms of the projected electron operators, the con- Lλ − i iα αβ iβ
X
i
straintofnodouble occupancy(NDO)takesonthe form
1
˜ = d2~r[(∂ na)2+v2(∂ na)2]. (10)
(c˜† c˜ )+c˜ c˜† =1. (7) Ln 2g Z τ s ~r
iα iα iα iα
Xα wherev =Jaisthespin-wavevelocity,andg =4Ja2/s.
s
In this model, the itinerant degrees of freedom are de-
It singlesout the physical 3don-site Hilbertspace. Only
scribed by the dopons whereas the localized ones – by
underthis conditionarethe projectedelectronoperators
the fluctuating AF order parameter, na. In the limit
isomorphic to the original Hubbard operators. Within
λ , the dopons and the lattice spins merge into the
the spin-dopon representation, the NDO constraint re-
→∞
gauge neutral objects (6) to represent the physical elec-
duces to a Kondo-type interaction,[5]
trons.
3 There are a few distinctions between ′ and ˜. To
S~i·s~i+ 4(d˜†i↑d˜i↑+d˜†i↓d˜i↓)=0, (8) startwith,themodel ˜isformulatedinteLrmsofthLehole
L
(dopon)operatorsd′s,ratherthantheelectronoperators
with~s = d˜† ~σ d˜ beingthedoponspinoperator. c′s. It might seem, at first sight, that the electron-like
i α,β iα αβ iβ
P
4
pockets should emerge in place of the hole-like pockets. Mottinsulatorisnotastablephaseinthephysicallywell
However, this is not the case, since in our model the defined regime of strong electron correlations in which
fermion dispersion changes sign, t t . Note also λ becomes boundless. This is hardly appropriate since
ij ij
→ −
that the renormalized hopping amplitude in ˜ contains those correlationsareinfact a keyingredientbehind the
L
thespin-fermioncouplingλ,asgivenbyEq.(10). Because Mott physics.
ofthis,inthephysicallimit,λ + ,themodelremains On the other hand, if we follow all the steps exposed
→ ∞
well defined and finite. Finally, the hopping amplitude in [4] to compute those quantities starting right from
in ˜ contains an important factor of 2 that reflects the the itinerant-localized model (10) we will obtain instead
L
fact that a vacancy is now represented as a spin-dopon E =∆ µ and E = µ+λ/2+2∆ λ2/4+4∆2.
G s
− − −
singlet (see, [5]). Bothexpressionsremainfinite, inthe largepλ limit. This
The physicalmeaning of the Kondo coupling λ is that occursdue to the factthat model(10) involvesthe extra
it takes care of the NDO constraint. As the quantum λ-dependent renormalization of the hopping term which
MonteCarlosimulationsofthemodel(9)explicitlyshow is missed in both and ′ SF models.
L L
[9], the NDO constraintplays the dominating role in the
destruction of the long-rangeAF state. The critical hole
concentrationis around xc =0.05 at J =0.2t. The spin- V. CONCLUSION
spincorrelationfunctions atfixeddopingbecome almost
identical to each other for λ 10t. This indicates that,
≥ To conclude, we show that a number of unsatisfac-
in principle, finite but large enough values of λ already
tory points that arisein an attempt to apply the SF-like
provide a reliable description of the existing strong cor-
modelstotreattheunderdopedcupratescanbesafelyre-
relations in the underdoped regime.
movedprovidedthe itinerant-localizedmodelofstrongly
correlatedelectronsisemployedintheirplace. Thisindi-
catesthatthismodelmaybethecorrectstartingpointto
IV. FURTHER COMPARISONS OF THE
dealwith the strongcorrelationphysics displayedby the
MODELS
underdoped cuprates. Within a framework of the quan-
tum Monte Carlo simulations, this model has recently
With this understanding of the physical meaning of
been shown to provide a few interesting new results. In
the Kondo coupling λ, let us compare the and ′ SF
models against the itinerant-localized ˜ moLdel. ALs the particular, the strong electron correlations are shown to
L play a key role in the abrupt destruction of the quasi-
conventional SF model (1) implies λ/t 1, it does not
≪ long-rangeAF order in the lightly doped regime.[9] This
capture strong correlations at all. Its phenomenological
modelwasalsoemployedtostudytheNagaoka(U = )
modification givenby (4) implies that λ t. This might ∞
∼ limit of the Hubbard model to demonstrate the absence
correspond to an optimally doped phase, provided the
ofthelong-rangeferromagneticorderingatfinitedoping.
extra λ-term in the hopping term is also taken into ac-
[10]
count. Infact,itsmagnitudeisofthesameorderasthat
Afurtherpossibleapplicationoftheitinerant-localized
of the t-term. In the 2 gauge theory based on ′ and
Z L model might be to theoretically explore an experimen-
described in detail in [4], the crucial statement is that
tally observed instability towards a d-wave formation of
the λ is the largestenergy scale. This, by definition, im-
achargeorderinthePGphase. Thischargeorderseems
plies that the theory must remain finite and well defined
to compete directly with the d-wave superconducting
at λ=+ , which physically corresponds to a regime of
∞ regime. There is a strong evidence that the observed
strong electron correlations.
charge order is due to strong electron correlations. This
There is an analogy with that and the conventional
work is already in progress and results will be presented
large U approximation to the Hubbard model referred
elsewhere.
to as the t J model. Namely, the Hubbard model re-
−
duces at U = to a well defined model that describes
∞
the so-called Nagaoka phase of strongly correlated elec-
trons. Theground-stateenergyisfinite andindependent
ofU inthelarge-U limit. Incontrast,thestatesobtained [1] Ar. Abanov, A.V. Chubukov, J. Schmalian, Adv. Phys.
in [4] in the leading in λ approximation do not exhibit 52, 119 (2003).
such a behaviour. For example, the ground state of the [2] Y. Qi, and S. Sachdev,Phys.Rev.B 81, 115129 (2010).
Mott insulator (x = 0) reads E = λ+∆ µ. The [3] M. Punk, and S. Sachdev, Phys. Rev. B 85, 195123
G
− − (2012).
energy of the spinon excitation above the Mott insula-
[4] S. Sachdev, E. Berg, S. Chaterjee, and Y. Schattner,
tor state takes the form E = µ+2∆ √λ2+4∆2.
s − − Phys. Rev.B 94, 115147 (2016).
Here µ and ∆ are certain asymptotically λ-independent
[5] A. Ferraz, E. Kochetov, and B. Uchoa, Phys. Rev. Lett.
parameters. In particular, the FL∗ state is favoured in 98, 069701 (2007); R.C. Pepino, A. Ferraz, and E. Ko-
the limit λ ∆ t J . [4] This implies that the chetov, Phys.Rev.B 77, 035130 (2008).
≫ ≫ | | ≫ | |
5
[6] T.C. Ribeiro, and X.-G. Wen, Phys. Rev. Lett. 95, (2015).
057001(2005);T.C.Ribeiro,andX.-G.Wen,Phys.Rev. [9] I. Ivantsov, A. Ferraz, and E. Kochetov, Phys. Rev. B
B 74, 155113 (2006). 94, 235118 (2016).
[7] A. Ferraz, and E. Kochetov, Eur. Phys. J. B 86, 512 [10] I. Ivantsov, A. Ferraz, and E. Kochetov, arXiv:
(2013). 1612.03646 [cond-mat. str-el].
[8] A. Ferraz and E. Kochetov, Europhys. Lett. 109, 37003