Hidden Fermi Liquid: Self-Consistent Theory for the Normal State of High-Tc Superconductors

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by  Philip A. Casey

Hidden Fermi Liquid: Self-Consistent Theory for
the Normal State of High-T Superconductors
c
Philip A. Casey and Philip W. Anderson
Princeton University, Department of Physics, Princeton, New Jersey 08544, USA
(Dated: January 20, 2011)
Hidden Fermi liquid theory explicitly accounts for the effects of Gutzwiller projection in the t-J
Hamiltonian, widely believed to contain the essential physics of the high-Tc superconductors. We
1 deriveexpressionsfortheentire“strangemetal”,normalstaterelatingangle-resolvedphotoemission,
1 resistivity, Hall angle, and by generalizing the formalism to include the Fermi surface topology -
0 angle-dependent magnetoresistance. We show this theory to be the first self-consistent description
2 for the normal state of the cuprates based on transparent, fundamental assumptions. Our well-
definedformalism also serves as a guide for furtherexperimental confirmation.
n
a
J
The anomalous “strange metal” properties of the nor- of v and k around the Fermi surface within HFL.
9 F F
1 mal, non-superconducting state of the high-Tc cuprate All data is described using only one main assumption:
superconductors have been extensively studied for over that the Hubbard energy, U, of excitations into doubly

] two decades.[1–6] The resistivity is robustly T-linear at occupied Wannier orbitals on a particular site is large.
n
high temperatures while at low T it appears linear near Antibound states, representing real double occupancy,
o
c optimal doping and is T2 at higher doping. The inverse are thus prohibited. This assumption requires us to use
- Hall angle is strictly T2 and hence has a distinct scat- the Gros-Ricecanonicaltransformationof the more gen-
r
p tering lifetime from the resistivity. The transport scat- eral Hubbard model to renormalize U to infinity[22] and
u tering lifetime is highly anisotropic as directly measured the HFL formalism starts with the t-J Hamiltonian,
s
by angle-dependent magnetoresistance (ADMR, or sim-
t. H =P[ t c† c ]P + J S ·S (1a)
a ilarly AMRO)[7–10] and indirectly in more traditional t−J ij iσ jσ ij i j
m transport experiments. The IR conductivity exhibits a iX,j,σ Xi,j
- non-integerpower-lawinfrequency[11,12],whichwetake P = (1−ni↑ni↓) (1b)
d as a defining characteristic of the “strange metal”. Yi
n
A phenomenologicaltheory of the transport and spec-
o where P is the Gutzwiller projector that will enforce the
troscopic properties at a self-consistent and predictive
c exclusionofantiboundstates. Inthenormalstate,above
[ level has been much sought after, yet elusive. We T∗, the temperature is sufficient to break the supercon-
demonstrate here that the hidden Fermi liquid theory
1 ducting pairs caused by the exchange, J. The system is
v (HFL)[13–18] is the effective low-energy theory for the thus gapless and J only enters into the calculation as an
9 normal state and no longer just a proposal. After re-
effective interaction that renormalizes the kinetic energy
0 viewing the theory, we derive well-defined expressions
in the Shankar sense - we have a HFL and not a hidden
6 relating ARPES, resistivity, Hall angle, and ADMR.
3 Fermi gas. We can write the effective Hamiltonian for
Self-consistency is shown in the one system where most
. the normal state as a projected kinetic energy,[23]
1 datasets are available, overdoped (OD) Tl Ba CuO .
2 2 6+δ
0 In the cuprate phase diagram, a line is usually drawn H = t Pc† c P = t cˆ† cˆ (2)
1 proj ij iσ jσ ij iσ jσ
1 upandtotherightfromtheedgeofthesuperconducting iX,j,σ iX,j,σ
: dome in the OD region to separate the “strange metal”
v
fromaconventionalFermiliquid(FL)athighdoping. We
i “Quasiparticles” cˆ† ,cˆ “Pseudoparticles”c† ,c
X argue there is no crossover from the “strange metal” for iσ iσ iσ iσ
Physicalelectronsofthesystem Trueexcitationsofthesystem
r highdoping. HFLisvalidfortheentirenormalstateand
a Excitationsofthe(reduced Excitationsoftheunprojected
we will show that the data only appears FL-like as the
dimension)projectedspace space,theHFL
HFL bandwidth, WHFL, becomes large at high doping. Automaticallyenforcetheprojection, HFLansatzstatesthattheseobey
Someproposals,whichhaveneitherdemonstratedself- lendnospectralweighttoUHB conventionalFermiliquidrules
consistencynorexplainedtheIRconductivity,invokethe Acceleratedbyanelectricfield, Magneticfieldcommuteswith
actasadecaybottleneck projection,soHalleffect,dHvA,
heuristic that there is single scattering lifetime in the
intheresistivity etc. willbethatoftheHFL
problemwith “hot”or“cold”spots[19–21]withdifferent
Donottransfermomentumtothe Transfermomentumtothelattice
temperaturedependencesindifferentregionsoftheFermi latticebutratherdecayintothe viaumklappscattering,
surface. However,themorerecentexperimentalprobeof HFL,Γdecay=pπT ΓHFL=T2/(2WHFL)
ADMR allows a direct test of the anisotropic scattering RenormalizedFermivelocity,vF,ren=vF,0gt(x)
AnisotropyinvF,kF givenbybandstructure
predicted by any theory. We show that the anisotropy
observed in the transport scattering rate in the ADMR TABLE I: Comparison of the two distinct excitations of the
is precisely reproduced by including the slight variation HFLtheory-thequasiparticlesandtheHFLpseudoparticles.
2
We introduce the projective quasiparticles (QPs), de- of ways, originally in ref. [24]. But the most physically
notedwith“hats”,whichenforcetheprojectionandhave insightful solution is in terms of Tomonaga bosons[25] -
no amplitude in the upper Hubbard band (UHB). These the solution being that the projected state decays as a
excitations span the projected Hilbert space of reduced power-law, G∗ (t) ∝ t−p. Using Nozieres’ theorem and
iσ
dimension. These are the physical electrons that a pho- Friedel’s sum rule, the doping dependent exponent was
toemission experiment would remove, for example. We foundtobep=(1−x)2/4. Thisdopingdependenceisin
can expand these in terms of three pseudoparticle oper- closeagreementoverawidedopingrangewiththepower-
ators (Eq. (3), see Table I), denoted as “hatless”. The law dependence of the IR conductivity.[14] This power-
HFL ansatz,[15]justifiedbyourself-consistencywithex- law decay extends to finite temperatures, replacing t−p
periment, states that these obey conventional FL rules - by e−pπTt (for Tt≫ 1)[26] and we define Γ = pπT.
decay
for example, they have a non-zero residue, Z. WecomparedthisprojectiveGreen’sfunctionwiththe
nodallaser-ARPESinBi Sr CaCu O [27]andfound
2 2 2 8+δ
cˆiσ =ciσP =ciσci−σc†i−σ (3a) excellent agreement.[14] The predicted ω−1+p decay of
cˆ† =Pc† =c c† c† (3b) the EDC’s accounted for the significant spectral weight
iσ iσ i−σ i−σ iσ
at high energies, historically mistaken as experimental
The pseudoparticles (PPs) span the unprojected background. The ARPES scattering rate was extracted
Hilbert space of many-body wavefunctions. Individually as a sum of two rates, CHFL the only free parameter,
they do not represent the physical electrons, rather they Γ =pπT +C v2(k−k )2 (5a)
ARPES HFL F F
are the true excitations of the system. To see this, we
3
start by noting that the solution for the wavefunctionof W = (5b)
HFL 2π2k C
the problem is projective in nature, Ψ(r ,r ,...,r ) = B HFL
1 2 N
PΦ(r ,r ,...,r ) - where Φ is the completely general We nowdescribethese twoscatteringratesinthe con-
1 2 N
many-body wavefunction. We can write the Schrodinger text of transport. The (k −kF)2 rate is the umklapp
equation as, H Ψ = ǫΨ. But to represent the exci- scattering that transfers momentum of the PPs in the
t−J
tations of the system at their most fundamental level is unprojected space to the lattice, τH−F1L = T2/WHFL.
not to consider this to be an equation for Ψ but to ob- WHFL is about an order of magnitude smaller than ǫF
serve that because P2 = P and ǫ commutes with P, the (vF,ren =vF,0gt(x)wheregt(x)=2x/(1+x)).[15]Wecal-
Schrodingerequationcanbere-written,Ht−JΦ=P(ǫΦ). culate the T2 expectation value of the (k−kF)2 rate by
Indeed P acts to the right on (ǫΦ), but regardless the weighting it by the density of states, given by the Fermi
true excitations are fundamentally of the unprojected, function derivative as PPs obey Fermi-Dirac statistics.
Φ-space - these are the HFL PPs. This equation is un- We thus relate WHFL (Eq. (5b)) to ΓARPES. This does
dercomplete and does not havea unique solution. In the notcoincidewithFLtheory,norshoulditsinceZ=0. The
normal state however, we can choose the ground state, lack of T2 term in ΓARPES is due to PP “protection” in
Φ , to be the Hartree-Fock product form of a Fermi sea. the unphysical space. The non-FL QPs, which are di-
0
Theresultingmean-fieldHartree-Fockequationsgivethe rectly probed in ARPES, are entangled and must break
energyspectrumofthePPsandmomentumconservation up into disentangled PPs prior to umklapp scattering.
then makes it unique.[13, 23] The QPs of the system are Applying an electric field will excite QPs of the pro-
then given using Eq. (3), but there is no adiabatic con- jectedspace,whichwillconsequentlydecayintotheHFL
tinuation between the QPs and PPs - as is the basis in (with rate Γdecay, above). Physically the decay process
FL theory of relating the “dressed” excitations back to correspondstotheQPdecayingintoonePPandcoupling
the “bare” ones. The metric of this correspondence, Z, to the Tomonaga bosons in the unprojected space.[15]
vanishes and the physical space is a non-Fermi Liquid. The G∗ (t) correction previously discussed results in a
iσ
The Green’s function of the QPs was calculated, ex- predicted zero-bias anomaly in the tunneling density of
plicitly accounting for projection.[14, 23] The result is a states,[23] which broadens as Γdecay, so the number of
product of two terms; the first is the conventional FL states available for the QPs to decay into increases lin-
termthatwouldresultinaLorentzianspectralfunction. earlywith T.[17] This decayprocess acts asa bottleneck
that must occur before momentum can be transferredto
Gholes =h0|cˆ† (t)cˆ (0)|0i∼=h0|c† (t)c (0)|0iG∗ (t) the lattice by the PPs. When calculating the transport
proj,iσ iσ iσ iσ iσ iσ
(4) lifetime, τ , we must therefore add the decay lifetime
tr
Thecorrection,G∗ (t)=h0|(1−n )(t)(1−n )(0)|0i, withtheHFLscatteringlifetime,τ =τ +τ .[15]
iσ i−σ i−σ tr decay HFL
represents a canonical inverse X-ray edge problem if we The resistivity is thus given in terms of W as,
HFL
treat the Gutzwiller projection at a singly-occupied site
2π2¯hp T2
as a infinite, repulsive potential. G∗ (t) acts to project ρ(T)= +ρ (6)
iσ e2ǫ T +2πpW res
out a -σ electron at t = 0, then measures the overlap F HFL
of this decaying state with the originally projected state Fitting Eq. (6) to ρ(T) for OD Tl Ba CuO , yields
2 2 6+δ
at time, t. This problem has been solved in a variety W = 800 K (see Fig. 1).
HFL
3
Applying a magnetic field does not excite projective dotheytransfermomentumviaumklapp. Butintheim-
QPs,whichwoulddecayintotheHFL,ratheronlycauses purity scattering channel, QP momentum is transferred.
their trajectories to deviate. Hence, the magnetic field The magnetic field deviates the trajectory of both PPs
commuteswithPanditseffectactsdirectlyontheunpro- andQPs,soQPswillcontributetocotθ asanimpurity
H
jectedspace,Φ. TheHallscatteringlifetimeishencethat term. Impurity scattering is not a many-body effect and
of the HFL, τ =τ .[15] Since the inverse Hall angle does not impact the unphysical PPs. It is a property of
H HFL
is given by cotθ = (ω τ )−1 and τ−1 = T2/W , thegeometryofthetransmissionchannelsandnotatrue
H c H HFL HFL
we can deduce W from the Hall angle. Ref. [29] de- dissipative process, a result of the Landauer formula for
HFL
rived a bandwidth in another context but we can follow conductivity, and is unaffected by the bottleneck.
their arguments for estimating the cyclotron frequency, Anelectron-phonon(e-ph)interactionexistswithboth
ω , leading to a relation between W and cotθ , PPs and QPs but it is too weak for umklapp scattering,
c HFL H
evidenced by ρ(T) as there is neither high-T saturation
nπ¯h
nor indication of the Debye frequency. A temperature
W = (7)
HFL rαeB gradientthermallyskewstheFermisurface(FS)sosmall
q, normal e-ph scattering can inelastically thermally re-
with n = k2/2π being the carrier concentration and α
F lax entropy carriers. The FS is not displaced so the gra-
the slope of cotθ versus T2. We estimate n as one car-
H
dient will not disrupt the equilibrium between QPs and
rier per unit cell, consistent with k from both ARPES
F
PPsandtheentropytransportisjustthatoftheHFL.[18]
and ADMR at slightly different doping.[7, 30] Using the
cotθ data[28] on Tl Ba CuO (x = 0.26), yields
H 2 2 6+δ
W = 800 K (Fig. 1), in excellent agreement with
HFL
25
W determinedfromthe resistivity. Evenhigherpre-
HFL
cision of W could have been achieved by accounting
HFL 20
for the variation of kF and vF about the Fermi surface. )
To balance precision with transparency in our solution 1 (15
- )
however,weinvokethefactthatthebulktransportprop- tr
c10
ertieswillbedominantalong(π,π),wherev ismaximal.
F (
QPs may impurity scatter prior to decay, leading to
5
ρ and, less obviously,the T=0 offset in cotθ . As for
res H
the latter, there is no intrinsic contribution of the QPs 0
0 40 80 120 160 200
to cotθ as the magnetic field neither creates QPs nor
H Temperature (K)
FIG. 2: ADMR (ωcτtr)−1 versus T showing excellent agree-
mentwithHFLusingnofreeparameters (solidlines)andthe
T2 ( K2) dataofref. [10]forODTl2Ba2CuO6+δ (x=0.26). Shownare
0.0 2.0x104 4.0x104 6.0x104 8.0x104 1.0x105 theisotropiccontribution(Eq. (9a),blue)andtheanisotropic
200 300
term(Eq. (9b),red)thatvanishesatthenode. WHFL=800K
)m160 250 Tesla dweapseunsdeedntfolyrftrhoemHbFoLthbρa(nTd)waidntdhcoatloθnHg,(aπs,dπi)s,cudsesteedrm. Tinoeedvianl--
c 200 7 uatethepre-factorofEqns. (9a)and(9b),wedeterminedthe
(sistivity 12800 110500 ()cot in H na((rrcoeecdffos.au.lnω[[t39c0,ffo])r3ro0mat])hn.ed(ωTaacnτhniHeisso)oTt−trr=1oopi0pni,iccFimkivgFpF.uw1rwiiwttiyhtihthokffFvτFs(H−eπ(t1π,0,=in)0/)Ttk/h2Fve/F(Wπ(bπ,lHπu,Fπe)=L)c=.9u/1Wr1/v02ee
e
R 40 is arbitrary as it is sample specific. (Color online)
50
0 0 We now extend the “bottleneck” form for τ , con-
0 50 100 150 200 250 300 tr
necting the projected space with the HFL, to include
Temperature (K)
the anisotropic k (φ) and v (φ) (φ is the angle with
F F
FIG.1: Self-consistent HFLfits(black)ofρversusTandin-
respect to (0,0) - (π,0) in the Brillouin zone). The in-
verseHallangleversusT2. ThesampleisODTl2Ba2CuO6+δ verse HFL lifetime, τ−1 (φ)= T2/W (φ), will share
(x=0.26). The resistivity is given by Eq. (6) (pre-factor is a HFL HFL
free parameter) with WHFL = 800 K.This linear best fitfor the anisotropy of (kF2(φ)/m∗(φ))−1 = (vF(φ)kF(φ))−1
cotθH wasusedtodetermineαforEq. (7),resultinginWHFL (recall WHFL ≈ ǫFgt2(x), which would essentially be an
=800K.DatafortheinverseHallangle(red)andresistivity equalityifnotforJ).Theanisotropick andv willalso
F F
(blue) were extracted from ref. [28]. The magnetic field of be evident in ω (φ) = eBv (φ)/¯hk (φ). Evaluating the
c F F
7 T in the Hall experiment is not expected to fully suppress
quantity measured in the ADMR[7–10] experiments,
superconductivityatthelowest temperaturessothatportion
oftheHalldatashouldbeignoredwhilethe16Tusedinthe 1 ¯hkF 1
(φ)= + (8)
ρ(T) experiment should be sufficient. (Color online) ω τ v eBτ ω τ
c tr F tr c res
4
= 2πph¯ kF(φ) T2 + h¯kF(φ) maldoping.[15]Sogiventheself-consistentWHFL=800K
(cid:20) eB (cid:21)(cid:20)vF(φ)(cid:21)T+2πpW(π,π) kF(φ) vF(φ) eBλres(φ) forthisODsample,wemakeawell-definedpredictionfor
HFL "kF(π,π)#"vF(π,π)# the laser-ARPES scattering rate. Using the projective
where τ and λ are the T=0, residual lifetime and lineshape(Eq. (1)ofref. [14])tofitthenodalEDC’s(or
res res
mean-free-path. Theresidualtermwillexhibitthe slight MDC’s butonly for |vF(k−kF)|<∼WHFL), shouldyield
anisotropy of k /λ , for simplicity we’ll assume it’s Eq. (5a)withπp=0.43andC =2.2eV−1. Thelinear-
F res HFL
isotropic and discuss this subtle point elsewhere. Along T term should be quite isotropic but the anisotropic v
F
(π,π), ADMR exhibits only the isotropic contribution. and k will increase C off the node, as discussed
F HFL
This isotropic term, Eq. (9a), corresponds to Eq. (8) C ∝ (k (φ)v (φ))−1. The predicted T-linear pre-
HFL F F
with k (φ) = k (π,π) and v (φ) = v (π,π). Along factor of 0.43 may differ slightly than measured as we
F F F F
(π,0), both the isotropic and anisotropic contributions found at optimal doping,[14] in as much as the Tt ≫
are present - corresponding to Eq. (8) with k (φ) = 1 requirement of the finite temperature extension[26] is
F
k (π,0) and v (φ) = v (π,0). The anisotropic term, upheldandtheslightexperimentalresolutionisincluded.
F F F
Eq. (9b), measured in ADMR is just the difference of
Within HFL, expressions were derived (Eq. (5)-
Eq. (8) evaluated at (π,0) and (π,π). This subtrac-
(7),(9); refs. [14, 23] for IR conductivity; ref. [18]
tion yields an anisotropic term that appears somewhat
for entropy transport) relating the normal state trans-
T-linear,interpretedintheexperimentalpapers[8,10]as
port and spectroscopic properties with self-consistency
havingaphysicallysignificantT-linearcontribution,even
in OD Tl Ba CuO . The framework is amply de-
asT →0. The physicalmotivationforthis contribution, 2 2 6+δ
tailedtoguidefurtherexperimentalconfirmationinother
however,was left as an open question. Decomposing the
cuprates and dopings, particularly for laser-ARPES,
data into these two terms is simply a convenient man-
ADMR, and ρ(T→0) in magnetic fields exceeding H .
ner to compareour theory with the data - we stressthat c2
there is no physical significance of the anisotropic term.
Generally speaking, since k (π,0)/k (π,π) and
F F
v (π,0)/v (π,π) are less than unity, HFL predicts that
F F
(ω τ )−1 is more linear (W (φ) decreases) and in-
c tr HFL [1] P.W.Anderson,TheTheoryofSuperconductivityinHigh
creases in magnitude off the node. Further consistent
withtheexperiment,bothTandT2 termsarepresentat TcCuprates (PrincetonUniv.Press,PrincetonNJ,1997).
[2] C. M. Varma et al., Phys. Rev.Lett. 63, 1996 (1989).
allanglesaboutthe Fermisurface. InFig. 2, wefind ex-
[3] J. Orenstein and A.J. Millis, Science 288, 468 (2000).
cellent agreementwith the ADMR data[10]that extends
[4] Z. X. Shen and D.S. Dessau, Phys. Rep.253, 1 (1995).
to the highest temperature currently available of 110 K
[5] A. Damascelli et al., Rev.Mod. Phys.75, 473 (2003).
- we use Eqns. (9a) and (9b) with no free parameters.
[6] N. E. Hussey,J. Phys.: Condens. Matter 20, 1 (2008).
1 2πph¯ k(π,π) T2 h¯k(π,π) [7] N. E. Hussey et al., Nature415, 814 (2003).
(ωcτtr)iso =(cid:20) eB (cid:21)"vFF(π,π)#T+2πpWH(πF,πL) + eBλFr(πes,π) (9a) [[89]] MJ..GA.bAdneal-lJyatwisaedteatl.a,lP.,hNysa.tuRreevP.Bhy7si6cs, 120,4852213((22000067))..
1 = 1 − 1 (9b) [10] M. M. J. French et al., New J. Phys. 11, 055057 (2009).
(ωcτtr)aniso ωc(π,0)τt(rπ,0) (ωcτtr)iso [11] Z. Schlesinger et al., Phys. Rev.Lett. 65, 801 (1990).
2πph¯ k(π,0) T2 [12] D. van derMarel et al., Nature425, 271 (2003).
= F
(cid:20) eB (cid:21)(cid:18)"vF(π,0)#T+2πpWH(πF,πL)"kkF(F(ππ,,π0))#"vvF(F(ππ,,π0))# [[1143]] PP.. AW..CAansdeyerestona,l.P,hNyast.uRreevP.hBys7ic8s,41,72415005(2(020080)8.).
k(π,π) T2 [15] P. W. Anderson et al., Phys.Rev.B 80, 094508 (2009).
− F [16] P. W. Anderson et al., arXiv:0902.1980v1 (2009).
"vF(π,π)#T+2πpWH(πF,πL)(cid:19) [17] P.W.Andersonetal.,J.Phys.: C.M.22,164201(2010).
At optimal doping, focusing on Bi Sr La CuO , [18] P. A. Casey, Princeton Physics PhD Thesis (2010).
2 2−x x 6+δ
ρ(T) is more linear than expected - W is about half [19] R.HlubinaandT.M.Rice,Phys.Rev.B51,9253(1995).
HFL [20] A. Zheleznyak et al., Phys. Rev.B 57, 3089 (1998).
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H [21] L. B. Ioffeet al., Phys. Rev.B 58, 11631 (1998).
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[22] T. M. Rice,Heavy Fermions, Frontiers & Borderlines in
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c2
H inevitably depends on somewhat arbitrary criteria. [25] K. D. Schotteet al., Phys.Rev.182, 479 (1969).
c2
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[28] A. P. Mackenzie et al., Phys.Rev.B 53, 5848 (1996).
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F H