Electromagnetic Response and Approximate SO(5) Symmetry in High-Tc Superconductors

4 Pages·0.15 MB·English

by  C. Burgess

McGill 97-37
Electromagnetic Response and Approximate SO(5) Symmetry
T
in High- c Superconductors
C.P. Burgess, J.M. Cline
8 Physics Department, McGill University, 3600 University Street, Montr´eal, Qu´ebec, Canada H3A 2T8.
9
9 C.A. Lu¨tken
1 Physics Department, University of Oslo, P.O. Box 1048, Blindern, N-0316 Norway.

n
a
It has been proposedthat the effective Hamiltonian describing high T superconductivity in cuprate materialshas an
J c
approximateSO(5)symmetryrelatingthesuperconducting(SC)andantiferromagnetic(AF)phasesofthesesystems.
8
2 We show that robust consequences of this proposal are potentially large optical conductivities and Raman scattering
ratesintheAFphase,duetotheelectromagneticresponseofthedoubly-chargedpseudoGoldstonebosonswhichmust

] exist there. This provides strong constraints on the properties of the bosons, such as their mass gap and velocity.
n
o
PACS numbers: 78.30.-j, 72.20.-i, 74.72.-h
c
- 2
r It has recently been proposed [1] that the antiferro- L=|(∂ −2ie(µ+A ))φ|2− vφ ∇ − 2ieA φ
p magnetic (AF) and superconducting (SC) phases of the t 0 i i c i
su high-Tc cuprates are related by an SO(5) symmetry, a +(∂ ~n)2− (vn∇~n)2−XiV((cid:12)(cid:12)(cid:12)φ,~n(cid:0)). (cid:1) ((cid:12)(cid:12)(cid:12)1)
. suggestionwhichhasbothstimulatedconsiderableinter- t i i
t
a est [2–5] and drawn sharp criticism [6,7]. Both the in- Xi
m
terest and the criticism are inspired by the central role Here µis the chemicalpotentialwhichdescribes the sys-
- played by the symmetry relating the two phases. tem’s doping, A is the electromagnetic gauge potential,
d i
n TheSO(5)picturehasmuchtorecommendit. Itgives and vφ and vn are the pGB velocities, which can dif-
i i
o asimpleandconcretequalitativeunderstandingofmany fer along the three principal directions of the medium.
c features of the cuprates, as well as predicting brand new In the limit of exact SO(5) symmetry, we would have
[
phenomena,suchassuperconductingvorticeshavingan- vn =vφ ≡v , and the scalar potential V(φ,~n) satisfying
i i i
1 tiferromagnetic cores [3] and persistent superconducting V =V(|φ|2+~n2).
v phasecorrelationswithintheAFphase[4]. Onthenega- Therearetwoimportant,butconceptuallyverydiffer-
3
tive side, Anderson and Baskaranhave argued that such ent, regimes to which this lagrangianapplies.
0
3 afinite-dimensionalsymmetrydescriptionofthecuprates 1. Deep within the AF or SC phases, where fluctu-
1 isinconsistentwiththeelectronlocalizationpropertiesof ations in the modulus |φ|2 + ~n2 are negligible, V be-
0 the two phases [7]. comes a constant in the SO(5) symmetry limit, and
8 In this work we exploit another strength of the SO(5) φ and ~n describe the dynamics of the Goldstone and
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theory – namely its predictive power – to bring further pseudo-Goldstonequasiparticles. ForinstancefortheAF
/
t evidence to the discussion. A model-independent conse- phase there are four such modes: two gapless magnons
a
m quence of the SO(5) picture is the existence in the AF with dispersion relation E2(p) = (vnp )2, and two
n i i i
phase of an electrically chargedpseudo-Goldstone boson pGB’s with charge±2eand dispersion(E (p)∓2eµ)2 =
- φ
d (pGB) quasiparticle, whose dispersion relation and low- ε2+ (vφp )2. ApproximateSO(5P)invarianceamounts
n energy couplings are tightly constrained by the SO(5) g i i i
o symmetry [1,?]. We use these to compute the contri- to thPe statement that |viφ−vin| is small compared to vi,
c and the gap, ε , is much smaller than the typical micro-
bution of these quasiparticles to the electromagnetic re- g
: scopic scale of the system: J ∼ 0.1 eV. ‘Much smaller’
v sponse of the cuprate materials in the AF phase. In
i here means of comparable size to the experimentally-
X particular we find the Raman scattering rate and the
measured41meVgap[8]whichis interpretedwithinthe
realpart of the far-infraredconductivity, which turn out
r SO(5) context as a pGB of the SC phase. Here the two-
a to be comparable to or larger than what is experimen-
dimensionalnatureof the cupratesdictates the sumon i
tally observed. We emphasize that our calculation re-
runsonlyoverthetwospatialderivatives(xandy)which
lies almost exclusively on the assumed SO(5) symmetry,
label the copper-oxygenplanes.
and depends only minimally on the microscopic details
2. Near the critical boundaries between the various
of these systems.
phases,L plays the roleof a Ginzburg-Landau(GL) free
ThekeytoolintheanalysisistheeffectiveLagrangian
energy, and (in mean field theory) V may be expanded
density which describes the electromagnetic couplings of
to quartic order: V = −m2|φ|2 − 1m2|~n|2 +λ |φ|4 +
the SO(5) pGB’s. This can be written as φ 2 n φ
2λ |φ|2|~n|2+λ |~n|2. Inthiscasethemodelcanbetwo-
φn n
1
or three-dimensional (but anisotropic, v 6=v =v ) de- space integral can be done exactly, giving
z x y
pending on how close one is to the critical limit. SO(5)
e2 v v 2 ω
invariance implies in this case m2φ =m2n and λφ = λn = σ1,i = 16π2¯ha viazω 1+NB 2 Gi(η), (3)
λnφ, and so approximate SO(5) invariance is the state- (cid:18) ⊥ (cid:19) h (cid:16) (cid:17)i
mentthatthedeviationsfromtheserelationsaresystem-
where G (η) = 1−η2 − ηcos−1η and G (η) =
atically small. Phenomenology requires m2 > m2 when ⊥ z
n φ 1(cos−1η−η 1−η2) ( these become respectively −πη
the doping is sufficiently small, in order that the ground 2 p
and π/2 for η < −1). The continuum approximation
stateofthe systembe AF,withφ=0and~n6=0. Inthis p
on which the 3D equation (2) is based is valid when
regime the pseudo-Goldstone gap in the AF phase may
a → 0 (so η → 1). For the parameters of interest, we
be computed, giving ε2 = −m2 + λφnm2, which clearly
g φ λn n find that the 3D formula is practically indistinguishable
vanishes in the SO(5) limit, as required.
from eq. (3), although the difference starts to become
The existence of charged bosons with a small gap has
apparent for frequencies greater than 800 cm−1.
strong observable consequences for the optical conduc-
tivity andRaman scatteringproperties in the AF phase.
Thesepredictionsarequiterobust. Theinteractionofthe
photons with the bosons, the first two terms of eq. (1), 3.0
is completely fixed by electromagnetic gauge invariance.
The self-interactions described by the other terms, in-
cluding possible higher powers of |φ| not shown, must
m) 2.0
vanish for Goldstone bosons in the limit of zero energy c
Ω
and exact SO(5) symmetry, and so may be treated per- 1/
σ (1 (1)
turbatively for low energies and approximate symmetry.
These also ensure the small size of the gap, ε .
g 1.0
Conductivity: We start with the contribution of the
charged pseudo-Goldstone quasiparticle to the real part (2)
oftheconductivity,σ ,withintheAFphase. Thisispro-
1
portionaltotheimaginarypartofthephotonself-energy 0.0
650.0 700.0 750.0 800.0
due to a virtual pseudo-Goldstone boson loop, and re-
frequency (1/cm)
lated to the rate at which the pGB’s are pair-produced
byincidentphotons. Ourresultagreesintheappropriate
limitswithcloselyanalogousformulasfortheelectromag- Figure 1: The optical conductivity σ1 as a function of the
photonenergyω,inunitsofΩ−1cm−1,fortwodifferentpGB
netic response of a hot pion gas [10]. For pGB’s which
velocities, (1)v/c=2×10−4,and(2)v/c=10−3. Theother
canmoveinthreedimensions,andwhosevelocityismuch
less than that of light, we obtain relevantparametersareεg =0.04eV,T =300 K,andµ=0.
3/2
e2 v2c 4ε2 ω How big are these results? If the gapis approximately
σ1,i = 24π¯hλ(cid:18)vxviyvz(cid:19) 1− ¯h2ωg2! h1+NB(cid:16)2(cid:17)i, εdgata=, t0h.0e4inetVerpalsanseugsgpeasctiendg bisyath=e 0n.e5utnrmon, ascnadttuersiinngg
(2) the magnon speed v = 2×10−4c [11], then for ω = 0.1
i
eV (frequency = 800 cm−1) and T = 0.025 eV (300K),
where the absorbed light of wavelength λ is polarized σ =2.75Ω−1cm−1. Thisismorethanthreetimeslarger
1
in the ‘i’ direction, taken to be one of the crystal axes. than the measured values of σ < 1 Ω−1cm−1 for the
1
(Henceforthwe dropthe superscriptφ fromthe pGB ve- undoped cuprate Sr CuO Cl , as reported in reference
2 2 2
locities vi.) NB(ω) = n+(ω)+n−(ω), where n±(ω) = [12]; moreover the frequency dependence disagrees with
1/(exp[(h¯ω±µ)/kT]−1)istheusualBose-Einsteinstatis- thedata,whichhasσ decreasinginthisfrequencyrange.
1
tics factor, and e2/¯h = 2.4341×10−4 Ω−1. This contri- AlthoughtheperturbativetreatmentofthepGB’sbegins
bution to the conductivity vanishes for photons whose to break down at higher energies, if we can trust our re-
energy h¯ω is below the threshold 2εg for producing two sults at somewhathigher energiesofω =0.5eV (=4000
pGB’s. Thecorrespondingabsorptioncoefficientisgiven cm−1), then σ grows to 100 Ω−1cm−1, in even greater
1
by σ1,i/(ǫ0c), where ǫ0c=2.654×10−3Ω−1. conflict with measured values near 1 Ω−1cm−1 in the
To take into account that the pGB’s may be con- materials Gd CuO and YBa Cu O [12]. (Reference
2 4 2 3 6
fined to move in the superconducting planes, we have [13] has also measured conductivities, but their maxi-
also computed σ1 using the lattice dispersion relation mumfrequency of700cm−1 may be below the threshold
E = (v⊥2p2⊥ +(2vzsin(apz/2)/a)2)+ε2g)1/2, integrating 2εg needed for production of pGB pairs.)
pz between ±π/a, where a is the spacing between the Although the magnitude and the shape of σ1 suggest
planes. Defining η =1− 21(a/v)2(ω2−ε2g/¯h2), the phase aconflictbetweenSO(5)andexperiments,therearesev-
2
∆ω ∆ω
eral caveats. One is that σ is inversely proportional to
1 E = +ω v f(p ); p =−ω − f(p )
the pGB velocity, vi, which is not precisely known. The i 2 0 z ⊥ i,z 0 2vz ⊥
other is that our computation is only valid for energies ∆ω ∆ω
E =− +ω v f(p ); p = ω − f(p ), (7)
within the domain of approximation of the low-energy f 2 0 z ⊥ f,z 0 2v ⊥
z
pGB lagrangian used here, i.e. for ω ≪ J ∼ 0.1eV.
where ∆ω =ω −ω , ω =(ω +ω )/2, and
Above these energiesthe pGB’s neednotcontribute as a f i 0 i f
weakly-coupledandcomparativelynarrowstate. Theex-
1/2
ε2+v2p2
perimentalconstraintshappentobestrongestjustinthe f(p )= 1+ g ⊥ ⊥ . (8)
region where our long-wavelength approximation starts ⊥ |vz2ω02−∆ω2/4|!
to break down. Thus we turn to another possible signal
of the pGB electric response. Care must be takenwith these expressions,however,be-
Raman Scattering: We now consider the contribution causethequantityAcandivergeforsomevalueofp⊥. In
totheRamanscatteringrate,comingfromCompton-like ageneralgaugethis comesaboutwhenthevirtualboson
photonscatteringfromthepseudo-Goldstonequasiparti- indiagram(2b)goesonitsmassshell. (Equivalently,the
cles in the sample. The Feynman graphs for this process gauge transformation which we used to remove diagram
are those of Figure 2. (2b) becomes singular for these momenta.) We handle
this situation, when it arises, by including the width of
the pGB itself. This may be done by adding a small
imaginary part, iΓ/2, to the boson energies in the defi-
nition of A, giving a finite scattering rate. We find our
results to be insensitive to reasonable values such that
Γ≤ε , for relevant temperatures and energies.
(a) (b) (c) g
The observable of interest is the differential scatter-
Figure 2: The Feynman graphs which describe photon-pGB ing rate per unit sample thickness, l, per unit incident
scattering. laser power, I: R = (1/I)(dΓ/dl). This is the quantity
which does not depend on the details of the target or
We work in a gauge for which the expression for the
of the incident photon flux. We compute the differen-
scattering amplitude, M, is particularly simple. That
is, if pµ and kµ denote the initial and final pGB and tial rate of such scattering into a solid angle dΩ (cen-
i,f i,f tred about 180◦), and into a final energy interval dω ,
photon four-momenta, respectively, and we choose the f
photon polarizations, ǫ˜µ , to satisfy ǫ˜ · (2p˜ + k˜ ) = R = dR/dωfdΩ. We then do the thermal average over
i,f i i i
the initial and final pGB’s, sum over the final photon
ǫ˜ · (2p˜ − k˜ ) = 0, then diagrams (2a) and (2b) of
f i f polarizations and average over the initial ones. This as-
the Figure vanish, leaving an amplitude of the form
sumptionthatthe incidentphotonsareunpolarized,and
M =4e2ε˜(a)·ε˜(b). Weuseheretheconvenientnotation
ab i f scattered-photon polarizations are not detected, simpli-
for any four-vector,in which a ‘˜’ indicates the multipli-
fies our expressions,but is not crucial for the result. We
cation of the spatial components by the corresponding
find
velocity, v . That is, p˜ =p , and p˜ =v p .
i 0 0 i i i
Let us assume that the initial photon is moving in the ωf
R≡ dp n (E )(1+n (E ))S
zdirection,perpendiculartothesuperconductingplanes, (4π)4v2ω2|∆ω| z ± i ± f
⊥ i ± Z
and the final one is scattered by 180◦, as is the case in X
theexperimentstowhichwecompare. Assumingthex-y where n± is the Bose-Einsteindistribution function with
plane to be isotropic, v = v = v , and averaging over chemical potential ±µ. The limits of integration are
x y ⊥
initial and summing over final polarizations leads to the found by varying p2 between 0 and ∞ in eq. (7) for
⊥
following expression: pi,z, and p2⊥ in the integrand is determined by pi,z by
inverting the same equation. As a function of the final
S ≡ 1 |M |2 =8e4v4 1+(1−A)2 , (4) photon energy, the rate is peaked near ωf =ωi, with an
2 ab ⊥ approximate maximum value of
Xa,b h i
2
where A=(1+vz2)v⊥2p2⊥/(DiDf), with Rmax ∼= αv⊥ Te−β(εg−|µ|), (9)
πv ω
(cid:18) z (cid:19)
1
Di =Ei−vz2pi,z + 2ωi(1−vz2), (5) using the Boltzmann approximation for the distribution
functions. Kinematics constrains the scattered photon
1
Df =Ei+vz2pi,z − 2ωf(1−vz2). (6) energy to lie in the narrow range 11+−vvzz < ωωfi < 11+−vvzz.
For visible light (500 nm) and v = 10−3c, this gives a
Conservation of four-momentum constrains the initial half-width of 40 cm−1 in the frequency ν. In figure 3
and final boson energies and momenta according to we show R for representative input values ω = 2 eV,
i
3
T = 93 K, ε = 0.04 eV, µ = 0 and 0.01 eV, Γ = 0.1ε , 2D and 3D Raman intensities are small.
g g
and optimistically large velocities v = v = 10−3c. For In conclusion, we have used SO(5) symmetry to com-
z ⊥
these numbers we find rates of order R ∼ (150−300) pute the low-energy contribution of the electrically-
cps/mW/˚A/steradian/eV, with a smooth, featureless chargedpseudo-Goldstonebosonstothe electromagnetic
lineshape. For optical photons the penetration depth response of cuprates doped to be antiferromagnets. We
of typical samples is of order 100 ˚A [14], leading to a find measurably large conductivities and Raman scat-
prediction of 104 events/mW/˚A/sr/eV in the backward tering rates, due to the presence of electrically-charged
direction. Experimentally,theobservedrateforthecom- states with a comparatively small gap. At present the
poundBi2212isonly15cps/mW/˚A/sr/eV.Howeverthis dispersion relation of the putative bosons is not suffi-
must be corrected [15] for detector efficiency (0.1) and ciently well known to rule out SO(5) for the cuprates
surfacelosses (0.1−0.01),whichin the worstcase would based on the data. It is encouraging, however that the
bringthe predictedvalue intoagreementwiththe exper- conductivity and Raman intensities have a complemen-
iments. tary dependence on the pGB velocity, so that one or the
We point out that the predicted rate is exponentially other should show evidence for the pGB’s, especially if
dependentontheratioofthepGBgaporchemicalpoten- the experiments are improved, e.g. Raman scattering at
tial to the temperature because of the Boltzmann factor frequency shifts less than 10/cm.
in eq. (9). If (ε −|µ|) ≤ kT, there is a further gain by We thank R. Hackl and T. Timusk for information
g
a factor of 150 in the rate, which would cause a serious about the experiments, and C. Gale and A. Berlinksy
discrepancy between the predicted and observed values. for useful discussions.
On the other hand, the width depends linearly on the
pGB velocity. If v < 2×10−4c, this width becomes less
than the experimental resolution of ref. [14], which only [1] S.-C. Zhang, Science 275, 1089 (1997).
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preprintcond-mat/9707027(1997);C.L.Henley,preprint
cond-mat/9707275 (1997); C.P. Burgess, J. Cline,
300.0
R.B. MacKenzie and R. Ray, cond-mat/ 9707290,
to appear in Phys. Rev. B. (1998); E. Demler,
H. Kohno, S.-C. Zhang, preprint cond-mat/9710139
V) (1997); D. Scalapino, S.-C. Zhang and W. Hanke,
oA/e 200.0 (2) preprint cond-mat/9711117 (1997);
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m
s/ preprint cond-mat/ 9704048, submitted to Phys. Rev.
p
y (c Lett. (1997).
sit [4] E. Demler, A.J. Berlinsky, C. Kallin, G.B. Arnold,
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I
(1) [5] C.P. Burgess and C.A. Lu¨tken, preprints cond-
mat/ 9611070; cond-mat/9705216, to appear in
Phys. Rev.B. (1998).
0.0 [6] M. Greiter, preprints cond-mat/9705049, cond-mat/
-40.0 -20.0 0.0 20.0 40.0
9705282; S.C. Zhang, preprintcond-mat/9705191.
frequency shift (1/cm)
[7] G. Baskaran and P.W. Anderson, preprint cond-
Figure3: ThedifferentialRamanscatteringrateRasafunc- mat/9706076; R. Laughlin, preprint cond-mat/9709195
tion of the photon frequency shift νi−νf, in units of cm−1. (1997); P.W. Anderson and G. Baskaran, preprint cond-
The curves correspond to: (1) the parameters given in the mat/9711197 (1997).
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tential, µ=0.01 eV. [9] E. Demler and S.-C. Zhang, Phys. Rev. Lett. 75, 4126
(1995), and preprint cond-mat/9705191.
Similarly to the conductivity, the Raman scattering [10] C. Gale and J.I. Kapusta, Nucl. Phys. B357 (1991)
rate near zero shift depends only weakly on whether the 65; A.I. Titov, T.I. Gulamov and B. K¨ampfer,
pGB’s are allowed to move in three dimensions or con- Phys. Rev.D53 (1996) 3770.
finedtothe2Dplanes,fortheparametersofinterest. By [11] J.M. Tranquada and G. Shirane, Phys. Rev. B40 1989)
repeating the above calculation using the lattice disper- 4503; S.M.Haydenet al.,Phys.Rev.B54 (1996) R6905;
sionrelationforthepGB’s,onefindsthatthecontinuum [12] D.B. Tanner et al.,SPIE Vol. 2696 (1996) 13.
[13] C.C. Homes andT. Timusk, cond-matt/9509128 (1995).
version is a good approximation when ωa ≪ c, rather
[14] R.Hackl,G.Krug,R.Nemetschek,M.OpelandB.Stad-
thantheconditionωa≪v whichappliedfortheconduc-
lober, SPIE Vol. 2696, (1996) 194.
tivity. Although we are interested in larger frequencies
[15] R. Hackl, privatecommunication.
inRamanscatteringthanforthe conductivity,the latter
conditionisstillsatisfied,andthedifferencesbetweenthe
4