Considerations on the symmetry of loop order in cuprates

0.36 MB·English

by  A. Shekhter

Considerations on the symmetry of loop order in cuprates and some effects
accompanying it.
A. Shekhter1 and C. M. Varma2
1National High Magnetic Field Lab, Tallahassee, Florida 32310
2Department of Physics and Astronomy, University of California, Riverside, California 92521
(Dated: October 14, 2009)
The loop-current state discovered in the pseudogap phase of cuprates breaks time reversal sym-
metry and lowers the point group symmetry of the crystal. The order parameter and the magnetic
structure within each unit cell which is associated with it can be described by a toroidal moment
paralleltothecopper-oxideplanes. Wediscusslatticepointgroupsymmetryofthemagneticstruc-
9
ture. As an application, we discuss a few effects that necessarily accompany order parameter in
0
0 the pseudogap phase. The magnitude estimated for these specific effects makes them hard to ob-
2 serve because they rely on the small magnetic fields associated with the order parameter. Effects,
associatedwiththeelectronicenergiesaremuchlarger. Someofthemhavealreadybeendiscussed.
t
c
O PACSnumbers: 74.72.-h,75.25.+z,72.55.+s

4
1 I. INTRODUCTION. energies and energies of magnetic fields associated with
spin-moment order. For example, the spin-splitting in

n] Polarized neutron scattering experiments, dichroic an- ferromagnetic iron (ordered moment of ≈ 2.5µB/unit-
cell) is about 1 eV, while the magnetostrictive distor-
o gle resolved photoemission experiments and magnetiza-
c tion measurements1,2,3,4,5 reveal the predicted line of tionduetothemagneticfieldsisalatticedistortiononly
r- phase transition6 in cuprates at pseudogap temperature about 1 part in 103.
p T∗(x). At the present time four classes of cuprates have
u
beenstudied,allofwhichshowconsistencywiththesame
s
. form of order. The purpose of this paper is to present a II. LOOP ORDER PARAMETER: TOROIDAL
at symmetryanalysisoftheorderparameter,whichisneces- MOMENT.
m saryforfurtherwork. Theneutronspin-flipintensityob-
- served in the experiment in the pseudogap phase is con- The polarized neutron scattering experiments1,2 re-
d sistentwiththemagneticstructurethattransformsunder veal an elastic spin-flip intensity on top of the subset
n
the two-dimensional irreducible representation E of the of Bragg peaks in pseudogap phase in cuprates. No new
o u
point group of the copper-oxide lattice, D . The order Bragg peaks appear; the magnetic structure responsible
c 4h
[ parameter L that can be used to characterize the mag- forspin-flipintensitydoesnotbreaktranslationalinvari-
neticstructureofsuchsymmetryisapolartime-reversal- ance. However,itfollowsfromtheanalysisoftheneutron
2
odd vector parallel to the copper-oxide planes. Such a data that the symmetry of the observed magnetic struc-
v
vectorhasbeentermeda”toroidal”oran”anapole”mo- ture is lower than the point group symmetry of the lat-
7
ment in the literature. The relation of such an order pa- tice. The copper-oxide lattice has tetragonal symmetry
8
9 rameter to the magnetic structure and the loop-currents 4/mmm(D4h)(theeffectsofthesmallorthorhombicdis-
1 in the unit cell of cuprates is specified. tortion due to ordering in Cu-O chains in some cuprates
. Several experiments other than those which measure will be mentioned). In the pseudogap phase the symme-
5
0 the order parameter directly1,2,3,4,5 have already been try is lowered to mmm or D2h(C2v)5,8. Such lowering
9 proposed which lead to unusual effects due to the cou- of the symmetry follows if the magnetic structure trans-
0 plingoftheorderparametertoexternalprobes. Thesein- forms under irreducible representation E of the point
u
: clude several forms of dichroism in x-ray scattering8 and groupof thelattice. Foran orderparameter inthe pseu-
v
i secondharmonicgenerationinopticalexperiments21. As dogap phase one can choose a polar in-plane (parallel
X
anapplicationofthesymmetryanalysispresentedforthe to copper-oxide plane) vector L = (L ,L ) which is re-
y x
r firsttimeinthispaper,wediscussfewotherphysicalphe- stricted to four crystalline directions because it trans-
a
nomenathatnecessarilyaccompanytheorderparameter forms the same way, E , under the operations of point
u
L. These are crystal lattice distortions which are second groupofthelattice. OrderparameterListime-reversal-
order in the order parameter, crystal distortion linear in odd because the magnetic structure which it represents
an applied uniform field and linear in the order parame- is odd under time reversal. In the literature polar time-
ter, and magneto-electric phenomena in transport prop- reversal-odd vector is called a toroidal moment as it is
erties. These are effects related to the small energies as- a symmetry of a magnetic field in a solenoid bent into a
sociatedwiththemagneticfieldduetotheorderparame- torus9; intheparticlephysicsobjectofthesamesymme-
ter. Muchlargereffectsareassociatedwiththeelectronic try is known as anapole moment10.
energies associated with the order parameter. This dis- A natural way to relate the order parameter L to
tinctionissimilartothedifferencebetweentheexchange the pattern of the magnetization M(r) in the pseu-
2
dogap phase is as follows. Since the observed mag- ture in the pseudogap phase can be an arbitrary combi-
netic structure retains lattice translation symmetries it nation of , M and M ; the precise balance between
⊥ (cid:107)
is enough to consider magnetization M(r) within sin- thetwodependsonthemicroscopicdetailsandisdecided
gle unit cell. In general, the function M(r) can be de- experimentally, see discussion in the end of this section.
composed into spatial harmonics of point group of the Experimental data and theoretical calculations indi-
(cid:80)
lattice (D ); M(r) = c M (r) . Here har- cate that microscopic orbital currents within the unit
4h λα λα λα
monic M (r) transforms under the irreducible repre- cell are responsible for magnetic structure in pseudogap
λα
sentation λ of the group D and α is internal index of phase of cuprates. Here we discuss symmetry aspects of
4h
the representation λ if it is not one-dimensional; c is this relation. In general, magnetic structure in the crys-
λα
a set of coefficients. The mathematical representation talcanbeequivalentlydescribedintermsoftheperiodic
of the fact that the magnetic structure M(r) belongs magnetization function, M(r), or a pattern of periodic
to the irreducible representation E is that it can be microscopic currents j closed within the unit cell (loop
u m
written as M(r)=c M (r)+c M (r) where M (r) currents). The two are related by
1 1 2 2 1,2
aretwoorthogonalharmonicsthattransformundertwo-
dimensionalirreduciblerepresentationEu. Itcanbeseen jm =c∇×M, (3)
that under all group operations a pair of vectors L
α=1,2 seeRef.13. Intheabsenceofmacroscopiccurrents,H =
defined by
B+4πM =0, the magnetization M(r) is proportional
(cid:90) to the magnetic field within the unit cell, B = −4πM.
Lα = drMα(r)×r (1) Themagnetizationcanbesolvedintermsofmicroscopic
currents j (r) by inverting equation j =c∇×M :
unitcell m m
also transforms under E representation. The two vec- 1 (cid:90) j (r(cid:48))×(r−r(cid:48))
u M(r)= dr(cid:48) m (4)
tors Lα are parallel to the copper-oxide plane and are 4πc |r−r(cid:48)|3
orthogonal to each other; we can use them as a basis to
defineatoroidalmomentL=c L +c L whichisasso- whereintegraldr(cid:48)isoverthewholevolumeofthecrystal.
1 1 2 2
ciated with the magnetic structure M(r) = c M (r)+ If the magnetization M(r) is a periodic function on the
1 1
c2M2(r) in the unit cell. Conversely, if the magnetiza- lattice, so is jm(r), and vice-versa.
tion M(r) does not have a component that transforms We can relate directly the toroidal order parameter
under E , the integral in Eq. (1) vanishes. We also note L to the microscopic current distribution within unit
u
thattheout-ofplanecomponentoftheintegralinEq.(1), cell. Current pattern is restricted to the unit cell,
(cid:82)
if non-zero, transforms under A2u representation; it van- unitcelldrjm(r) = 0. Consider second moment of cur-
ishesifM (r)transformsunderE representation. We rent distribution
1,2 u
conclude that the vector L defined by (cid:90)
drr2j (r). (5)
(cid:90) m
unitcell
L= drM(r)×r (2)
Using definition of microscopic current in terms of mag-
unitcell netization, j =c∇×M, we write this as
m
can be used as an order parameter to characterize the (cid:90)
magnetic structure in the pseudogap phase of cuprates. drcr2∇×M(r). (6)
The simplest structure that has non-zero toroidal mo- unitcell
mentisapairofmomentsatfiniteoffset;eachmomentis Using identity
directed perpendicular to the line connecting them; the
moments are of equal magnitude and opposite in direc- r2∇×M =∇×(Mr2)+2M(r)×r (7)
tion. However, function M(r) may have more elabo-
and the definition of the toroidal moment in Eq. (2) we
rate structure; see the end of this section for discussion
find
of the experimental situation. It is instructive to sep-
arate M(r) into into planar and perpendicular compo- (cid:90) 1(cid:90)
nents, M(r) = M⊥(r)+M(cid:107)(r); here M⊥ = zMz and L= dr(1/2c)r2jm(r)− v ds×M(r)r2/2.
M(cid:107) = xMx + yMy and x,y is the basis in the plane unitcell (8)
parallel to the copper-oxide plane and z is in the direc-
tion perpendicular to it. Fig. 2 schematically represents The second integral is over the surface of the unit cell,
this decomposition with the red arrows understood as a ds is surface element. Using the fact that the surface is
magnetization direction M(r). Under the point group shared between adjacent unit-cells, the surface integral
operations each component, M and M , transforms vanishes. Therefore, one can define a toroidal moment
⊥ (cid:107)
independently under its own E representation. Conse- by
u
quently, each component may contribute independently
(cid:90)
to the toroidal moment L = (Lx,Ly) via Eq. (2). We L= dr(1/2c)r2j (r). (9)
m
conclude that on symmetry grounds the magnetic struc- unitcell
3
We conclude that the toroidal order parameter is pro- example to an in-plane polarization of electron spin in
portional to the planar (parallel to copper-oxide plane) the unit cell11. In mercury compound this mechanism is
componentofthesecondmomentofmicroscopiccurrents notallowedbecauseeachcopper-oxideplaneseescentro-
within the unit cell. symmetric environment. A horizontal component how-
The magnitude of the spin-flip intensity observed in ever naturally appears if one considers loop currents via
experiments for the most underdoped samples studied is apical oxygen2,12 as well as via planar copper. It is then
consistent with a pair of magnetic moments ∼ 0.1µ in reasonable that the major part of the horizontal com-
B
the centroids of the two triangles; the centroids are lo- ponent in YBCO is also due to loop-currents via apical
catedatdistancex ∼a/2fromthecenteroftheunitcell oxygens.
0
(a is unit cell size in the CuO plane). The magnetic mo-
ment M is given by a volume integral of magnetization,
(cid:82) drM(r). Integrating over half of unit cell (the full in- YBCO
tegral over unit cell vanishes) we must obtain magnetic
moment of order of 0.1µB; we conclude that the magne- = +
tization in the unit cell is estimated as |M| ∼ 0.1µ /a3
B
and the integral in Eq. (2) is L∼|M|a4 ∼0.1µ a.
B
HgBaCuO
d
C'' = +
v 2 Lx+y Lx-y
C'
2
(a) (b) FIG. 2: Schematic representation ofthe magnetic flux distri-
FIG.1: (a)YBCOunitcell,blackdots-copperatoms;open butionobservedintheexperiment. Thediagramsymbolically
circles - oxygen atoms. Operations of the point group are represents decomposition of the magnetic structure into hor-
indicated. (b)Representationsofthepointgroupcanbecon- izontal and vertical components. The actual composition of
structed on the space of patterns of currents on the inter- the two into the physical order parameter depends on micro-
atomic links in the unit cell. Two diagrams indicate the pair scopic details and is not dictated by symmetry.
of current patterns that transforms under inversion-odd two
dimensional representation E . The loop order parameter
u
transforms under the same representation and can be repre-
sented by these pictures.
The simplest loop-current pattern that leads to mag-
netic structure consistent with an in-plane toroidal mo-
mentconsistsofapairofplanarcurrentloopspassingvia
III. LATTICE DISTORTIONS THAT
planar oxygens and planar copper, see Fig. 1(b). Such ACCOMPANY LOOP-CURRENT ORDER.
distribution of microscopic currents generates two oppo-
site magnetic fluxes directed along z axis, see Fig. 2;
as has been mentioned earlier, this is consistent with As mentioned earlier7, mmm symmetry does not al-
the pattern of the magnetization which has non-zero lowpiezo-magnetism, i.e. nodistortionchangingsponta-
toroidal moment. However, polarized neutron scatter- neously the symmetry of the unit-cell to linear order in
ing experiments indicate a finite horizontal component the order parameter is allowed. Here we discuss lattice
of the magnetization M(r) in the unit cell1,2. If the deformations that accompany current-loop order to sec-
assumption is made in the analysis of the experiments ond order. To do so and for the other results derived in
that the structure factor for the in-plane and the out this paper, one must specify the irreducible representa-
of plane components is the same, the magnitude of the tions of the crystal symmetry in the absence of the loop-
horizontal and the vertical components are similar. To current order. The copper-oxide lattice has tetragonal
understand the presence of both horizontal and verti- symmetry 4/mmm (D ). The operations in the group
4h
cal components in the pattern of magnetization one has fall into 10 equivalence classes: π/2-rotations around
to consider also spin-orbit interactions and/or loop cur- z axis C , π-rotations around z axis C , π-rotations
4 2
rentsinvolvingtheapicaloxygen. InYBCOeachcopper- around x axis C(cid:48), π-rotations around x + y axis C(cid:48)(cid:48),
2 2
oxideplaneisseeingnon-centro-symmetricenvironment; π/2-rotations around z axis followed by reflection S ,
4
consequently, symmetry allows a Dzialoshinskii-Moriya reflections σ ,σ σ in the planes perpendicular to axes
h v d
type interaction which in the presence of vertical com- z,x,x + y respectively, spatial inversion i and identity
ponentofthemagnetization (planarloop currents)leads operation E, see Fig. 1(a). Under symmetry operations
to a horizontal component of the magnetization due for in D , any physical object transforms under one of the
4h
4
10 irreducible representations : bic distortion. At low enough temperatures the crys-
tal structure of YBCO has intrinsic orthorhombic dis-
A1g 1orz2 orx2+y2 A1u xyz(x2−y2) tortion u(B ) of the B symmetry due to the order-
1g 1g
A2g xy(x2−y2) A2u z inginthecopper-oxigenchains(thelatticeconstantsare
B1g x2−y2 , B1u xyz . a = 3.82˚A, b = 3.89˚A). Within D group the follow-
4h
B2g xy B2u (x2−y2)z ing relation holds B B = A (this can be deduced
1g 2g 2g
Eg (zx,zy) Eu (x,y) from the character table). Therefore, there exists a term
(10) u(A )u(B )[L2 − L2 ] which will create a distortion
2g 1g y(cid:48) x(cid:48)
u(A ) of A symmetry in the presence of finite or-
The second column illustrates each irreducible represen- 2g 2g
thorombic lattice distortion u(B ) and loop order pa-
tations by a polynomial of the same symmetry. Within 1g
D thevectorrepresentationoftherotationsinspaceis rameter Lx(cid:48) or Ly(cid:48). Another effect of finite orthorhom-
4h bic distortion u(B ) in YBCO is to introduce a mixing
no more irreducible; instead, the the polar vector E and 1g
axial vector M break into irreducible representations of between Ly(cid:48) and Lx(cid:48) via the term u(B1g)Ly(cid:48)Lx(cid:48). Along
the same line of analysis one concludes that no distor-
D as follows :
4h tion of A symmetry is allowed in mercury compound
2g
A M in the pseudogap phase since its lattice has tetragonal
2g z
Eg (My(cid:48),Mx(cid:48)) (11) symmetry.
A E The magnitude of the distortions may be estimated
2u z
Eu (Ey(cid:48),Ex(cid:48)) from the free energy
Thebasisx,y ischosenalongthecopper-oxidelinks. We δF =γ(u/a)L2+E(u/a)2/2 (13)
use notation x(cid:48),y(cid:48) for a basis√rotated by 45◦ with√re-
spect to x,y, i.e., x(cid:48) =(x−y)/ 2 and y(cid:48) =(x+y)/ 2. whereuisdistortioninaparticularsymmetrychannel(a
Representations Eu and Eg are two-dimensional repre- isin-planelatticeconstant),L2 standforoneoftheloop-
sentations, all other irreducible representations are one- orderbilinearsinEq.(12)), andE istheelasticmodulus
dimensional. (in energy units) for the distortion in the correspond-
The term in the free energy that can couple loop or- ing symmetry channel. Minimizing the Free energy we
der and lattice distortion has to be of at least a second obtain u/a = γL2/E. It is quite difficult to get an esti-
orderinthecurrent-looporderparametersinceListime- mate of the magnitude of the γ’s directly. For an order
reversal-odd; such a term has a structure LLu where u of magnitude estimate, we proceed as follows: The A
1g
describesdistortion. Asaconsequence,thelatticedistor- distortion arises from the same symmetry arguments as
tionisofasecondorderinloop-orderparameter,u∝L2. thechangeinthevolumeofferromagneticsubstancesdue
To obtain the symmetry of allowed distortions we have to their order and we may use experimental results ob-
tobreakbilinearproductsofLintoirreduciblerepresen- tained for them to get an idea of the order of magnitude
tations. Within D4h the product of two polar in-plane of the effect to be expected. For example, detailed mea-
vectors (transforming under Eu representation) breaks surements are available for Fe14. The lattice constant
into four irreducible representations : changes from above the transition at over 800 degrees C
to 300 degrees C due to ferromagnetism with a moment
A L2 +L2
1g y(cid:48) x(cid:48) ofabout2.5µ peratombylessthanapartin103. With
A2g Ly(cid:48)L(cid:48)x(cid:48) −Lx(cid:48)L(cid:48)y(cid:48) (12) the ordered mBoment an order of magnitude smaller, we
B1g Ly(cid:48)Lx(cid:48) would therefore expect for similar coupling constant and
B L2 −L2
2g y(cid:48) x(cid:48) Bulk modulus, a change in similar temperature region of
only a part in 105. This estimate is unlikely to be in-
Letusconsidereachofthetermsabove. Onekindofdis-
correct by more than an order of magnitude. One would
tortionsthatbelongstoA varietyamountstoachange
1g expect that similar magnitude of distortion is to be ex-
in the size of the cell and is obviously allowed to cou-
pected in the interesting case of the B distortion and
ple to the square of the order parameter. Examples of 2g
much smaller for the A distortion since that must rely
the distortions in the other irreducible representations 2g
also on the orthorhombicity of the original structure.
are shown in Fig 3 where black arrows represent atom
displacements. The most interesting case is that corre-
sponding to the third and fourth lines above: the free
energy contains a term γ u (L2 −L2 ) where u IV. DEFORMATIONS INDUCED BY
is the distortion that belBo2nggsB2tgo By(cid:48) irrexd(cid:48)ucible repBr2eg- MAGNETIC FIELD.
2g
sentation and γ is a coupling constant; similarly for
B2g
B . Fromthepointofviewofsymmetry,thisdistortion Wenowconsiderpossibleunitcelldistortionsthatcan
1g
is reminiscent of how finite polarization is generated in couplelinearlytomagneticfieldH andlinearlytotheor-
improper ferroelectrics13. der parameter. Such distortion is allowed by symmetry,
In YBCO distortion of the A symmetry is allowed since the order parameter L is polar time-reversal-odd
2g
in the presence of loop order and intrinsic orthorhom- vector. So a term in the free-energy ∝ uLH is allowed
5
A
1u
A1g A1g
A
2u
A
2g
B
1u
B
1g
B
2u
B
2g
E
u
E E E
g g g
FIG. 4: Inversion-odd representations of the group D .
4h
FIG. 3: Inversion-even representations of the group D .
4h
Blackarrowsrepresentapolarvector,suchasatomdisplace-
ment; red arrow represent an axial vector, such as magneti-
zation. introduces additional contribution to magnetic suscepti-
bility via the term CL2M2 which is obtained upon min-
imization with respect to u/a; here C = β2/E. The
with the lattice distortion u being odd under space in-
change in susceptibility in the presence of loop order
version. To find the symmetry of all allowed distortions
is δχ ∼ −χ2CL2, see Ref 16 for details. Both L and
we must decompose the product of L and H into irre-
M can be measured in units of Bohr magnetons per
ducible representations. For vector H = (Hx(cid:48),Hy(cid:48)) in- unit cell, with L interpreted as staggered magnetiza-
plane (transforming under E ), the product LH breaks
g tion within unit cell; the convenient energy units are
into four irreducible representations set with µ2/˚A3 ∼ 1K. In the experiment the suscep-
B
tibility is measured 3mm3/Mol (or about 3×10−5 in
A1u Ly(cid:48)Hy(cid:48) +Lx(cid:48)Hx(cid:48) dimensionless units for YBCO) at temperatures about
A2u Ly(cid:48)Hx(cid:48) −Lx(cid:48)Hy(cid:48) (14) pseudogap temperature. Since the loop order saturates
B1u Ly(cid:48)Hy(cid:48) −Lx(cid:48)Hx(cid:48) ratherquicklybelowthepseudogaptemperature, wecan
B2u Ly(cid:48)Hx(cid:48) +Lx(cid:48)Hy(cid:48) attributeexperimentalvalueδχ∼0.1χtosaturatedloop
order magnitude L ∼ 0.1µ /Unit cell. We estimate
For magnetic field out of plane H = H (which trans- s B
z C ∼ (δχ/χ2)(1/L2) ∼ 108K−1/Unit cell; the unit cell
formsunderA2g)thedistortionu=(ux(cid:48),uy(cid:48))transforms volume of YBCOsis ∼ 170˚A3. Using C = β2/E with
under E . Indeed, the product uL contains irreducible
u E ∼ 10eV/Unit cell we estimate β ∼ 5×106 in dimen-
representation A :
2g sionlessunits(whereL2andM2areinunitsofenergyper
volumeofunit-cell). Wecannowestimatethemagnitude
A2g Ly(cid:48)ux(cid:48) −Lx(cid:48)uy(cid:48) (15) of the distortion. In the experiment the magnetization
at external field of 1T corresponds to magnetic moment
We conclude that depending on the orientation of the
M ∼ 10−5µ /Unit cell. In this situation the distortion
magnetic field and the order parameter, distortion can B
is estimated as u/a=βLM/E ∼10−6.
appear in each of the five inversion-odd representations,
see Fig 4.
To estimate the magnitude of the distortion we con-
sider free energy
δF =β(u)LM +E(u)2/2 A. Magnetic field dependence of elastic neutron
a a scattering intensity.
where u is distortion in one of the possible irreducible
representations and LM is the corresponding bilinear. We now discuss a possibility of detecting lattice dis-
Minimization gives u/a = βLM/E. We can estimate tortions in the present of external magnetic field by the
the magnitude of the coefficient β as well as typical dis- usual x-ray or neutron crystallographic techniques. The
tortion magnitude using data from recent magnetization dominant contribution to elastic neutron scattering in-
measurements4. The cubic term of the form β(u)LM tensity comes from contact interaction between neutron
a
6
andatomicnucleus. Onecananalyzethesestartingwith We rewrite these in a matrix form :
  
I (q)∝|A (q)|2 0 0 −λ2Ly(cid:48) Ex(cid:48)
N N (cid:2) (cid:3)
(cid:88) Hx(cid:48) Hy(cid:48) Hz  0 0 λ2Lx(cid:48)  Ey(cid:48)  .
AN(q)= fαeiq(R+rα) (16) −λ1Ly(cid:48) λ1Lx(cid:48) 0 Ez
R,α (21)
where q is (neutron) momentum transfer. rα is position Thetwocouplingconstantsλ1,2havedifferentmagnitude
of atom α within unit cell R; fα are nuclear scattering since they describe coupling in two different irreducible
amplitudes. Let us assume that due to lattice distortion representations. Assuming a single domain phase, say
atomsshifttoanewpositionsrα →rα+uα withinunit L = (Lx(cid:48),0), we obtain a magneto-electric tensor in the
cell and ask whether there is a change in intensity to symmetry-broken phase of the form :
linear order in u :
α
 
0 0 0
δIN(q)=δA†N(q)AN(q)+c.c (17) 0 0 α2  . (22)
0 α 0
1
For centro-symmetric crystal the sum in Eq. (16) evalu-
atestoarealnumber(thenuclearamplitudesfα arereal where α1,2 = λ1,2Lx. General symmetry consideration
for q typical of slow neutrons), i.e., the amplitude A (q) allow axial time-reversal-odd tensor of the second rank
N
isreal. Tolinearorderinuα thechangeintheamplitude of this form in the crystal class mmm, see Ref. 15.
is Wenowdiscussthemagnitudeofthemagneto-electric
susceptibility α in cuprates that is associated with the
(cid:88)
δA (q)= f iqu eiq(R+rα) (18) magnetic structure in the pseudogap phase. In spin sys-
N α α
R,α tems the magneto-electric coefficient is limited by the
magnitude of the spin-orbit coupling18; a typical value
In loop-ordered phase, external magnetic field can pro- α ∼ 10−4 to 10−3 (dimensionless in cgs units) is several
duce several types of distortions, all of which are odd ordersofmagnitudesmallerthantheupperboundsetby
under spatial inversion, see Eqs. (14) and (15). For the thermodynamics20. In the cuprates the magnetic struc-
inversion-odd distortion uα the sum in the right-hand ture in the pseudogap phase has an orbital origin. How-
side of Eq. (18) is pure imaginary; intensity variation everduetofiniteresistivityofcupratesinthepseudogap
δIN(q) given by Eq. (17) vanishes. We conclude that phase the observation of the magneto-electric phenom-
the lattice distortion associated with external magnetic ena may not be possible. To estimate the magnitude of
field, though non-zero, does not change intensity of the the magneto-electric coefficient α we start with thermo-
Bragg peaks in elastic neutron scattering to linear order dynamic identity (see, e.g., Ref. 19)
in magnetic field.
∂2F ∂M ∂P
α =− = j = i (23)
ij
∂E ∂H ∂E ∂H
i j i j
V. MAGNETO-ELECTRIC EFFECTS IN THE
PSEUDOGAP PHASE We use the last equality, i.e., we apply magnetic field H
and estimate the induced polarization. It has been esti-
mated earlier that in the experiments in Ref. 4 the dis-
Copper-oxide metal in the pseudogap phase exhibits
placement of copper and oxygen atoms relative to each
magneto-electric effects. Magnetic field out of plane be-
other is of order u/a ∼ 10−6 at external field of 1T; the
longs to A irreducible representation. The product of
2g symmetry of the displacement has been discussed earlier
an in-plane electric field E and loop order L has a com-
inthepaper. Thepolarizationwhichcorrespondstosuch
ponent E×L that belongs to the same, A irreducible
2g displacementisestimatedasP ∼(e/a2)(u/a)∼10−3esu
representation(andsimilarlyforout-ofplaneelectricfield
(here a is the ab plane unit cell size ∼ 4˚A; we have as-
which belongs to A ) :
2u sumedthatcopperandoxygeninthecopper-oxideplane
have charge ±e). We find that magneto-electric sus-
A2g Hz E×L=Ey(cid:48)Lx(cid:48) −Ex(cid:48)Ly(cid:48) (19) ceptibility is α = P/H ∼ 10−5, (which is dimension-
A2u Ez H ×L=Hy(cid:48)Lx(cid:48) −Hx(cid:48)Ly(cid:48) less in cgs units). This corresponds to about 3×10−8
Gauss/(Volt/cm). This value could be observed exper-
Corresponding to the two lines in this table, there are
imentally in an insulator but not in the metallic pseu-
two terms in the free energy that couple electric and
dogap state of the cuprates where final voltage arises in
magnetic fields in the presence of loop order parameter
the bulk only due to finite resistivity. We should men-
L=(Lx(cid:48),Ly(cid:48)) : tion for completeness that the orbital ordered phase also
supports a zero-field Hall effect. But for a field applied
δFME =λ1Hz(Ey(cid:48)Lx(cid:48) −Ex(cid:48)Ly(cid:48)) of 10 volts/cm, the Hall effect would correspond only to
+λ2Ez(Hy(cid:48)Lx(cid:48) −Hx(cid:48)Ly(cid:48)). (20) that due to a magnetic field of about 10−6 Gauss.
7
Notalleffectsaccompanyingtheobservedorbitalorder Bourges and Steve Kivelson are greatly acknowledged.
are so small. For example, x-ray dichroism is predicted The suggestion of a distortion in the ordered phase pro-
to be observable8. So is the predicted second harmonic portionaltoanappliedmagneticfieldswasmadetousby
generation21. All the effects investigated here come due P.W.Anderson. Thesuggestionofthemonoclinicdistor-
to the periodic magnetic fields generated by the orbital- tionduetotheorbitalorderworkedouthereisoriginally
ordered phase, which are always small. The effective due to Steve Kivelson. A.S. would like to acknowledge
electronic energy of such phases is two to three orders hospitality of the Aspen Center of Physics where part of
of magnitudes larger than the magnetic energies. This is the work was done.
similar to the difference between exchange energies and
magnetic field energies in magnetic order due to spin-
moments.
Acknowledgments
Numerous discussions with Vivek Aji, Philippe
1 B. Fauqu´e, Y. Sidis, V. Hinkov, S. Pailh`es, C. T. Lin, 17 T. R. Chien, Z. Z. Wang, and N. P. Ong,
X. Chaud, and P. Bourges, Phys. Rev. Lett. 96, 197001 Phys. Rev. Lett. 67, 2088 (1991).
(2006);H.Mook,inBulletin of American Physical Society 18 G. T. Rado, Phys. Rev 128, 2546 (1962).
March Meeting (2008). 19 J. P. Rivera, Ferroelectrics 161, 165 (1994).
2 Y. Li, V. Baledent, N. Barisic, P. Bourges, Y. Cho, 20 W. F. Brown, R. M. Hornreich, and S. Shtrikman,
B. Fauque, Y. Sidis, G. Yu, X. Zhao, and M. Greven, Na- Phys. Rev. 168, 574 (1968);
ture 455, 372 (2008). 21 M. E. Simon, and C. M. Varma, Phys. Rev. B 67, 054511
3 A. Kaminski, S. Rosenkranz, H. M. Fretwell, J. C. Cam- (2003).
puzano, Z. Li, H. Raffy, W. G. Cullen, H. You, C. G. Ol-
son,C.M.Varma,andH.Ho¨chst,Nature 416,610(2002);
A.Kaminski,H.M.Fretwell,M.R.Norman,M.Randeria,
S.Rosenkranz,U.Chatterjee,J.C.Campuzano,J.Mesot,
T.Sato,T.Takahashi,T.Terashima,M.Takano,K.Kad-
owaki, Z. Z. Li, and H. Raffy, Phys. Rev. B 71, 014517
(2005).
4 B. Leridon, P. Monod, and D. Colson, arXiv:0806.2128
(2008).
5 M. E. Simon, and C. M. Varma, Phys. Rev. Lett. 89,
247003 (2002).
6 C. M. Varma, Phys. Rev. B 55, 14554 (1997).
7 C. M. Varma, Phys. Rev. B 73, 155113 (2006).
8 S.DiMatteo,andC.M.Varma,Phys. Rev. B 67,134502
(2003).
9 V. M. Dubovik, and V. V. Tugushev, Phys. Rep., North-
Holland 187, 145 (1990).
10 Ya. B. Zeldovich, Zh. Eksp. Teor. Fiz. 6, 1184 (1958).
11 V. Aji, and C. M. Varma, Phys. Rev. Lett 99, 067003
(2007).
12 C´edric Weber, Ph.D. Thesis, (E´cole Polytechnique Fed-
eraleDeLausanne,2007);C.Weber,A.Laeuchli,F.Mila,
and T. Giamarchi, arXiv:0803.3983 (2008).
13 L. D. Landau, and E. M. Lifshitz, Course of Theoreti-
cal Physics, Vol. 8, E. M. Lifshitz, and L. P. Pitaevskii,
ElectrodynamicsofContinuousMedia (Pergamon,Oxford,
1984).
14 H. Stuart, and N. Ridley, Brit. J. Appl. Phys.
(J. Phys. D) 2, 485 (1969).
15 R.R.Birss,SymmetryandMagnetism,(Wiley-Interscience
Inc., New York, 1964).
16 M.S.Grønsleth,T.B.Nilssen,E.K.Dahl,E.B.Stiansen,
C. M. Varma, and A. Sudbø, Phys. Rev. B 79, 094506
(2009).