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Tunneling gap as evidence for time-reversal symmetry breaking at surfaces of high-temperature superconductors
ELSEVIER
Physiea C 234 (1994) 280-284
Tunneling gap as evidence for time-reversal symmetry breaking at surfaces of high-temperature superconductors R.B.
Laughlin
Department of Physics, Stanford University, Stanford, CA 94305, USA
Received 8 August 1994; revised manuscript received 27 September 1994
Abstract
It is argued that recent Josephson junction and point-contact tunneling experiments, interpreted as intended by their authors, indicate that time-reversal symmetry breaking occurs at surfaces of cuprate superconductors. The variation among experiments and the failure of previous searches to find T violation are ascribed to disorder and effects of three-dimensionality. The anyon approach to the t-J model is shown to predict a conventional BCS order parameter of dx2_r2+i~d~r symmetry, with E roughly 3 times the doping fraction & which is consistent with these experiments but not demonstrated by them.
The purpose of this Letter is to point out that timereversal symmetry breaking [ 1 ], the key prediction o f the anyon approach to the high-temperature superconductivity problem, may already have been demonstrated in a series o f recent experiments conducted for other purposes. While it is wise to be cautious, particularly in light o f previous failures to detect T violation [2 ] and the possibility that one or more of these experiments may later prove to be wrong or misinterpreted, the implication of the experiments as they now stand is highly suggestive. The relevant experiments are the photoemission [ 3 ] and light-scattering [ 4 ] measurements of the gap anisotropy, the microwave measurements o f the low-temperature conductivity [ 5 ], the Josephson phase coherence experiments o f Wollman et al. [ 6 ], Brawner and Ott [ 7 ] and Tsuei et al. [ 8 ], and the scanning tunneling microscope measurements o f Hasegawa et al. [9 ] reporting an intrinsic energy gap. The essence of the argument is that all o f these except the last point to the occurrence o f a conventional BCS order parameter ofd~2_y2 symmetry, the behavior c o m m o n l y
found in theories based on incipient antiferromagnetism o f the conducting electrons [ 10]. This, however, is fundamentally incompatible with the last experiment unless the order parameter is complex, which is impossible unless the ground state violates T. Thus the simultaneous occurrence o f d wave superconductivity and an energy gap in any part of the sample, if true, constitutes definitive evidence for T violation. A dx2_y2 order parameter Ak is distinguished from a conventional s wave order parameter by sign reversals. In high-To superconductors this is most easily discussed in terms o f the idealized electron energy band E ° = -2to[COs(kxb) + c o s ( k y b ) ] ,
where to is an energy parameter and b is the b o n d length of a two-dimensional square lattice. One imagines creating a Fermi sea by filling the states with Ek < 0. Allowing the electrons to interact through weak near-neighbor spin exchange [ 10 ] then leads to a su-
0921-4534/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSD10921-4534 (94) 00631-8
(1)
R.B. Laughlin / PhysicaC 234 (1994)280-284 perconducting state for which the quasiparticles have energies Ek=
-t-4(E°)2+ IA, I2 ,
(2)
where Ak=zt° [cos(kxb) - cos(krb) ] .
(3)
The gap parameter Ak is positive in the x-direction, negative in the y-direction, and zero at the nodes in between. In an s wave superconductor Ak is the constant A°. Of the experiments listed above, only the Josephson experiments can directly sense the sign reversal of Ak, even in principle. The experiments of Wollman et al. [ 6 ] and Brawner and O t t [ 7 ] accomplish this by detecting electric currents spontaneously generated in a loop of Pb wire connected between the xand y-faces of a high-To superconductor. The experiment of Tsuei et al. [ 8 ] detects similar currents in a small high-To loop consisting of three single crystals with different orientations. Although these experiments agree with each other and with the large body of circumstantial evidence for d wave pairing, they are still quite controversial. In what follows we shall assume that they are right. It must be emphasized that they constitute the only direct evidence for sign reversal of Ak. These experiments are supported by considerable indirect evidence for d wave superconductivity with nodes at k = ( +~/2b, +rc/2b), in particular by the presence in a//samples of low-energy excitations in the superconducting state. The evidence is, unfortunately, complicated by the disorder effects that plague these materials. For example, Giaver tunneling finds states in the gap so commonly that the claim of Hasegawa et al. [ 9 ] to have observed a clean gap, the experiment motivating this Letter, is widely questioned. At the same time, the large zero-bias conductance seen in most tunneling experiments is commonly dismissed as an artifact of disorder at the tunnel contact [ 1 1 ]. The T 2 deviation of the penetration depth from its zero-temperature value, and its crossover to linear Tbehavior above about 5 K, seen in high-quality samples [12] is consistent with dx2_y2 superconductivity only if disorder that is different or impossible to eliminate from the samples sample assumed to exist. The same is true of the zeromechanism microwave conductivity [ 5,12 ]. The in-
281
trinsic heat capacity below 10 K is not known for any high-To superconductor because the signal is always swamped by a large Schottkey-like heat capacity [ 14 ] similar to that expected of a spin glass. That at least some of these excitations are intrinsic and attributable to a d wave node is indicated by several less accurate or model-dependent experiments, the most accessible of which is the angle-resolved photoemission work of Shen et al. [ 3 ]. This reports measurable changes to the quasipartide energies Ek resulting from cooling the sample through its superconducting transition except near this special value of k. The complex temperature and polarization dependence of inelastic light scattering is accounted for quantitatively by dx2_y2 superconductivity if reasonable assumptions are made about the relevant matrix elements [4]. The same is true for the extensive magnetic resonance data [ 15 ]. While the situation is still confusing and controversial, it is clear that a large body of experimental evidence is consistent with the simultaneous presence in all samples of both disorder and a d wave node. Let us now consider the tunneling experiment of Hasegawa et al. [ 9 ]. It is very important for our argument that this experiment was performed with a scanning tunneling microscope tip, that it reported a well-developed energy gap only at certain places on the sample, and that it was inconsistent with conventional thin-film tunnel junction experiments, which never reveal a full gap [ 11 ]. Barring the possibility that it is an experimental artifact, such as Coulomb blockade [ 16 ] or an effect of anisotropic tunneling, the cleanliness of this result implies that a genuine and full energy gap in the quasiparticle spectrum occasionally develops at some places on the surface. The intermittency of this effect is an obvious worry, but it could easily be caused by surface disorder, particularly if the order parameter had sign reversals [ 13 ]. The absence of the gap in the bulk is a different matter. Since the gaplessness of the sample interior does not appear to be a disorder effect, we must speculate that the second symmetry breaking required to open a gap occurs only at the surface, due perhaps to a surface term in the fee energy. However, regardless of the mechanism by which the gap is destroyed, its existence anywhere in the sample indicates local T violation, since an energy gap is forbidden as a matter of
282
R.B. Laughlin /Physica C 234 (1994) 280-284
principle in a d wave superconductor with a real order parameter. Let us now consider the specific complex order parameter
(4)
where ~ is a number, implicit in the anyon technique. That there should be such an order parameter was first suggested by Rokhsar [ 17 ], who also pointed out that it was potentially inconsistent with the known phase diagram of these materials. Since the real and imaginary parts of d~hi-a1 lie in different irreducible representations of the lattice point group, they cannot mix in the Ginzberg-Landau functional, and thus must develop as separate order parameters as the temperature is lowered. This would give two phase transitions, rather than the observed one. However, the suppression of the smaller dxy order parameter almost everywhere in the sample would resolve this paradox by preventing the acquisition of long-range order and killing the lower phase transition. It would also kill all effects of global T violation, and account for the failure of previous searches to find them [2 ]. Let us now show that this order parameter is implicit in the anyon approach to the t-J hamiltonian
•gt~t_j= ~ {-t~oc~ack~+Js#'Sk}, El,k)
(5)
and estimate its magnitude. It should be noted such an order parameter is not uniquely obtained this way, but also arises in certain simple BCS hamiltonians [ 18 ]. As usual, Sj denotes the spin operator for the jth site,=
~
II ,...,IM
at,,...,tMIll,...,lM),
J N
X n ( 1 - n k , , - n l , , ~ ) I $tn,x > ,
(7)
19l
/t~him =3°[cos(kxb) - cos (kyb) +i~ sin(k~b) sin(kyb) ] ,
N
IIi .... ,IM>= l--[ (1--n~,nj,)
(6)
where I11..... IM) is a basis wavefunction describing "holons" at sites Ii .... , IM. This is given explicitly by [191
where nj~=c],% is the number operator for an electron of spin a at site j, and 17'n~,> is the ground state of the commensurate flux hamiltonian
I2 2 e x p
× (Yj + Yk)/b2}c],Ck,,
(8)
with N - M electrons. Yru~ violates T, as is required for the holon basis to be defined. The holons defined by Eq. (7) obey ½fractional statistics in that varying the expansion coefficients in Eq. (6) to minimize the expected energy ( ~PI~t-JI ~> is equivalent, when J is small, to solving the fermion problem [ 19 ]
~yo.
=
h2 2m* IPJ+ c.4Jl e 2,
.4j= l_hc S" ~X (rj--rk) 2 e ~'¢j Irj--rkl 2 '
(9)
with isospin. The latter corresponds to the valley degeneracy of the holon band structure, which in the J--,0 limit is described by Eq. (8) without spin. Since the holon dispersion relation in this limit is given by E ~ °1°n _~ +
2tx/cos2( kxb ) +cos2(kyb),
(10)
we find that the valley minima occur at k= (0, O) and (n/b, O) and are characterized by the mass m*~h2/
( x/C2b2t). The first step in the order parameter calculation is to compute the non-local order parameter of the continuum anyon gas described by Eq. (9) [ 20 ]. We will take the ground state to be [ ~)anyon ) ~-~ H (½ e x p ( - I q l / ~ / - ~ p , p _ ¢ ) ) l ~ l ~ F > , q
(11) where p=J/b 2 is the particle density, pq= Yjexp(iq-rj) is the density operator, and [~HF> is the ground state of the Hamiltonian obtained by substituting the mean-field vector potential <.4)=he~ (2e)py~ for .4j in Eq. (9). The prefactor in this
R.B. Laughlin / PhysicaC 234 (1994)280--284 expression is the usual modification of the HartreeFock ground state implicit in the random phase approximation. Let 9/+ (z) and ~_ (z) denote the operators annihilating holons with "up" and "down" isospin, respectively, at z = x + iy in the Fermi representation. The nonlocal order parameter is given in terms of these by [20 ]
283
(l~, ..., l~lCk,,q,¢C]~C~, I/i, "", l u )
M Otm3 (z[~*+z~*)/2-z~ (Zj+Zk)/2--Z,~ x I(Z]+Z'k)/2--Z,~I I(Zj+Zk)/2--Z,~I × (j', k'lcj,rCk,, 10) (OIc]tc~, IJ, k)
~ < ~anyon I ~'+ (ZI)~'/~-"(Zl) {~ff~+(Z) ~/+ (z)
+ cyclic permutations,
+ ~t (z) ~,_ (z))~,+ (z~) ¢/_ (z~)l ¢'.n,o. > X (z~--z*__.......~)(z2--z) d2z
Iz~-zl Iz2-zl
~
(1.9p)2.
I~,-~1-®
(12) The numerical value of 1.9p reported here for the first time is obtained using the hypernetted chain technique [19]. We will adopt the notation (V___(z) ~,(z) ) = 1.9p as shorthand for this result. The second step is to convert this continuum order parameter to the site basis. The unitary transformation relating ¥+ (z) and ~u_ (z) to ~u(]), the Fermi operator to annihilate a holon at site rj= (lj, mj)b, is simply the matrix of one-body eigenstates of Eq. (8) at the two valley minima. We thus have ~,(j) ~ x/~b{cos [ ( ~ _ i m2) n] ~v+(z) + ( - 1)~sin[ ( / + ½m])n]~u_ (z)}.
(13)
In obtaining this expression, we have imagined the sample to be divided into four-site cells, and that z defines the center of the cell containingj. Then, substituting Eq. ( 13 ) into Eq. (12), and using the fact that V+ (z) and ¥_ (z) anticommute with themselves and each other, we obtain
where 10) denotes the state with no holons and zj= (lj+imj) b. The product onot in this expression is the same "unwinding" factor appearing in Eq. ( 1 2 ) and is the microscopic justification for its inclusion in Eq. (12). The remaining factor is given approximately by (OIclrc~, Ij, k) ~ (zl -z2_______) ith_12)(mt+m2) (
IZl-z~l
-
-
1 )ll+ml
q
['( 1 - ~ ) exp{- n/4( 1 - J ) } ,
I and 2 near neighbors,
l
l and2 ~ond neighbors,J
x[
a~xp~-n6~, O,
otherwise. (16)
The final step is to combine Eqs. (15) and (16) with Eq. (14) to obtain an expression for (c]~ c~ ). This is accomplished by multiplying together the right sides of Eqs. (14) and (16) and then dividing out the factor (z,-z2)/Izl-z21. This latter step accounts for the transformation of the basis functions defined by Eq. (7), which are symmetric under interchange of the holon positions, to their Fermi representation. We obtain finally
(c~tc~,) ~ i"'-1~)~"2- m') ( -- 1 ),~,--m~ ['V/2( 1-- ~) exp{-- 7r/4( 1-- J) },1and 2 near neighbors, ] x1.9 t~/ - 2J exp{- ~t~}, 1and 2 secondneighbors,| l J O, otherwise.
(¢t( 1 )V(2) > - 1.9 J( - 1 ) ll+m2+(ll+12")r¢~l X { ~ 2 [1--(--1
( 15 )
/
)h+12+ml+m2]
(17) This is equivalent to Eq. (4) with
- [1 - ( - 1 ) u,+z,)¢m,+,-n ] ) .
(14) ~= l~8~e O-5'~)m4
The third step is to evaluate the matrix element of cttc[ , to annihilate a holon pair into the vacuum. This matter has been studied extensively in previous papers [ 19 ] and is too involved to discuss in detail here. We shall simply quote the result
(18)
Let us now make some comments about this result. It is quite crude and should be compared with experiment carefully. For example, it makes the unphysical prediction that superconductivity occurs in the t-
284
R.B. Laughlin /Physica C 234 (1994)280-284
J m o d e l at any value o f J, t, a n d ~. A m o r e formal d e v e l o p m e n t o f the a n y o n a p p r o a c h using gauge-theory techniques [ 21 ] cures p r o b l e m s o f this kind, b u t is too technical to discuss here. Let us simply state the m a i n results. ( 1 ) F o r realistic values of J / t the o r d e r p a r a m e t e r is significantly smaller than Eq. ( 1 7 ) predicts, due to r e t a r d a t i o n effects. ( 2 ) The equations have an antiferromagnetic instability for ~ < 0.05 that d e p e n d s weakly on J/t. ( 3 ) The commensurate flux b a n d structure o f Eq. ( 6 ) has a gap collapse at 6--- ] which causes calculations for doping fractions larger t h a n this to be unreliable. T h a t the calculation finds c o n v e n t i o n a l superconducting o r d e r a n d d wave s y m m e t r y is not surprising. The anyone a p p r o a c h is a legitimate variational technique for the t-J model, which is k n o w n by m o r e reliable m e t h o d s to have a tendency to d wave pairing [ 22,23 ]. Also, it has been known for several years that " f l u x " vacua are f u n d a m e n t a l l y related to d wave superconducting states [24 ]. The calculation is significant m a i n l y because it predicts that superconducting pairing by m e a n s o f " s p i n fluctuations" tends naturally to an o r d e r p a r a m e t e r with dx2_r2 + i¢dxy symmetry, with ~ significantly large.
Acknowledgements I wish to t h a n k M.R. Beasley, A. Kapitulnik, T.H. Geballe, S. Doniach, Z. Zou, A.A. Abrikosov and D.J. Scalapino for numerous helpful discussions. The work was s u p p o r t e d p r i m a r i l y b y the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t No. DMR-91-20361. Additional support was p r o v i d e d by the N S F M R L Program through the Center for Materials Research at Stanford University.
References [ 1] X.-G. Wen, F. Wilczek and A. Zee, Phys. Rev. B 39 (1989) 11413.
[2] S. Spielman et al., Phys. Rev. Lett. 69 (1992) 123; T.W. Lawrence,A. Sz~Skeand R.B. Laughlin, Phys. Rev. Lett. 69 (1992) 439. [3] Z.-X. Shen et al., Phys. Rev. Lett. 70 (1993) 1553; J. Phys. Chem. Solids 53 (1992) 1583; D.S. Dessau et al., J. Phys. Chem. Solids 52 ( 1991 ) 1401; B.O. Wells et al., Phys. Rev. B 46 (1992) 11830. [4] T.P. Devereaux et al., Phys. Rev. Lett. 72 (1994) 396; T. Stanfer et al., Phys. Rev. Len. 68 (1992) 1069. [ 5 ] D.A. Bonn et al., Phys. Rev. B 47 (1993) 11314. [ 6 ] D.A. Wollman et al., Phys. Rev. Lett. 72 (1993) 2134. [ 7 ] D.A. Brawner and H.R. Ott, Phys. Rev. B, to be published. [8] C.C. Tsuei et al., Phys. Rev. Left. 73 (1994) 593. [ 9 ] T. Hasegawa et al., J. Phys. Chem. Solids 54 (1993) 1351. [ 10] S.R. White et al., Phys. Rev. B 39 (1989) 839; P. Monthoux and D. Pines, Phys. Rex,.Left. 69 (1992) 961. [ 11 ] J.M. Valles Jr. et al., Phys. Rev. B 44 ( 1991 ) 11986; D, Mandrus et al., Nature (London) 351 ( 1991 ) 460. [ 12 ] Z. Ma et al., Phys. Rev. Lett 71 ( 1993 ) 781; W. Hardy et al., Phys. Rev. Lett. 70 (1993) 3999; S. Anlage et al., Phys. Rev. B 44 ( 1991 ) 9764. [ 13 ] M. Prohammer and J.P. Carbotte, Phys. Rev. B 43 ( 1991 ) 5370; J. Annett, N. Goldenfeld and S.R. Renn, Phys. Rev. B 43 (1991) 2778; R.J. Radtke, K. Levin, H.-B. Schiittler and M.R. Norman, Phys. Rev. B 48 (1993) 653; F. Gross et al., Z. Phys. B 64 (1986) 175; P.J. Hirschfeld, W.O. Putikka and D.J. Scalapino, Phys. Rev. Lett. 71 (1993) 3705. [ 14 ] N.E. Phillips, R.A. Fisher and J.E. Gordon, Prog. Low Temp. Phys. 13 (1992) 267. [ 15 ] N. Bulut and D.J. Scalapino, Phys. Rev. Lett. 68 (1992) 706; J.A. Martindale et al., Phys. Rev. Lett. 68 (1992) 702. [ 16 ] P.A. Lee, Phys. Rev. Lett. 71 (1993) 1887. [17] D.S. Rokhsar, Phys. Rev. Lett. 70 (1993) 493. [ 18] F. Wenger and S. (~stlund, Phys. Rev. B 47 (1993) 5977; F. Wenger, Licentiate Thesis, U. of Gothenburg (1993). [ 19] Z. Zou, J.L. Levy and R.B. Laughlin, Phys. Rev. B 45 (1992) 993; Z. Zou and R.B. Laughlin, Phys. Rev. B 42 (1990) 4073; R.B. Laughlin and Z. Zou, Phys. Rev. B 41 (1989) 664; P. B6ran and R.B. Laughlin, Phys. Rev. B 48 (1993) 10382. [20] S.M. Girvin et al., Phys. Rev. Lett. 65 (1990) 1671. [21 ] A.M. Tikofsky, R.B. Laughlin and Z. Zou, Phys. Rev. Lett. 69 (1992) 3670. [22] E. Dagotto and J. Riera, Phys. Rev. Lett. 70 (1993 ) 682. [23 ] Y. Ohta, T. Shimozato, R. Eder and S. Maekawa, Phys. Rev. Lett. 73 (1994) 324. [24] F.C. Zhang, C. Gros, T.M. Rice and H. Shiba, Supercond. Sci. Teehnol. 1 (1989) 36.
Physiea C 234 (1994) 280-284
Tunneling gap as evidence for time-reversal symmetry breaking at surfaces of high-temperature superconductors R.B.
Laughlin
Department of Physics, Stanford University, Stanford, CA 94305, USA
Received 8 August 1994; revised manuscript received 27 September 1994
Abstract
It is argued that recent Josephson junction and point-contact tunneling experiments, interpreted as intended by their authors, indicate that time-reversal symmetry breaking occurs at surfaces of cuprate superconductors. The variation among experiments and the failure of previous searches to find T violation are ascribed to disorder and effects of three-dimensionality. The anyon approach to the t-J model is shown to predict a conventional BCS order parameter of dx2_r2+i~d~r symmetry, with E roughly 3 times the doping fraction & which is consistent with these experiments but not demonstrated by them.
The purpose of this Letter is to point out that timereversal symmetry breaking [ 1 ], the key prediction o f the anyon approach to the high-temperature superconductivity problem, may already have been demonstrated in a series o f recent experiments conducted for other purposes. While it is wise to be cautious, particularly in light o f previous failures to detect T violation [2 ] and the possibility that one or more of these experiments may later prove to be wrong or misinterpreted, the implication of the experiments as they now stand is highly suggestive. The relevant experiments are the photoemission [ 3 ] and light-scattering [ 4 ] measurements of the gap anisotropy, the microwave measurements o f the low-temperature conductivity [ 5 ], the Josephson phase coherence experiments o f Wollman et al. [ 6 ], Brawner and Ott [ 7 ] and Tsuei et al. [ 8 ], and the scanning tunneling microscope measurements o f Hasegawa et al. [9 ] reporting an intrinsic energy gap. The essence of the argument is that all o f these except the last point to the occurrence o f a conventional BCS order parameter ofd~2_y2 symmetry, the behavior c o m m o n l y
found in theories based on incipient antiferromagnetism o f the conducting electrons [ 10]. This, however, is fundamentally incompatible with the last experiment unless the order parameter is complex, which is impossible unless the ground state violates T. Thus the simultaneous occurrence o f d wave superconductivity and an energy gap in any part of the sample, if true, constitutes definitive evidence for T violation. A dx2_y2 order parameter Ak is distinguished from a conventional s wave order parameter by sign reversals. In high-To superconductors this is most easily discussed in terms o f the idealized electron energy band E ° = -2to[COs(kxb) + c o s ( k y b ) ] ,
where to is an energy parameter and b is the b o n d length of a two-dimensional square lattice. One imagines creating a Fermi sea by filling the states with Ek < 0. Allowing the electrons to interact through weak near-neighbor spin exchange [ 10 ] then leads to a su-
0921-4534/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSD10921-4534 (94) 00631-8
(1)
R.B. Laughlin / PhysicaC 234 (1994)280-284 perconducting state for which the quasiparticles have energies Ek=
-t-4(E°)2+ IA, I2 ,
(2)
where Ak=zt° [cos(kxb) - cos(krb) ] .
(3)
The gap parameter Ak is positive in the x-direction, negative in the y-direction, and zero at the nodes in between. In an s wave superconductor Ak is the constant A°. Of the experiments listed above, only the Josephson experiments can directly sense the sign reversal of Ak, even in principle. The experiments of Wollman et al. [ 6 ] and Brawner and O t t [ 7 ] accomplish this by detecting electric currents spontaneously generated in a loop of Pb wire connected between the xand y-faces of a high-To superconductor. The experiment of Tsuei et al. [ 8 ] detects similar currents in a small high-To loop consisting of three single crystals with different orientations. Although these experiments agree with each other and with the large body of circumstantial evidence for d wave pairing, they are still quite controversial. In what follows we shall assume that they are right. It must be emphasized that they constitute the only direct evidence for sign reversal of Ak. These experiments are supported by considerable indirect evidence for d wave superconductivity with nodes at k = ( +~/2b, +rc/2b), in particular by the presence in a//samples of low-energy excitations in the superconducting state. The evidence is, unfortunately, complicated by the disorder effects that plague these materials. For example, Giaver tunneling finds states in the gap so commonly that the claim of Hasegawa et al. [ 9 ] to have observed a clean gap, the experiment motivating this Letter, is widely questioned. At the same time, the large zero-bias conductance seen in most tunneling experiments is commonly dismissed as an artifact of disorder at the tunnel contact [ 1 1 ]. The T 2 deviation of the penetration depth from its zero-temperature value, and its crossover to linear Tbehavior above about 5 K, seen in high-quality samples [12] is consistent with dx2_y2 superconductivity only if disorder that is different or impossible to eliminate from the samples sample assumed to exist. The same is true of the zeromechanism microwave conductivity [ 5,12 ]. The in-
281
trinsic heat capacity below 10 K is not known for any high-To superconductor because the signal is always swamped by a large Schottkey-like heat capacity [ 14 ] similar to that expected of a spin glass. That at least some of these excitations are intrinsic and attributable to a d wave node is indicated by several less accurate or model-dependent experiments, the most accessible of which is the angle-resolved photoemission work of Shen et al. [ 3 ]. This reports measurable changes to the quasipartide energies Ek resulting from cooling the sample through its superconducting transition except near this special value of k. The complex temperature and polarization dependence of inelastic light scattering is accounted for quantitatively by dx2_y2 superconductivity if reasonable assumptions are made about the relevant matrix elements [4]. The same is true for the extensive magnetic resonance data [ 15 ]. While the situation is still confusing and controversial, it is clear that a large body of experimental evidence is consistent with the simultaneous presence in all samples of both disorder and a d wave node. Let us now consider the tunneling experiment of Hasegawa et al. [ 9 ]. It is very important for our argument that this experiment was performed with a scanning tunneling microscope tip, that it reported a well-developed energy gap only at certain places on the sample, and that it was inconsistent with conventional thin-film tunnel junction experiments, which never reveal a full gap [ 11 ]. Barring the possibility that it is an experimental artifact, such as Coulomb blockade [ 16 ] or an effect of anisotropic tunneling, the cleanliness of this result implies that a genuine and full energy gap in the quasiparticle spectrum occasionally develops at some places on the surface. The intermittency of this effect is an obvious worry, but it could easily be caused by surface disorder, particularly if the order parameter had sign reversals [ 13 ]. The absence of the gap in the bulk is a different matter. Since the gaplessness of the sample interior does not appear to be a disorder effect, we must speculate that the second symmetry breaking required to open a gap occurs only at the surface, due perhaps to a surface term in the fee energy. However, regardless of the mechanism by which the gap is destroyed, its existence anywhere in the sample indicates local T violation, since an energy gap is forbidden as a matter of
282
R.B. Laughlin /Physica C 234 (1994) 280-284
principle in a d wave superconductor with a real order parameter. Let us now consider the specific complex order parameter
(4)
where ~ is a number, implicit in the anyon technique. That there should be such an order parameter was first suggested by Rokhsar [ 17 ], who also pointed out that it was potentially inconsistent with the known phase diagram of these materials. Since the real and imaginary parts of d~hi-a1 lie in different irreducible representations of the lattice point group, they cannot mix in the Ginzberg-Landau functional, and thus must develop as separate order parameters as the temperature is lowered. This would give two phase transitions, rather than the observed one. However, the suppression of the smaller dxy order parameter almost everywhere in the sample would resolve this paradox by preventing the acquisition of long-range order and killing the lower phase transition. It would also kill all effects of global T violation, and account for the failure of previous searches to find them [2 ]. Let us now show that this order parameter is implicit in the anyon approach to the t-J hamiltonian
•gt~t_j= ~ {-t~oc~ack~+Js#'Sk}, El,k)
(5)
and estimate its magnitude. It should be noted such an order parameter is not uniquely obtained this way, but also arises in certain simple BCS hamiltonians [ 18 ]. As usual, Sj denotes the spin operator for the jth site,
~
II ,...,IM
at,,...,tMIll,...,lM),
J N
X n ( 1 - n k , , - n l , , ~ ) I $tn,x > ,
(7)
19l
/t~him =3°[cos(kxb) - cos (kyb) +i~ sin(k~b) sin(kyb) ] ,
N
IIi .... ,IM>= l--[ (1--n~,nj,)
(6)
where I11..... IM) is a basis wavefunction describing "holons" at sites Ii .... , IM. This is given explicitly by [191
where nj~=c],% is the number operator for an electron of spin a at site j, and 17'n~,> is the ground state of the commensurate flux hamiltonian
I2 2 e x p
× (Yj + Yk)/b2}c],Ck,,
(8)
with N - M electrons. Yru~ violates T, as is required for the holon basis to be defined. The holons defined by Eq. (7) obey ½fractional statistics in that varying the expansion coefficients in Eq. (6) to minimize the expected energy ( ~PI~t-JI ~> is equivalent, when J is small, to solving the fermion problem [ 19 ]
~yo.
=
h2 2m* IPJ+ c.4Jl e 2,
.4j= l_hc S" ~X (rj--rk) 2 e ~'¢j Irj--rkl 2 '
(9)
with isospin. The latter corresponds to the valley degeneracy of the holon band structure, which in the J--,0 limit is described by Eq. (8) without spin. Since the holon dispersion relation in this limit is given by E ~ °1°n _~ +
2tx/cos2( kxb ) +cos2(kyb),
(10)
we find that the valley minima occur at k= (0, O) and (n/b, O) and are characterized by the mass m*~h2/
( x/C2b2t). The first step in the order parameter calculation is to compute the non-local order parameter of the continuum anyon gas described by Eq. (9) [ 20 ]. We will take the ground state to be [ ~)anyon ) ~-~ H (½ e x p ( - I q l / ~ / - ~ p , p _ ¢ ) ) l ~ l ~ F > , q
(11) where p=J/b 2 is the particle density, pq= Yjexp(iq-rj) is the density operator, and [~HF> is the ground state of the Hamiltonian obtained by substituting the mean-field vector potential <.4)=he~ (2e)py~ for .4j in Eq. (9). The prefactor in this
R.B. Laughlin / PhysicaC 234 (1994)280--284 expression is the usual modification of the HartreeFock ground state implicit in the random phase approximation. Let 9/+ (z) and ~_ (z) denote the operators annihilating holons with "up" and "down" isospin, respectively, at z = x + iy in the Fermi representation. The nonlocal order parameter is given in terms of these by [20 ]
283
(l~, ..., l~lCk,,q,¢C]~C~, I/i, "", l u )
M Otm3 (z[~*+z~*)/2-z~ (Zj+Zk)/2--Z,~ x I(Z]+Z'k)/2--Z,~I I(Zj+Zk)/2--Z,~I × (j', k'lcj,rCk,, 10) (OIc]tc~, IJ, k)
~ < ~anyon I ~'+ (ZI)~'/~-"(Zl) {~ff~+(Z) ~/+ (z)
+ cyclic permutations,
+ ~t (z) ~,_ (z))~,+ (z~) ¢/_ (z~)l ¢'.n,o. > X (z~--z*__.......~)(z2--z) d2z
Iz~-zl Iz2-zl
~
(1.9p)2.
I~,-~1-®
(12) The numerical value of 1.9p reported here for the first time is obtained using the hypernetted chain technique [19]. We will adopt the notation (V___(z) ~,(z) ) = 1.9p as shorthand for this result. The second step is to convert this continuum order parameter to the site basis. The unitary transformation relating ¥+ (z) and ~u_ (z) to ~u(]), the Fermi operator to annihilate a holon at site rj= (lj, mj)b, is simply the matrix of one-body eigenstates of Eq. (8) at the two valley minima. We thus have ~,(j) ~ x/~b{cos [ ( ~ _ i m2) n] ~v+(z) + ( - 1)~sin[ ( / + ½m])n]~u_ (z)}.
(13)
In obtaining this expression, we have imagined the sample to be divided into four-site cells, and that z defines the center of the cell containingj. Then, substituting Eq. ( 13 ) into Eq. (12), and using the fact that V+ (z) and ¥_ (z) anticommute with themselves and each other, we obtain
where 10) denotes the state with no holons and zj= (lj+imj) b. The product onot in this expression is the same "unwinding" factor appearing in Eq. ( 1 2 ) and is the microscopic justification for its inclusion in Eq. (12). The remaining factor is given approximately by (OIclrc~, Ij, k) ~ (zl -z2_______) ith_12)(mt+m2) (
IZl-z~l
-
-
1 )ll+ml
q
['( 1 - ~ ) exp{- n/4( 1 - J ) } ,
I and 2 near neighbors,
l
l and2 ~ond neighbors,J
x[
a~xp~-n6~, O,
otherwise. (16)
The final step is to combine Eqs. (15) and (16) with Eq. (14) to obtain an expression for (c]~ c~ ). This is accomplished by multiplying together the right sides of Eqs. (14) and (16) and then dividing out the factor (z,-z2)/Izl-z21. This latter step accounts for the transformation of the basis functions defined by Eq. (7), which are symmetric under interchange of the holon positions, to their Fermi representation. We obtain finally
(c~tc~,) ~ i"'-1~)~"2- m') ( -- 1 ),~,--m~ ['V/2( 1-- ~) exp{-- 7r/4( 1-- J) },1and 2 near neighbors, ] x1.9 t~/ - 2J exp{- ~t~}, 1and 2 secondneighbors,| l J O, otherwise.
(¢t( 1 )V(2) > - 1.9 J( - 1 ) ll+m2+(ll+12")r¢~l X { ~ 2 [1--(--1
( 15 )
/
)h+12+ml+m2]
(17) This is equivalent to Eq. (4) with
- [1 - ( - 1 ) u,+z,)¢m,+,-n ] ) .
(14) ~= l~8~e O-5'~)m4
The third step is to evaluate the matrix element of cttc[ , to annihilate a holon pair into the vacuum. This matter has been studied extensively in previous papers [ 19 ] and is too involved to discuss in detail here. We shall simply quote the result
(18)
Let us now make some comments about this result. It is quite crude and should be compared with experiment carefully. For example, it makes the unphysical prediction that superconductivity occurs in the t-
284
R.B. Laughlin /Physica C 234 (1994)280-284
J m o d e l at any value o f J, t, a n d ~. A m o r e formal d e v e l o p m e n t o f the a n y o n a p p r o a c h using gauge-theory techniques [ 21 ] cures p r o b l e m s o f this kind, b u t is too technical to discuss here. Let us simply state the m a i n results. ( 1 ) F o r realistic values of J / t the o r d e r p a r a m e t e r is significantly smaller than Eq. ( 1 7 ) predicts, due to r e t a r d a t i o n effects. ( 2 ) The equations have an antiferromagnetic instability for ~ < 0.05 that d e p e n d s weakly on J/t. ( 3 ) The commensurate flux b a n d structure o f Eq. ( 6 ) has a gap collapse at 6--- ] which causes calculations for doping fractions larger t h a n this to be unreliable. T h a t the calculation finds c o n v e n t i o n a l superconducting o r d e r a n d d wave s y m m e t r y is not surprising. The anyone a p p r o a c h is a legitimate variational technique for the t-J model, which is k n o w n by m o r e reliable m e t h o d s to have a tendency to d wave pairing [ 22,23 ]. Also, it has been known for several years that " f l u x " vacua are f u n d a m e n t a l l y related to d wave superconducting states [24 ]. The calculation is significant m a i n l y because it predicts that superconducting pairing by m e a n s o f " s p i n fluctuations" tends naturally to an o r d e r p a r a m e t e r with dx2_r2 + i¢dxy symmetry, with ~ significantly large.
Acknowledgements I wish to t h a n k M.R. Beasley, A. Kapitulnik, T.H. Geballe, S. Doniach, Z. Zou, A.A. Abrikosov and D.J. Scalapino for numerous helpful discussions. The work was s u p p o r t e d p r i m a r i l y b y the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t No. DMR-91-20361. Additional support was p r o v i d e d by the N S F M R L Program through the Center for Materials Research at Stanford University.
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