Some general remarks on superconductivity

Physica A 281 (2000) 442– 449

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Some general remarks on superconductivity A. Zee ∗ Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, USA

Abstract We make some general remarks about high-temperature superconductivity. A review of duality c 2000 Elsevier Science B.V. All rights reserved. in (2 + 1) dimensional eld theory is given. Keywords: Superconductivity; Eective theory; Duality

It has been more that 10 years since the discovery of high-temperature superconductivity, and there have been no lack of theories purporting to explain the observations. But “the” theory is still missing. Indeed, it is not even clear that we can rule out the nasty possibility that there might be dierent theories for dierent materials. The physics may be in the details, to paraphrase a well-known saying. A detailed explanation may involve “metallurgical” or “engineering” considerations, words that the typical theoretical physicist does not particularly like. That some sort of pairing between electrons is responsible is not in doubt, but while the phonon mechanism appears to be unlikely, there are still “too” many possible pairing mechanisms. The speci c mechanism may well depend on the material. It may thus be useful not to be tied to a speci c mechanism, but to pose some questions of general principle, make some general remarks, etc. It is in this spirit that I present this talk. Let us start with the familiar phase diagram in the x–T plane, where x denotes the concentration of holes and T the temperature. At low T we have an antiferromagnetic Mott insulator for small x, and d-wave superconductivity above some critical x. Over the last 10 years, theorists have gradually realized that the novel physics might be in the intermediate region, rather than in the superconducting phase. Many exotic states of matter have been proposed for the intermediate region, including “strange” metal, nodal liquid, and striped phases. Fisher et al. (see Ref. [1]) start with the d-wave superconductor, add quantum disorder to destroy the superconductivity, and study the resulting phase, dubbed the nodal liquid. In the striped phase, described by Zaanen, ∗

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Emery, Kivelson, and others see for e.g., Refs. [2,3], the two-dimensional planes in the superconductor are supposed to “self-organize” into slowly uctuating structures consisting of stripes of antiferromagnetic domains, separated by regions rich in holes. The self-organization is supposed to be driven by the desire of the antiferromagnetic domains to expel the doped holes. There is by now considerable experimental evidence for striped phases, but the details appear to vary from material to material. The direction of the stripes seem to vary from plane to plane, and may point along the “diagonal”. Thus, even the eective dimension of high-temperature superconductor is not established. The dimension may be 3 (favored by those who think that interlayer hopping is important), 2 (preferred by many theorists), or 1 (which may be eectively the case if the formation of stripes drives the superconductivity). In this last case, we might have a quasi-one-dimensional gas of holes in an “active antiferromagnetic environment”, whatever that means. Perhaps one should think about a 1+1 dimensional eld theory coupled to a 2 + 1 dimensional “background” eld theory, with an action of the form Z Z dt d x [Lholes (’(x; t)) S= dt d x dyLAF (n(x; y; t)) + + Lint (n(x; y = 0; t); ’(x; t))] ;

(1)

where n(x; y; t) represents the dynamics of the anti-ferromagnetic ordering parameter living in (2 + 1) dimensions and ’(x; t) the dynamics of the holes living in (1 + 1) dimensions. Field theorists have not studied this type of “mixed dimensions” eld theory much, although eld theories with this sort of avor have already appeared in the soft condensed matter literature. The theory of metals and a substantial portion of solids-state physics is founded on the fact that in many systems the kinetic energy totally dominates the potential or interaction energy, to the point that the latter may be neglected entirely and that a single electron theory suces. That the electron–electron interaction can be neglected can be traced back to the Pauli principle. In momentum space, the electrons form a Fermi sphere and thus electron–electron scattering is severely limited by phase space. Cooper had the insight of considering the problem of a Fermi sphere plus two electrons. In the resulting BCS theory of superconductivity, the size of the Cooper pairs, namely the coherence length, comes out to be much bigger than the separation between electrons, and so mean eld approximation works very well. This fortunate circumstance does not hold in high-temperature superconductors. The diculty of nding “the” theory may be attributed directly to this fact. The challenging problem in contemporary condensed-matter physics is thus precisely to understand the opposite situations, in which the potential or interaction energy completely dominates the kinetic energy. The canonical example of this phenomenon is the basic odd denominator quantum Hall uid for which there is essentially “no kinetic energy”, as the electrons all belong to the lowest Landau level. The strong correlation between the electrons is due entirely to the interaction between them. High-temperature superconductivity appears to belong to this class of problem, in which the interaction energy completely dominates the kinetic energy.

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Let us go back then to the foundation and ask what is in fact superconductivity. The standard answer is that superconductivity equals the Meissner eect. Is it possible to question this equality? (By Meissner eect we of course do not insist on the complete expulsion of the magnetic eld from the superconductor. As Abrikosov emphasized, the magnetic eld could be expelled everywhere except along the cores of vortex lines where superconductivity is destroyed. If the vortex lines are pinned by impurities, a weak enough current could pass through the system without dissipation.) By Meissner eect we mean the phenomenon whereby the imposition of an external magnetic eld costs an energy E that goes to in nity faster than the volume V as V goes to in nity: E ˙ V 1+ , with  positive. In the standard textbook treatment, the eective Lagrangian of a superconductor when probed with an external electromagnetic gauge potential Ai , after all other degrees of freedom have been integrated out, is supposed to have the form Le = (s =m)A2i with s some eective super uid density. Thus, for a constant external magnetic eld B; for which A ∼ x; the exponent  = 23 : Is it possible for  to take on other values? Are there other possibilities for Le ? For example, can Le be proportional to (A2 ) ; or A2 log A2 ? What about Ai Bi ; which violates parity? In two dimensions, we can have Ai B (with B the magnetic perpendicular to the system) which violates rotational invariance, but perhaps that is not as nasty a possibility as one would think, if the formation of a striped phase is in fact what drives superconductivity. By de nition, the low-energy Lagrangian density L = A J  + · · · with the current  J constructed out of the relevant low-energy degrees of freedom. To obtain superconductivity, we would like the eective Lagrangian Le to exhibit the Meissner eect when the low-energy degrees of freedom contained in J  are integrated out. Can we list or classify all the possibilities? The simplest possibility is that the low-energy degrees of freedom are represented by a scalar eld. For a charged scalar eld ; gauge invariance requires that L ∼ A {i(@ †  − †   @ ) + A † }: The current J  depends on A : This represents of course the Landau–Ginzburg solution of superconductivity: upon replacing  by v; we obtain Meissner Le ∼ v2 A2 : What if the low-energy degrees of freedom are represented by a vector eld a ? In (2 + 1) dimensional spacetime gauge invariance is automatically assured if L ∼ A ( @ a ) + · · · where  dentes the Levi–Civita antisymmetric symbol. Under the gauge transformation A → A + @ , we have upon integration by parts L → L − @ ( @ a ) = L by the Bianchi identity. This appealing possibility is realized, but not tied to, anyon or semion superconductivity [4], which unfortunately appears to be ruled out by experiments. Of all the pairing mechanisms proposed, I nd that the most conceptually novel mechanism is the one provided by semion superconductivity. Let me state this mechanism in a nutshell. In (2 + 1) dimensional spacetime, we can have fractional statistics [5–8]: upon the interchange of two particles carrying fractional statistics, their wave function acquires the phase factor ei ; with  = 0 for bosons,  for fermions, and =2

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Table 1

Meissner Chern–Simons Maxwell

Eective L

Scaling dimension

A  A A  @ A F F 

2 3 4

for semions, particles half-way between bosons and fermions. If by some mysterious “statistics transmutation” the electron becomes semionic, then electrons would want to be pair. Consider a bound pair of semions. When two such bound pairs are interchanged the wave function acquires the phase factor (ei )(2×2) = e4i = +1 if  = =2: In other words, a pair of semions acts like a boson. Semions want to pair and condense, that is, relax into the ground state. Let us go back to the long distance eective theory which we now take to be quadratic in A and which we classify by powers of derivatives, from 0 to 2. For simplicity of notation, we write the eective long-distance theory in relativistic form (Table 1). The Chern–Simons term [8,9] is possible only in (2 + 1)-dimensional spacetime, violates P and T; and is explicitly realized in quantum Hall systems. In the Maxwell term, F = @ A − @ A of course. A quick review of Gaussian integration: Given that RL ∼ ’K’ + ’h; then upon −1 integrating ’ we obtain Le ∼ hK −1 h: In other words, D’e−(’K’+’h) ∼ e−hK h : We are now ready to go back to (2 + 1) dimensions and L = A ( @ a ) + · · · = a ( @ A ) + · · · (upon integration by parts.) Applying the formula for Gaussian integration just given with ’ → a and h → @A; we immediately know the eective low-energy dynamics of A given the eective low-energy dynamics of a (that is, the kernel K:) To summarize, given L ∼ aKa;

(2)

we have

  1 1 L ∼ (@A) (@A) ∼ A @ @ A : K K

(3)

For our schematic purposes there is no need to keep track of indices. For example, given L = f f with f = @ a − @ a : We can write L ∼ a@2 a and so K = @2 : Thus, the eective dynamics of the external electromagnetic gauge potential is given by (3) as L ∼ A(@(1=@2 )@)A ∼ A2 , the Meissner Lagrangian. In this “quick and dirty” way of doing Gaussian integral, we can simply set  ∼ 1 and cancel factors of @ in the numerator against those in the denominator. The reader can easily check that keeping track of the indices and doing everything carefully give the same result. Proceeding in this way, we can construct Table 2. Thus, we see that Meissner begets Maxwell, Chern–Simons begets Chern–Simons, and Maxwell begets Meissner. I nd this beautiful and fundamental result, which

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Table 2 Dynamics of a

K

Eective Lagrangian L ∼ A(@ K1 @)A

Dynamics of the external probe A

Meissner a2 Chern–Simons a@a

1 @

Maxwell F 2 Chern–Simons A@A

Maxwell f2 ∼ a@2 a

@2

A(@@)A ∼ A@2 A 1 A(@ @ @)A ∼ A@A A(@ @12 @)A ∼ AA

Meissner A2

represents a form of duality, very striking. We say that Chern–Simons is self-dual: it begets itself. Duality is a profound and far reaching concept in theoretical physics, with origin in electromagnetism and statistical mechanics. The emergence of duality in recent years in several areas of modern physics, ranging from the quantum Hall uids to string theory, represents a major development. The concept of duality however, appears to be under-appreciated in the condensed matter community and thus I will go over the essence of duality in (2 + 1) dimensions [10–14] in the remainder of this talk. Start with a scalar- eld theory in (2 + 1) dimensions: 1 (4) L = |(@ − iqA )’|2 − V (’) : 2 I have inserted the electric charge q for later convenience. Again, we treat a relativistic theory for notational simplicity. Minimizing the potential V (’) sets the eld to ’=vei ; and gives L = 12 v2 (@  − qA )2 ;

(5)

which upon absorbing  into A by a gauge transformation we recognize as the Meissner Lagrangian. We rewrite L as 1 2  +  (@  − qA ) : (6) 2v2  Upon integrating out the auxiliary eld  we recover Eq. (5). The existence of vortices amounts to the statement that there are places where ’ vanishes, around which
vortex = 2: Around an anti-vortex,
vortex = −2: Let me remind the reader that around a vortex sitting at rest, the electromagnetic gauge potential has to go as L=−

qAi → @i  ;

(7)

at spatial in nity in order for the energy of the vortex to be nite as we can see from Eq. (5). In turn this implies that the magnetic ux I Z (8) q d 2 xij @i Aj = q dx A =
vortex = 2 is quantized in units of 2, as is well known.

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Heuristically, we can treat vortices by writing @  = @ smooth + @ vortex : (A rigorous treatment would require a proper short distance cut-o by putting the system on a lattice, etc.) Let us pause to think physically for a minute. On a distance scale large compared to the size of the vortex, vortices and anti-vortices appear as points. The interaction energy of a pair of vortex and anti-vortex separated by a distance R is given by simply plugging into Eq. (5) and ignoring the probe eld A which we can take to be as weak RR as possible ∼ a dr r(∇)2 ∼ log(R=a) where a is some short distance cut-o. The logarithmic dependence corresponds to the Coulomb interaction in two dimensional space. Thus, a gas of vortices and anti-vortices appears as a gas of point “charges” with a Coulomb interaction between them. The concept of duality is driven by our desire to treat vortices as point “charges” of some as yet unknown gauge eld. Remarkably, this can be accomplished in just a few simple steps. Integrating over smooth we nd the constraint @  = 0 on the auxiliary eld, which can be solved by writing  =  @ a : (The preceding statement is nothing but the well-known theorem in three-dimensional space from elementary physics that if a vector eld is divergence free it is the curl of another vector eld, transcribed to (2 + 1) dimensional spacetime.) At this point a is merely a symbol de ned by the equation  =  @ a : But we immediately recognize a as a gauge potential since the change a → a + @ u does not change  : Plugging into Eq. (6), we nd L=−

1 2 f +  @ a (@ vortex − qA ) ; 8v2 

(9)

where f = @ a − @ a : Note for later use that the electromagnetic current, J  ; de ned as the coecient of A ; is determined in terms of the gauge potential a to be J  = q @ a :

(10)

Let us rewrite the term coupling a to vortex upon integrating by parts as a  @ @ vortex : According to Newton and Leibniz, @ commutes with @ ; and so apparently we get zero. But @ and @ commute only when acting on a globally de ned function, and vortex is precisely not globally de ned since it changes by 2 when we go around a vortex. In particular, consider a vortex sitting at rest, and look at the quantity a0 couples physics. The to, namely ij @i @j vortex = ∇ × (∇vortex ) in the notation of elementary R 2 integral of this over a region where the vortex is sitting gives d x∇ × (∇vortex ) = H ij dx∇vortex = 2: Thus, we recognize 1=2  @i @j vortex as the density of vortices. By  Lorentz invariance, we can write the vortex current as jvortex = (1=2)  @ @ vortex . Thus, we can now write Eq. (9) as 1 2  f + (2)a jvortex − A (q @ a ) : (11) 8v2  Lo and behold, we have accomplished what we set out to do. We have rewritten the theory so that the vortex appears as an “electric charge” for the gauge potential a : L=−

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Sometimes this is called a dual theory, but strictly, it is more accurate to refer to this as the dual representation of the original theory (4). Since there are vortices and anti-vortices, we have to introduce a complex scalar eld ; which we will refer to as the vortex eld, to describe them. In other words, we can “elaborate” the description in (11) and write L=−

1 1 2 f + |(@ − i(2)a )|2 − W () − A (q @ a ) : 2 8v 2

(12)

The potential W () contains terms like († )2 describing the short distance interaction of two vortices (or a vortex and an anti-vortex.) In principle, if we had mastered all the short-distance physics contained in the original theory (4) then these terms are all determined in terms of parameters in the original theory. Now, we come to the most fascinating aspect of the duality representation and the reason why the word “duality” is used in the rst place. The vortex eld  is a complex scalar eld, just like the eld ’ we started with, and so we can perfectly well form a vortex out of ; namely a region where  vanishes and where the phase of  goes trough 2: Amusingly, we are forming a vortex of a vortex, so to speak. The duality theorem states that the vortex of a vortex is nothing but the original charge, described by the eld ’ we started out with! The proof is remarkably simple. The vortex in the theory (12) carries “magnetic

ux”. By exactly the same manipulation as in (8) we have I Z (13) 2 d 2 xij @i aj = 2 dx a = 2 : Note that we puts quotation marks around the term “magnetic ux” since as is evident we are talking about the ux associated with the gauge potential a and not the ux associated with the electromagnetic potential A : But remember the electromagnetic current J  = q @ a from (10), and in particular J 0 = qij @i aj : Hence,R the electric charge (notice no quotation marks) of this vortex of a vortex is equal to d 2 x J 0 = q; precisely the charge of the original complex scalar eld ’: This proves the assertion. Acknowledgements I thank C.K. Hu and F.Y. Wu for inviting me to speak at the Symposium on “Equilibrium and Non-equilibrium Phase Transitions” Stat-Phys 1999 and M.P.A.F. Fisher and L. Balents for a helpful conversation on the manuscript and encouragement. References [1] L. Balents, M.P.A.F. Fisher, C. Nayak, cond-mat=9903294. [2] J. Zaanen, O. Gunnarson, Phys. Rev. B 40 (1989) 7391. [3] V. Emery, S. Kivelson, J. Tranquada, cond-mat=9907228.

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[4] A. Zee, in: K. Bedell, D. Coey, D.E. Meltzer, D. Pines, J.R. Schrieer (Eds.), Semionics: A Theory of High Temperature Superconductivity, in High Temperature Superconductivity, Addison-Wesley, Reading, MA, 1990. [5] J.M. Leinaas, J. Myrheim, Nuovo Cimento B 37 (1977) 1. [6] F. Wilczek, Phys. Rev. Lett. 48 (1982) 1144. [7] F. Wilczek, Phys. Rev. Lett. 49 (1982) 957. [8] F. Wilczek, A. Zee, Phys. Rev. Lett. 51 (1983) 2250. [9] A. Zee, in: Y.M. Cho (Ed.), Physics in (2 + 1)-Dimension, World Scienti c, Singapore, 1992, p. 53. [10] M.P.A.F. Fisher, D.H. Lee, Phys. Rev. Lett. 63 (1989) 903. [11] M.P.A.F. Fisher, D.H. Lee, Int. J. Mod. Phys. B 5 (1991) 2671. [12] X.G. Wen, A. Zee, Phys. Rev. Lett. 62 (1989) 1937. [13] X.G. Wen, A. Zee, Int. J. Mod. Phys. B 4 (1990) 437. ˜ [14] A. Zee, Quantum Hall Fluids, in: H.B. Geyer (Ed.), Field Theory, Topology and Condensed Matter Physics, sections 6 and 10, Springer, Berlin, 1995.