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We are interested in the low-energy dynamics of long- To see that K-theory controls the stability of Fermi sur- wavelength excitations in a Fermi liquid, described micro- faces, we study the exact propagator in Euclidean time scopically by a complex fermion ψiσ(x,t). σ is the spinor (and in the obvious notation), index of the SO(d) rotation group, and i is an internal in- iσ iσ † dex of an n-dimensional representation of some compact G i′σ′ (k,ω)= ψ (0, 0) ψ ′ ′ (k,ω) . (2) h i σ i symmetry G. For simplicity, we consider i the fundamen- Gapless excitations correspond to a pole in G along some tal of SU(n), the generalization to any compact G being submanifold Σ, of some dimension d p, in the (k,ω)- straightforward. Since the irreducible spinor representa- space. We will analyze the stability− of such poles un- tion of SO(d) is2[d/2]-dimensional (with [k] denoting the der small perturbations in the theory, i.e., perturbations integral part of k), ψ has N 2[d/2] n complex com- which do not change G qualitatively far from Σ. (The ponents. Real fermions will be≡ considered· in the closing analysis of large perturbations, corresponding to possi- part of the paper. ble instabilities due to relevant deformations of the fixed The theory is microscopically described by the follow- point, is outside of the scope of this paper.) ing non-relativistic Lagrangian, It will be convenient to introduce a collective index (iσ) a, a = 1,...N with N = 2[d/2] n, and consider d † iσ 1 † iσ † iσ ≡ · S = dt d x iψ ∂tψ + ψ ∆ψ µ ψ ψ the inverse exact propagator − iσ 2m iσ − iσ Z   ′ ′ a −1 a + . . . a (G )a (k,ω) G ≡ ′ ′ = δa iω k2/2m + µ + Π a (k,ω), (3) Here ∆ is the Laplace operator in d dimensions, µ is a − a the chemical potential, and “. . .” denotes all interactions, ′ a
such as those with an order parameter, four-Fermi inter- with Πa (k,ω) the exact polarization tensor. The ques- actions, impurities, etc. tion of stability of the manifold Σ of gapless modes re- duces to the classification of the zeros of the matrix We wish to identify possible universality classes of the ′ a G RG behavior and the corresponding RG fixed points. The that cannot be lifted by small perturbations of Πa . Our arguments will be topological, and our results thus inde- essence of the Landau theory of Fermi liquids [12,13] is in ′ a the observation that the low-energy excitations of the sys- pendent of the precise details of Πa . tem populate a narrow shell near the Fermi surface, sug- Assume that has a zero (i.e., det vanishes) along a submanifold Σ ofG dimension d p in theG d+1-dimensional gesting a coarse-graining procedure whereby momenta − are scaled towards the Fermi surface. Free fermions rep- (k,ω) space. Σ lies within the subspace of zero frequency, resent the simplest fixed point of this scaling. The Fermi ω = 0. Pick a point kF on the Fermi surface Σ, and con- surface supports gapless coarse-grained fermionic excita- sider the p + 1 dimensions k⊥ transverse to Σ in the tions χ, with the characteristic linear dispersion relation (k,ω)-space at kF . A small perturbation of the sys- tem can either move the zero of slightly away from ω k + . . . at low energies. G ∼Once a free-field fixed point has been identified, a full kF along k⊥, or eliminate the zero altogether. In the RG analysis reveals possible instabilities of the system, latter case, the purported Fermi surface is unstable, and due to (marginally) relevant interactions. The degener- a small perturbation will either produce a gap or will acy of the gapless modes χ can be lifted completely, and at least further reduce the dimension of the Fermi sur- the system can develop a gap (such as in the case of su- face. In order to classify stable zeros, consider a sphere p perconductivity, or in the B-phase of 3He). Alternatively, S wrapped around Σ at some small distance ǫ in the the gapless excitations can be only partially lifted, leav- transverse p + 1 dimensions k⊥, k⊥ kF = ǫ. We as- | p− | ing a submanifold of some lower dimension (d p) in the sume ǫ small enough so that this S does not intersect (k,ω)-space where gapless fermionic excitations− are sup- any other components of the Fermi surface, a situation ported. Since such submanifolds with gapless excitations that can always be arranged in a generic point on the generalize the concept of the Fermi surface, we will refer Fermi surface. The matrix is nondegenerate along this p G to them as “generalized Fermi surfaces” (and drop the S , and therefore defines a map word “generalized” from now on). : Sp GL(N, C) (4) Perhaps the standard example of a partially lifted G → Fermi surface is the A phase of 3He in d = 3, where from Sp to the group of non-degenerate complex N N the interactions with the order parameter cause the fermi matrices. If this map represents a nontrivial class× in modes to become massive outside of isolated points. Such the p-th homotopy group πp(GL(N, C)), the zero along Fermi points are stable under small perturbations of the Σ cannot be lifted by a small deformation of the the- system. Thus, in d = 3, we can have stable Fermi sur- ory. The Fermi surface is then stable under small per- faces (as in the free fermion system) or Fermi points (as in turbations, and the corresponding nontrivial element of 3 He-A); however, Fermi lines appear to be (generically) πp(GL(N, C)) represents the topological invariant (or unstable (see e.g. [14]). We will now argue that this is “winding number”) responsible for the stability of the the beginning of a pattern, explained as a consequence Fermi surface.

2 One may worry that this could result in a very com- coarse grained fermions χα can in principle be lower than plicated pattern, dependent on the specific values of p N. The ABS construction determines the universal value [p/2] and N. Fortunately, this is not the case, and the pattern of N to be N =2 . If pµ (µ =0,...p) are the dimen- that emerges is quite simple. The key observation is that sions transverse to Σ in (k,ω), we first define the gamma the values of N and p are always in the so-called stable matricese Γµeof SO(p, 1) to satisfy Γµ, Γν = 2ηµν with { } regime [7], of N large enough so that πp(GL(N, C)) is ηµν a (Lorentz-signature) metric. The ABS construction independent of N. This stable regime lies at N > p/2 then gives the leading expression for the topological in- [7]. It is easy to check that in our setting, the stability variant e, and hence for the inverse propagator of the condition is always satisfied. coarse-grained fermions χα near the Fermi surface,D It is a deep mathematical result that in this stable µ regime, the homotopy groups of GL(N, C) define a gen- =Γ pµ + .... (7) D eralized cohomology theory, known as K-theory [7]. In K-theory, any smooth manifold X is assigned an abelian The “. . .” in (7) refers to higher-order corrections to the group K(X). (For X noncompact, K(X) is to be in- leading low-energy term. The precise form of the metric terpreted as compact K-theory [7].) For example, for ηµν is determined by the microphysics of ψ. This estab- X = Rk this group is given by lishes our second result: At low energies, the dispersion relation of the coarse- k α K(R )= πk−1(GL(N, C)) (5) grained gapless fermions χ near the Fermi surface is governed by the Atiyah-Bott-hapiro construction of K- with N in the stable regime. The corresponding groups theory. 2ℓ 2ℓ+1 are known to be K(R ) = Z and K(R ) = 0. This The ABS construction has determined the unversal periodicity by two is known as Bott periodicity [7]. value of the range N of α, making χα automatically Our analysis of the Fermi surface can now be rein- a spinor of the Lorentz group SO(p, 1). (Notice that terpreted as a statement about K-theory: The map (4) N 2[p/2] N, withe the equality only when n = 1 and defines an element of the K-theory group K(Rk) of the p =≡d.) The≤ equation of motion of χα in the low-energy transverse space k ; the Fermi surface Σ is stable if this ⊥ regime is the relativistic Dirac equation, Γµ∂ χ = 0+. . ., element is nontrivial. We have established our first re- e µ with “. . .” now denoting possible nonrelativistic correc- sult: tions at higher energies. In the free theory, we get one Stable Fermi surfaces in Fermi liquids are classified by copy of the Dirac fermion for each point (or patch) θ on K-theory; in the case of complex fermions, Fermi surfaces the Fermi surface. This is our third result: of codimension in the k -space are stable for p +1 ( ,ω) p The low-energy modes exhibit an emergent relativistic odd, and unstable for p even. In d = 3, this reproduces dispersion relation, in the p +1 dimensions transverse to the observed pattern of stability mentioned above. the Fermi surface. As our first application of this picture, we will use K- Again, some low-dimensional examples of this behavior theory to determine the dispersion relation of the gapless are well-known. Here we have extended the statement to modes near a general stable Fermi surface Σ. Such modes arbitrary dimensions, and found the emergent relativistic will be described by coarse-grained fermions χα(ω, p,θ), invariance as a simple consequence of the ABS construc- with θ denoting coordinates on Σ, and p being the spa- tion. Notice that both the gamma matrices and the fact tial momenta normal to Σ. The index α goes over some that the coarse-grained fermions χα transform as spinors subset, to be determined below, of the range of a. The of SO(p, 1) are emergent properties. In particular, the leading quadratic part of the action is spin-statistics theorem familiar from relativistic quantum field theory emerges naturally. S = dµ(ω, p,θ) χ† α (ω, p,θ)χβ + . . . . (6) αD β The ABS construction played a crucial role in string Z
theory, in Sen’s picture [16] of stable D-branes as topo- dµ(ω, p,θ) is the flat measure dωddk written in terms of logical defects in the tachyon on higher-dimensional D- the new coordinates (ω, p,θ), and “. . .” refers to interac- branes. In the string theory literature, this construction tions of χα, to be studied by RG methods in the vicinity is sometimes referred to as the “Γ x construction,” since of the free-field fixed point given by (6). is the operator x are the spacetime dimensions transverse· to the soliton we now wish to determine. D inside the higher-dimensional branes. In the theory of K-theory provides an explicit construction of the gen- Fermi liquids, we have put the Γ x construction where erator e in K(R2ℓ), known as the Atiyah-Bott-Shapiro it naturally belongs: in the momentum· space. (ABS) construction [15,7,8]. Consider a stable Fermi sur- So far we have analyzed the theory locally near a point face Σ, of codimension p +1 2ℓ, with winding num- kF on the Fermi surface. K-theory provides a natural ex- ber one. Any Σ with a higher≡ winding number n can tension of our arguments globally, to a Fermi surface of be perturbed into n separated Fermi surfaces of winding arbitrary topology. Our construction demonstrates that number one. The range N of the index a carried by the a Fermi surface Σ is stable precisely when it carries a microscopic fermion is in the stable regime (and therefore topological charge in K-theory. Fermi surfaces are ob- quite large), but the range N of the index α carried by the jects in K-theory and not just submanifolds in the (k,ω)-

e 3 space. K-theory thus provides the natural arena for un- ACKNOWLEDGMENTS derstanding the structure of Fermi surfaces. We expect it to be particularly useful in the case of Fermi surfaces with This paper is based on work largely done in Fall 2000 complicated topologies and/or singularities. It could also at Rutgers University. I wish to thank Tom Banks, Mike be a natural tool for understanding the ideas of topolog- Douglas, Greg Moore, and Nick Read for valuable dis- ical order in Fermi systems [17]. cussions at that time. This work has been supported in The case of real fermions is perhaps even more interest- part by NSF grant PHY-0244900, by the Berkeley Center iσ ing. Repeating the above steps for ψ satisfying a reality for Theoretical Physics, and by DOE grant DE-AC03- condition ψ∗ ψ, we are naturally led to the classifica- ∼ 76SF00098. Any opinions, findings, and conclusions or tion of stable Fermi surfaces in terms of Real KR-theory recommendations expressed in this material are those of [7]. The simplest reality condition in K-theory would the author and do not necessarily reflect the views of the define what is known as KO-theory, related to the homo- National Science Foundation. topy groups of GL(N, R). The subtlety here is that the involution that defines the reality condition on ψ acts simultaneously as the time reversal symmetry, so as to preserve the equation of motion i∂tψ +∆ψ/2m+. . . = 0. Consequently, the stable Fermi surfaces of dimension d p are now classified by groups KR(Rp,1). These groups are− periodic in p with periodicity 8; for low values of p, one [1] P. Hoˇrava, Nucl. Phys. B327 (1989) 461, Phys. Lett. B231 finds Z for p =1, 5, Z2 for p =2, 3, and 0 for p =4, 6, 7, 8. (1989) 251; Dai, Leigh and Polchinski, Mod. Phys. The ABS construction of the low-energy dispersion rela- Lett. A4 (1989) 2073. tion is again given by (7), and leads to coarse-grained rel- [2] J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724, hep- ativistic Majorana fermions; the case of pseudo-Majorana th/9510017. fermions can be similarly incorporated. [3] for elements of string theory, see: J. Polchinski, String The novel phenomenon for real fermions is the exis- Theory, Vols. 1 & 2 (Cambridge U., 1998). [4] R. Minasian and G. Moore, JHEP 9711 (1997) 002, hep- tence of Fermi surfaces with a Z2 charge. For exam- ple, in 2 + 1 dimensions our framework predicts a stable th/9710230. [5] E. Witten, JHEP 9812 (1998) 019, hep-th/9810188. Fermi line due to the Z charge in KR(R1,1), but also [6] P. Hoˇrava, Adv. Theor. Math. Phys. 2 (1999) 1373, hep- a stable Fermi point carrying a Z charge in KR(R2,1). 2 th/9812135. The low-energy dispersion relation is that of a relativistic [7] M. Karoubi, K-Theory. An Introduction (Springer, 1978). SO(2, 1) Majorana fermion. Since the charge takes val- [8] H.B. Lawson, Jr. and M.-L. Michelsohn, Spin Geometry ues in Z2, two such Fermi points when brought together (Princeton, 1989). would annihilate, and a the system would form a gap. [9] E. Witten, JHEP 0005 (2000) 031, hep-th/9912086; Such behavior has been observed [18]. Similarly, for real G. Moore and E. Witten, JHEP 0005 (2000) 032, hep- fermions in d = 3 we can have a Fermi surface carrying a th/9912279. Z charge, but also a Fermi line and a Fermi point, each [10] D.-E. Diaconescu, G. Moore and E. Witten, hep- supported by a Z2 charge in KR-theory. th/0005090, hep-th/0005091. Having analyzed the stability of the Fermi surface in [11] D.S. Freed and M.J. Hopkins, JHEP 0005 (2000) 044, the ground state, one can extend our framework to in- hep-th/0002027; D.S. Freed, hep-th/0011220. clude the classification of topological defects in Fermi liq- [12] R. Shankar, Rev. Mod. Phys. 66 (1994) 129, cond- uids. In the semiclassical regime of small ¯h, one can con- mat/9307009. sider slowly-varying Fermi surfaces (in the spirit of [19]) [13] J. Polchinski, in: Recent Directions in Particle Theory, and defects [20] as submanifolds in (x, t, k,ω). Repeat- TASI ’92, eds.: J.A. Harvey and J. Polchinski (World ing the analysis of this paper will now lead to a classifi- Scientific, 1993), hep-th/9210046. 351 cation of the spectrum of stable topological defects, and [14] G.E. Volovik, Phys. Rep. (2001) 195, gr-qc/0005091. 3 the dispersion relations of their low-energy modes, again [15] M.F. Atiyah, R. Bott and A. Shapiro, Topology Suppl. 1 in terms of K-theory. This pattern is indeed very remi- (1964) 3. niscent of how K-theory controls the spectrum of stable [16] see e.g. A. Sen, hep-th/9904207 and references therein. defects (in particular, D-branes) in string theory. It re- [17] X.G. Wen and A. Zee, cond-mat/0202166. [18] N. Read and D. Green, cond-mat/9906453. mains to be seen whether this analogy between Fermi [19] F.D.M. Haldane, in: Perspectives in Many-Particle liquids and string theory runs deeper than suggested by Physics, eds.: R.A. Broglia and J.R. Schrieffer (North the results of this paper. Holland, 1994). [20] G.E. Volovik and V.P. Mineev, Sov. Phys. JETP 56 (1982) 579; P.G. Grinevich and G.E. Volovik, J. Low Temp. Phys. 72 (1988) 371.

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