Symmetry Classes 0 1 0 2 January 5, 2010 n a J 5 Martin R. Zirnbauer ] h Institut fu¨r Theoretische Physik, Universit¨at zu K¨oln, p Zu¨lpicher Straße 77, 50937 K¨oln, Germany - h t a Abstract m [ Physical systems exhibiting stochastic or chaotic behavior are often amenable 1 to treatment by random matrix models. In deciding on a good choice of model, v random matrix physics is constrained and guided by symmetry considerations. 2 2 The notion of ‘symmetry class’ (not to be confused with ‘universality class’) 7 expresses the relevance of symmetries as an organizational principle. Dyson, 0 in his 1962 paper referred to as The Threefold Way, gave the prime classifica- . 1 tion of random matrix ensembles based on a quantum mechanical setting with 0 0 symmetries. In this article we review Dyson’s Threefold Way from a modern 1 perspective. We then describe a minimal extension of Dyson’s setting to in- : v corporate the physics of chiral Dirac fermions and disordered superconductors. i X In this minimally extended setting, where Hilbert space is replaced by Fock r space equipped with the anti-unitary operation of particle-hole conjugation, a symmetry classes are in one-to-one correspondence with the large families of Riemannian symmetric spaces. 1 Introduction In Chapter 2 of this handbook1, the historical narrative by Bohigas and Wei- denmu¨llerdescribeshowrandommatrixmodelsemergedfromquantumphysics, more precisely from a statistical approach to the strongly interacting many- body system of the atomic nucleus. Although random matrix theory is nowa- days understood to be of relevance to numerous areas of physics, mathematics, and beyond, quantum mechanics is still where many of its applications lie. 1ThepresentarticleistobeChapter3oftheOxfordHandbookofRandomMatrixTheory. 1 Quantum mechanics also provides a natural framework in which to classify random matrix ensembles. In this thrust of development, a symmetry classification of random matrix ensembles was put forth by Dyson in his 1962 paper The Threefold Way: alge- braicstructureofsymmetrygroupsandensemblesinquantummechanics,where he proved (quote from the abstract of [Dys62]) “that the most general matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of which belongs to one of the three known types”. The three types known to Dyson wereensemblesofmatriceswhichareeithercomplexHermitian,orrealsymmet- ric, or quaternion self-dual. It is widely acknowledged that Dyson’s Threefold Way has become fundamental to various areas of theoretical physics, includ- ing the statistical theory of complex many-body systems, mesoscopic physics, disordered electron systems, and the field of quantum chaos. Overthelastdecade,anumberofrandommatrixensemblesbeyondDyson’s classificationhavecometotheforeinphysicsandmathematics. Onthephysics side these emerged from work [Ver94] on the low-energy Dirac spectrum of quantum chromodynamics, and also from the mesoscopic physics of low-energy quasi-particles in disordered superconductors [AZ97]. In the mathematical re- search area of number theory, the study of statistical correlations of the values of Riemann zeta and related L-functions has prompted some of the same gen- eralizations [KS99]. It was observed early on [AZ97] that these post-Dyson ensembles, or rather the underlying symmetry classes, are in one-to-one corre- spondence with the large families of symmetric spaces. The prime emphasis of the present handbook article will be on describing Dyson’sfundamentalresultfromamodernperspective. Asecondgoalwillbeto introduce the post-Dyson ensembles. While there seems to exist no unanimous view on how these fit into a systematic picture, here we will follow [HHZ05] to demonstrate that they emerge from Dyson’s setting upon replacing the plain structure of Hilbert space by the more refined structure of Fock space.2 The reader is advised that some aspects of this story are treated in a more leisurely manner in the author’s encyclopedia article [Zir04]. To preclude any misunderstanding, let us issue a clarification of language right here: ‘symmetry class’ must not be confused with ‘universality class’! Indeed, inside a symmetry class as understood in this article various types of physicalbehaviorarepossible. (Forexample, randommatrixmodelsforweakly disordered time-reversal invariant metals belong to the so-called Wigner-Dyson symmetry class of real symmetric matrices, and so do Anderson tight-binding models with real hopping and strong disorder. The former are believed to 2We mention in passing that a classification of Dirac Hamiltonians in two dimensions has been proposed in [BL02]. Unlike ours, this is not a symmetry classification in Dyson’s sense. 2 exhibit the universal energy level statistics given by the Gaussian Orthogo- nal Ensemble, whereas the latter have localized eigenfunctions and hence level statistics which is expected to approach the Poisson limit when the system size goes to infinity.) For this reason the present article must refrain from writing down explicit formulas for joint eigenvalue distributions, which are available only in certain universal limits. 2 Dyson’s Threefold Way Dyson’s classification is formulated in a general and simple mathematical set- ting which we now describe. First of all, the framework of quantum theory calls for the basic structure of a complex vector space V carrying a Hermitian scalar product , : V V C. (Dyson actually argues [Dys62] in favor of h· ·i × → working over the real numbers, but we will not follow suit in this respect.) For technical simplicity, we do join Dyson in taking V to be finite-dimensional. In applications, V Cn will usually be the truncated Hilbert space of a family of ≃ disordered or quantum chaotic Hamiltonian systems. The Hermitian structure of the vector space V determines a group U(V) of unitary transformations of V. Let us recall that the elements g U(V) are ∈ C-linear operators satisfying the condition gv,gv′ = v,v′ for all v,v′ V. h i h i ∈ Building on the Hermitian vector space V, Dyson’s setting stipulates that V be equipped with a unitary group action G V V, (g,v) ρ (g)v , ρ (g) U(V) . (2.1) 0 V V × → 7→ ∈ In other words, there is some group G whose elements g are represented on V 0 by unitary operators ρ (g). This group G is meant to be the group of joint V 0 (unitary) symmetries of a family of quantum mechanical Hamiltonian systems with Hilbert space V. We will write ρ (g) g for short. V ≡ Now, not every symmetry of a quantum system is of the canonical unitary kind. The prime counterexample is the operation, T, of inverting the time direction, called time reversal for short. It is represented on Hilbert space V by an anti-unitary operator T ρ (T), which is to say that T is complex anti- V ≡ linear and preserves the Hermitian scalar product up to complex conjugation: T(zv) = zTv, Tv,Tv′ = v,v′ (z C; v,v′ V) . (2.2) h i ∈ ∈ (cid:10) (cid:11) Anotheroperationofthiskindischargeconjugationinrelativistictheoriessuch as the Dirac equation for the electron and its anti-particle, the positron. Thus in Dyson’s general setting one has a so-called symmetry group G = G G where the subgroup G is represented on V by unitaries, while G (not 0 1 0 1 ∪ a group) is represented by anti-unitaries. By the definition of what is meant 3 by a ‘symmetry’, the generator of time evolution, the Hamiltonian H, of the quantum system is fixed by conjugation gHg−1 = H with any g G. ∈ The set G may be empty. When it is not, the composition of any two 1 elements of G is unitary, so every g G can be obtained from a fixed element 1 1 ∈ of G , say T, by right multiplication with some U G : g = TU. The 1 0 ∈ same goes for left multiplication, i.e., for every g G there also exists U′ 1 ∈ ∈ G so that g = U′T. In other words, when G is non-empty, G G is a 0 1 0 ⊂ proper normal subgroup and the factor group G/G Z consists of exactly 0 2 ≃ two elements, G and TG = G . For future use we record that conjugation 0 0 1 U TUT−1 =: a(U) by time reversal is an automorphism of G . 0 7→ Following Dyson [Dys62] we assume that the special element T represents an inversion symmetry such as time reversal or charge conjugation. T must then be a (projective) involution, i.e., T2 = z Id with 0 = z C, so that V × 6 ∈ conjugation by T2 is the identity operation. Since T is anti-unitary, z must have modulus z = 1, and by the C-antilinearity of T the associative law | | zT = T2 T = T T2 = Tz = zT (2.3) · · forces z to be real, which leaves only two possibilities: T2 = Id . V ± Let us record here a concrete example of some historical importance: the Hilbert space V might be the space of totally anti-symmetric wave functions of n particles distributed over the shell-model space of an atom or an atomic nucleus, and the symmetry group G might be G = O TO , the full rotation 3 3 ∪ group O (including parity) together with its translate by time reversal T. 3 In summary, Dyson’s setting assumes two pieces of data: a finite-dimensional complex vector space V with Hermitian structure, • agroupG = G TG actingonV byunitaryandanti-unitaryoperators. 0 0 • ∪ It should be stressed that, in principle, the primary object is the Hamiltonian, andthesymmetriesGaresecondaryobjectsderivedfromit. However,adopting Dyson’s standpoint we now turn tables to view the symmetries as fundamental andgivenandtheHamiltoniansasderivedobjects. Thus, fixinganypair(V,G) our goal is to elucidate the structure of the space of all compatible Hamiltoni- ans, i.e., the self-adjoint operators H on V which commute with the G-action. Such a space is reducible in general: the G-compatible Hamiltonian matrices decompose as a direct sum of blocks. The goal of classification is to enumerate the irreducible blocks that occur in this setting. WhilethemainobjectstoclassifyarethespacesofcompatibleHamiltonians H, we find it technically convenient to perform some of the discussion at the integrated level of time evolutions U = e−itH/~ instead. This change of focus t results in no loss, as the Hamiltonians can always be retrieved by linearization 4 in t at t = 0. The compatibility conditions for U U read t ≡ U = g Ug−1 = g U−1g−1 (for all g G ) . (2.4) 0 0 1 1 σ ∈ σ The strategy will be to make a reduction to the case of the trivial group G = Id . The situation with trivial G can then be handled by enumeration 0 0 { } of a finite number of possibilities. 2.1 Reduction to the case of G = Id 0 { } To motivate the technical reduction procedure below, we begin by elaborating the example of the rotation group O acting on a Hilbert space of shell-model 3 states. Any Hamiltonian which commutes with G = O conserves total an- 0 3 gular momentum, L, and parity, π, which means that all Hamiltonian matrix elements connecting states in sectors of different quantum numbers (L,π) van- ish identically. Thus, the matrix of the Hamiltonian with respect to a basis of states with definite values of (L,π) has diagonal block structure. O -symmetry 3 further implies that the Hamiltonian matrix is diagonal with respect to the orthogonal projection, M, of total angular momentum on some axis in position space. Moreover, for a suitable choice of basis the matrix will be the same for each M-value of a given sector (L,π). To put these words into formulas, we employ the mathematical notions of orthogonal sum and tensor product to decompose the shell-model space as V V , V = Cm(L,π) C2L+1 , (2.5) (L,π) (L,π) ≃ ⊗ L≥0M;π=±1 where m(L,π) is the multiplicity in V of the O -representation with quantum 3 numbers (L,π). The statement above is that all symmetry operators and com- patible Hamiltonians are diagonal with respect to this direct sum, and within a fixed block V the Hamiltonians act on the first factor Cm(L,π) and are (L,π) trivial on the second factor C2L+1 of the tensor product, while the symmetry operators act on the second factor and are trivial on the first factor. Thus we may picture each sector V as a rectangular array of states where the (L,π) Hamiltonians act, say, horizontally and are the same in each row of the array, while the symmetries act vertically and are the same in each column. This concludes our example, and we now move on to the general case of any group G acting reductively on V. To handle it, we need some language 0 and notation as follows. A G -representation X is a C-vector space carrying a 0 G -actionG X X by(g,x) ρ (g)x. IfX andY areG -representations, 0 0 X 0 × → 7→ then by the space Hom (X,Y) of G -equivariant homomorphisms from X to G0 0 Y one means the complex vector space of C-linear maps ψ : X Y with → the intertwining property ρ (g)ψ = ψρ (g) for all g G . If X is an ir- Y X 0 ∈ reducible G -representation, then Schur’s lemma says that Hom (X,X) is 0 G0 5 one-dimensional, being spanned by the identity, Id . For two irreducible G - X 0 representations X and Y, the dimension of Hom (X,Y) is either zero or one, G0 by an easy corollary of Schur’s lemma. In the latter case X and Y are said to belong to the same isomorphism class. Using the symbol λ to denote the isomorphism classes of irreducible G - 0 representations,wefixforeachλastandardrepresentationspaceR . Notethat λ dimHom (R ,V)countsthemultiplicityinV oftheirreduciblerepresentation G0 λ of isomorphism class λ. In our shell-model example with G = O we have 0 3 λ = (L,π), R = C2L+1, and dimHom (R ,V) = m(L,π). λ G0 λ The following statement can be interpreted as saying that the example ad- equately reflects the general situation. Lemma 2.1 Let G act reductively on V. Then 0 Hom (R ,V) R V, (ψ r ) ψ (r ) λ G0 λ ⊗ λ → λ λ⊗ λ 7→ λ λ λ M M X is a G -equivariant isomorphism. 0 Remark. The decomposition offered by this lemma perfectly separates the unitary symmetry multiplets from the dynamical degrees of freedom and thus gives an immediate view of the structure of the space of G -compatible Hamil- 0 tonians. Indeed, the direct sum over isomorphism classes (or ‘sectors’) λ is preserved by the symmetries G as well as the compatible Hamiltonians H; 0 and G is trivial on Hom (R ,V) while the Hamiltonians are trivial on R . 0 G0 λ λ Next, we remove the time-evolution trivial factors R from the picture. To λ do so, we need to go through the step of transferring all given structure to the spaces E := Hom (R ,V). λ G0 λ 2.1.1 Transfer of structure. We first transfer the Hermitian structure of V. In the present setting of a unitary G -action, the Hermitian scalar product of V reduces to a Hermitian 0 scalar product on each sector of the direct-sum decomposition of Lemma 2.1, byorthogonality ofthesum. Hence, wemayfocusattention onadefinite sector E R E R . Fixing a G -invariant Hermitian scalar product , on λ λ 0 R ⊗ ≡ ⊗ h· ·i R = R we define such a product , : E E C by λ E h· ·i × → ψ,ψ′ := ψ(r),ψ′(r) / r,r , (2.6) E V R h i h i h i which is easily checked to be independent of the choice of r R, r = 0. ∈ 6 Before carrying on, we note that for any Hermitian vector space V there exists a canonically defined C-antilinear bijection C : V V∗ to the dual V → vectorspaceV∗ by C (v) := v, . (InDirac’slanguagethis istheconversion V V h ·i from ‘ket’ vector to ‘bra’ vector.) By naturalness of the transfer of Hermitian structure we have the relation C = C C . E⊗R E R ⊗ 6 Turning to the more involved step of transferring time reversal T, we begin withapreparation. IfL : V W isalinearmappingbetweenvectorspaces,we → denote by Lt : W∗ V∗ the canonical transpose defined by (Ltf)(v) = f(Lv). → Let now V be our Hilbert space with ket-bra bijection C C . Then for any V ≡ g U(V) we have the relation CgC−1 = (g−1)t because ∈ C(gv) = gv, = v,g−1 = (g−1)tC(v) (v V) . (2.7) h ·i h ·i ∈ Moreover,recallingtheautomorphismG g a(g) = TgT−1 ofG weobtain 0 0 ∋ 7→ CT g = a(g−1)tCT (g G ) . (2.8) 0 ∈ Thus, since C and T are bijective, the C-linear mapping CT : V V∗ is a → G -equivariant isomorphism interchanging the given G -representation on V 0 0 with the representation on V∗ by g a(g−1)t. In particular, it follows that T 7→ stabilizes the decomposition V = V (E R ) of Lemma 2.1. λ λ λ λ λ ⊕ ≃ ⊕ ⊗ If T exchanges different sectors V , the situation is very easy to handle (see λ below). The more challenging case is TV = V , which we now assume. λ λ Lemma 2.2 Let TV = V . Under the isomorphism V E R the time- λ λ λ λ λ ≃ ⊗ reversal operator transfers to a pure tensor T = α β , α : E E , β : R R , λ λ λ λ ⊗ → → with anti-unitary α and β. Proof. Writing E E and R R for short, we consider the transferred λ λ ≡ ≡ mapping CT : E R E∗ R∗, which expands as CT = φ ψ with C- i i ⊗ → ⊗ ⊗ linearmappingsφ ,ψ . SinceCT isknowntobeaG -equivarPiantisomorphism, i i 0 so is every map ψ : R R∗. By the irreducibility of R and Schur’s lemma, i → there exists only one such map (up to scalar multiples). Hence CT is a pure tensor: CT = φ ψ. Using C = C = C C we obtain T = α β E⊗R E R ⊗ ⊗ ⊗ with C-antilinear α = C−1φ and β = C−1ψ. Since the tensor product lets E R you move scalars between factors, the maps α and β are not uniquely defined. We may use this freedom to make β anti-unitary. Because T is anti-unitary, it then follows from the definition (2.6) of the Hermitian structure of E that α is anti-unitary. Remark. By an elementary argument, which was spelled out for the anti- unitary operator T in Eq. (2.3), it follows that α2 = ǫ Id and β2 = ǫ Id α E β R with ǫ ,ǫ 1 . Writing T2 = ǫ Id we have the relation ǫ ǫ = ǫ . Thus α β T V α β T ∈ {± } when ǫ = 1 the parity ǫ = ǫ of the transferred time-reversal operator α β α T − − is opposite to that of the original time reversal T. This change of parity occurs, e.g., in the case of G = SU . Indeed, let 0 2 R R be the irreducible SU -representation of dimension n + 1. It is a n 2 ≡ 7 standard fact of representation theory that R is SU -equivariantly isomorphic n 2 to R∗ by a symmetric isomorphism ψ = ψt for even n and skew-symmetric n isomorphism ψ = ψt for odd n. From ( 1)nψt = ψ = C β and R − − ψ(v)(v′) = βv,v′ = β2v,βv′ = βv′,β2v = ψ(v′)(β2v) , (2.9) h iR h iR h iR we conclude that β2 = ( 1)nId . − Rn 2.2 Classification BythedecompositionofLemma2.1thespaceZ (G )ofG -compatibletime U(V) 0 0 evolutions in U(V) is a direct product of unitary groups, Z (G ) U(E ) . (2.10) U(V) 0 λ ≃ λ Y We now fix a sector V E R and run through the different situations λ λ λ ≃ ⊗ (of which there exist three, essentially) due to the absence or presence of a transferred time-reversal symmetry α : E E . λ λ → 2.2.1 Class A The first type of situation occurs when the set G of anti-unitary symmetries is 1 eitheremptyorelsemapsVλ Eλ Rλ toadifferentsectorVλ′,λ = λ′. Inboth ≃ ⊗ 6 cases, the G-compatible time-evolution operators restricted to V constitute a λ unitary group U(E ) U with N = dimE being the multiplicity of the λ N λ ≃ irreducible representation R in V. The unitary groups U or to be precise, λ N their simple parts SU , are symmetric spaces (cf. Section 3.4) of the A family N or A series in Cartan’s notation – hence the name Class A. In random matrix theory, the Lie group U equipped with Haar measure is commonly referred to N as the Circular Unitary Ensemble, CUE [Dys62a]. N TheHamiltoniansH inClassAarerepresentedbycomplexHermitianN N × matrices. By putting a U -invariant Gaussian probability measure N N dµ(H) = c e−TrH2/2σ2dH , dH = dH dH dH , (2.11) 0 ii jk kj Yi=1 jY<k on that space, one gets what is called the GUE – the Gaussian Unitary Ensem- ble – defining the Wigner-Dyson universality class of unitary symmetry. An important physical realization of that class is by electrons in a disordered metal with time-reversal symmetry broken by the presence of a magnetic field. 2.2.2 Classes AI and AII We now turn to the cases where T is present and TV = V E R . We λ λ λ λ ≃ ⊗ abbreviate E E. From Lemma 2.1 we know that T = α β is a pure tensor λ ≡ ⊗ with anti-unitary α, and we have α2 = ǫ Id with parity ǫ = ǫ ǫ . α E α T β 8 Using conjugation by α to define an automorphism τ : U(E) U(E) , u αuα−1, (2.12) → 7→ we transfer the conditions (2.4) to V and describe the set Z (G) of G- λ U(E) compatible time evolutions in U(E) as Z (G) = x U(E) τ(x) = x−1 . (2.13) U(E) { ∈ | } Now let U U(E) for short and denote by K U the subgroup of τ-fixed ≡ ⊂ elements k = τ(k) U. The set Z (G) is analytically diffeomorphic to the U ∈ coset space U/K by the mapping U/K Z (G) U , uK uτ(u−1) , (2.14) U → ⊂ 7→ which is called the Cartan embedding of U/K into U. The remaining task is to determine K. This is done as follows. Recalling the definition C α = φ and using C k = (k−1)tC we express E E E thefixed-pointconditionk = τ(k) = αkα−1 as(k−1)t = φkφ−1 or, equivalently, φ = ktφk, which means that the bilinear form associated with φ : E E∗, → Q : E E C , (e,e′) φ(e)(e′) , (2.15) φ × → 7→ is preserved by k K. By running the argument around Eq. (2.9) in reverse ∈ order (with the obvious substitutions ψ φ and β α), we see that the → → non-degenerate form Q is symmetric if ǫ = +1 and skew if ǫ = 1. In the φ α α − former case it follows that K = O(E) O is an orthogonal group, while in N ≃ the latter case, which occurs only if N 2N, K = USp(E) USp is unitary ∈ ≃ N symplectic. InbothcasesthecosetspaceU/K isasymmetricspace(cf.Section 3.4) – a fact first noticed by Dyson in [Dys70]. Thus in the present setting of TV = V we have the following dichotomy λ λ for the sets of G-compatible time evolutions Z (G) U/K : U(E) ≃ Class AI : U/K U /O (ǫ = +1) , N N α ≃ Class AII : U/K U /USp (ǫ = 1, N 2N) . ≃ N N α − ∈ Again we are referring to symmetric spaces by the names they – or rather their simple parts SU /SO and SU /USp – have in the Cartan classification. N N N N In random matrix theory, the symmetric space U /O (or its Cartan embed- N N ding into U as the symmetric unitary matrices) equipped with U -invariant N N probability measure is called the Circular Orthogonal Ensemble, COE , while N the Cartan embedding of U /USp equipped with U -invariant probability N N N measure is known as the Circular Symplectic Ensemble, CSE [Dys62a]. (Note N the confusing fact that the naming goes by the subgroup which is divided out.) 9 Examples for Class AI are provided by time-reversal invariant systems with symmetryG = (SU ) . Indeed,bythefundamentallawsofquantumphysics 0 2 spin timereversalT squaresto( 1)2S timestheidentityonstateswithspinS. Such − states transform according to the irreducible SU -representation of dimension 2 2S + 1, and from β2 = ( 1)2S (see the Remark after Lemma 2.1) it follows − that ǫ = ǫ ǫ = ( 1)2S( 1)2S = +1 in all cases. A historically important α T β − − realization of Class AI is furnished by the highly excited states of atomic nuclei as observed by neutron scattering just above the neutron threshold. By breaking SU -symmetry (i.e., by taking G = Id ) while maintaining 2 0 { } T-symmetry for states with half-integer spin (say single electrons, which carry spinS = 1/2),onegetsǫ = ǫ = ( 1)2S = 1,therebyrealizingClassAII.An α T − − experimental observation of this class and its characteristic wave interference phenomena was first reported in the early 1980’s [Ber84] for disordered metallic magnesium films with strong spin-orbit scattering caused by gold impurities. The Hamiltonians H, obtained by passing to the tangent space of U/K at unity, are represented by Hermitian matrices with entries that are real num- bers (Class AI) or real quaternions (Class AII). The simplest random matrix modelsresultfromputtingK-invariantGaussianprobabilitymeasuresonthese spaces; they are called the Gaussian Orthogonal Ensemble and Gaussian Sym- plectic Ensemble, respectively. Their properties delineate the Wigner-Dyson universality classes of orthogonal and symplectic symmetry. 3 Symmetry Classes of Disordered Fermions WhileDyson’sThreefoldWayisfundamentalandcompleteinitsgeneralHilbert space setting, the early 1990’s witnessed the discovery of various new types of strong universality, which were begging for an extended scheme: The introduction of QCD-motivated chiral random matrix ensembles (re- • viewedbyVerbaarschotinChapter32ofthishandbook)mimickedDyson’s scheme but also transcended it. Number theorists had introduced and studied ensembles of L-functions • akin to the Riemann zeta function (see the review by Keating and Snaith in Chapter 24 of this handbook). These display random matrix phenom- ena which are absent in the classes A, AI, or AII. The proximity effect due to Andreev reflection, a particle-hole conversion • process in mesoscopic hybrid systems involving metallic as well as su- perconducting components, was found [AZ97] to give rise to post-Dyson mechanisms of quantum interference (cf. Chapter 35 by Beenakker). By the middle of the 1990’s, it had become clear that there exists a unifying mathematicalprinciplegoverningthesepost-Dysonrandommatrixphenomena. 10