Comparison of the superbosonization formula and the generalized Hubbard-Stratonovich transformation
aa r X i v : . [ m a t h - ph ] J un Comparison of the superbosonization formula andthe generalized Hubbard–Stratonovichtransformation
Mario Kieburg † , Hans-J¨urgen Sommers and Thomas Guhr Universit¨at Duisburg-Essen, Lotharstraße 1, 47048 Duisburg, GermanyE-mail: † [email protected] Abstract.
Recently, two different approaches were put forward to extend thesupersymmetry method in random matrix theory from Gaussian ensembles to generalrotation invariant ensembles. These approaches are the generalized Hubbard–Stratonovich transformation and the superbosonization formula. Here, we prove theequivalence of both approaches. To this end, we reduce integrals over functions ofsupersymmetric Wishart–matrices to integrals over quadratic supermatrices of certainsymmetries.PACS numbers: 02.30.Px, 05.30.Ch, 05.30.-d, 05.45.Mt
Submitted to:
J. Phys. A: Math. Gen. uperbosonization formula and generalized Hubbard–Stratonovich transformation
1. Introduction
The supersymmetry technique is a powerful method in random matrix theory anddisordered systems. For a long time it was thought to be applicable for Gaussianprobability densities only [1, 2, 3, 4]. Due to universality on the local scale of themean level spacing [5, 6, 7, 8], this restriction was not a limitation for calculatingin quantum chaos and disordered systems. Indeed, results of Gaussian ensembles areidentical for large matrix dimension with other invariant matrix ensembles on this scale.In the Wigner–Dyson theory [9] and its corrections for systems with diffusive dynamics[10], Gaussian ensembles are sufficient. Furthermore, universality was found on largescale, too [11]. This is of paramount importance when investigating matrix models inhigh–energy physics.There are, however, situations in which one can not simply resort to Gaussianrandom matrix ensembles. The level densities in high–energy physics [12] and finance[13] are needed for non-Gaussian ensembles. But these one–point functions stronglydepend on the matrix ensemble. Other examples are bound–trace and fixed–traceensembles [14], which are both norm–dependent ensembles [15], as well as ensemblesderived from a non-extensive entropy principle [16, 17, 18]. In all these cases one isinterested in the non-universal behavior on special scales.Recently, the supersymmetry method was extended to general rotation invariantprobability densities [15, 19, 20, 21]. There are two approaches. The first one is thegeneralized Hubbard–Stratonovich transformation [15, 21]. With help of a proper Dirac–distribution in superspace an integral over rectangular supermatrices was mapped to asupermatrix integral with non-compact domain in the Fermion–Fermion block. Thesecond approach is the superbosonization formula [19, 20] mapping the same integralover rectangular matrices as before to a supermatrix integral with compact domain inthe Fermion–Fermion block.In this work, we prove the equivalence of the generalized Hubbard–Stratonovichtransformation with the superbosonization formula. The proof is based on integralidentities between supersymmetric Wishart–matrices and quadratic supermatrices. Theorthogonal, unitary and unitary-symplectic classes are dealt with in a unifying way.The article is organized as follows. In Sec. 2, we give a motivation and introduceour notation. In Sec. 3, we define rectangular supermatrices and the supersymmetricversion of Wishart-matrices built up by supervectors. We also give a helpful corollary forthe case of arbitrary matrix dimension discussed in Sec. 7. In Secs. 4 and 5, we presentand further generalize the superbosonization formula and the generalized Hubbard–Stratonovich transformation, respectively. The theorem stating the equivalence of bothapproaches is given in Sec. 6 including a clarification of their mutual connection. InSec. 7, we extend both theorems given in Secs. 4 and 5 to arbitrary matrix dimension.Details of the proofs are given in the appendices. uperbosonization formula and generalized Hubbard–Stratonovich transformation
2. Ratios of characteristic polynomials
We employ the notation defined in Refs. [22, 21]. Herm ( β, N ) is either the set of N × N real symmetric ( β = 1), N × N hermitian ( β = 2) or 2 N × N self-dual ( β = 4) matrices,according to the Dyson–index β . We use the complex representation of the quaternionicnumbers H . Also, we define γ = ( , β ∈ { , } , β = 1 , γ = ( , β ∈ { , } , β = 4 (2.1)and ˜ γ = γ γ .The central objects in many applications of supersymmetry are averages over ratiosof characteristic polynomials [23, 24, 25] Z k k ( E − ) = Z Herm ( β,N ) P ( H ) k Q n =1 det ( H − ( E n − ıε ) γ N ) k Q n =1 det ( H − ( E n − ıε ) γ N ) d [ H ]= Z Herm ( β,N ) P ( H )Sdet − / ˜ γ (cid:0) H ⊗ ˜ γ ( k + k ) − γ N ⊗ E − (cid:1) d [ H ] (2.2)where P is a sufficiently integrable probability density on the matrix set Herm ( β, N )invariant under the groupU ( β ) ( N ) = O( N ) , β = 1U ( N ) , β = 2USp (2 N ) , β = 4 . (2.3)Here, we assume that P is analytic in its real independent variables. We use thesame measure for d [ H ] as in Ref. [22] which is the product over all real independentdifferentials, see also Eq. (4.11). Also, we define E = diag ( E , . . . , E k , E , . . . , E k ) ⊗ ˜ γ and E − = E − ıε ˜ γ ( k + k ) .The generating function of the k –point correlation function [26, 27, 15, 21] R k ( x ) = γ − k Z Herm ( β,N ) P ( H ) k Y p =1 tr δ ( x p − H ) d [ H ] (2.4)is one application and can be computed starting from the matrix Green function andEq. (2.2) with k = k = k . Another example is the n –th moment of the characteristicpolynomial [28, 29, 27] b Z n ( x, µ ) = Z Herm ( β,N ) P ( H )Θ( H )det n ( H − E γ k ) d [ H ] , (2.5)where the Heavyside–function for matrices Θ( H ) is unity if H is positive definite andzero otherwise. [21]With help of Gaussian integrals, we get an integral expression for the determinantsin Eq. (2.2). Let Λ j be the Grassmann space of j –forms. We consider a complex uperbosonization formula and generalized Hubbard–Stratonovich transformation γ Nk L j =0 Λ j with γ N k pairs { ζ jn , ζ ∗ jn } , 1 ≤ n ≤ k , ≤ j ≤ γ N , of Grassmann variables and use the conventions of Ref. [22] for integrations overGrassmann variables. Due to the Z –grading, Λ is a direct sum of the set of commutingvariables Λ and of anticommuting variables Λ . The body of an element in Λ lies in Λ while the Grassmann generators are elements in Λ .Let ı be the imaginary unit. We take γ N k pairs { z jn , z ∗ jn } , 1 ≤ n ≤ k , ≤ j ≤ γ N , of complex numbers and find for Eq. (2.2) Z k k ( E − ) = (2 π ) γ N ( k − k ) ı γ Nk Z C F P ( K ) exp (cid:0) − ı Str BE − (cid:1) d [ ζ ] d [ z ] (2.6)where d [ z ] = k Q p =1 γ N Q j =1 dz jp dz ∗ jp , d [ ζ ] = k Q p =1 γ N Q j =1 ( dζ jp dζ ∗ jp ) and C = C γ k N × Λ γ Nk . Thecharacteristic function appearing in (2.6) is defined as F P ( K ) = Z Herm ( β,N ) P ( H ) exp ( ı tr HK ) d [ H ] . (2.7)The two matrices K = 1˜ γ V † V and B = 1˜ γ V V † (2.8)are crucial for the duality between ordinary and superspace. While K is a γ N × γ N ordinary matrix whose entries have nilpotent parts, B is a ˜ γ ( k + k ) × ˜ γ ( k + k )supermatrix. They are composed of the rectangular γ N × ˜ γ ( k + k ) supermatrix V † | β =2 = ( z , . . . , z k , Y z ∗ , . . . , Y z ∗ k , ζ , . . . , ζ k , Y ζ ∗ , . . . , Y ζ ∗ k ) ,V | β =2 = ( z ∗ , . . . , z ∗ k , Y z , . . . , Y z k , − ζ ∗ , . . . , − ζ ∗ k , Y ζ , . . . , Y ζ k ) T ,V † | β =2 = ( z , . . . , z k , ζ , . . . , ζ k ) ,V | β =2 = ( z ∗ , . . . , z ∗ k , − ζ ∗ , . . . , − ζ ∗ k ) T . (2.9)The transposition “ T ” is the ordinary transposition and is not the supersymmetricone. However, the adjoint “ † ” is the complex conjugation with the supersymmetrictransposition “ T S ” σ T S = " σ σ σ σ T S = " σ T σ T − σ T σ T , (2.10)where σ is an arbitrary rectangular supermatrix. We introduce the constant γ N × γ N matrix Y = ( N , β = 1 Y Ts ⊗ N , β = 4 , Y s = " − . (2.11)The crucial duality relation [15, 21]tr K m = Str B m , m ∈ N , (2.12) uperbosonization formula and generalized Hubbard–Stratonovich transformation F P inherits the rotationinvariance of P , the duality relation (2.12) yields Z k k ( E − ) = (2 π ) γ N ( k − k ) ı γ Nk Z C Φ( B ) exp (cid:0) − ı Str BE − (cid:1) d [ ζ ] d [ z ] . (2.13)Here, Φ is a supersymmetric extension of a representation F P of the characteristicfunction, Φ( B ) = F P (Str B m | m ∈ N ) = F P (tr K m | m ∈ N ) = F P ( K ) . (2.14)The representation F P is not unique [31]. However, the integral (2.13) is independentof a particular choice [21].The supermatrix B fulfills the symmetry B ∗ = ( e Y B e Y T , β ∈ { , } , e Y B ∗ e Y T , β = 2 (2.15)with the supermatrices e Y | β =1 = k k Y s ⊗ k , e Y | β =4 = Y s ⊗ k k k (2.16)and e Y | β =2 = k + k and is self-adjoint for every β . Using the π/ U | β =1 = 1 √ k k − ı k ı k
00 0 √ k , U | β =4 = 1 √ √ k k k − ı k ı k (2.17)and U | β =2 = k + k , b B = U BU † lies in the well-known symmetric superspaces [32], e Σ ( † ) β,γ k ,γ k = (cid:26) σ ∈ Mat(˜ γk / ˜ γk ) (cid:12)(cid:12)(cid:12)(cid:12) σ † = σ,σ ∗ = ( b Y γ k ,γ k σ b Y Tγ k ,γ k , β ∈ { , } b Y k k σ ∗ b Y Tk k , β = 2 )) (2.18)where b Y pq (cid:12)(cid:12)(cid:12) β =1 = " p Y s ⊗ q , b Y pq (cid:12)(cid:12)(cid:12) β =2 = p + q and b Y pq (cid:12)(cid:12)(cid:12) β =4 = " Y s ⊗ p q . (2.19)The set Mat( p/q ) is the set of ( p + q ) × ( p + q ) supermatrices on the complex Grassmannalgebra pq L j =0 Λ j . The entries of the diagonal blocks of an element in Mat( p/q ) lie in Λ whereas the entries of the off-diagonal block are elements in Λ .The rectangular supermatrix b V † = V † U † is composed of real, complex orquaternionic supervectors whose adjoints form the rows. They are given byΨ † j = (cid:16)(cid:8) √ z jn , √ z jn (cid:9) ≤ n ≤ k , (cid:8) ζ jn , ζ ∗ jn (cid:9) ≤ n ≤ k (cid:17) , β = 1 , (cid:16) { z jn } ≤ n ≤ k , { ζ jn } ≤ n ≤ k (cid:17) , β = 2 , ( z jn − z ∗ j + N,n z j + N,n z ∗ jn ) ≤ n ≤ k , ( ζ ( − ) jn ζ (+) jn ζ ( − ) ∗ jn ζ (+) ∗ jn ) ≤ n ≤ k ! , β = 4 , (2.20) uperbosonization formula and generalized Hubbard–Stratonovich transformation ζ ( ± ) jn = ı (1 ± / ( ζ jn ± ζ ∗ j + N,n ) / √
2. Then, the supermatrix b B acquiresthe form b B = 1˜ γ N X j =1 Ψ j Ψ † j . (2.21)The integrand in Eq. (2.13) F (cid:16) b B (cid:17) = Φ (cid:16) b B (cid:17) exp (cid:16) − ı Str E b B (cid:17) (2.22)comprises a symmetry breaking term, ∃ U ∈ U ( β ) ( γ k /γ k ) that F (cid:16) b B (cid:17) = F (cid:16) U b BU † (cid:17) , (2.23)according to the supergroupU ( β ) ( γ k /γ k ) = UOSp (+) (2 k / k ) , β = 1U ( k /k ) , β = 2UOSp ( − ) (2 k / k ) , β = 4 . (2.24)We use the notation of Refs. [33, 22] for the representations UOSp ( ± ) of the supergroupUOSp . These representations are related to the classification of Riemannian symmetricsuperspaces by Zirnbauer [32]. The index “+” in Eq. (2.24) refers to real entries in theBoson–Boson block and to quaternionic entries in the Fermion–Fermion block and “ − ”indicates the other way around.
3. Supersymmetric Wishart–matrices and some of their properties
We generalize the integrand (2.22) to arbitrary sufficiently integrable superfunctions onrectangular ( γ c + γ d ) × ( γ a + γ b ) supermatrices b V on the complex Grassmann–algebraΛ = ad + bc ) L j =0 Λ j . Such a supermatrix b V = (cid:16) Ψ (C)11 , . . . , Ψ (C) a , Ψ (C)12 , . . . Ψ (C) b (cid:17) = (cid:16) Ψ (R) ∗ . . . , Ψ (R) ∗ c , Ψ (R) ∗ , . . . Ψ (R) ∗ d (cid:17) T S (3.1)is defined by its columnsΨ (C) † j = (cid:16) { x jn } ≤ n ≤ c , (cid:8) χ jn , χ ∗ jn (cid:9) ≤ n ≤ d (cid:17) , β = 1 , (cid:16) { z jn } ≤ n ≤ c , { χ jn } ≤ n ≤ d (cid:17) , β = 2 , ( z jn − z ∗ jn z jn z ∗ jn ) ≤ n ≤ c , ( χ jn χ ∗ jn ) ≤ n ≤ d ! , β = 4 , (3.2)Ψ (C) † j = ( ζ jn ζ ∗ jn ) ≤ n ≤ c , ( ˜ z jn − ˜ z ∗ jn ˜ z jn ˜ z ∗ jn ) ≤ n ≤ d ! , β = 1 , (cid:16) { ζ jn } ≤ n ≤ c , { ˜ z jn } ≤ n ≤ d (cid:17) , β = 2 , (cid:16)(cid:8) ζ jn , ζ ∗ jn (cid:9) ≤ n ≤ c , { y jn } ≤ n ≤ d (cid:17) , β = 4 , (3.3) uperbosonization formula and generalized Hubbard–Stratonovich transformation (R) † j = (cid:16) { x nj } ≤ n ≤ a , (cid:8) ζ ∗ nj , − ζ nj (cid:9) ≤ n ≤ b (cid:17) , β = 1 , (cid:16)(cid:8) z ∗ nj (cid:9) ≤ n ≤ a , (cid:8) ζ ∗ nj (cid:9) ≤ n ≤ b (cid:17) , β = 2 , ( z ∗ nj z ∗ nj − z nj z nj ) ≤ n ≤ a , ( ζ ∗ nj − ζ nj ) ≤ n ≤ b ! , β = 4 , (3.4)Ψ (R) † j = ( − χ ∗ nj χ nj ) ≤ n ≤ a , ( ˜ z ∗ nj ˜ z ∗ nj − ˜ z nj ˜ z nj ) ≤ n ≤ b ! , β = 1 , (cid:16)(cid:8) − χ ∗ nj (cid:9) ≤ n ≤ a , (cid:8) ˜ z ∗ nj (cid:9) ≤ n ≤ b (cid:17) , β = 2 , (cid:16)(cid:8) − χ ∗ nj , χ nj (cid:9) ≤ n ≤ a , { y nj } ≤ n ≤ b (cid:17) , β = 4 (3.5)which are real, complex and quaternionic supervectors. We use the complex Grassmannvariables χ mn and ζ mn and the real numbers x mn and y mn . Also, we introduce thecomplex numbers z mn , ˜ z mn , z mnl and ˜ z mnl . The ( γ c + γ d ) × ( γ c + γ d ) supermatrix b B = ˜ γ − b V b V † can be written in the columns of b V as in Eq. (2.21). As this supermatrixhas a form similar to the ordinary Wishart–matrices, we refer to it as supersymmetricWishart–matrix. The rectangular supermatrix above fulfills the property b V ∗ = b Y cd b V b Y Tab . (3.6)The corresponding generating function (2.2) is an integral over a rotation invariantsuperfunction P on a superspace, which is sufficiently convergent and analytic in itsreal independent variables, Z abcd ( E − ) = Z e Σ ( − ψ ) β,ab P ( σ )Sdet − / ˜ γ (cid:16) σ ⊗ b Π (C)2 ψ − γ a + γ b ⊗ E − (cid:17) d [ σ ] , (3.7)where E − = diag ( E ⊗ γ , . . . , E c ⊗ γ , E ⊗ γ , . . . , E d ⊗ γ ) − ıε γ c + γ d . (3.8)Let e Σ † ) β,ab be a subset of e Σ ( † ) β,ab . The entries of elements in e Σ † ) β,ab lie in Λ and Λ . Theset e Σ ( − ψ ) β,ab = b Π (R) − ψ e Σ † ) β,ab b Π (R) − ψ is the Wick–rotated set of e Σ † ) β,ab by the generalized Wick–rotation b Π (R) − ψ = diag ( γ a , e − ıψ/ γ b ). As in Ref. [21], we introduce such a rotation forthe convergence of the integral (3.7). The matrix b Π (C)2 ψ = diag ( γ c , e ıψ γ d ) is also aWick–rotation.In the rest of our work, we restrict the calculations to a class of superfunctions.These superfunctions has a Wick–rotation such that the integrals are convergent. Wehave not explicitly analysed the class of such functions. However, this class is very largeand sufficient for physical interests. We consider the probability distribution P ( σ ) = f ( σ ) exp( − Str σ m ) , (3.9)where m ∈ N and f is a superfunction which does not increase so fast as exp(Str σ m )in the infinty, in particularlim ǫ →∞ P ( ǫe ıα σ ) = 0 ⇔ lim ǫ →∞ exp (cid:0) − ǫe ıα Str σ m (cid:1) = 0 (3.10) uperbosonization formula and generalized Hubbard–Stratonovich transformation α ∈ [0 , π ]. Then, a Wick–rotation exists for P .To guarantee the convergence of the integrals below, let b V ψ = b Π (C) ψ b V , b V †− ψ = b V † b Π (C) ψ and b B ψ = b Π (C) ψ b B b Π (C) ψ . Considering a function f on the set of supersymmetric Wishart–matrices, we give a lemma and a corollary which are of equal importance for thesuperbosonization formula and the generalized Hubbard–Startonovich transformation.For b = 0, the lemma presents the duality relation between the ordinary and superspace(2.12) which is crucial for the calculation of (2.2). This lemma was proven in Ref. [20]by representation theory. Here, we only state it. Lemma 3.1
Let f be a superfunction on rectangular supermatrices of the form (3.1) and invariantunder f ( b V ψ , b V †− ψ ) = f (cid:16) b V ψ U † , U b V †− ψ (cid:17) , (3.11) for all b V and U ∈ U ( β ) ( a/b ) . Then, there is a superfunction F on the U ( β ) ( c/d ) –symmetric supermatrices with F ( b B ψ ) = f ( b V ψ , b V †− ψ ) . (3.12)The U ( β ) ( c/d )–symmetric supermatrices are elements of e Σ ( † ) β,ab . The invariancecondition (3.11) implies that f only depends on the rows of b V ψ by Ψ (R) † nr Ψ (R) ms for arbitrary n, m, r and s . These scalar products are the entries of the supermatrix b V ψ b V †− ψ whichleads to the statement.The corollary below is an application of integral theorems by Wegner [34] workedout in Refs. [35, 36] and of the Theorems III.1, III.2 and III.3 in Ref. [22]. It states thatan integration over supersymmetric Wishart–matrices can be reduced to integrationsover supersymmetric Wishart–matrices comprising a lower dimensional rectangularsupermatrix. In particular for the generating function, it reflects the equivalence ofthe integral (3.7) with an integration over smaller supermatrices [22]. We assume that˜ a = a − b − ˜ b ) /β ≥ b = ( , β = 4 and b ∈ N + 10 , else . (3.13) Corollary 3.2
Let F be the superfunction of Lemma 3.1, real analytic in its real independent entriesand a Schwartz–function. Then, we find Z R F ( b B ψ ) d [ b V ] = C Z e R F ( e B ψ ) d [ e V ] (3.14) where e B = ˜ γ − e V e V . The sets are R = R βac +4 bd/β × Λ ad + bc ) and e R = R β ˜ ac +4˜ bd/β × Λ ad +˜ bc ) , the constant is C = h − γ i ( b − ˜ b ) c h γ i ( a − ˜ a ) d (3.15) uperbosonization formula and generalized Hubbard–Stratonovich transformation and the measure d [ b V ] = Y ≤ m ≤ a ≤ n ≤ c ≤ l ≤ β dx mnl Y ≤ m ≤ b ≤ n ≤ d ≤ l ≤ /β dy mnl Y ≤ m ≤ b ≤ n ≤ c dζ mn dζ ∗ mn Y ≤ m ≤ a ≤ n ≤ d dχ mn dχ ∗ mn . (3.16) The ( γ c + γ d ) × ( γ ˜ a + γ ˜ b ) supermatrix e V and its measure d [ e V ] is defined analogous to b V and d [ b V ] , respectively. Here, x mna and y mna are the independent real components of thereal, complex and quaternionic numbers of the supervectors Ψ (R) j and Ψ (R) j , respectively. Proof:
We integrate F over all supervectors Ψ (R) j and Ψ (R) j except Ψ (R)11 . Then, Z R ′ F ( V ψ V †− ψ ) d [ b V =11 ] (3.17)only depends on Ψ (R) † Ψ (R)11 . The integration set is R ′ = R βa ( c − bd/β × Λ ad + b ( c − and the measure d [ b V =11 ] is d [ b V ] without the measure for the supervector Ψ (R)11 . Withhelp of the Theorems in Ref. [34, 35, 36, 22], the integration over Ψ (R)11 is up to aconstant equivalent to an integration over a supervector e Ψ (R)11 . This supervector is equalto Ψ (R)11 in the first ˜ a –th entries and else zero. We repeat this procedure for all othersupervectors reminding that we only need the invariance under the supergroup actionU ( β ) (cid:16) b − ˜ b/b − ˜ b (cid:17) on f as in Eq. (3.11) embedded in U ( β ) ( a/b ). This invariance ispreserved in each step due to the zero entries in the new supervectors. (cid:3) This corollary allows us to restrict our calculation on supermatrices with b = 1 onlyto β = 4 and b = 0 for all β . Only the latter case is of physical interest. Thus, we givethe computation for b = 0 in the following sections and consider the case b = 1 in Sec.7. For b = 0 we omit the Wick–rotation for b B as it is done in Refs. [15, 21] due to theconvergence of the integral (3.7).
4. The superbosonization formula
We need for the following theorem the definition of the setsΣ ,pq = σ = σ η η ∗ − η † σ σ (1)22 η T σ (2)22 σ T ∈ Mat( p/ q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ † = σ ∗ = σ with positivedefinite body , σ (1) T = − σ (1)22 , σ (2) T = − σ (2)22 o , (4.1)Σ ,pq = ( σ = " σ η − η † σ ∈ Mat( p/q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ † = σ with positive definite body ) , (4.2)Σ ,pq = σ = σ σ η − σ ∗ σ ∗ η ∗ − η † η T σ ∈ Mat(2 p/q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ † = σ = " σ σ − σ ∗ σ ∗ with positive definite body , σ = σ T (cid:27) . (4.3) uperbosonization formula and generalized Hubbard–Stratonovich transformation ( † ) β,pq = n σ ∈ Σ β,pq | σ † = σ o = e Σ ( † ) β,pq ∩ Σ β,pq (4.4)and Σ (c) β,pq = n σ ∈ Σ β,pq | σ ∈ CU (4 /β ) ( q ) o (4.5)where CU ( β ) ( q ) is the set of the circular orthogonal (COE, β = 1), unitary (CUE, β = 2) or unitary-symplectic (CSE, β = 4) ensembles,CU ( β ) ( q ) = A ∈ Gl( γ q, C ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A = A T ∈ U (2) ( q ) , β = 1 A ∈ U (2) ( q ) , β = 2 A = ( Y s ⊗ q ) A T ( Y Ts ⊗ q ) ∈ U (2) (2 q ) , β = 4 (4.6)The index “ † ” in Eq. (4.4) refers to the self-adjointness of the supermatrices and theindex “c” indicates the relation to the circular ensembles. We notice that the set classespresented above differ in the Fermion–Fermion block. In Sec. 6, we show that this is thecrucial difference between both methods. Due to the nilpotence of B ’s Fermion–Fermionblock, we can change the set in this block for the Fourier–transformation. The sets ofmatrices in the sets above with entries in Λ and Λ are denoted by Σ β,pq , Σ † ) β,pq andΣ β,pq , respectively.The proof of the superbosonization formula [19, 20] given below is based onthe proofs of the superbosonization formula for arbitrary superfunctions on realsupersymmetric Wishart–matrices in Ref. [19] and for Gaussian functions on real,complex and quaternionic Wishart–matrices in Ref. [37]. This theorem extends thesuperbosonization formula of Ref. [20] to averages of square roots of determinants overunitary-symplectically invariant ensembles, i.e. β = 4, b = c = 0 and d odd in Eq. (3.7).The proof of this theorem is given in Appendix A. Theorem 4.1 (Superbosonization formula)
Let F be a conveniently integrable and analytic superfunction on the set of ( γ c + γ d ) × ( γ c + γ d ) supermatrices and κ = a − c + 1 γ + d − γ . (4.7) With a ≥ c , (4.8) we find Z R F ( b B ) exp (cid:16) − ε Str b B (cid:17) d [ b V ] = C ( β ) acd Z Σ β,cd F ( ρ ) exp ( − ε Str ρ ) Sdet ρ κ d [ ρ ] , (4.9) where the constant is C ( β ) acd = ( − πγ ) − ad (cid:18) − πγ (cid:19) cd − c ˜ γ βac/ Vol (cid:16) U ( β ) ( a ) (cid:17) Vol (cid:16) U ( β ) ( a − c ) (cid:17) × uperbosonization formula and generalized Hubbard–Stratonovich transformation × d Y n =1 Γ ( γ κ + 2( n − d ) /β ) ı n − /β π n − /β . (4.10) We define the measure d [ b V ] as in Corollary 3.2 and the measure on the right hand sideis d [ ρ ] = d [ ρ ] d [ ρ ] d [ η ] where d [ ρ ] = c Y n =1 dρ nn × Q ≤ n 2) (4.14) and the ratio of volumes of the group flag manifold and the permutation group FU (4 /β ) d = 1 d ! d Y j =1 π j − /β Γ(2 /β )Γ(2 j/β ) . (4.15) The absolute value of the Vandermonde determinant ∆ d ( e ıϕ j ) = Q ≤ n 5. The generalized Hubbard–Stratonovich transformation The following theorem is proven in a way similar to Refs. [15, 21]. The proof is givenin Appendix B. We need the Wick–rotated set Σ ( ψ ) β,cd = b Π (C) ψ Σ † ) β,cd b Π (C) ψ , particularlyΣ (0) β,cd = Σ † ) β,cd . The original extension of the Hubbard–Stratonovich transformation[15, 21] was only given for γ c = γ d = ˜ γk . Here, we generalize it to arbitrary c and d . Theorem 5.1 (Generalized Hubbard–Stratonovich transformation) Let F and κ be the same as in Theorem 4.1. If the inequality (4.8) holds, we have Z R F ( b B ) exp (cid:16) − ε Str b B (cid:17) d [Ψ] == e C ( β ) acd Z Σ ( ψ ) β,cd F ( ˆ ρ ) exp ( − ε Str ˆ ρ ) det ρ κ (cid:16) e − ıψd D (4 /β ) dr (cid:17) a − c δ ( r ) | ∆ d ( e ıψ r ) | /β e − ıψcd d [ ρ ] == e C ( β ) acd Z Σ (0) β,cd det ρ κ δ ( r ) | ∆ d ( r ) | /β (cid:16) ( − d D (4 /β ) dr (cid:17) a − c F ( ˆ ρ ) exp ( − ε Str ˆ ρ ) | ψ =0 d [ ρ ] (5.1) with ˆ ρ = " ρ e ıψ/ ρ η − e ıψ/ ρ † η e ıψ (cid:0) ρ − ρ † η ρ − ρ η (cid:1) . (5.2) The variables r are the eigenvalues of the supermatrix ρ . The measure d [ ρ ] = d [ ρ ] d [ ρ ] d [ η ] is defined by Eqs. (4.11) and (4.13). For the measure d [ ρ ] we take thedefinition (4.11) for /β . The differential operator in Eq. (5.1) is an analog of theSekiguchi–differential operator [38] and has the form [21] D (4 /β ) dr = 1∆ d ( r ) det (cid:20) r d − mn (cid:18) ∂∂r n + ( d − m ) 2 β r n (cid:19)(cid:21) ≤ n,m ≤ d . (5.3) The constant is e C ( β ) acd = 2 − c (2 πγ ) − ad (cid:18) πγ (cid:19) cd ˜ γ βac/ Vol (cid:16) U ( β ) ( a ) (cid:17) Vol (cid:16) U ( β ) ( a − c ) (cid:17) FU (4 /β ) d . (5.4)Since the diagonalization of ρ yields an | ∆ d ( r ) | /β in the measure, the ratio ofthe Dirac–distribution with the Vandermonde–determinant is for Schwartz–functionson Herm (4 /β, d ) well–defined. Also, the action of D (4 /β ) dr on such a Schwartz–functionintegrated over the corresponding rotation group is finite at zero.The distribution in the Fermion–Fermion block in Eq. (5.1) takes for β ∈ { , } thesimpler form [15, 21] (cid:16) D (4 /β ) dr (cid:17) a − c δ ( r ) | ∆ d ( r ) | /β == FU (4 /β ) d d Y n =1 Γ ( a − c + 1 + 2( n − /β )( − π ) n − /β Γ ( γ κ ) d Y n =1 ∂ γ κ − ∂r γ κ − n δ ( r n ) . (5.5) uperbosonization formula and generalized Hubbard–Stratonovich transformation β = 4, we do not find such a simplification due to the term | ∆( r ) | as the Jacobian inthe eigenvalue–angle coordinates. 6. Equivalence of and connections between the two approaches Above, we have argued that both expressions in Theorems 4.1 and 5.1 are equivalentfor β ∈ { , } . Now we address all β ∈ { , , } . The Theorem below is proven inAppendix C. The proof treats all three cases in a unifying way. Properties of theordinary matrix Bessel–functions are used. Theorem 6.1 (Equivalence of Theorems 4.1 and 5.1) The superbosonization formula, 4.1, and the generalized Hubbard–Stratonovichtransformation, 5.1, are equivalent for superfunctions which are Schwartz–functions andanalytic in the fermionic eigenvalues. The compact integral in the Fermion–Fermion block of the superbosonizationformula can be considered as a contour integral. In the proof of Theorem 6.1, wefind the integral identity Z [0 , π ] d e F ( e ıϕ j ) | ∆ d ( e ıϕ j ) | /β d Y n =1 e ı (1 − γ κ ) ϕ n dϕ n π == d Y n =1 ı n − /β Γ(1 + 2 n/β )Γ(2 /β + 1)Γ( γ κ − n − /β ) (cid:16) D (4 /β ) dr (cid:17) a − c e F ( r ) (cid:12)(cid:12)(cid:12)(cid:12) r =0 (6.1)for an analytic function e F on C d with permutation invariance. Hence, we can relateboth constants (4.10) and (5.4), e C ( β ) acd C ( β ) acd = ( − d ( a − c ) d Y n =1 ı n − /β Γ(1 + 2 n/β )Γ(2 /β + 1)Γ( γ κ − n − /β ) . (6.2)The integral identity (6.1) is a reminiscent of the residue theorem. It is the analog ofthe connection between the contour integral and the differential operator in the cases β ∈ { , } , see Fig. 1. Thus, the differential Operator with the Dirac–distribution inthe generalized Hubbard–Stratonovich transformation restricts the non-compact integralin the Fermion–Fermion block to the point zero and its neighborhood. Thereforeit is equivalent to a compact Fermion–Fermion block integral as appearing in thesuperbosonization formula. 7. The general case for arbitrary positive integers a , b , c , d and arbitraryDyson–index β ∈ { , , } We consider an application of our results. The inequality (4.8) reads N ≥ γ k (7.1) uperbosonization formula and generalized Hubbard–Stratonovich transformation Figure 1. In the superbosonization formula, the integration of the fermioniceigenvalues is along the unit circle in the complex plane (dotted circle). The eigenvalueintegrals in the generalized Hubbard–Stratonovich transformation are integrations overthe real axis (bold line) or on the Wick–rotated real axis (thin line at angle ψ ) if thedifferential operator acts on the superfunction or on the Dirac–distribution at zero(bold dot, 0), respectively. for the calculation of the k –point correlation function (2.4) with help of the matrix Greenfunction. For β = 1 , a N × N real symmetric matrix has in the absence of degeneracies N different eigenvalues. However, we can only calculate k –point correlation functionswith k < N/ 2. For N → ∞ , this restriction does not matter. But for exact finite N calculations, we have to modify the line of reasoning.We construct the symmetry operator S ( σ ) = S " σ σ σ σ = " − σ − σ σ σ (7.2)from ( m + m ) × ( n + n ) supermatrix to ( m + m ) × ( n + n ) supermatrix. Thisoperator has the properties S ( σ † ) = S ( σ ) † , (7.3) S ( σ ∗ ) = S ( σ ) ∗ , (7.4) S ( σ ) = − σ, (7.5) S " σ σ σ σ ρ ρ = S " σ σ σ σ S " ρ ρ . (7.6)Let a, b, c , d be arbitrary positive integers and β ∈ { , , } . Then, the equation(7.6) reads for a matrix product of a ( γ c + γ d ) × (0 + γ b ) supermatrix with a uperbosonization formula and generalized Hubbard–Stratonovich transformation γ b ) × ( γ c + γ d ) supermatrix " ζ † ˜ z † ζ ˜ z i = S " ˜ z † − ζ † S (cid:16)h ˜ z ζ i(cid:17) = S " ˜ z † − ζ † ˜ z ζ i! . (7.7)With help of the operator S , we split the supersymmetric Wishart–matrix b B into twoparts, b B = b B + S ( b B ) (7.8)such that b B = ˜ γ − a X j =1 Ψ (C) j Ψ (C) † j and b B = ˜ γ − b X j =1 S (cid:16) Ψ (C) j (cid:17) S (cid:16) Ψ (C) j (cid:17) † . (7.9)The supervectors S (cid:16) Ψ (C) j (cid:17) are of the same form as Ψ (C) j . Let σ be a quadraticsupermatrix, i.e. m = n and m = n . Then, we find the additional propertySdet S ( σ ) = ( − m Sdet − σ. (7.10)Let b Σ (0) β,pq = S (cid:16) Σ (0) β,pq (cid:17) , b Σ β,pq = S (cid:16) Σ β,pq (cid:17) and the Wick–rotated set b Σ ( ψ ) β,pq = b Π (C) ψ b Σ (0) β,pq b Π (C) ψ . Then, we construct the analog of the superbosonization formula andthe generalized Hubbard–Stratonovich transformation. Theorem 7.1 Let F be the superfunction as in Theorem 4.1 and κ = a − c + 1 γ − b − d + 1 γ . (7.11) Also, let e ∈ N and ˜ a = a + γ e and ˜ b = b + γ e (7.12) with ˜ a ≥ c ˜ b ≥ d. (7.13) We choose the Wick–rotation e ıψ such that all integrals are convergent. Then, we have Z R F ( b B ψ ) exp (cid:16) − ε Str b B ψ (cid:17) d [ b V ] == (cid:18) − γ (cid:19) γ ec (cid:18) γ (cid:19) γ ed Z e R F ( e B ψ ) exp (cid:16) − ε Str e B ψ (cid:17) d [ e V ] == C SF Z Σ β,cd Z b Σ /β,dc d [ ρ (2) ] d [ ρ (1) ] F ( ρ (1) + e ıψ ρ (2) ) exp (cid:2) − ε Str ( ρ (1) + e ıψ ρ (2) ) (cid:3) ×× Sdet κ +˜ b/γ ρ (1) Sdet κ − ˜ a/γ ρ (2) = (7.14)= C HS Z Σ (0) β,cd Z b Σ (0)4 /β,cd d [ ρ (2) ] d [ ρ (1) ] δ (cid:16) r (1)2 (cid:17)(cid:12)(cid:12)(cid:12) ∆ d (cid:16) r (1)2 (cid:17)(cid:12)(cid:12)(cid:12) /β δ (cid:16) r (2)1 (cid:17)(cid:12)(cid:12)(cid:12) ∆ c (cid:16) r (2)1 (cid:17)(cid:12)(cid:12)(cid:12) β det κ + b/γ ρ (1)1 det a/γ − κ ρ (2)2 × uperbosonization formula and generalized Hubbard–Stratonovich transformation × (cid:18) D (4 /β ) dr (1)2 (cid:19) ˜ a − c (cid:18) D ( β ) cr (2)1 (cid:19) ˜ b − d F ( ˆ ρ (1) + e ıψ ˆ ρ (2) ) exp (cid:2) − ε Str ( ˆ ρ (1) + e ıψ ˆ ρ (2) ) (cid:3) , (7.15) where the constants are C SF = ( − c ( b − d ) e ıψ (˜ ad − ˜ bc ) (cid:18) γ (cid:19) γ ec (cid:18) γ (cid:19) γ ed C ( β )˜ acd C (4 /β )˜ bdc , (7.16) C HS = ( − d ( a − c ) e ıψ (˜ ad − ˜ bc ) (cid:18) − γ (cid:19) γ ec (cid:18) − γ (cid:19) γ ed e C ( β )˜ acd e C (4 /β )˜ bdc . (7.17) Here, we define the supermatrix ˆ ρ (1) + e ıψ ˆ ρ (2) = " ρ (1)1 + e ıψ (cid:16) ρ (2)1 − ρ (2)˜ η ρ (2) − ρ (2) † ˜ η (cid:17) ρ (1) η + e ıψ ρ (2)˜ η − ρ (1) † η − e ıψ ρ (2) † ˜ η ρ (1)2 − ρ (1) † η ρ (1) − ρ (1) η + e ıψ ρ (2)2 (7.18) The set e R is given as in corollary 3.2. The measures d [ ρ (1) ] = d [ ρ (1)1 ] d [ ρ (1)2 ] d [ η ] and d [ ρ (2) ] = d [ ρ (2)1 ] d [ ρ (2)2 ] d [˜ η ] are given by Theorem 4.1. The measures (4.11) for β and /β assign d [ ρ (1)1 ] and d [ ρ (2)2 ] in Eqs. (7.14) and (7.15), respectively. In Eq. (7.14), d [ ρ (1)2 ] and d [ ρ (2)1 ] are defined by the measure (4.12) for the cases β and /β , respectively,and, in Eq. (7.15), they are defined by the measure (4.11) for the cases /β and β ,respectively. The measures d [ η ] and d [˜ η ] are the product of all complex Grassmann pairsas in Eq. (4.13). Since this Theorem is a consequence of corollary 3.2 and Theorems 4.1 and 5.1, theproof is quite simple. Proof: Let e ∈ N as in Eq. (7.12). Then, we use corollary 3.2 to extend the integral over b V toan integral over e V . We split the supersymmetric Wishart–matrix b B as in Eq. (7.8). BothWishart–matrices b B and b B fulfill the requirement (4.8) according to their dimension.Thus, we singly apply both Theorems 4.1 and 5.1 to b B and b B . (cid:3) Our approach of a violation of inequality (4.8) is quite different from the solutiongiven in Ref. [31]. These authors introduce a matrix which projects the Boson–Bosonblock and the bosonic side of the off-diagonal blocks onto a space of the smallerdimension a . Then, they integrate over all of such orthogonal projectors. This integralbecomes more difficult due to an additional measure on a curved, compact space. Weuse a second symmetric supermatrix. Hence, we have up to the dimensions of thesupermatrices a symmetry between both supermatrices produced by S . There is noadditional complication for the integration, since the measures of both supermatricesare of the same kind. Moreover, our approach extends the results to the case of β = 4and odd b which is not considered in Ref. [31]. 8. Remarks and conclusions We proved the equivalence of the generalized Hubbard–Stratonovich transformation[15, 21] and the superbosonization formula [19, 20]. Thereby, we generalized bothapproaches. The superbosonization formula was proven in a new way and is now uperbosonization formula and generalized Hubbard–Stratonovich transformation β = 4 and odd b which has not been considered in Ref. [31].The generalized Hubbard–Stratonovich transformation and the superbosonizationformula reduce in the absence of Grassmann variables to the ordinary integral identityfor ordinary Wishart–matrices. [29, 20] In the general case with the restriction (4.8),both approaches differ in the Fermion–Fermion block integration. Due to the Dirac–distribution and the differential operator, the integration over the non-compact domainin the generalized Hubbard–Stratonovich transformation is equal with help of the residuetheorem to a contour integral. This contour integral is equivalent to the integration overthe compact domain in the superbosonization formula. Hence, we found an integralidentity between a compact integral and a differentiated Dirac–distribution. Acknowledgements We thank Heiner Kohler for fruitful discussions. This work was supported by DeutscheForschungsgemeinschaft within Sonderforschungsbereich Transregio 12 “Symmetries andUniversality in Mesoscopic Systems”. Appendix A. Proof of Theorem 4.1 (Superbosonization formula) First, we consider two particular cases. Let d = 0 and a ≥ c be an arbitrary positiveinteger. Then, we find b B ∈ Σ β,c = Σ † ) β,c = Σ β,c ⊂ Herm ( β, c ) . (A.1)We introduce a Fourier–transformation Z R βac F ( b B ) exp (cid:16) − ε tr b B (cid:17) d [ b V ] == (cid:16) γ π (cid:17) c (cid:16) γ π (cid:17) βc ( c − / Z Herm ( β,c ) Z R βac F F ( σ ) exp (cid:16) ı tr b Bσ +1 (cid:17) d [ b V ] d [ σ ] (A.2)where the measure d [ σ ] is defined as in Eq. (4.10) and σ +1 = σ + ıε γ c . The Fourier–transform is F F ( σ ) = Z Herm ( β,c ) F ( ρ ) exp ( − ı tr ρ σ ) d [ ρ ] . (A.3) uperbosonization formula and generalized Hubbard–Stratonovich transformation Z R βac exp (cid:16) ı tr b Bσ +1 (cid:17) d [ b V ] = det (cid:18) σ +1 ıγ π (cid:19) − a/γ . (A.4)The Fourier–transform of this determinant is an Ingham–Siegel integral [40, 41] Z Herm ( β,c ) exp ( − ı tr ρ σ ) det (cid:0) − ıσ +1 (cid:1) − a/γ d [ σ ] = G ( β ) a − c,c det ρ κ exp ( − ε tr ρ ) Θ( ρ ) , (A.5)where the constant is G ( β ) a − c,c = (cid:16) γ π (cid:17) γ cκ a Y j = a − c +1 π βj/ Γ( βj/ 2) (A.6)and the exponent is κ = a − c + 1 γ − γ . (A.7)Γ( . ) is Euler’s gamma–function. This integral was recently used in random matrix theory[29] and is normalized in our notation as in Ref. [21]. Thus, we find for Eq. (A.2) Z R βac F ( b B ) exp (cid:16) − ε tr b B (cid:17) d [ b V ] = C ( β ) ac Z Σ β,c F ( ρ ) exp ( − ε tr ρ ) det ρ κ d [ ρ ] , (A.8)which verifies this theorem. The product in the constant C ( β ) ac = 2 − c ˜ γ βac/ Vol (cid:16) U ( β ) ( a ) (cid:17) Vol (cid:16) U ( β ) ( a − c ) (cid:17) (A.9)is a ratio of group volumes.In the next case, we consider c = 0 and arbitrary d . We see that b B ∈ Σ ( † ) β, d (A.10)is true. We integrate over Z Λ ad F ( b B ) exp (cid:16) ε tr b B (cid:17) d [ b V ] , (A.11)where the function F is analytic. As in Ref. [19], we expand F ( b B ) exp (cid:16) ε tr b B (cid:17) in theentries of b B and, then, integrate over every single term of this expansion. Every termis a product of b B ’s entries and can be generated by differentiation of (cid:16) tr A b B (cid:17) n withrespect to A ∈ Σ † ) β, d for certain n ∈ N . Thus, it is sufficient to proof the integral theorem Z Λ ad (cid:16) tr A b B (cid:17) n d [ b V ] = C ( β ) a d Z Σ β, d (tr Aρ ) n det ρ − κ d [ ρ ] . (A.12) uperbosonization formula and generalized Hubbard–Stratonovich transformation † ) β, d is generated of Σ β, d by analytic continuation in the eigenvalues, it isconvenient that A ∈ Σ β, d . Then, A − / is well-defined and A − / ρ A − / ∈ Σ β, d .We transform in Eq. (A.13) b V → A − / b V , b V † → b V † A − / and ρ → A − / ρ A − / . (A.13)The measures turns under this change into d [ b V ] → det A a/γ d [ b V ] and (A.14) d [ ρ ] → det A − κ + a/γ d [ ρ ] , (A.15)where the exponent is κ = a + 1 γ + d − γ . (A.16)Hence, we have to calculate the remaining constant defined by Z Λ ad (cid:16) tr b B (cid:17) n d [ b V ] = C ( β ) a d Z Σ β, d (tr ρ ) n det ρ − κ d [ ρ ] . (A.17)This equation holds for arbitrary n . Then, this must also be valid for F ( b B ) = ε = 1 inEq. (A.11). The right hand side of Eq. (A.11) is Z Λ ad exp (cid:16) tr b B (cid:17) d [ b V ] = ( − π ) − ad . (A.18)On the left hand side, we first integrate over the group U (4 /β ) ( d ) and get Z Σ β, d exp (tr ρ ) det ρ − κ d [ ρ ] = (A.19)= FU (4 /β ) d Z [0 , π ] d | ∆ d ( e ıϕ j ) | /β d Y n =1 exp ( γ e ıϕ n ) e − ıϕ n ( γ κ − dϕ n π . (A.20)We derive this integral with help of Selberg’s integral formula [14]. We assume that˜ β = 4 /β and γ κ are arbitrary positive integers and ˜ β is even. Then, we omit theabsolute value and Eq. (A.20) becomes Z Σ β, d exp (tr ρ ) det ρ − κ d [ ρ ] = FU ( β ) d ∆ ˜ βd (cid:18) γ ∂∂λ j (cid:19) d Y n =1 ( γ λ n ) γ κ − Γ ( γ κ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 . (A.21)We consider another integral which is the Laguerre version of Selberg’s integral [14] Z R d + ∆ ˜ βd ( x ) d Y n =1 exp ( − γ x n ) x ξn dx n = ∆ ˜ βd (cid:18) − γ ∂∂λ j (cid:19) d Y n =1 Γ( ξ + 1) ( γ λ n ) − ξ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 == d Y n =1 Γ (cid:16) n ˜ β/ (cid:17) Γ (cid:16) ξ + 1 + ( n − 1) ˜ β/ (cid:17) γ ξ +1+ ˜ β ( d − / Γ (cid:16) β/ (cid:17) , (A.22) uperbosonization formula and generalized Hubbard–Stratonovich transformation ξ is an arbitrary positive integer. Since ˜ β is even the minus sign in theVandermonde determinant vanishes. The equations (A.21) and (A.22) are up to theGamma–functions polynomials in κ and ξ . We remind that (A.22) is true for everycomplex ξ . Let Re ξ > 0, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ ˜ βd (cid:18) − γ ∂∂λ j (cid:19) d Y n =1 ( γ λ n ) − ξ − Γ (cid:16) ξ + 1 + ( n − 1) ˜ β/ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const . < ∞ and (A.23) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ − d ( ξ +1+ ˜ β ( d − / d Y n =1 Γ (cid:16) n ˜ β/ (cid:17) Γ( ξ + 1)Γ (cid:16) β/ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const . < ∞ . (A.24)The functions are bounded and regular for Re ξ > ξ = − γ κ and find Z Σ β, d exp (tr ρ ) det ρ − κ d [ ρ ] == γ ad FU (4 /β ) d d Y n =1 Γ (cid:16) n ˜ β/ (cid:17) Γ (cid:16) − γ κ + ( n − 1) ˜ β/ (cid:17) Γ (cid:16) β/ (cid:17) Γ ( γ κ ) Γ (1 − γ κ ) . (A.25)Due to Euler’s reflection formula Γ( z )Γ(1 − z ) = π/ sin( πz ), this equation simplifies to Z Σ β, d exp (tr ρ ) det ρ − κ d [ ρ ] = γ ad FU (4 /β ) d d Y n =1 ı n − /β Γ (1 + 2 n/β )Γ (1 + 2 /β ) Γ ( γ κ − n − /β ) (A.26)or equivalent2 ˜ βd ( d − / Z [0 , π ] d Y ≤ n 1) ˜ β/ (cid:17) . (A.27)Since a is a positive integer for all positive and even ˜ β , the equations above are true forall such ˜ β . For constant natural numbers a , d , γ and complex ˜ β with Re ˜ β > 0, theinequalities (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z [0 , π ] d Y ≤ n 1) ˜ β/ (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ uperbosonization formula and generalized Hubbard–Stratonovich transformation ≤ const . − Re ˜ βd ( d − / < ∞ (A.29)are valid and allow us with Carlson’s theorem to extend Eq. (A.27) to every complex ˜ β ,in particular to ˜ β = 1. Thus, we find for the constant in Eq. (A.17) C a d = ( − πγ ) − ad " d Y n =1 ı n − /β π n − /β Γ( a + 1 + 2( n − /β ) − . (A.30)Now, we consider arbitrary d and a ≥ c and splitΨ (C) j = " x j χ j (A.31)and b B = 1˜ γ a X j =1 Ψ (C) j Ψ (C) † j = a X j =1 x j x † j ˜ γ a X j =1 x j χ † j ˜ γ a X j =1 χ j x † j ˜ γ a X j =1 χ j χ † j ˜ γ = " B B B B (A.32)such that x j contains all commuting variables of Ψ (C) j and χ j depends on all Grassmannvariables. Then, we replace the sub-matrices B , B and B by Dirac–distributions Z R F ( b B ) exp (cid:16) − ε Str b B (cid:17) d [ b V ] == C Z Herm (4 /β,d ) Z R Z (Λ cd ) d [ η ] d [˜ η ] d [ b V ] d [ ˜ ρ ] d [ σ ] F " B ρ η − ρ † η ˜ ρ ×× exp h − ε Str B − ı (cid:16) tr( ρ † η + B ) σ ˜ η + tr σ † ˜ η ( ρ η − B ) − tr( ˜ ρ − B ) σ (cid:17)i , (A.33)where C = (cid:18) π ˜ γ (cid:19) cd (cid:16) γ π (cid:17) d ( d − /β (cid:16) γ π (cid:17) d . (A.34)The matrices ρ η and σ ˜ η are rectangular matrices depending on Grassmann variablesas in the Boson–Fermion and Fermion–Boson block in the sets (4.1-4.3). Shifting χ j → χ j + (cid:0) σ +2 (cid:1) − σ † ˜ η x j and χ † j → χ † j − x † j σ ˜ η (cid:0) σ +2 (cid:1) − , we get Z R F ( b B ) exp (cid:16) − ε Str b B (cid:17) d [ b V ] == C Z Herm (4 /β,d ) Z R Z (Λ cd ) d [ η ] d [˜ η ] d [ b V ] d [ ˜ ρ ] d [ σ ] F " B ρ η − ρ † η ˜ ρ ×× exp h − ε Str B − ı (cid:16) tr σ † ˜ η B σ ˜ η (cid:0) σ +2 (cid:1) − + tr ρ † η σ ˜ η + tr σ † ˜ η ρ η − tr( ˜ ρ − B ) σ (cid:17)i . (A.35) uperbosonization formula and generalized Hubbard–Stratonovich transformation B and B . Thus, we apply the first case of this proofand replace B . We find Z R F ( b B ) exp (cid:16) − ε Str b B (cid:17) d [ b V ] == C ( β ) ac C Z Herm (4 /β,d ) Z R Z (Λ cd ) d [ χ ] d [ η ] d [˜ η ] d [ ρ ] d [ ˜ ρ ] d [ σ ] F " B ρ η − ρ † η ˜ ρ det ρ ˜ κ ×× exp h ε (tr B − tr ρ ) + ı (cid:16) tr σ † ˜ η ρ σ ˜ η (cid:0) σ +2 (cid:1) − − tr ρ † η σ ˜ η − tr σ † ˜ η ρ η + tr( ˜ ρ − B ) σ (cid:17)i (A.36)with the exponent˜ κ = a − c + 1 γ − γ . (A.37)After another shifting σ ˜ η → σ ˜ η − ρ − ρ η σ +2 and σ † ˜ η → σ † ˜ η − σ +2 ρ † η ρ − , we integrate over d [˜ η ] and B and have Z R F ( b B ) exp (cid:16) − ε Str b B (cid:17) d [ b V ] == C ( β ) ac C Z Σ β,c Z Herm (4 /β,d ) Z Λ cd d [ η ] d [ ρ ] d [ ˜ ρ ] d [ σ ] F " ρ ρ η − ρ † η ˜ ρ ×× det ρ κ det (cid:0) σ +2 (cid:1) ( a − c ) /γ exp (cid:2) − ε tr ρ + ı (cid:0) tr ρ † η ρ − ρ η σ +2 + tr ˜ ρ σ (cid:1)(cid:3) , (A.38)where the exponent is κ = a − c + 1 γ + d − γ (A.39)and the new constant is C = (cid:16) ı π (cid:17) ad (cid:18) π ˜ γı (cid:19) cd (cid:16) γ π (cid:17) d ( d − /β (cid:16) γ π (cid:17) d . (A.40)We express the determinant in σ +2 as in Sec. 2 as Gaussian integrals and define anew ( γ ( a − c ) + 0) × (0 + γ d ) rectangular supermatrix b V new and its corresponding(0 + γ d ) × (0 + γ d ) supermatrix b B new = ˜ γ − b V new b V † new . Integrating σ and ρ , Eq. (A.38)becomes Z R F ( b B ) exp (cid:16) − ε Str b B (cid:17) d [ b V ] = ˜ γ − cd C ( β ) ac Z Σ β,c Z Λ a − c ) d F " ρ ρ η − ρ † η b B new − ρ † η ρ − ρ η ×× exp (cid:16) − ε tr ρ + ε tr( b B new − η † ρ − η ) (cid:17) det ρ κ d [ b V new ] d [ η ] d [ ρ ] . (A.41)Now, we apply the second case in this proof and shift ρ → ρ + ρ † η ρ − ρ η by analyticcontinuation. We get the final result Z R F ( b B ) exp (cid:16) − ε Str b B (cid:17) d [ b V ] = C ( β ) acd Z Σ β,cd F ( ρ ) exp ( − ε Str ρ ) Sdet ρ κ d [ ρ ] (A.42) uperbosonization formula and generalized Hubbard–Stratonovich transformation C ( β ) acd = ˜ γ − cd C ( β ) ac C ( β ) a − c, d == ( − πγ ) − ad (cid:18) − πγ (cid:19) cd − c ˜ γ βac/ Vol (cid:16) U ( β ) ( a ) (cid:17) Vol (cid:16) U ( β ) ( a − c ) (cid:17) d Y n =1 Γ ( γ κ + 2( n − d ) /β ) ı n − /β π n − /β == ı − d ( d − /β (2 π ) d ˜ γ βac/ − cd ( − ( c − a ) d c d Vol (cid:16) U (1) ( a ) (cid:17) Vol (cid:16) U (1) ( a − c + 2 d ) (cid:17) , β = 1Vol(U (2) ( a )Vol (cid:16) U (2) ( a − c + d ) (cid:17) , β = 22 − (2 a +1 − c ) c Vol (cid:16) U (1) (2 a + 1) (cid:17) Vol (cid:16) U (1) (2( a − c ) + d + 1) (cid:17) , β = 4 . (A.43) Appendix B. Proof of Theorem 5.1 (Generalized Hubbard–Stratonovichtransformation) We choose a Wick–rotation e ıψ that all calculations below are well defined. Then, weperform a Fourier transformation Z R F ( b B ) exp (cid:16) − ε Str b B (cid:17) d [ b V ] = e C Z e Σ ( − ψ ) β,cd Z R F F ( σ ) exp (cid:16) ı Str b Bσ + (cid:17) d [ b V ] d [ σ ] , (B.1)where σ + = σ + ıε γ c + γ d , F F ( σ ) = Z e Σ ( ψ ) β,cd F ( ρ ) exp ( − ı Str ρσ ) d [ ρ ] , (B.2)and the constant is e C = (cid:18) π ˜ γ (cid:19) cd (cid:16) γ π (cid:17) c (cid:16) γ π (cid:17) βc ( c − / (cid:16) γ π (cid:17) d (cid:16) γ π (cid:17) d ( d − /β . (B.3)The integration over b V yields Z R F ( b B ) exp (cid:16) − ε Str b B (cid:17) d [ b V ] = e C Z e Σ ( − ψ ) β,cd F F ( σ )Sdet − a/γ σ + d [ σ ] (B.4)with e C = (cid:18) π ˜ γ (cid:19) cd (cid:16) γ π (cid:17) c (cid:16) γ π (cid:17) βc ( c − / (cid:16) γ π (cid:17) d (cid:16) γ π (cid:17) d ( d − /β (cid:16) ı π (cid:17) ad ( γ πı ) βac/ . (B.5)We transform this result back by a Fourier–transformation Z R F ( b B ) exp (cid:16) − ε Str b B (cid:17) d [ b V ] = e C Z e Σ ( ψ ) β,cd F ( ρ ) I ( β,a ) cd ( ρ ) exp ( − ε Str ρ ) d [ ρ ] , (B.6) uperbosonization formula and generalized Hubbard–Stratonovich transformation I ( β,a ) cd ( ρ ) = Z e Σ ( − ψ ) β,cd exp (cid:0) − ı Str ρσ + (cid:1) Sdet − a/γ σ + d [ σ ] . (B.7)This distribution is rotation invariant under U ( β ) ( c/d ). The ordinary version, d = 0, ofEq. (B.6) is Eq. (A.5).After performing four shifts σ → σ − σ ˜ η (cid:0) σ + ıe ıψ ε γ d (cid:1) − σ † ˜ η , (B.8) σ ˜ η → σ ˜ η − ρ − ρ η (cid:0) σ + ıe ıψ ε γ d (cid:1) , (B.9) σ † ˜ η → σ † ˜ η − (cid:0) σ + ıe ıψ ε γ d (cid:1) ρ † η ρ − , (B.10) ρ → ρ − ρ † η ρ − ρ η , (B.11)and defining ˆ ρ = " ρ e ıψ/ ρ η − e ıψ/ ρ † η e ıψ (cid:0) ρ − ρ † η ρ − ρ η (cid:1) , (B.12)we find Z R F ( b B ) exp (cid:16) − ε Str b B (cid:17) d [ b V ] = e C Z e Σ ( ψ ) β,cd F ( ˆ ρ ) e I ( ρ ) exp ( − ε Str ˆ ρ ) d [ ρ ] , (B.13)where e I ( ρ ) = Z e Σ ( − ψ ) β,cd exp h ε Str ρ − ı (cid:16) tr ρ σ − tr ρ σ + tr σ † ˜ η ρ σ ˜ η ( σ + ıe ıψ ε γ d ) − (cid:17)i ×× (cid:18) det( e − ıψ σ + ıε γ d )det( σ + ıε γ c ) (cid:19) a/γ d [ σ ] . (B.14)We integrate over d [˜ η ] and apply Eq. (A.5) for the d [ σ ]–integration. Then, Eq. (B.14)reads e I ( ρ ) = e C e − ıψcd det ρ κ Θ( ρ ) ×× Z Herm (4 /β,d ) exp (cid:0) − ı tr ρ ( σ + ıe ıψ ε γ d ) (cid:1) det( e − ıψ σ + ıε ) ( a − c ) /γ d [ σ ] (B.15)with the constant e C = ı − βac/ (cid:18) ˜ γ πı (cid:19) cd G ( β ) a − c,c , (B.16)see Eq. (A.6). The exponent κ is the same as in Eq. (4.7). As in Ref. [21], we decompose σ in angles and eigenvalues and integrate over the angles. Thus, we get the ordinarymatrix Bessel–function ϕ (4 /β ) d ( r , s ) = Z U (4 /β ) ( d ) exp (cid:0) ı tr r U s U † (cid:1) dµ ( U ) (B.17) uperbosonization formula and generalized Hubbard–Stratonovich transformation β and d explicitly known. However, the analogof the Sekiguchi differential operator for the ordinary matrix Bessel–functions D (4 /β ) dr ,see Eq. (5.3), fulfills the eigenvalue equation D (4 /β ) dr ϕ (4 /β ) d ( r , s ) = ( ıγ ) d det s /γ ϕ (4 /β ) d ( r , s ) . (B.18)Since the determinant of σ stands in the numerator, we shift σ → σ − ıe ıψ ε γ d andreplace the determinants in Eq. (B.15) by D (4 /β ) dr . After an integration over σ , we have e I ( ρ ) = e C e − ıψcd det ρ κ Θ( ρ ) (cid:16) e − ıψd D (4 /β ) dr (cid:17) a − c δ ( r ) | ∆ d ( e ıψ r ) | /β . (B.19)The constant is e C = ı − βac/ (cid:18) ˜ γ πı (cid:19) cd G ( β ) a − c,c ( ıγ ) ( c − a ) d (cid:18) πγ (cid:19) d ( d − /β (cid:18) πγ (cid:19) d (4 /β ) d . (B.20)Summarizing the constants (B.5) and (B.20), we get e C ( β ) acd = e C e C = 2 − c (2 πγ ) − ad (cid:18) πγ (cid:19) cd ˜ γ βac/ Vol (cid:16) U ( β ) ( a ) (cid:17) Vol (cid:16) U ( β ) ( a − c ) (cid:17) FU (4 /β ) d . (B.21)Due to the Dirac–distribution, we shift D (4 /β ) dr from the Dirac–distribution to thesuperfunction and remove the Wick–rotation. Hence, we find the result of the Theorem. Appendix C. Proof of Theorem 6.1 (Equivalence of both approaches) We define the function e F ( r ) = Z U /β ( d ) Z Herm ( β,c ) Z Λ cd F " ρ ρ η − ρ † η U r U † − ρ † η ρ − ρ η ×× exp (cid:2) − ε (tr ρ − tr( r − ρ † η ρ − ρ η ) (cid:3) det κ ρ d [ η ] d [ ρ ] dµ ( U ) . (C.1)Then, we have to prove C ( β ) acd Z [0 , π ] d e F ( e ıϕ j ) | ∆ d ( e ıϕ j ) | /β d Y n =1 e ı (1 − κ ) ϕ n dϕ n π = e C ( β ) acd (cid:16) ( − d D (4 /β ) dr (cid:17) a − c e F ( r ) (cid:12)(cid:12)(cid:12)(cid:12) r =0 . (C.2)Since e F is permutation invariant and a Schwartz–function, we express e F as an integralover ordinary matrix Bessel–functions, e F ( r ) = Z R d g ( q ) ϕ (4 /β ) d ( r , q ) | ∆ d ( q ) | /β dq, (C.3)where g is a Schwartz–function. The integral and the differential operator in Eq. (C.2)commute with the integral in Eq. (C.3). Thus, we only need to prove C ( β ) acd Z [0 , π ] d ϕ (4 /β ) d ( e ıϕ j , q ) | ∆ d ( e ıϕ j ) | /β d Y n =1 e ı (1 − γ κ ) ϕ n dϕ n π == e C ( β ) acd (cid:16) ( − d D (4 /β ) dr (cid:17) a − c ϕ (4 /β ) d ( r , q ) (cid:12)(cid:12)(cid:12)(cid:12) r =0 (C.4) uperbosonization formula and generalized Hubbard–Stratonovich transformation q ∈ S d where S is the unit–circle in the complex plane. The right hand side ofthis equation is with help of Eq. (B.18) (cid:16) D (4 /β ) dr (cid:17) a − c ϕ (4 /β ) d ( r , q ) (cid:12)(cid:12)(cid:12)(cid:12) r =0 = ( − ıγ ) d ( a − c ) det q ( a − c ) /γ . (C.5)The components of q are complex phase factors. The integral representation of theordinary matrix Bessel–functions (B.17) and the d [ ϕ ]–integral in Eq. (C.4) form theintegral over the circular ensembles CU (4 /β ) ( d ). Thus, q can be absorbed by e ıϕ j andwe find Z [0 , π ] d ϕ (4 /β ) d ( e ıϕ j , q ) | ∆ d ( e ıϕ j ) | /β d Y n =1 e ı (1 − γ κ ) ϕ n dϕ n π == det q ( a − c ) /γ Z [0 , π ] d ϕ (4 /β ) d ( e ıϕ j , | ∆ d ( e ıϕ j ) | /β d Y n =1 e ı (1 − γ κ ) ϕ n dϕ n π . (C.6)The ordinary matrix Bessel–function is at q = 1 the exponential function ϕ (4 /β ) d ( e ıϕ j , 1) = exp ıγ d X n =1 e ıϕ n ! . (C.7)With Eq. (A.27), the integral on the left hand side in Eq. (C.6) yields with thisexponential function Z [0 , π ] d | ∆ d ( e ıϕ j ) | /β d Y n =1 e ı (1 − γ κ ) ϕ n exp ( ıγ e ıϕ n ) dϕ n π == ( ıγ ) ( a − c ) d d Y n =1 ı n − /β Γ (1 + 2 n/β )Γ (1 + 2 /β ) Γ ( a − c + 1 + 2( n − /β ) == ( ıγ ) ( a − c ) d FU (4 /β ) d d Y n =1 ı n − /β π n − /β Γ ( a − c + 1 + 2( n − /β ) . (C.8)Hence, the normalization on both sides in Eq. (C.2) is equal. References [1] K.B. Efetov. Adv. Phys. , 32:53, 1983.[2] J.J.M. Verbaarschot and M.R. Zirnbauer. J. Phys. , A 17:1093, 1985.[3] J.J.M. Verbaarschot, H.A. Weidenm¨uller, and M.R. Zirnbauer. Phys. Rep. , 129:367, 1985.[4] K.B. Efetov. Supersymmetry in Disorder and Chaos . Cambridge University Press, Cambridge,1st edition, 1997.[5] E. Brezin and A. Zee. Nucl. Phys. , B 402:613, 1993.[6] E. Brezin and A. Zee. C.R. Acad. Sci. , 17:735, 1993.[7] G. Hackenbroich and H.A. Weidenm¨uller. Phys. Rev. Lett. , 74:4118, 1995.[8] T. Guhr, A. M¨uller-Groeling, and H.A. Weidenm¨uller. Phys. Rep. , 299:189, 1998.[9] C.W.J. Beenakker. Rev. Mod. Phys. , 69:733, 1997.[10] A.D. Mirlin. Phys. Rep. , 326:259, 2000.[11] J. Ambjørn, J. Jurkiewicz, and M. Makeenko Yu. Phys. Lett. , B 251:517, 1993. uperbosonization formula and generalized Hubbard–Stratonovich transformation [12] E. Brezin, C. Itzykson, G. Parisi, and J. Zuber. Commun. Math. Phys. , 59:35, 1978.[13] L. Laloux, P. Cizeau, J.P. Bouchard, and M. Potters. Phys. Rev. Lett. , 83:1467, 1999.[14] M.L. Mehta. Random Matrices . Academic Press Inc., New York, 3rd edition, 2004.[15] T. Guhr. J. Phys. , A 39:13191, 2006.[16] F. Toscano, R.O. Vallejos, and C. Tsallis. Phys. Rev. , E 69:066131, 2004.[17] A.C. Bertuola, O. Bohigas, and M.P. Pato. Phys. Rev. , E 70:065102(R), 2004.[18] Y.A. Abul-Magd. Phys. Lett. , A 333:16, 2004.[19] H.-J. Sommers. Acta Phys. Pol. , B 38:1001, 2007.[20] P. Littelmann, H.-J. Sommers, and M.R. Zirnbauer. Commun. Math. Phys. , 283:343, 2008.[21] M. Kieburg, J. Gr¨onqvist, and T. Guhr, 2008. J. Phys. , A 42:275205, 2009.[22] M. Kieburg, H. Kohler, and T. Guhr. J. Math. Phys. , 50:013528, 2009.[23] G. Akemann and Y.V. Fyodorov. Nucl. Phys. , B 664:457, 2003.[24] G. Akemann and A. Pottier. J. Phys. , A 37:L453, 2004.[25] A. Borodin and E. Strahov. Commun. Pure Appl. Math. , 59:161, 2005.[26] E. Brezin and S. Hikami. Commun. Math. Phys. , 214:111, 2000.[27] M.R. Zirnbauer. The Supersymmetry Method of Random Matrix Theory, Encyclopedia ofMathematical Physics, eds. J.-P. Franoise, G.L. Naber and Tsou S.T., Elsevier, Oxford , 5:151,2006.[28] M.L. Mehta and J.-M. Normand. J. Phys. A: Math. Gen. , 34:1, 2001.[29] Y.V. Fyodorov. Nucl. Phys. , B 621:643, 2002.[30] F.A. Berezin. Introduction to Superanalysis . D. Reidel Publishing Company, Dordrecht, 1stedition, 1987.[31] J.E. Bunder, K.B. Efetov, K.B. Kravtsov, O.M. Yevtushenko, and M.R. Zirnbauer. J. Stat. Phys. ,129:809, 2007.[32] M.R. Zirnbauer. J. Math. Phys. , 37:4986, 1996.[33] H. Kohler and T. Guhr. J. Phys. , A 38:9891, 2005.[34] F. Wegner, 1983. unpublished notes.[35] F. Constantinescu. J. Stat. Phys. , 50:1167, 1988.[36] F. Constantinescu and H.F. de Groote. J. Math. Phys. Math. Res. Letters , 4:69, 1997.[39] F. Basile and G. Akemann. JHEP , page 0712:043, 2007.[40] A.E. Ingham. Proc. Camb. Phil. Soc. , 29:271, 1933.[41] C.L. Siegel.