Lyapunov Spectra for All Ten Symmetry Classes of Quasi-one-dimensional Disordered Systems of Non-interacting Fermions

Abstract

A random phase property is proposed for products of random matrices drawn from any one of the classical groups associated with the ten Cartan symmetry classes of non-interacting disordered Fermion systems. It allows to calculate the Lyapunov spectrum explicitly in a perturbative regime. These results apply to quasi-one-dimensional random Dirac operators which can be constructed as representatives for each of the ten symmetry classes. For those symmetry classes that correspond to two-dimensional topological insulators or superconductors, the random Dirac operators describing the one-dimensional boundaries have vanishing Lyapunov exponents and almost surely an absolutely continuous spectrum, reflecting the gapless and conducting nature of the boundary degrees of freedom.

Appendix:  Moments for the Haar Measures Over Compact Groups

Section A.1 of this Appendix collects formulas for certain averages over the compact classical groups that are used in the main part of this paper. These formulas can be derived rather easily from results scattered over the literature, and those involving averages over the unitary group were already listed in [27]. Nevertheless, it is emphasized in the remaining sections of the Appendix that all these formulas follow from a useful calculus involving the so-called Weingarten function [10, 11, 37], which may be viewed as a systematic approach to invariant theoretic arguments in the physics literature on averages, as in Sect. VIII of [2]. To illustrate how invariant theory is applied, Sect. A.2 deals explicitly with the fourth moments over the symplectic group (averages of a product of four matrix entries) and provides the corresponding value of the Weingarten function. This allows in Sect. A.3 to deduce Lemma 3(ii), which is actually the most complicated of all formulas in Sect. A.1. All others follow similarly from prior results on the orthogonal group [10]. Higher moments can also be calculated explicitly, provided that the matrix size is sufficiently large compared to the order of the moment (that is, the number of matrix entries that are being considered). Section A.4 presents the general version of the Weingarten integration formula for the unitary, orthogonal, and symplectic groups. These results are not really used in this paper, but they are definitely of independent interest and are provided for further reference.

1.1 A.1 Collection of Formulas Used in Sect. 2

Lemma 1

Let $A,B,C,D\in\mathrm{Mat}(N\times N,{\mathbb{C}})$. The following holds for averages over U(N):

• (i)
  $$\bigl\langle \operatorname{Tr}\bigl(U^*AUB\bigr)\bigr\rangle = \frac{1}{N} \operatorname{Tr}(A)\operatorname{Tr}(B), $$
• (ii)
  $$\bigl\langle \operatorname{Tr}(\overline{U}AUB)\bigr\rangle = \frac{1}{N} \operatorname{Tr}\bigl(AB^t\bigr), \qquad \bigl\langle\operatorname{Tr} \bigl(U^tAUB\bigr)\bigr\rangle = 0, $$
• (iii)
  $$\begin{aligned}[t] \bigl\langle \operatorname{Tr}\bigl(U^*AUBU^*CUD \bigr) \bigr\rangle = & \frac{1}{N^2-1} \bigl[\operatorname{Tr}(A) \operatorname{Tr}(C)\operatorname{Tr}(BD)+\operatorname{Tr}(AC)\operatorname{Tr}(B) \operatorname{Tr}(D) \bigr] \\ &{}- \frac{1}{N(N^2-1)} \bigl[\operatorname{Tr}(AC) \operatorname{Tr}(BD) \\ &{}+ \operatorname{Tr}(A)\operatorname{Tr}(B)\operatorname{Tr}(C) \operatorname{Tr} (D) \bigr] , \end{aligned} $$
• (iv)
  $$\begin{aligned}[t] \bigl\langle\operatorname{Tr}\bigl(U^*AUBU^tC \overline{U}D\bigr)\bigr\rangle = & \frac{1}{N^2-1} \bigl[ \operatorname{Tr}(A)\operatorname{Tr}(C)\operatorname{Tr}(BD)+\operatorname{Tr} \bigl(AC^t\bigr)\operatorname{Tr}\bigl(BD^t\bigr) \bigr] \\ & {}- \frac{1}{N(N^2-1)} \bigl[\operatorname{Tr}\bigl(AC^t\bigr) \operatorname{Tr}(BD)+ \operatorname{Tr}(A)\operatorname{Tr}(C)\operatorname{Tr} \bigl(BD^t\bigr) \bigr] , \end{aligned} $$
• (v)
  $$\begin{aligned}[t] \bigl\langle\operatorname{Tr}\bigl(U^*A \overline{U}BU^tCUD\bigr)\bigr\rangle = & \frac{1}{N^2-1} \bigl[\operatorname{Tr}\bigl(AC^t\bigr)\operatorname{Tr} \bigl(BD^t\bigr)+ \operatorname{Tr}(AC)\operatorname{Tr}(B) \operatorname{Tr}(D) \bigr] \\ & {}- \frac{1}{N(N^2-1)} \bigl[\operatorname{Tr}(AC)\operatorname{Tr} \bigl(BD^t\bigr)+ \operatorname{Tr}\bigl(AC^t\bigr) \operatorname{Tr}(B)\operatorname{Tr} (D) \bigr] . \end{aligned} $$

All formulas remain valid if all U’s are replaced by their complex conjugates, e.g.

$$\bigl\langle\operatorname{Tr}\bigl(U^tA\overline{U}BU^tC \overline{U}D\bigr)\bigr\rangle = \bigl\langle\operatorname{Tr} \bigl(U^*{A}U{B}U^*{C} {U} {D}\bigr)\bigr\rangle . $$

Lemma 2

Let $A,B,C,D\in\mathrm{Mat}(N\times N,{\mathbb{C}})$. The following holds for averages over O(N):

• (i)
  $$\bigl\langle \operatorname{Tr}\bigl(O^tAOB\bigr)\bigr\rangle = \frac{1}{N} \operatorname{Tr}(A)\operatorname{Tr}(B) , \qquad \bigl\langle\operatorname{Tr}(OAOB)\bigr\rangle = \frac{1}{N}\operatorname{Tr}\bigl(A B^t\bigr) . $$
• (ii)
  $$\begin{aligned}[t] & \bigl\langle \operatorname{Tr}\bigl(O^tAOBO^tCOD \bigr)\bigr\rangle \\ &\quad = \frac{N+1}{N(N-1)(N+2)} \bigl[\operatorname{Tr}(A)\operatorname{Tr}(C) \operatorname{Tr}(BD)+\operatorname{Tr}\bigl(AC^t\bigr)\operatorname{Tr} \bigl(BD^t\bigr) \\ &\qquad{}+\operatorname{Tr}(AC)\operatorname{Tr}(B) \operatorname{Tr} (D) \bigr] \\ &\qquad{} -\frac{1}{N(N-1)(N+2)} \bigl[\operatorname{Tr}(A)\operatorname{Tr}(C) \operatorname{Tr}\bigl(BD^t\bigr)+ \operatorname{Tr}(A)\operatorname{Tr}(C) \operatorname{Tr}(B)\operatorname{Tr}(D) \\ & \qquad{}+ \operatorname{Tr} \bigl(AC^t \bigr)\operatorname{Tr}(BD) + \operatorname{Tr} \bigl(AC^t\bigr)\operatorname{Tr}(B)\operatorname{Tr}(D) + \operatorname{Tr}(AC)\operatorname{Tr}(BD) \\ &\qquad{}+\operatorname{Tr}(AC)\operatorname{Tr} \bigl(BD^t\bigr) \bigr] . \end{aligned} $$
• (iii)
  $$\begin{aligned}[t] & \bigl\langle \operatorname{Tr}\bigl(O^tAO^tBOCOD \bigr)\bigr\rangle \\ &\quad = \frac{N+1}{N(N-1)(N+2)} \bigl[\operatorname{Tr}\bigl(A^tBC^tD \bigr)+\operatorname{Tr}(DCBA) +\operatorname{Tr}(AC)\operatorname{Tr} (B) \operatorname{Tr} (D) \bigr] \\ & \qquad{}-\frac{1}{N(N-1)(N+2)} \bigl[\operatorname{Tr}\bigl(A^tBC^tD^t \bigr)+\operatorname{Tr}\bigl(A^tBC^t\bigr) \operatorname{Tr}(D) +\operatorname{Tr}\bigl(CBAD^t\bigr) \\ & \qquad +\operatorname{Tr}(CBA) \operatorname{Tr}(D) +\operatorname{Tr}\bigl(CAD^t\bigr)\operatorname{Tr}(B)+ \operatorname{Tr}\bigl(CAD^t\bigr)\operatorname{Tr}(B) \bigr] . \end{aligned} $$

Lemma 3

Let $A,B,C,D\in\mathrm{Mat}(2N\times 2N,{\mathbb{C}})$. The following holds for averages over SP(2N):

• (i) One has

  

  (27)

  and

  $$\bigl\langle\operatorname{Tr}(UAUB)\bigr\rangle = \bigl\langle \operatorname{Tr}\bigl(U^*AU^*B\bigr)\bigr\rangle = \frac{1}{2N} \operatorname{Tr}\bigl(AIB^tI\bigr), \qquad \bigl\langle\operatorname{Tr}( \overline{U}AUB)\bigr\rangle = \frac{1}{2N} \operatorname{Tr} \bigl(AB^t\bigr) . $$
• (ii)
  $$\begin{aligned}[t] & \bigl\langle \operatorname{Tr}\bigl(U^*AUBU^*CUD\bigr)\bigr \rangle \\ & \quad = \frac{1}{4N(N-1)(2N+1)} \bigl[ (2N-1)\operatorname{Tr}(A)\operatorname{Tr}(C) \operatorname{Tr}(BD) +\operatorname{Tr}(A)\operatorname{Tr}(C)\operatorname{Tr} \bigl(I^*B^tID\bigr) \\ & \qquad{} -\operatorname{Tr}(A)\operatorname{Tr}(C)\operatorname{Tr}(B) \operatorname{Tr}(D) -\operatorname{Tr}\bigl(I^*A^tIC\bigr) \operatorname{Tr}(BD) +\operatorname{Tr}\bigl(I^*A^tIC\bigr) \operatorname{Tr}(B)\operatorname{Tr}(D) \\ & \qquad{} -\operatorname{Tr}(AC)\operatorname{Tr}(BD) -(2N-1)\operatorname{Tr} \bigl(I^*A^tIC\bigr)\operatorname{Tr}\bigl(I^*B^tID\bigr) \\ & \qquad{} -\operatorname{Tr}(AC)\operatorname{Tr}\bigl(I^*B^tID\bigr) +(2N-1)\operatorname{Tr}(AC)\operatorname{Tr}(B)\operatorname{Tr}(D) \bigr] . \end{aligned} $$

Let us close this section by stating two results that follow from the above and are also used in Sect. 1. Recall from Sect. 1.5 the special form of $\mathcal{U}^{ \mathrm{CI}}=\mathcal{U}^{\mathrm{DIII}}\subset\mathrm{U}(2N)$, which is isomorphic to U(N). The following lemma presents no general formula for second and fourth moments of this group, but only those moments which are needed in Sect. 2.6. The proof of the lemma is a calculation based on the identities in Lemma 1.

Lemma 4

Let $A,B\in\mathrm{Mat}(2N\times 2N,{\mathbb{C}})$. Then the averages over $\mathcal{U}^{\mathrm{CI}}=\mathcal{U}^{ \mathrm{DIII}}$ lead to the following:

• (i) With and

  $$\bigl\langle \operatorname{Tr}\bigl(U^*AUB\bigr) \bigr\rangle = \frac{1}{N} \operatorname{Tr} ( \varPi_+A ) \operatorname{Tr} (\varPi_+B ) + \frac{1}{N} \operatorname{Tr} (\varPi_-A ) \operatorname{Tr} ( \varPi_-B ) . $$

• (ii) Let A and B be of the form

  

  where all entries are of size N×N. Then

  

• (iii) If, moreover, a ^t=−a, then

  $$\bigl\langle \operatorname{Tr}\bigl(U^*AUBU^*AUB\bigr)\bigr\rangle = \frac{\operatorname{Tr}(A^2)(\operatorname{Tr}(b)\operatorname{Tr}(c)- \operatorname{Tr}(bc^t))}{N(N-1)}. $$

Recall from Sect. 1.5 the special form of $\mathcal{U}^{ \mathrm{CII}}=\mathrm{SP}(2N)\times\mathrm{SP}(2N)$. The following lemma presents no general formula for second and fourth moments of this group, but only those moments which are needed in Sect. 2.8. The proof of the lemma is a somewhat tedious calculation based on the identities in Lemma 3(i).

Lemma 5

Let $A,B\in\mathrm{Mat}(4N\times 4N,{\mathbb{C}})$. Then the averages over $\mathcal{U}^{\mathrm{CI}}$ lead to the following:

• (i) With and as above,

  $$\bigl\langle\operatorname{Tr}\bigl(U^*AUB\bigr)\bigr\rangle = \frac{1}{2N} \operatorname{Tr} ( \varPi_+A ) \operatorname{Tr} (\varPi_+B ) + \frac{1}{2N} \operatorname{Tr} (\varPi_-A ) \operatorname{Tr} ( \varPi_-B ) . $$

• (ii) Let A and B be of the form

  

  where all entries are of size 2N×2N and furthermore $I\overline {a}I=a$ and B ^∗=B ^t=B. Then

  $$\bigl\langle\operatorname{Tr}\bigl(U^*AUBU^*AUB\bigr)\bigr\rangle = \frac{\operatorname{Tr}(A^2) (\operatorname{Tr}(e)^2+\operatorname{Tr}(I^*fIf) )}{4N^2}. $$

1.2 A.2 The Symplectic Weingarten Function for Fourth Moments

As already pointed out, the formulas of Sect. A.1 can be deduced from [8, 10] except for averages involving the symplectic group. In the symplectic case a sign factor is missing in [10]. This was corrected in [11], but no explicit value of the Weingarten function was given. This section treats the fourth moment needed for the proof of Lemma 3(ii) in detail. This also indicates how in principle the general results stated in Sect. A.4 can be proved.

Let e _j , j=1,…,2N, denote the standard basis of the complex vector space $V={\mathbb{C}}^{2N}$. For any linear map T on V, the matrix entries are denoted by $T_{i,j}=e_{i}^{*}Te_{j}$. Associated to I defined in (2), let us introduce on V the skew symmetric bilinear form a(v,w)=v ^t Iw where $v^{t}=\overline{v}^{*}$ denotes the transpose of v∈V. Then for i,j=1,…,N,

Thus, for i ₁,j ₁,…,i ₄,j ₄∈{1,…,N} and α ₁,β ₁,…,α ₄,β ₄∈{0,1},

(28)

where a is extended to V ^⊗4 via

$$a(v_1 \otimes v_2 \otimes v_3 \otimes v_4, w_1 \otimes w_2 \otimes w_3 \otimes w_4) = a(v_1, w_1) a(v_2, w_2)a(v_3, w_3) a(v_4, w_4). $$

Note that this yields a symmetric bilinear form on V ^⊗4. By the definition of Haar measure, the integral

$$ \int_{\operatorname {SP}(2N)} dU\, U e_{j_1 + \beta_1 N} \otimes \cdots\otimes U e_{j_4 + \beta_4 N} $$

(29)

is an $\operatorname{SP}(2N)$-invariant in V ^⊗4, specifically, it is fixed under the action of $U \in\operatorname {SP}(2N)$ on V ^⊗4 which on decomposable tensors is given via U(v ₁⊗v ₂⊗v ₃⊗v ₄)=(Uv ₁⊗Uv ₂⊗Uv ₃⊗Uv ₄). If N≥2, then by the symplectic case of Weyl’s First Fundamental Theorem for tensor invariants [17, Theorem 5.3.3] and the results of Sect. 3.4 in [33], a basis of the (three-dimensional) subspace of invariant tensors in V ^⊗4 can be described as follows. Write ${\mathfrak{p}}_{1} = \{ \{1, 2\}, \{3, 4\}\}$, ${\mathfrak{p}}_{2} = \{ \{1,3\}, \{2, 4\} \}$ and ${\mathfrak{p}}_{3} = \{ \{1, 4\}, \{2, 3\}\}$. These are the pair partitions of the set {1,2,3,4}. To control the sign factors that arise from the skew symmetry of the form a on V, let us keep track of the natural order on the individual blocks by using the following notation. For ${\mathfrak{m}}\in\{{\mathfrak{p}}_{1}, {\mathfrak{p}}_{2}, {\mathfrak{p}}_{3}\}$, let us write ${\mathfrak{m}}= \{ (m_{1}({\mathfrak{m}}), n_{1}({\mathfrak{m}})), (m_{2}({\mathfrak{m}}), n_{2}({\mathfrak{m}})) \}$, where $\{m_{1}({\mathfrak{m}}), n_{1}({\mathfrak{m}})\}, \{m_{2}({\mathfrak{m}}), n_{2}({\mathfrak{m}})\}$ are the blocks of ${\mathfrak{m}}$, and $m_{1}({\mathfrak{m}}) < n_{1}({\mathfrak{m}}), m_{2}({\mathfrak{m}}) < n_{2}({\mathfrak{m}})$. To ${\mathfrak{p}}_{1}$ is now associated the tensor

$$\theta_1 = \theta_{{\mathfrak{p}}_1} = \sum _{\eta_1, \eta_2 = 1}^{N} \sum_{\epsilon_1, \epsilon_2 = 0}^1 e_{\eta_1 + \epsilon_1 N} \otimes e_{\eta_1 + (1 - \epsilon_1)N} (-1)^{\epsilon_1} \otimes e_{\eta_2 + \epsilon_2 N} \otimes e_{\eta_2 + (1 - \epsilon_2)N} (-1)^{\epsilon_2} . $$

Here, the maps r↦η _r and r↦ϵ _r are constant on the block $\{ m_{1}({\mathfrak{p}}_{1}), n_{1}({\mathfrak{p}}_{1})\}$ as well as on $\{ m_{2}({\mathfrak{p}}_{1}), n_{2}({\mathfrak{p}}_{1})\}$, and to the m component of a block corresponds a vector of the form $e_{\eta_{r} + \epsilon_{r} N}$, while to the n component corresponds a vector of the form $(-1)^{\epsilon_{r}} e_{\eta_{r} + (1 - \epsilon_{r})N}$. Analogously,

Then {θ ₁,θ ₂,θ ₃} is a basis of the space of symplectic invariants in V ^⊗4, and it is a straightforward (tedious) computation to verify that the Gram matrix of the form a with respect to this basis is given by

$$\left ( \begin{array}{c@{\quad}c@{\quad}c} 4 N^2 & 2N & - 2N\\ 2N & 4 N^2 & 2N\\ - 2N & 2N & 4 N^2 \end{array} \right ) . $$

As long as N≥2, this matrix is invertible with inverse

$$ \frac{1}{4N (N-1) (2N+1)} \left ( \begin{array}{c@{\quad}c@{\quad}c} 2N - 1 & -1 & 1 \\ -1 & 2N - 1 & -1 \\ 1 & -1 & 2N - 1 \end{array} \right ) . $$

(30)

By definition, the (i,j)-entry of this inverse is the value $\operatorname {Wg}_{\operatorname {SP}(2N)}({\mathfrak{p}}_{i}, {\mathfrak{p}}_{j})$ of the symplectic Weingarten function. In Sect. A.4, the general Weingarten function $\operatorname {Wg}_{\operatorname {SP}(2N)}$ is defined by the matrix entries of the inverse of the Gram matrix of a w.r.t. a basis of the invariants, which in turn is in bijection with the pair partitions (of 2k points if 2kth moments are considered). In order to evaluate (28), one now expresses the invariant (29) in terms of the basis {θ ₁,θ ₂,θ ₃}. Noting that $a( e_{i_{1} + \alpha_{1}N} \otimes\cdots\otimes e_{i_{4} + \alpha_{4} N}, \theta_{{\mathfrak{p}}_{j}})$ vanishes unless $i_{m_{r}({\mathfrak{p}}_{j})} = i_{n_{r}({\mathfrak{p}}_{j})}$ and $\alpha_{m_{r}({\mathfrak{p}}_{j})} = 1 - \alpha_{n_{r}({\mathfrak{p}}_{j})}$ for r=1,2, one obtains

(31)

1.3 A.3 Proof of Lemma 3(ii)

Let us begin by writing out the l.h.s. explicitly:

The next step is to expand this average $\langle.\rangle=\int_{\operatorname {SP}(2N)} dU$ according to (31). Then one interchanges the sum over pairs of pair partitions with the sums over the i and α indices and then determines the contribution of each pair $({\mathfrak{p}}_{i}, {\mathfrak{p}}_{j})$. Let us illustrate this procedure for the pair $({\mathfrak{p}} _{1}, {\mathfrak{p}}_{2})$. One reads off from (31) that for a choice of i and α indices to give a non-vanishing contribution, one must have that i ₃=i ₂, α ₃=α ₂, i ₇=i ₆, α ₇=α ₆, i ₅=i ₁, α ₅=1−α ₁, i ₈=i ₄, α ₈=1−α ₄. Furthermore, the sign factor corresponding to each choice of i and α indices is

$$(-1)^{(1-\alpha_2) + (1-\alpha_6) + (1 - \alpha_1) + \alpha_4} = (-1)^{\alpha_1 + \alpha_2 + \alpha_4 + (1-\alpha_6)} . $$

Consequently, the coefficient of $\operatorname {Wg}_{\operatorname {SP}(2N)}({\mathfrak{p}}_{1}, {\mathfrak{p}} _{2})$ in the sum over pairs of pair partitions is

Observing that I _{i+αN,i+(1−α)N} =(−1)^1−α and I _{i+(1−α)N,i+αN} =(−1)^α, one can group the factors as follows:

Hence, $\operatorname {Wg}_{\operatorname {SP}(2N)}({\mathfrak{p}}_{1}, {\mathfrak{p}}_{2})$ comes with the coefficient $\operatorname {Tr}(A) \operatorname {Tr}(C) \operatorname {Tr}(BID^{t}I)$. Going in this manner through all nine cases yields

and then the claim follows from (30).

1.4 A.4 The General Weingarten Integration Formulas

This Appendix collects the general results on the integration over the orthogonal, symplectic and unitary group, as they are given in (or can be deduced from) [10], [11] and [8] respectively. The following notation will be used. For $m, n \in\mathbb{N}$ denote by ${\mathcal{F}}(m, n)$ the set of all maps from {1,…,m} to {1,…,n}. If ${\mathfrak{m}}$ is a partition of {1,…,m}, then ${\mathcal{F}}({\mathfrak{m}}, n)$ denotes the set of those elements of ${\mathcal{F}}(m, n)$ which are constant on the blocks of ${\mathfrak{m}}$. Denote by $\mathcal{P}\mathcal{P}(k)$ the set of all pair partitions of {1,…,k}. In particular, $\mathcal{P} \mathcal{P}(k) = \emptyset$ if k is odd. If k=2l is even, we write $\mathcal{P} \mathcal{P} (k) \ni{\mathfrak{m}}= \{ \{m_{\nu}({\mathfrak{m}}), n_{\nu }({\mathfrak{m}})\}\mid \nu= 1, \ldots , l\}$, where the numbering of the blocks is arbitrary, but m _ν <n _ν holds for all ν. For maps $\alpha\in{\mathcal{F}}(k, \{ 0, 1\})$ from {1,…,k} to {0,1}, let us write $\alpha\in {\mathcal{A}} ({\mathfrak{m}}, \{0, 1\})$ if for all ν holds $\alpha(m_{\nu }({\mathfrak{m}})) = 1 - \alpha(n_{\nu}({\mathfrak{m}}))$.

Now let us begin with the orthogonal case. If N≥l, then the space of $\operatorname {O}(N)$-invariants in V ^⊗2l admits a basis $\{\theta _{{\mathfrak{m}}}\mid {\mathfrak{m}}\in\mathcal{P}\mathcal{P}(2l)\}$ given by

$$\theta_{{\mathfrak{m}}} = \sum_{\phi\in{\mathcal{F}}({\mathfrak{m}}, N)} e_{\phi(1)} \otimes \cdots\otimes e_{\phi(2l)}, $$

where as above e _i , i=1,…,N, is the standard orthonormal basis on $V={\mathbb{C}}^{N}$ which is furnished with the symmetric bilinear form b(v,w)=v ^t w. The Weingarten function $\operatorname {Wg}_{\operatorname {O}(N)}({\mathfrak{m}}, {\mathfrak{n}} )$ is the $({\mathfrak{m}}, {\mathfrak{n}})$-entry of the inverse of the Gram matrix of the extension of b via b(v ₁⊗⋯⊗v _2l ,w ₁⊗⋯⊗w _2l )=b(v ₁,w ₁)⋅…⋅b(v _2l ,w _2l ) with respect to the basis $\{\theta_{{\mathfrak{m}}}\mid {\mathfrak{m}}\in\mathcal{P} \mathcal{P} (2l)\}$. For example, on $\mathcal{P}\mathcal{P}(2) \times\mathcal{P}\mathcal{P}(2)$ one obtains $\operatorname {Wg}_{\operatorname {O}(N)} = \frac{1}{N}$, while on $\mathcal{P}\mathcal{P}(4)\times\mathcal{P}\mathcal{P}(4)$ the orthogonal Weingarten function $\operatorname {Wg}_{\operatorname {O}(N)}$ is given by the matrix entries

$$\frac{1}{N (N-1) (N+2)} \left ( \begin{array}{c@{\quad}c@{\quad}c} N+1 & -1 & -1 \\ -1 & N+1 & -1 \\ -1 & -1 & N+1 \end{array} \right ) . $$

The general orthogonal Weingarten integration formula now reads as follows.

Proposition 1

Let $\phi, \psi\in{\mathcal{F}}(k, N)$. Then

$$\int_{\mathrm{O}(N)} dU \prod_{j=1}^k U_{\phi(j), \psi(j)} = \sum_{{\mathfrak{m}}, {\mathfrak{n}}\in\mathcal{P}\mathcal{P}(k)} 1_{{\mathcal{F}}({\mathfrak{m}}, n)}(\phi) 1_{{\mathcal{F}}({\mathfrak{n}}, n)}(\psi) \operatorname {Wg}_{\mathrm{O}(N)}({\mathfrak{m}}, {\mathfrak{n}}) . $$

Next let us consider the symplectic case, hence $V={\mathbb{C}}^{2N}$ with a as in Sect. A.2. If N≥l, then the space of $\operatorname {SP}(2N)$-invariants in V ^⊗2l admits a basis $\{\theta _{{\mathfrak{m}} }\mid {\mathfrak{m}}\in\mathcal{P}\mathcal{P}(2l)\}$, where

$$\theta_{{\mathfrak{m}}} = \sum_{\eta\in{\mathcal{F}}(l, n)} \sum _{\varepsilon\in{\mathcal{F}}(l, \{ 0, 1\})} \bigotimes_{j=1}^{2l} v( \eta, \epsilon, j) , $$

with

$$v(\eta, \epsilon, j) = \left \{ \begin{array}{l@{\quad}l} e_{\nu(\eta+ n \epsilon)}, & \mbox{if } j = m_{\nu}, \\ e_{\nu(\eta+ n(1-\epsilon))} (-1)^{\nu\epsilon}, & \mbox{if } j = n_{\nu}. \end{array} \right . $$

The Weingarten function $\operatorname {Wg}_{\operatorname {SP}(2N)}({\mathfrak{m}}, {\mathfrak{n}})$ is the $({\mathfrak{m}}, {\mathfrak{n}})$-entry of the inverse of the Gram matrix of the extension of a (as above) with respect to the basis $\{\theta_{{\mathfrak{m}}}\mid {\mathfrak{m}} \in\mathcal{P}\mathcal{P}(2l)\}$. On $\mathcal{P}\mathcal{P}(2)\times\mathcal{P}\mathcal{P}(2)$ one obtains $\operatorname {Wg}_{\operatorname {SP}(2N)} = \frac{1}{2N}$, and on $\mathcal{P}\mathcal{P}(4)\times\mathcal{P}\mathcal{P}(4)$ it is given by (30) above.

Proposition 2

Let $\phi, \psi\in{\mathcal{F}}(k, N)$, $\alpha, \beta\in {\mathcal{F}} (k, \{ 0, 1\})$. Then

Finally let us turn to the unitary case. Let $\operatorname {S}_{k}$ denote the full symmetric permutation group of the set {1,…,k}, and $e_{1}^{*}, \ldots, e_{N}^{*}$ the dual to the basis e ₁,…,e _N on $V={\mathbb{C}}^{N}$ w.r.t. the standard scalar product c. If N≥k, then the space of $\operatorname {U}(N)$-invariants of V ^⊗k⊗(V ^∗)^⊗k (with the contragradient representation on the dual space) admits a basis $\{ C_{\pi}\mid \pi\in\operatorname {S}_{k}\}$ where

$$C_{\pi} = \sum_{\phi\in{\mathcal{F}}(k, N)} e_{\phi(\pi^{-1}(1))} \otimes\cdots\otimes e_{\phi(\pi^{-1}(k))} \otimes e^*_{\phi(1)} \otimes\cdots \otimes e^*_{\phi(k)} . $$

The Weingarten function $\operatorname {Wg}_{\operatorname {U}(N)}(\sigma, \tau)$ is the (σ,τ)-entry of the inverse of the Gram matrix of the extension of the scalar product c w.r.t. the basis $\{ C_{\pi}\mid \pi\in\operatorname {S}_{k}\}$. Then on $\operatorname {S}_{1} \times\operatorname {S}_{1}$ one has $\operatorname {Wg}_{\operatorname {U}(N)} = \frac {1}{N}$, while on $\operatorname {S}_{2} \times \operatorname {S}_{2}$ the unitary Weingarten function is given by

$$\frac{1}{N(N^2 - 1)} \left ( \begin{array}{c@{\quad}c} N & -1 \\ -1 & N \end{array} \right ) . $$

Proposition 3

Let $\phi, \psi, \phi^{\prime}, \psi^{\prime}\in {\mathcal{F}}(k, N)$. Then

$$\int_{\mathrm{U}(N)} dU \prod_{j = 1}^k U_{\phi(j), \psi(j)}\overline{U}_{\phi^{\prime} (j), \psi ^{\prime}(j)} = \sum _{\sigma, \tau\in\mathrm{S}_k} \operatorname {Wg}_{\mathrm{U}(N)}(\sigma, \tau ) \delta\bigl(\phi, \sigma\phi^{\prime}\bigr) \delta\bigl(\psi, \tau \psi^{\prime}\bigr) . $$

Let us conclude with a comment on higher order Weingarten functions. The definitions given here have the merit that they can be easily motivated in an invariant theoretic context. Their calculation, in particular, for higher orders, can be simplified by remarking, e.g., that $\operatorname{Wg}_{\operatorname {U}(N)}(\sigma,\tau)$ depends only on the cycle type of στ ⁻¹. This combined with harmonic analysis on the symmetric group leads to sophisticated and computationally feasible expansions of $\operatorname{Wg}$, see [9] for the unitary and orthogonal cases. The recent preprint [24] which contains an expansion in the symplectic case was posted only after the present paper had been submitted.