Unitary Group Integration

This section details the integration of polynomial functions over the unitary group $U(d)$ with respect to the Haar measure. It covers the theoretical foundations based on Weingarten Calculus and provides practical examples using IntU.jl.

Overview

IntU.jl allows for evaluating integrals of the form:

\[\int_{U(d)} U_{i_1 j_1} \dots U_{i_n j_n} \bar{U}_{k_1 l_1} \dots \bar{U}_{k_n l_n} dU\]

where $U$ is a $d \times d$ unitary matrix ($U U^\dagger = I$). The integral is taken with respect to the Haar measure, which is the unique translation-invariant probability measure on the compact Lie group $U(d)$.

The result is expressed in terms of the dimension $d$ and Kronecker deltas matching the indices.

Theory: Weingarten Calculus

The integration relies on the Weingarten Calculus, a combinatorial method for evaluating integrals over compact groups.

The Integration Formula

The general formula for the integral is given by [Collins & Śniady, 2006]:

\[\int_{U(d)} U_{i_1 j_1} \dots U_{i_n j_n} \bar{U}_{k_1 l_1} \dots \bar{U}_{k_n l_n} dU = \sum_{\sigma, \tau \in S_n} \delta_{i, k_\sigma} \delta_{j, l_\tau} \text{Wg}(\sigma \tau^{-1}, d)\]

where:

  • $S_n$ is the symmetric group of permutations of $\{1, \dots, n\}$.
  • $\delta_{i, k_\sigma} = \prod_{m=1}^n \delta_{i_m, k_{\sigma(m)}}$ contracts row indices according to $\sigma$.
  • $\delta_{j, l_\tau} = \prod_{m=1}^n \delta_{j_m, l_{\tau(m)}}$ contracts column indices according to $\tau$.
  • $\text{Wg}(\pi, d)$ is the Weingarten function, which depends only on the cycle structure of the permutation $\pi$ and the dimension $d$.

The Weingarten Function

The Weingarten function $\text{Wg}(\pi, d)$ is a rational function of $d$. It is defined via the inverse of the Gram matrix of Schur functions or via character theory.

For small $n$, the values are:

  • n=1: $\text{Wg}([1], d) = \frac{1}{d}$
  • n=2:
    • Identity $\text{Wg}([1,1], d) = \frac{1}{d^2-1}$
    • Transposition $\text{Wg}([2], d) = -\frac{1}{d(d^2-1)}$
  • n=3:
    • $\text{Wg}([1,1,1], d) = \frac{d^2-2}{d(d^2-1)(d^2-4)}$
    • $\text{Wg}([2,1], d) = -\frac{1}{(d^2-1)(d^2-4)}$
    • $\text{Wg}([3], d) = \frac{2}{d(d^2-1)(d^2-4)}$

IntU.jl computes these values symbolically for any $n$, using the Murnaghan-Nakayama rule to evaluate characters of the symmetric group.

Implementation Details

IntU.jl automates the following steps:

  1. Index Identification: It parses the symbolic expression to identify which variables correspond to elements of $U$ and $\bar{U}$, extracting their indices ($i, j, k, l$).
  2. Degree Matching: It checks if the number of $U$ factors matches the number of $\bar{U}$ factors. If they differ ($n \neq m$), the integral vanishes (returns 0) due to phase invariance ($U \to e^{i\theta}U$).
  3. Symbolic Summation: It generates the sum over permutations $\sigma, \tau \in S_n$, computing the Kronecker delta products symbolically.
  4. Weingarten Evaluation: It computes the values of $\text{Wg}(\pi, d)$ using Combinatorics.jl characters.

Symbolic Dimension

A key feature of IntU.jl is the ability to leave the dimension $d$ as a symbolic variable. This is achieved through the SymbolicMatrix type, which represents a matrix of arbitrary (symbolic) size.

Examples

1. Basic Integration using @integrate

The @integrate macro provides a convenient way to integrate expressions without manually declaring variables. It uses heuristics to identify random unitaries (usually U) and dimensions.

 using IntU, Symbolics
 @variables d
 # E[|U_11|^2]
 @integrate abs(U[1, 1])^2 dU(d)
 # Output: 1/d
 ```
 
 ### 2. Manual Integration
 
 For more control, or when dealing with multiple matrices, you can declare symbols explicitly.

julia using IntU, Symbolics @variables d U = SymbolicMatrix(:U, :U)

1. Norm of a matrix element

Integral of |U_{11}|^2

integrate(abs(U[1, 1])^2, dU(d))

Output: 1/d


### 3. Higher Unitary Moments

julia using IntU, Symbolics @variables d U = SymbolicMatrix(:U, :U)

2. Fourth moment

Integral of |U_{11}|^4

integrate(abs(U[1, 1])^4, dU(d))

Output: 2 / (d*(d + 1))


### 4. Trace Moments

julia using IntU, Symbolics @variables d U = SymbolicMatrix(:U, :U)

3. Trace moments

Integral of |Tr(U)|^2

integrate(abs(tr(U))^2, dU(d))

Output: 1


### 5. Matrix Integration

You can integrate matrix-valued expressions directly. The function `integrate` will element-wise integrate any `AbstractArray` (including `SymbolicMatrix` and `SymbolicMatrixProduct`) passed to it.

julia using IntU, Symbolics @variables d U = SymbolicMatrix(:U, :U)

E[tr(U A U' B)] = tr(A) * tr(B) / d

A = SymbolicMatrix(:A) B = SymbolicMatrix(:B) integrate(tr(U * A * U' * B), dU(d))


### 6. HCIZ Integrals

IntU.jl provides direct support for **Harish-Chandra-Itzykson-Zuber (HCIZ)** integrals.

julia using IntU, LinearAlgebra

Define matrices

A = diagm([1.0, 2.0]) B = diagm([0.5, 1.5])

Compute ∫ dU exp(Tr(A U B U'))

res = hciz(A, B) println(res)

Output: 20.9329...

```

See the API Reference for more details.

Potential Pitfalls

  • Symbolic vs Numeric Dimension: The dimension $d$ can be symbolic. However, the Weingarten function has poles at small integers ($d < n$). The symbolic result assumes $d$ is generic/large. Substituting discrete values $d < n$ into the rational function may result in division by zero, although the integral itself is well-defined.
  • Removable Singularities: When using evaluate to substitute numeric values into symbolic results, $0/0$ forms may appear (e.g., at $d=1$ for some expressions). IntU.jl automatically detects when a denominator evaluates to zero and simplifies the expression to attempt to resolve these removable singularities.
  • Computational Complexity: The sum involves $(n!)^2$ terms. While optimized to group cycles, integrals with high degrees ($n > 6$) can become computationally expensive. The number of terms grows factorially.

References

  1. Collins, B. (2003). Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability. International Mathematics Research Notices, 2003(17), 953-982.
  2. Collins, B., & Śniady, P. (2006). Integration with respect to the Haar measure on unitary, orthogonal and symplectic groups. Communications in Mathematical Physics, 264(3), 773-795. arXiv:math-ph/0402073
  3. Puchala, Z., & Miszczak, J. A. (2017). Symbolic integration with respect to the Haar measure on the unitary groups. Bulletin of the Polish Academy of Sciences. Technical Sciences, 65(1), 21-27.