Statement of the chGinOE model

As indicated earlier, random matrix models are constructed using only global sym-metries, and so we introduce a random matrix iD, which will model the Euclidean QCD Dirac operator D_QCD= γ_k(∂_k+ iA_k) + µγ₀, where theγ_k (for space-time index k = 0, 1, 2, 3) are the Euclidean Dirac gamma matrices, and the A_k are the gluon

2.2 Random matrices and QCD

fields (with various other labels suppressed). For two-colour QCD with a chemi-cal potential, the model that we will adopt is the non-Hermitian extension of the chGOE model [Ver94b; Ver94c] using a prescription similar to that given in [Osb04].

The matrix D is constructed as follows:

• The QCD Dirac operator possesses axial U_A(1)symmetry (i.e. {γ₅, D_QCD} = 0 where γ₅ ≡ γ₀γ₁γ₂γ₃ is the chirality operator) which implies that all non-zero eigenvalues come in pairs ±Λ. We wish to retain this property, and hence we write D as an off-diagonal block (‘chiral’) matrix.

• By the Atiyah-Singer index theorem, in a gauge (gluon) field of fixed topo-logical charge ν the Dirac operator will have precisely |ν| eigenvalues that are exactly zero. Therefore, we will work in sectors of fixed topological charge, and useN × (N + ν) matrices as the blocks in D, whereN corresponds to the volume of the system in QCD, and ν to the topological charge of the gluon fields.

• With no chemical potential present,D_QCDis anti-Hermitian, and soDmust be a Hermitian random matrix. For a non-zero chemical potentialµ > 0, we then add µ times a second random matrix (see the earlier discussion concerning universality) which must be anti-Hermitian; µ is assumed to be the same for all quark flavours.

• Since the fields in two-colour QCD are in representations of the Lie group SU_c(2) which is pseudo-real, we can choose to write D in a basis with real matrix elements. Therefore, when µ = 0, D will be symmetric, and forµ > 0, we add an anti-symmetric matrix, so that D becomes asymmetric.

The matrix D is therefore given by

D ≡ Ã

0 A

B^T 0

!

≡ Ã

0 P + µQ

P^T − µQ^T 0

!

(2.5) where A, B, P and Q are all real-valued matrices of size N × (N + ν), and µ ∈ [0, 1]

is the non-Hermiticity parameter.

Universality arguments imply that the choice of distribution for matrix elements is arbitrary, and so we choose the simplest mathematically, which is the Gaussian

distribution. The partition function (from which we can read off the JPDF of the matrix elements of D) is consequently given by

Z_N,ν^(Nf)= µ 1

√2π

¶_{2N (N +ν)}Z

dP dQ exp£

−¹₂Tr(P P^T + QQ^T)¤

×

Nf

Y

f =1

det(D + m_fI), (2.6)

wheredP ≡Q

i,jP_ij, and similarly fordQ. The integral is over theN (N +ν)elements of each ofP andQ, and is overRfor each element. In order to simplify the notation, we have not included any N-dependency in the weight function at this stage (the N-dependent scaling is introduced in §6.2).

The product over the determinants in eq. (2.6) models the effects of virtual quarks, and corresponds to a similar factor in the QCD partition function once the (Grassmann-valued) quark fields have been integrated out at fixed gauge field configuration. N_f ≥ 0 is the number of virtual quark flavours, and the{m_f} are the individual quark masses (1 ≤ f ≤ N_f). Note that, numerically, when N_f = 0, we have Z_N,ν = 1; however, when N_f > 0, we haveZ_N,ν ≡ Z_N,ν({m_f})which in general will not equal unity.

The partition function eq. (2.6) is valid in a sector of fixed topological charge ν, and so the complete QCD partition function at vacuum angle θ is given by

Z_N^(Nf)(θ) = X∞ ν=−∞

e^iνθZ_N,ν^(Nf). (2.7)

All the results in this thesis are given for a fixed topology ν, and so we drop the subscript ν in the remainder of this thesis, always considering it to be fixed.

From a physics point of view, we will be interested in the limit N → ∞. This limit can be taken in different ways, depending on how we scale the eigenvalues, quark masses and chemical potential with N.

The term ‘massless’ is somewhat ambiguous. Throughout this document we will use the term ‘quenched’ for the case where N_f = 0(corresponding to large m_f when N_f > 0), and the term ‘zero mass’ for m_f = 0. The unquenched case is the more general case when N_f > 0. In the specific examples of the unquenched case given in Chapter 9, we will generally choose N_f = 2 (corresponding to the lightest u and d quarks). We will often consider the case of degenerate masses m₁ = m₂ ≡ m, so that the JPDF is real and positive for all values of the matrix elements. However, we will also consider non-degenerate masses, since this is physically realistic, and leads to interesting phenomena. Also, since µ = 0 corresponds to a Hermitian (and

2.2 Random matrices and QCD

not an anti-Hermitian) matrix in our convention, we will mostly be interested in imaginary m_f for applications. However, in this document, we will also show some graphs where the m_f are taken to be real.

From matrix representation to eigenvalue JPDF

Starting with the partition function of the chGinOE expressed as an integral over the matrix elements, we now show how to integrate out the angular degrees of freedom to arrive at the joint probability density function (JPDF) of the eigenvalues. We specified the ensemble for the general unquenched case (N_f > 0) in §2.2.3. However, the quenched case (N_f = 0) is easier to solve, since the elements of P and Q are independent, normally-distributed random variables ∼ N (0, 1). Furthermore, many of the unquenched results can actually be expressed in terms of the quenched results.

Therefore it makes sense to solve the quenched case first.

Because the chGinOE is non-Hermitian, we cannot use a simple spectral de-composition to obtain a diagonal matrix of eigenvalues, but instead we must use a variant of the Schur transformation. Since the model has a chiral form involving two matrices, and not one, we actually require theQZ decomposition. And because the matrices in question all have real elements, we use the real QZ decomposition.

After making the transformation, we can integrate out all of the degrees of freedom apart from the eigenvalues themselves, leaving the partition function expressed as an integral over the eigenvalue JPDF.

3.1 JPDF for quenched case

For N_f = 0, the partition function in eq. (2.6) reads

Z_N = µ 1

√2π

¶_{2N (N +ν)}Z

dP dQ exp£

−¹₂Tr(P P^T + QQ^T)¤

. (3.1)

3.1 JPDF for quenched case

Here, and throughout this thesis, we will usually drop the superscript ‘(N_f)’ on all quantities for the case when N_f = 0. We wish to solve for the eigenvalue JPDF of the Dirac matrix D defined in eq. (2.5):

D ≡

In practice, it is easier to solve for the eigenvalues of the so-called Wishart matrix W ≡ AB^T ≡ (P + µQ)(P^T − µQ^T), which is of size N-by-N. The eigenvalues of W are the squares of the (non-zero) eigenvalues of D. This can be understood by considering the characteristic equation

0 = det[Λ − D] = Λ^νdet[Λ²− AB^T] = Λ^ν YN j=1

(Λ²− Λ²_j). (3.2) The matrix W ≡ AB^T has real elements, and so its eigenvalues can be grouped into pairs: either both eigenvalues in a pair will be real, or they will both be complex valued (∈ C\R) and complex conjugates of each other. ForN odd, there will also be a final unpaired real eigenvalue. Hence, the non-zero eigenvalues of D come in real pairs (Λ²_j > 0), pure imaginary pairs (Λ²_j < 0), or complex quadruplets (±Λ_j, ±Λ^∗_j).