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Moreover, the charges Q and
Q satisfy the bracket
algebra
Q;
Q
¼2iH; Q; H
fg
¼
Q; H
¼ 0 ½41
Thus, the action [35] is the classical counterpart of
the quantum theory [9]–[17] in the correspondence
limit i{A, B} ! [A, B]
= AB BA. For these the-
ories, supersymmetry is rooted in the classical
transformations [36].
Supersymmetric Quantum Mechanics
The construction for the supersymmetric oscillator
can be generalized to other dynamical systems in
two ways. First, the nature of the interactions as
represented by the potential can be modified.
Second, the number of degrees of freedom can be
varied. This section presents a generalization of the
supersymmetric oscillator to anharmonic interac-
tions, obtained by modification of the supercharges
[37] with a general function (x) as follows:
Q ¼ðp iðxÞÞ;
Q ¼ðp þ i ðxÞÞ
½42
The brackets [39] imply the supersymmetry algebra
[41] with the Hamiltonian
H ¼
i
2
Q;
Q
¼
1
2
p
2
þ
1
2
2
ðxÞþ
1
2
0
ðxÞ
½43
In quantum mechanics, the supercharges become
operators Q and Q
y
upon reinterpretation of (x, p)
as canonically conjugate operato rs, and the replace-
ment ! f
y
and
! f ; this procedure involves no
ordering ambiguity. The Hamiltonian operator
defined by the anticommutator of Q and Q
y
then
takes the operator form associated with [43]. With
the identification
A ¼
1
ffiffiffi
2
p
ðp iÞ; A
y
¼
1
ffiffiffi
2
p
ðp þ iÞ½44
and making use of the (anti)commutation relations
AA
y
A
y
A ¼
0
ðxÞ; ff
y
þ f
y
f ¼ 1 ½45
this Hamilton operator can be written in normal-
ordered form as
H ¼
1
2
QQ
y
þ Q
y
Q
¼ A
y
A þ
0
ðxÞf
y
f ½46
It is positive-semi definite by constructio n. All results
for the supersymmetric oscillator are reproduced
upon taking (x) = !x.
As the Hamiltonian commutes with the fermion
number operator N
f
, we can label all stationary
states jE, n
f
i by the energy E and the fermion
number n
f
= (0, 1). Moreover, all states of positive
energy are degenerate with respect to fermion
number, as they form pairs related by
supersymmetry:
QjE; 0i¼
ffiffiffiffiffiffi
2E
p
jE; 1i;
QjE; 1i¼
ffiffiffiffiffiffi
2E
p
jE; 0i½47
Only ground states with E
0
= 0 can occur as singlets
under supersymmetry. The existence of such a
ground state with fermion number n
f
amounts to
the existence of a state j0, n
f
i satisfying
A
y
f j0; n
f
i¼Af
y
j0; n
f
i¼0 ½48
The corresponding wave functions are of the form
j0; 0i!
0
ðx;Þ¼
ðxÞ
j0; 1i!
1
ðx;Þ¼
þ
ðxÞ
½49
where
(x) are solutions of the equations
A
¼ 0; A
y
þ
¼ 0 ½50
These functions are formally given by the
expressions
ðxÞ¼C
e
R
x
0
ðyÞdy
½51
For a zero-energy ground state to exist, one of these
functions must be normalizable. For example, if
(x) is a polynomial of positive odd degree 2k 1,
then, depending on the sign of the coefficient of
x
2k1
, one of the exponents is bounded, approaching
zero for x !1, and as a result becomes square
integrable.
If no normalizable wave functions of the form
[51] exist, the ground state cannot have zero energy
(E
0
> 0) and all states necessarily belong to
superdoublets.
Spinning-Particle Mechanics
Minimal supersymmetric classical or quantum
mechanics requires equal number of bosonic and
fermionic coordinates in configuration space (x
i
,
i
),
rather than equal number of bosonic and fermionic
degrees of freedom in phase space. Specifically,
minimal free supersymmetric particle mechanics in
n dimensions is described by the classical
Lagrangian
L ¼
1
2
_
x
2
i
þ
i
2
i
_
i
; i ¼ 1; ...; n ½52
It is invariant modulo a total time derivative under
infinitesimal supersymm etry transformations
x
i
¼i
i
;
i
¼
_
x
i
½53
148 Supersymmetric Quantum Mechanics
The canonical phase-space formulation is phrased
in terms of the free-particle momentum and
Hamiltonian
p
i
¼
_
x
i
; H ¼
1
2
p
2
i
½54
and the brackets
A; B
fg
¼
@A
@x
i
@B
@p
i
@A
@p
i
@B
@x
i
þ ið1Þ
A
@A
@
i
@B
@
i
½55
The supersymmetry transformations are generated
by the supercharge
Q ¼ p
i
i
;A ¼ i Q; A
fg
½56
with the supersymmetry algeb ra
i Q; Q
fg
¼ 2H; Q; H
fg
¼ 0 ½57
An important quantity in these models is the bilinear
(Grassmann-even) antisym metric tensor
ij
¼ i
i
j
½58
For a free particle, it is a set of constants of motion
forming a representation of so(n), the Lie algebra of
n-dimensional rotations:
ij
;
kl
¼
jk
il
jl
ik
ik
jl
þ
il
jk
½59
Therefore, the physical interpretation of
ij
is that it
represents the particle spin. For this reason, super-
symmetric particle mechanics is often called spin-
ning-particle mechanics.
Quantum mechanics of the spinning particle has
the same algebraic structure, with (x
i
, p
i
) the
standard canonically conjuga te operators, and the
fermionic coordinates
i
represented by the genera-
tors of a Clifford algebra; the irreducble representa-
tion in term s of Pauli–Dirac mat rices of dimension
2
[n=2]
2
[n=2]
is
i
!
1
ffiffiffi
2
p
i
;
i
j
þ
j
i
¼ 2
ij
½60
It follows that the wave functions have 2
[n=2]
components, describing different polarization states.
Furthermore, in minimal supersymmetric quantum
mechanics, the supersymmetry operator is repre-
sented by the Dirac operator:
Q !
1
ffiffiffi
2
p
p; ð pÞ
2
¼ p
2
i
¼ 2H ½61
Hence, the stationary states of the system solve the
Dirac equation
p ¼
ffiffiffiffiffiffi
2E
p
½62
The models can, without difficulty, be extended to
include interactions with external fields. As an
example, we consider the coupling to a magnetic
field described by a vector potential A
i
(x). An
extension of the free-particle action [52], invariant
under the same supersymmetry transformations
[53],is
S ¼
Z
dt
1
2
_
x
2
i
þ
i
2
i
_
i
þ qA
i
ðxÞ
_
x
i
iq
2
F
ij
ðxÞ
i
j
½63
where F
ij
= r
i
A
j
r
j
A
i
is the field strength. The
canonical momentum in this model is
p
i
¼
_
x
i
þ qA
i
ðxÞ½64
with the result that the canonical expressions for the
Hamiltonian and supercharge become
H ¼
1
2
ðp
i
qA
i
ðxÞÞ
2
; Q ¼ðp
i
qA
i
ðxÞÞ
i
½65
In the quantum theory, these constants of motion
become the covariant Laplacian and Dirac operator
in an external vector potential A
i
(x). Observe that
supersymmetry requires the spin to couple to the
magnetic field with gyromagnetic ratio g = 2. Expli-
citly, the equation of motion for can be trans-
formed into an equation for the spin precession:
_
i
¼ qF
ij
j
) _
ij
¼ qðF
ik
kj
ik
F
kj
Þ½66
In three dimensi ons, this is equivalent to an equation
in terms of axial vectors:
F
ij
¼ "
ijk
B
k
;
ij
¼ "
ijk
s
k
)
_
s ¼qB s ½67
showing that the precession rate of s is given by
twice the Larmor frequency.
Extended Supersymmetry
It is possible to construct theories with more
supersymmetries by associating with every bosonic
coordinate several fermionic coordinates. An exam-
ple is the supersymmetric oscillator and its general-
izations considered earlier, which has equal number
of bosonic and fermionic degrees of freedom in
phase space, rather than equal number of bosonic
and fermionic coordinates in configuration space.
The classical phase space, spanned by variables
(x
i
, p
i
;
i
,
i
) with i = 1, ..., n, then has double the
number of fermionic variables compared to the
minimal supersymmetric particle models. Such mod-
els can be constructed for systems with an
n-dimensional boson ic configuration space. Their
supercharges take the form
Q ¼ðp
i
i
i
ðxÞÞ
i
;
Q ¼ðp
i
þ i
i
ðxÞÞ
i
x ¼ðx
1
; ...; x
n
Þ
½68
Supersymmetric Quantum Mechanics 149
whilst the Hamiltonian becomes
H ¼
1
2
p
2
i
þ
1
2
2
i
ðxÞ
þ
1
4
ðr
j
i
þr
i
j
Þ
i
j
i
j
½69
The supercharges are conserved if the curl of
i
(x)
vanishes: r
i
j
r
j
i
= 0. It follows that at least
locally there exists a single function W(x) such that
i
ðxÞ¼r
i
WðxÞ½70
W(x) is called the superpotential. Defining the
operators
A
i
¼ p
i
i
i
ðxÞ; A
y
i
¼ p
i
þ i
i
ðxÞ
A
i
A
y
j
A
y
j
A
i
¼r
i
j
þr
j
i
½71
the supersymmetric quantum theory is defined by
Q ¼ A
i
f
y
i
; Q
y
¼ A
y
i
f
i
H ¼
1
2
QQ
y
þ Q
y
Q
½72
The Hamiltonian is the direct operator translation of
the classical expression [69]; its normal-ordered form is
H ¼ A
y
i
A
i
þ
1
2
r
i
j
þr
j
i
f
y
i
f
j
½73
The total fermion number operator
N
f
¼ f
y
i
f
i
½74
(summed over i) satisfying the commutation
relations
N
f
; f
y
j
hi
¼ f
y
j
; N
f
; f
j
¼f
j
½75
commutes with the Hamiltonian. Hence, the station-
ary states can be labeled by the energy E and the total
fermion number n
f
= (0, ..., n). The energy spectrum
being positive semidefinite, all positive-energy states
occur in pairs of fermion number (n
f
, n
f
þ 1); zero-
energy states exist only if the equations
A
i
f
y
i
j0; n
f
i¼A
y
i
f
i
j0 ; n
f
i¼0 ½76
admit a normalizable solution. In this context, the
vanishing of the curl of
i
(x) is important, as it is a
necessary condition for the formal solutions
ðxÞ¼C
exp
Z
x
0
ðyÞdy
¼ C
0
e
WðxÞ
½77
to be single-valued. If one of them is normalizable,
there exists a zero-energy ground state with n
f
= 0
or n
f
= n, represented by a wave function:
j0; 0i!
0
ðx; xÞ¼
ðxÞ
j0; ri!
r
ðx; xÞ¼
þ
ðxÞ
1
...
n
½78
Alternatively, we can represent the wave functions
as spinors of dimension 2
n
, on which the fermion
operators f
y
i
and f
i
act as a 2
n
-dimensional matrix
representation of the Clifford algebra with gen era-
tors
a
, a = 1, ...,2n, defined by
i
¼ f
i
þ f
y
i
;
iþr
¼ i f
i
f
y
i
½79
These operators indeed satisfy the anticommutation
rule
a
b
þ
b
a
¼ 2
ab
½80
Thus, the wave functions have 2
n
components, as
compared to the 2
[n=2]
polarization states of the
minimal models.
The Witten Index
We have noted that for supersymmetric quantum
systems, like the harmonic and anharmonic super-
symmetric oscillator, states exist in pairs of different
fermion number, degenerate in energy, except for
possibly one or more zero-energy states which are
superinvariant in the sense that
Qj0; ni¼
Qj0; ni¼0 , Hj 0; ni¼0 ½81
In the Schro¨ dinger representation, these states are
characterized as zero modes of the Dirac operator:
D ¼ 0 ½82
where D
i
is an ordinary or field-dependent (e.g.,
covariant) derivative. Clearly, the existence of such
states can, in some cases, be guaranteed if there is no
state which can pair up with a given state to form a
superdoublet. Witten developed a topological char-
acterization of this condition, encoded in an index
defined by
I ¼ trð1Þ
N
f
¼ n
b
ðE ¼ 0Þn
f
ðE ¼ 0Þ½83
where N
f
is the fermion number operator, and
n
b, f
(E = 0) are the number of bosonic and fermionic
zero-energy states. The trace is taken over the
complete space of states, but as all nonzero energy
states occu r in pairs of a bosonic and a fermionic
state, their contributions to the trace cancel, having
opposite sign. Therefore, the trace is actually only
over the zero-energy states, and counts the number
of bosonic states with positive sign, and the number
of fermionic states with negative sign. If the index
vanishes, I = 0, then any zero- energy states necessa-
rily exist in equal number of bosonic and fermionic
states; under perturbations of the potential, these
states can form pairs and change their energy to a
positive value. However, if the index does not
vanish, I 6¼ 0, then there are states which ha ve no
150 Supersymmetric Quantum Mechanics
partner of complementary fermion number; these
states can never get a nonzero energy under changes
in the parameters of the potential, as long as the
changes respect sup ersymmetry. Such systems, there-
fore, necessarily possess exact zero-energy states
which are invariant under all supersymmetries.
Deformations of the potential respecting super-
symmetry are those obtained by changing the
parameters in the superpotential. The usefulness of
this concept is, therefore, that the index for models
with complicated superpotentials can be computed
by comparing them with models with simple super-
potentials having similar topological properties.
Counting the number of states is not always a
simple procedure, in particular when the spectrum
includes continuum states. Therefore, in practice one
often needs a regularization procedur e, by taking the
trace over the full state space of the exponentially
damped quantity
IðÞ¼trð1Þ
N
f
e
H
½84
and taking the limit ! 0. The quantity [84] can be
computed in terms of a path integral with periodic
boundary conditions for the fermionic degrees of
freedom.
Finally, as the wave function representation of
supersymmetric quantum mechanics [82] links the
Witten index to the space of zero modes of a Dirac
operator, in particular cases it can be used to
describe topological aspects of sigma models and
gauge theories, and related mathematical quantities
such as the Atiyah–Singer index.
More details and references to the original
literature can be found in the reviews listed in the
Further Rea ding sect ion.
See also: Path-Integrals in Non Commutative Geometry;
Supermanifolds.
Further Reading
Cooper F, Khare A, and Sukhatme U (1995) Supersymmetry and
quantum mechanics. Physics Reports 251: 267.
De Witt BS (1984, 1992) Supermanifolds. Cambridge: Cambridge
University Press.
Shifman MA (1999) ITEP Lectures on Particle Physics and Field
Theory, vol. 1, ch. 4. Singapore: World Scientific.
van Holten JW (1996) D = 1 Supergravity and spinning particles.
In: Jancewicz B and Sobezyk J (eds.) From Field Theory to
Quantum Groups, p. 173. Singapore: World Scientific.
Supersymmetry Methods in Random Matrix Theory
M R Zirnbauer, Universita
¨
tKo
¨
ln, Ko
¨
ln, Germany
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
A prominent theme of modern condensed matter
physics is electronic transport – in particular, the
electrical conductivity – of disordered metallic
systems at very low temperatures. From the Landau
theory of weakly interacting Fermi liquids, one
expects the essential aspects of the situation to be
captured by the single-electron approxim ation.
Mathematical models that have been proposed and
studied in this context include random Schro¨ dinger
operators and band random matrices.
If the physical system has infinite size, two distinct
possibilities exist: the quantum single-electron
motion may either be bounded or unbounded. In the
former case, the disordered electron system is an
insulator, in the latter case, a metal with finite
conductivity (if the electron motion is not critical
but diffusive). Metallic behavi or is expected for
weakly disordered systems in three dimensions;
insulating behavior sets in when the disorder strength
is increased or the space dimension reduce d.
The main theoretical tool used in the phy sics
literature on the subject is the ‘‘supersymmetry
method’’ pioneered by Wegner and Efetov (1979–83).
Over the past 20 years, physicists have applied the
method in many instances, and a rather complete
picture of weakly disordered metals has emerged.
Several excellent reviews of these developments are
available in print.
From the perspective of mathematics, however, the
method has not always been described correctly, and
what is sorely lacking at present is an exposition of
how to implement the method rigorously. (Unfortu-
nately, the correct exposition by Scha¨ fer and Wegner
(1980) was largely ignored or forgotten by later
authors.) In this article, an attempt is made to help
remedy the situation, by giving a careful review of
the Wegner–Efetov supersymmetry method for the
case of Hermitian band random matrices.
Gaussian Ensembles
Let V be a unitary vector space of finite dimension.
A Hermitian random matrix model on V is defined
Supersymmetry Methods in Random Matrix Theory 151
by some probability distribution on Herm(V), the
Hermitian linear operators on V. We may fix some
orthonormal basis of V and represent the elements
H of Herm(V) by Hermitian square matrices.
Quite generally, probability distributions are
characterized by their Fourier transform or char-
acteristic function. In the present case this is
ðKÞ¼ e
itrHK
where the Fourier variable K is some other linear
operator on V, and h...i denotes the expectation
value with respe ct to the probability distribution for
H. Later, it will be important that, if (K)isan
analytic function of K, the matrix entries of K need
not be from R or C but can be taken from the even
part of some exterior algebra.
The probability distributions to be considered in
this article are Gaussian with zero mean, hHi= 0.
Their Fourier transform is also Gaus sian:
ðKÞ¼e
ð1=2ÞJðK;KÞ
with J some quadratic form. We now describe J for a
large family of hierarchical models that includes the
case of band random matrices.
Let V be given a decomposition by orthogonal
vector spaces:
V ¼V
1
V
2
V
jj
We should imagine that every vector space V
i
corresponds to one site i of some lattice , and the
total number of sites is jj. For simplicity, we take
all dimensions to be equal: dim V
1
= = dim
V
jj
= N. Thus, the dimension of V is Njj. The
integer N is called the number of orbitals per site.
If
i
is the orthogonal projector on the linear
subspace V
i
V, we take the bilinear form J to be
JðK ; K
0
Þ¼
X
jj
i;j¼1
J
ij
trð
i
K
j
K
0
Þ
where the coefficients J
ij
are real, symmetric, and
positive. This choice of J implies invariance under
the group U of unitary transformations in each
subspace:
U ¼ UðV
1
ÞUðV
2
ÞU ðV
jj
Þ
Clearly, (K) =( UKU
1
) or, equivalently, the
probability distribution for H is invariant under
conjugation H 7!UHU
1
, for U 2 U .
If {e
a
i
}
a = 1,... , N
is an orthonormal basis of V
i
,we
define linear operators E
ab
ij
: V
j
!V
i
by E
ab
ij
e
b
j
= e
a
i
.
By evaluating J(E
ab
ij
, E
b
0
a
0
j
0
i
0
) = J
ij
ii
0
jj
0
aa
0
bb
0
, one sees
that the mat rix entries of H all are statistically
independent.
By varying the lattice , the number of orbitals N,
and the variances J
ij
, one obtains a large class of
Hermitian random matrix models, two prominent
subclasses of which are the following:
1. For jj= 1, one gets the Gaussian Unitary
Ensemble (GUE). Its symmetry group is U =
U(N), the largest one possible in dimension
N = dim V.
2. If ji jj denotes a distance function for , and f a
rapidly decreasing positive function on R
þ
of
width W, the choice J
ij
= f (ji jj) with N = 1
gives an ensemble of band random matrices with
bandwidth W and symmetry group U = U(1)
jj
.
Beyond being real, symmetric, and positive, the
variances J
ij
are required to have two extra proper-
ties in order for all of the following treatment to go
through:
They must be positive as a quadratic form. This is
to guarantee the existence of an inverse, which we
denote by w
ij
= (J
1
)
ij
.
The off-diagonal matrix entries of the inverse
must be nonpositive: w
ij
0 for i 6¼ j.
Basic Tools
Green’s Functions
A major goal of random matrix theory is to
understand the statistical behavior of the spectrum
and the eigenstates of a random Hamiltonian H.
Spectral and eigenstate information can be extracted
from the Green’s function, that is, from matrix
elements of the operator (z H)
1
with complex
parameter z 2 CnR. For the models at hand, the
good objects to consider are averages of U -invariant
observables such as
G
ð1Þ
i
ðzÞ¼ tr
i
ðz HÞ
1
DE
½1
G
ð2Þ
ij
ðz
1
; z
2
Þ¼ tr
i
ðz
1
HÞ
1
j
ðz
2
HÞ
1
DE
½2
The discontinuity of G
(1)
i
(z)acrosstherealz-axis
yields the local density of states. In the limit of
infinite volume (jj!1), the function G
(2)
ij
(z
1
, z
2
)
for z
1
= E þ i", z
2
= E i",realenergyE ,and">0
going to zero , gives information on tr ansport,
for example, the electrical conductivity by the
Kubo–Greenwood formula.
Mathematically speaking, if G
(2)
ij
(E þ i", E i")is
bounded (for infinite volume) in " and decreases
algebraically with distance ji jj at " = 0þ , the
152 Supersymmetry Methods in Random Matrix Theory
spectrum is absolut ely continuous and the eigen-
states are extended at energy E. On the other hand,
a pure point spectrum and localized eigenstates are
signaled by the behavior G
(2)
ij
"
1
e
jijj
with
positive Lyapunov exponent .
Green’s Functions from Determinants
For any pair of linear operators A, B on a finite-
dimensional vector space V, the following formula
from basic linear algebra holds if A has an inver se:
d
dt
detðA þ tBÞ
t¼0
¼ det ðAÞtrðA
1
BÞ
Using it with A = z H and z 2 CnR, all Green’s
functions can be expressed in terms of determinants;
for example,
G
ð2Þ
ij
ðw; zÞ
¼
X
N
a;b¼1
@
2
@s@t
detðw HÞdetðz H þ tE
ab
ij
Þ
detðw H sE
ba
ji
Þdetðz HÞ
*+
s¼t¼0
It is clear that, given a formula of this kind, what
one wants is a method to handle ensemble averages
of ratios of determinants. This is what is reviewed in
the sequel.
Determinants as Gaussian Integrals
Let the Hermitian scalar product of the unitary
vector space V be written as ’
1
, ’
2
7!(
’
1
, ’
2
),
and denote the adjoint or Hermitian conjugate
of a linear operator A on V by A
.If
ReA := (1/2)(A þ A
) > 0, the standard Lebesgue
integral of the Gaussian function ’ 7!e
(
’, A’)
makes sense and gives
Z
e
ð
’;A’Þ
¼ det A
1
½3
where it is understood that we are integrating with
the Lebesgue measure on (the norme d vector space)
V normalized by
R
e
(
’, ’)
= 1. The same integral
with anticommuting instead of the (commuting)
’ 2 V gives
Z
e
ð
;A Þ
¼ det A ½4
This basic formula from the field theory of
fermionic particles is a consequence of the integra-
tion over anticommuting variables actually being
differentiation:
Z
d
1
d
1
f ð
1
;
1
; ...Þ:¼
@
2
@
1
@
1
f ð
1
;
1
; ...Þ
Fermionic Variant
The supersymmetry method of random matrix
theory is a theme with many variations. The first
variation to be described is the ‘‘fermionic’’ one. To
optimize the notation, we now write d
N, J
(H) for
the density of the Gaussian probability distribution
of H:
hFðH Þi ¼
Z
FðHÞd
N; J
ðHÞ
All determinants and traces appearing below will be
taken over vector spaces that are clear from the
context.
Let z
1
, ..., z
n
be any set of n complex numbers,
put z := diag(z
1
, ..., z
n
) for later purposes, and
consider
ferm
n; N
ðz; JÞ¼
Z
Y
n
¼1
detðz
HÞd
N;J
ðHÞ½5
The supersymmetry met hod expresses this average
of a product of determinants in an alternative way,
by integrating over a ‘‘dual’’ measure as follows.
Introducing an auxiliary unitary vector space
C
n
, one associates with every site i of the lattice
an object Q
i
2 Herm(C
n
), the space of Hermitian
n n matrices. If dQ
i
for i = 1, ..., jj are Lebesgue
measures on Herm(C
n
), one puts DQ = const.
Q
i
dQ
i
and
d
n; J
ðQÞ :¼ e
ð1=2Þ
i;j
ðJ
1
Þ
ij
tr Q
i
Q
j
DQ ½6
The multiplicative constant in DQ is fixed by
requiring the density to be normalized:
R
d
n, J
(Q) = 1. By completing the square, this Gaussian
probability measure has the charac teristic function
Z
e
i
j
trQ
j
K
j
d
n; J
ðQÞ¼e
ð1=2Þ
ij
J
ij
tr K
i
K
j
where the Fourier variables K
1
, ..., K
jj
are n n
matrices with matrix entries taken from C or
another commutative algebra.
The key relation of the fermionic variant of the
supersymmetry method is that the expectation of the
product of determinants [5] has another expression as
ferm
n; N
ðz; JÞ¼
Z
Y
jj
j¼1
det
N
ðz iQ
j
Þd
n; J
ðQÞ½7
(i =
ffiffiffiffiffiffi
1
p
). The strategy of the proof is quite simple:
one writes the determinants in both expressions for
ferm
n, N
as Gaussian integrals ov er nNjj complex
fermionic variab les
1
, ...,
n
(each
is a vector in
V with anticommuting coefficients), using the basic
Supersymmetry Methods in Random Matrix Theory 153
formula [4]. The integrals then encountered are
essentially the Fourier transforms of the distribu-
tions d
N, J
(H) resp., d
n, J
(Q). The result is
Z
e
z
ð
;
Þ
e
ð1=2Þ
ij
J
ij
ð
;
i
Þð
;
j
Þ
for both expressions of
ferm
n, N
. In other words,
although the probability distributions d
N, J
(H) and
d
n, J
(Q) are distinct (they are defined on different
spaces), thei r characteristic functions coincide when
evaluated on the Fourier variables K =
P
(
, )
for H and (K
i
)
= (
,
i
) for Q
i
. This establishes
the claimed equality of the expressions [5] and [7] for
ferm
n, N
(z, J).
What is the advantage of passing to the alternative
expression by d
n, J
(Q)? The answer is that, while H
is made up of independent random variables, the new
variables Q
i
, called the Hubbard–Stratonovich field,
are correlated: they interact through the ‘‘exchange’’
constants w
ij
= (J
1
)
ij
. If that interaction creates
enough collectivity, a kind of mean-field behavior
results.
For the simple case of GUE (jj= 1, w
11
= N=
2
)
with z
1
= = z
n
= E, one gets the relation
det
n
ðE HÞ
hi
¼
Z
det
N
ðE iQÞe
ðN=2
2
Þtr Q
2
dQ
the right -hand side of which is easily analyzed by the
steepest descent method in the limit of large N.
For band random matrices in the so-called ergodic
regime, the physical behavior turns out to be governed
by the constant mode Q
1
= = Q
jj
– a fact that can
be used to establish GUE universality in that regime.
Bosonic Variant
The bosonic variant of the present method, due to
Wegner, computes averages of products of determi-
nants placed in the denominator:
bos
n;N
ðz; JÞ¼
Z
Y
n
¼1
det
1
ðz
HÞd
N ; J
ðHÞ½8
where we now require Im z
6¼ 0 for all = 1, ..., n.
Complications relative to the fermionic case arise
from the fact that the integrand in [8] has poles. If
one replaces the anticommuting vectors
by
commuting ones ’
, and then simply repeats the
previous calculation in a naive manner, one arrives at
bos
n;N
ðz; JÞ¼
?
Z
Y
jj
j¼1
det
N
ðz Q
j
Þd
n; J
ðQÞ½9
where the integral is still over Q
j
2 Herm(C
n
). The
calculation is correct, and relation [9] therefore
holds true, provided that the parameters z
1
, ..., z
n
all lie in the same half (upper or lower) of the
complex plane. To obtain information on transport
properties, however, one ne eds parameters in both
the upper and lower halves; see the paragraph
following [2]. The general case to be addressed
below is Im z
> 0 for = 1, ..., p, and Im z
< 0
for = p þ 1, ..., n. Careful inspection of the steps
leading to eqn [9] reveals a convergence problem for
0 < p < n. In fact, [9] with Q
j
in Herm( C
n
) turns
out to be false in that range. Learning how to
resolve this problem is the main step toward
mathematical mastery of the method. Let us there-
fore give the details.
If s
:= sgnIm z
, the good (meaning convergent)
Gaussian integral to consider is
Z
e
i
s
’
;ðz
HÞ ’
ðÞ
¼
Y
n
¼1
det
1
is
ðz
HÞðÞ
To avoid carrying around trivial constants, we now
assume i
(n2p)Nj j
= 1. Use of the characteristic
function of the distribution for H then gives
bos
n;N
ðz; JÞ¼
Z
e
i
s
z
ð
’
;’
Þ
e
1
2
ij
J
ij
s
ð
’
;
i
’
Þs
ð
’
;
j
’
Þ
½10
The difficulty of analyzing this expression stems
from the ‘‘hyperbolic’’ nature (due to the indefinite-
ness of the signs s
= 1) of the term quartic in the
’
,
’
.
Fyodorov’s Method
The integrand for
bos
is naturally expressed in
terms of n n matrices M
i
with matrix ele-
ments (M
i
)
= (
’
,
i
’
). These matrices lie in
Herm
þ
(C
n
), that is, they are non-negative as well
as Hermitian. Fyodorov’s idea was to introduce
them as the new variables of integrat ion. To do
that step, recall the basic fact that, giv en two
differentiable spaces X and Y and a smooth map
: X !Y, a distribution on X is pushed forward
to a distribution ()onY by ()[f ]:= [f ],
where f is any test function on Y.
We apply this universal principle to the case at
hand by identifying X with V
n
, and Y with
(Herm
þ
(C
n
))
jj
, and with the mapping that sends
ð’
1
; ...;’
n
Þ2X to ðM
1
; ...; M
jj
Þ2Y
by (M
i
)
= (
’
,
i
’
). On X = V
n
we are integrat-
ing with the product Lebesgue measure normalized
by
R
e
(’
, ’
)
= 1. We now want the push-forward
of this flat measure (or distribution) by the mapping
. In general, the push-forward of a measure is not
154 Supersymmetry Methods in Random Matrix Theory
guaranteed to have a density but may be singular
(like a Dirac -distribution). This is in fact what
happens if N < n. The matrices M
i
then have less
than the maximal rank, so they fail to be positive
but possess zero eigenvalues, which implies that
the flat measure on X is pus hed forward by into the
boundary of Y. For N n, on the other hand, the
push-forward measure does have a density on Y; and
that density is
Q
jj
i = 1
(det M
i
)
N n
dM
i
, as is seen by
transforming to the eigenvalue representation and
comparing Jacobians. The dM
i
are Lebesgue mea-
sures on Her m( C
n
), nor malized by the condition
Z
M
i
>0
e
trM
i
ðdet M
i
Þ
N n
dM
i
¼
Z
e
ð
’
;
i
’
Þ
¼ 1
Assembling the sign information for Im z
in a
diagonal matrix s := diag(s
1
, ..., s
n
), and pushing the
integral over X forward to an integral over Y with
measure DM :=
Q
i
dM
i
, we obtain Fyodorov’s
formula:
bos
n;N
ðz; JÞ¼
Z
Y
e
ð1=2Þ
ij
J
ij
trðsM
i
sM
j
Þ
e
k
tr iszM
k
þðN nÞln M
k
ðÞ
DM ½11
This formula has a number of attractive features.
One is ease of derivation, another is ready general-
izability to the case of non-Gaussian distributions.
The main disadvantage of the formula is that it does
not apply to the case of band random matrices
(because of the restriction N n); nor does it
combine nicely with the fermionic formula [7] to
give a supersymmetric formalism, as one formula is
built on J
ij
and the other on w
ij
.
Note that [11] clearly displays the dependence on
the signature of Im z: you cannot remove the s
1
, ..., s
n
from the integrand without changing the domain of
integration Y = (Herm
þ
(C
n
))
jj
. This important
feature is missing from the naive formula [9].
Setting q = n p, let U(p, q) be the pseudounitary
group of complex n n matrices T with inverse
T
1
= sT
s. Since jdet Tj= 1 for T 2 U(p, q), the
integration domain Y and density DM =
Q
i
dM
i
of
Fyodorov’s formula are invariant under U( p, q)
transformations M
i
7!TM
i
T
, and so is actually the
integrand in the limit where all parameters z
1
, ..., z
n
become equal. Thus, the elements of U(p, q) are
global symmetries in that limit. This observation
holds the key to ano ther method of transform ing the
expression [10].
The Method of Scha¨ fer and Wegner
To rescue the naive formula [9], what needs to be
abandoned is the integration domain Herm(C
n
) for
the matrices Q
i
. The good domain to use was
constructed by Scha¨ fer and Wegner, but was largely
forgotten in later physics work.
Writing (M
k
)
= (
’
,
k
’
) as before, consider
the function
F
M
ðQÞ¼e
ð1=2Þ
ij
w
ij
trðsQ
i
þizÞðsQ
j
þizÞ
k
trM
k
Q
k
½12
viewed as a holomorphic function of
Q ¼ðQ
1
; ...; Q
jj
Þ2EndðC
n
Þ
jj
If the Gaussian integral
R
F
M
(Q)DQ with holo-
morphic density DQ =
Q
i
dQ
i
is formally carried
out by completing the square, one gets the integrand
of [10]. This is just what we want, as it would allow
us to pass to a Q-matrix formulation akin to the one
of the previous section. But how can that formal
step be made rigorous? To that end, one needs to (1)
construct a domain on which jF
M
(Q)j decreases
rapidly so that the integral exists, and (2) justify
completion of the square and shifting of variables.
To begin, take the absolute value of F
M
(Q).
Putting (1=2)(Q
j
þ Q
j
) =: ReQ
j
and (1=2i)(Q
j
Q
j
) =: Im Q
j
, we have jF
M
j= e
(1=4)(f
1
þf
2
þf
3
)
with
f
1
ðQÞ¼
X
ij
w
ij
trðsIm Q
i
þ zÞðsIm Q
j
þ zÞþc:c:
f
2
ðQÞ¼2
X
ij
w
ij
trðsReQ
i
ÞðsReQ
j
Þ
f
3
ðQÞ¼4
X
i
tr M
i
þ sIm z
X
j
w
ij
!
ReQ
i
These expressions suggest making the following
choice of integration domain for Q
i
(i = 1, ..., jj).
Pick some real constant >0 and put
ReQ
i
¼ T
i
T
i
; Im Q
i
¼ P
i
:¼
P
þ
i
0
0 P
i
with T
i
2 U(p, q), P
þ
i
2 Herm(C
p
), P
i
2 Herm(C
q
).
The set of matrices Q
i
so defined is referred to as
the Scha¨fer–Wegner domain X
p, q
. The range of the
field Q = (Q
1
, ..., Q
jj
) is the direct product
X : = (X
p, q
)
jj
.
To show that this is a good choice of domain, we
first of all show convergence of the integral
R
X
F
M
(Q)DQ. The matrices P
i
commute with s,so
f
1
ðQÞj
X
¼2Re
X
ij
w
ij
trðP
i
þ szÞðP
j
þ szÞ
Since the coefficients w
ij
are positive as a quadratic
form, this expression is convex (w ith a positive
Hessian) in the Hermitian matrices P
i
. Second, the
function
f
2
ðQÞj
X
¼2
2
X
ij
w
ij
tr T
i
T
i
1
T
j
T
j
Supersymmetry Methods in Random Matrix Theory 155
is bounded from below by the constant 2
2
n
i
w
ii
.
This holds true because w
ij
is negative for i 6¼ j, and
because T
i
T
i
> 0 and the trace of a product of two
positive Hermitian matrices is always positive.
Third,
f
3
ðQÞj
X
¼4
X
i
tr M
i
þ sIm z
X
j
w
ij
!
T
i
T
i
is positive, as ( ...) is positive Hermitian. As long as
sIm z > 0, the function f
3
goes to infinity for all
possible dire ctions of taking the T
i
to infinity on
U(p, q).
Thus, when the matrices Q
i
are taken to vary on
the Scha¨fer–Wegner domain X
p, q
, the absolute value
jF
M
j= e
(1=4)(f
1
þf
2
þf
3
)
decreases rapidly at infinity.
This establishes the convergence of
R
X
F
M
(Q)DQ.
Next, let us count dimensions. The mapping
T 7!TT
for T 2 U(p, q) =: G is invariant under
right multiplication of T by elements of the unitary
subgroup H:= U(p) U(q) – it is called the ‘‘Cartan
embedding’’ of G=H into G. The real manifold G=H
has dimension 2pq and so does its image under the
Cartan embedding. Augmenting this by the dimen-
sion of Herm(C
p
) and Herm(C
q
) (from P
i
), one gets
dimX
p, q
= 2pq þ p
2
þ q
2
= (p þ q)
2
= n
2
, which is as
it should be.
Finally, why can one shift variables and do the
Gaussian integral over Q (with translation-invariant
DQ) by completing the square? This question is
legitimate as the Scha¨fer–Wegner domain X
p, q
lacks
invariance under the required shift, which is
Q
i
7!Q
i
isz þ
P
j
J
ij
sM
j
s.
To complete the square in [12], introduce a
parameter t 2 [0, 1] and consi der the family of shifts
Q
i
7!Q
i
þ t isz þ
j
J
ij
sM
j
s
For fixed t, this shift takes X = (X
p, q
)
jj
into another
domain, X (t). Inspection shows that the function
[12] still decreases rapidly (uniformly in the M
i
)on
X (t), as long as t < 1. Without changing the
integral, one can add pieces to X (t) (for t < 1) at
infinity to arrange for the chain X X (t)tobea
cycle. Because X (t) is homotopic to X (0) = X , this
cycle is a boundary: there exists a manifold Y (t)of
dimension dimX þ 1 such that @Y (t) = X X (t).
Viewed as a holomorphic differential form of degree
(n
2
jj, 0) in the complex space End(C
n
)
jj
, the
integrand ! := F
M
(Q)DQ is closed (i.e., d! = 0).
Therefore, by Stokes’ theorem,
Z
X
!
Z
X ðtÞ
! ¼
Z
@Y ðtÞ
! ¼
Z
Y ðtÞ
d! ¼ 0
which proves
R
X (t)
F
M
(Q)DQ =
R
X
F
M
(Q)DQ, inde-
pendent of t. (This argument does not go through
for the nonrigorous choice sQ
i
:= T
i
P
i
T
1
i
usually
made!)
In the limit t !1, one encounters the expression
Z
X ð1Þ
F
M
ðQÞDQ ¼
Z
X
d
n;J
ðisQÞ
e
ð1=2Þ
ij
J
ij
trðsM
i
sM
j
Þþi
k
trðszM
k
Þ
with d
n, J
as in [6]. The normalization integral over
X is defined by taking the Hermitian matrices P
i
to
be the inner variables of integration. The outer
integrals over the T
i
then demonstrably exist, and
one can fix the (otherwise arbitrary) normalization
of DQ by setting
R
X
d
n, J
(isQ) = 1. Making that
choice, and comparing with [10], one has proved
bos
n;N
¼
Z
’;
’
Z
X
F
ðM
i
Þ
¼ð’
;
i
’
Þ
ðQÞDQ
The final step is to change the order of integration
over the Q-and’-variables, which is permitted
since the Q-integral converges uniformly in ’.
Doing the Gaussian ’-integral and shifting Q
k
!
Q
k
isz, one arrives at the Scha¨fer–Wegner formula
for
bos
n, N
:
bos
n;N
ðz; w
1
Þ¼
Z
X
e
ð1=2Þ
ij
w
ij
trðsQ
i
sQ
j
Þ
e
N
k
tr lnðQ
k
iszÞ
DQ ½13
which is a rigorous version of the naive formula [9].
Compared to Fyodorov’s formula, it has the dis-
advantage of not being manifestly invariant under
global hyperbolic transformations Q
i
7!TQ
i
T
(the
integration domain X is not invariant). Its best
feature is that it does apply to the case of band
random matrices with one orbital per site (N = 1).
Supersymmetric Variant
We are now in a position to tackle the problem of
averaging ratios of determinants. For concreteness,
we shall discuss the case where the number of
determinants is two for both the numerator and the
denominator, which is what is needed for the
calculation of the function G
(2)
ij
(z
1
, z
2
) defined in
eqn [2]. We will consider the case of relevance for
the electrical conductivity: z
1
= E þ i, z
2
= E i,
with E 2 R and >0.
A Q-integral formula for G
(2)
ij
(z
1
, z
2
) can be
derived by combining the fermionic method for
detðz
1
HÞdet z
2
H þ t
2
E
ab
ij
DE
156 Supersymmetry Methods in Random Matrix Theory
with the Scha¨fer–Wegner bosonic formalism for
det
1
z
1
H t
1
E
ba
ji
det
1
ðz
2
HÞ
DE
and eventually differentiating with respect to t
1
, t
2
at
t
1
= t
2
= 0 and summing over a, b; see the subsection
‘‘Green’s functions from determinants.’’ All steps are
formally the same as before, but with traces and
determinants replaced by their supersymmetric
analogs. Having given a great many technical details
in the last two sections, we now just present the
final formula along with the necessary definitions
and some indication of what are the new elements
involved in the proof.
Let each of Q
BB
, Q
FF
, Q
BF
, and Q
FB
stand for a
2 2 matrix. If the first two matrices have
commuting entries and the last two anticommuting
ones, they combine to a 4 4 supermatrix:
Q ¼
Q
BB
Q
BF
Q
FB
Q
FF
Relevant operations on supermatrices are the
supertrace,
StrQ ¼ trQ
BB
trQ
FF
and the superdeterminant,
SdetQ ¼
detðQ
BB
Þ
detðQ
FF
Q
FB
Q
BB
1
Q
BF
Þ
These are related by the identity Sdet= exp Str ln
whenever the superdeterminant exists and is
nonzero.
In the process of applying the method described
earlier, a supermatrix Q
i
gets introduced at every
site i of the lattice . The domain of integration for
each of the matrix blocks (Q
i
)
BB
(i = 1, ..., jj)is
taken to be the Scha¨ fer–Wegner domain X
1, 1
(with
some choice of >0); the integration domain for
each of the (Q
i
)
FF
is the space of Hermitian 2 2
matrices, as before.
Let E
11
BB
be the 4 4 (super)matrix with unit entry
in the upper-left corner and zeros elsewhere; simi-
larly, E
22
FF
has unity in the lower-right corner and
zeros elsewhere. Putting s = diag(1, 1, 1, 1) and
z = diag(z
1
, z
2
, z
1
, z
2
), the supersymmetric Q-integral
formula for the generating function of G
(2)
ij
–
obtained by combining the Scha¨fer–Wegner bosonic
method with the fermionic variant – is written as
detðz
1
HÞdetðz
2
H þ t
2
E
ab
ij
Þ
detðz
1
H t
1
E
ba
ji
Þdetðz
2
HÞ
*+
¼
Z
DQe
ð1=2Þ
kl
w
kl
StrðsQ
k
sQ
l
Þ
e
Str ln
r;c
ðQ
r
iszÞE
cc
rr
þit
1
E
11
BB
E
ba
ji
it
2
E
22
FF
E
ab
ij
½14
where the second supertrace includes a sum over
sites and orbitals, and on setting t
1
= t
2
= 0 becomes
e
N
r
Str lnðQ
r
iszÞ
¼
Y
r
Sdet
N
ðQ
r
iszÞ
The superintegral ‘‘measure’’ DQ =
Q
r
DQ
r
is the
flat Berezin form, that is, the product of differentials
for all the commuting matrix entries in (Q
r
)
BB
and
(Q
r
)
FF
, times the product of derivatives for all the
anticommuting matrix entries in (Q
r
)
BF
and (Q
r
)
FB
.
To prove the formula [14], two new tools are
needed, a brief account of which is as follows.
Gaussian Superintegrals
There exists a supersymmetric generalization of the
Gaussian integration formulas given in the subsec-
tion ‘‘Determinants as Gaussian integrals’’: if
A, D(B, C) are linear operators or matrices with
commuting (resp., anticommuting) entries, and
ReA > 0, one has
Sdet
1
AB
CD
¼
Z
e
ð
’;A’Þð
’;B Þð
;C’Þð
;D Þ
Verification of this formula is straightforward.
Using it, one writes the last factor in [14] as a
Gaussian superintegral over four vectors: ’
1
, ’
2
,
1
,
and
2
. The integrand then becomes Gaussian in the
matrices Q
r
.
Shifting Variables
The next step in the proof is to do the ‘‘Gaussian’’
integral over the supermatrices Q
r
. By definition, in
a superintegral, one first carries out the Fermi
integral, and afterw ards the ordinary integrations.
The Gaussian integral over the anticommuting parts
(Q
r
)
BF
and (Q
r
)
FB
is readily done by completing the
square and shifting variables using the fact that
fermionic integration is differentiation:
Z
df ð
0
Þ¼
@
@
f ð
0
Þ¼
Z
df ðÞ
Similarly, the Gaussian integral over the Hermitian
matrices (Q
r
)
FF
is done by completing the square
and shifting. The inte gral over (Q
r
)
BB
, however, is
not Gaussian, a s the domain is not R
n
but the
Scha¨ fer–Wegner domain. Here, more advanced
calculus is required: these integrations are done by
using a supersymmetric change-of-variables theorem
due to Berezin to make the necessary shifts by
nilpotents. (There is not enough space to describe
this here, so please consult Berezin’s (1987) book.)
Without difficulty, one finds the result to agree with
the left-hand side of eqn [14], thereby establishing
that formula.
Supersymmetry Methods in Random Matrix Theory 157