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Random-Matrix Approach to RPA Equations
X. Barillier-Pertuisel,
IPN, Orsay O. Bohigas,
LPTMS, Orsay H. A. Weidenmüller,
MPI für Kernphysik, Heidelberg
Workshop on Random-Matrix Theory and
Applications From Number Theory to Mesoscopic
Physics, June 25 27, Orsay.
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Contents
1. Motivation 2. RPA Equations 3.
Random-Matrix Approach 4. Pastur Equation 5.
Solutions. Numerical Results 6. Critical
Strength 7. Summary
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1. Motivation
Random-phase approximation (RPA) is a standard
tool of many-body physics. First used in
condensed-matter physics but later also in other
areas and especially in nuclear-structure physics.
Study RPA equations in most general form by
taking matrix elements as random. Yields
unitarily invariant random-matrix model. Study
spectrum using (generalized) Pastur equation as
a function of coupling between states at positive
and at negative energies. When does the spectrum
become instable (i.e., when do eigenvalues become
complex)? Distribution of eigenvalues in complex
energy plane?
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2. RPA Equations
N-dimensional space for particle-hole pairs.
RPA equations have dimension 2N
Matrix A is Hermitean. Matrix C is
symmetric. Total matrix not Hermitean. Eigenvalues
not necessarily real. Non-real eigenvalues
Instability.
Two symmetries
If ? is eigenvalue then also (- ?) is an
eigenvalue.
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If ? is an eigenvalue then also ? is
an eigenvalue.
Result Real and purely imaginary eigenvalues
come in pairs with opposite signs. Complex
eigenvalues with non-vanishing real and imaginary
parts come in quartets arranged symmetrically
with respect to the real and to the
imaginary energy axis.
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Hermitean matrix A0 causes repulsion amongst
positive eigenvalues, matrix (A0) causes
repulsion amongst negative eigenvalues. For C 0
all eigenvalues real. Role of C Eliminate
negative-energy subspace and obtain Hermitean
effective operator - C (E (A(0))-1 C
in positive-energy subspace. Causes additional
level repulsion amongst positive and
amongst negative eigenvalues and level attraction
between positive and negative eigenvalues (Trace
of that operator is negative). With increasing
strength of C, pairs of real eigenvalues
with opposite signs coalesce at E 0 and then
move along the imaginary axis in opposite
directions. Instability of the RPA equations.
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3. Random-Matrix Approach
Degenerate particle-hole energies at r, A0 1N r
A where matrix A contains particle-hole
interaction matrix elements, belongs to GUE
with
and is invariant under A ? U A (U)T. Matrix C
obeys C is invariant under C ? U C UT. The
RPA matrix becomes
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Generalized unitary invariance Ensemble is
invariant under
The matrices are not Hermitian. Ensemble does not
belong to ten generic random-matrix ensembles
listed by Altland and Zirnbauer. We have also
looked at the orthogonal case. Aim To study
average spectrum for N ? 1. Model depends on two
dimensionless parameters ? ?2 / ?2 measures
relative strength of C versus A. For ? 0,
spectrum consists of two semicircles. And x r /
(2?) gives distance of centers of semicircles
from origin at E 0. Spectral fluctuations
not studied are very likely to be same as for
GUE in each branch. Two limiting cases C 0 and
A 0. Find value of ? for which RPA equations
become instable.
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Distribution of eigenvalues in the complex energy
plane
N 20, x 4
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4. Pastur Equation
Do this by calculating
Imaginary part of average Greens function yields
level density. Expand in powers of H.
where
Average each term in sum separately. For N large,
keep only nested contributions. Yields Pastur
equation
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Define for i 1,2 the spectral density of
subspace i in total spectrum (projection
operators Qi onto the two subspaces)
and take trace of Pastur equation. Yields two
coupled equations for ?1 and ?2, ?1
? / (E r - ?(?1 - ? ?2)), ?2 ?
/ (E r - ?(?2 - ? ?1)).

• For ? 0 these are two separate equations.
  Imaginary parts
• of solutions yield two semicircles of radius 2 ?
  centered at r
• and at r. Two dimensionless variables
• ? ?2 / ?2 ,
• x r / (2 ?) .
• Study spectrum as function of ? for fixed x.

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5. Solutions. Numerical Results

• To gain understanding of solutions, consider
  first ? 0.
• With ?i (E (-)i r) / (2 ?), spectrum given by
  
• Im(?i) 1 - ?2i1/2 .
• Usual semicircle law. Two branch points at ?
  1. Each
• ?i defined on Riemann surface with two sheets.
• For ? ? 0, solution defined on Riemann surface
  with four sheets.
• But which sheet to choose for physically relevant
  solution? Take
• very small, use perturbation theory, find that
  we need pair
• (?1, ?2) of solutions for which imaginary parts
  have opposite
• signs and Im(?1) gt 0. Then total level density
  ?(E) given by
• ?(E) (N / (? ?)) Im
  (?1 ?2) .

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Eliminate ?2 and obtain fourth-order equation for
?1. Then ?2 obtained from solution ?1 via linear
equation. Pairs of solutions ? Four pairs of
complex solutions Domain (I) ? Two pairs of
real and two pairs of complex solutions with
equal signs for Im(?1) and Im(?2) Domain (II) ?
Two pairs of real and two pairs of complex
solutions with opposite signs for Im(?1) and
Im(?2) Domain (III) ? Four pairs of real
solutions Domain (IV) Physically interesting
solutions only in domains (I) and (III).
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Case (a) x 1.2 Case (b) x 2.0 Case
(c) x 4.0
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Comparison of results for Pastur equation
with those of matrix diagonalization.
N 50, 100 realizations
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Evolution of two spectra with increasing ?
. Upper panels x 1.2 Lower panels x 4
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Normalization integral of total level density
taken over real energies versus ?. For ? 0,
integral is normalized to unity.
X 1.2
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6. Critical Strength
Find smallest value of ? for which average
spectra touch Imaginary parts of a pair of
physically acceptable solutions (?1, ?2) have
non-vanishing values at E 0. Determine ?crit
analytically from solutions of fourth-order equati
on for ?1 at E 0. Find ?crit x2
1.
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7. Summary
Random-Matrix model for RPA equations with
Generalized unitary invariance. Level
repulsion between levels of same sign, level
attraction between levels with opposite sign.
Latter causes coalescence of pairs of eigenvalues
with opposite signs at E 0 and instability of
RPA equations. Use Pastur equation to derive
two coupled equations for ?1, ?2. Surprisingly
simple structure. Criterion for physically
relevant solutions. Get average level density
from (?1, ?2). Two parameters x and ?. With
increasing ?, semicircles are deformed and move
toward each other. RPA instability for average
spectrum The deformed spectra touch. Critical
strength ?crit x2 1.
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