J.J.M. Verbaarschot

Grassmann integration in stochastic quantum physics: The case of compound-nucleus scattering

Using astochasticmodel for N compound—nucleusresonances coupledto thechannels,we calculatein the limit N—* theensembleaverageof theS-matrix (the “one-point function”),andof theproductof an S-matrixelementwith thecomplexconjugateof another,both takenat different energies(the “two-point function”). Using a generatingfunction involving both commutingand anticommutingintegrationvariables,we evaluate the ensembleaveragestrivially. The problem of carrying Out the remaining integrationsis solved with the help of theHubbard—Stratonovitch transformation.We put specialemphasison theconvergencepropertiesof this transformation,andon the underlyingsymmetriesof thestochastic model for thecompoundnucleus.Thesetwo featurestogethercompletelydefinetheparametrizationof thecompositevariablesin termsof agroup of transformations.Thisgroupis compactin the“Fermion—Fermionblock” andnon-compactin the“Boson—Bosonblock”. The limit N-. x is taken with thehelp of thesaddle-pointapproximation.After integrationover the“massive modes”,we showthat thetwo-point function can be expressed in terms of thetransmissioncoefficients. In this way we prove that thefluctuation propertiesof thenuclearS-matrix arethesameover theentire spectrumof the random Hamiltonian describingthe compoundnucleus.The integration over the saddle-pointmanifold is carried out using symmetry propertiesof the randomHamiltonian. We finally obtain a closed-formexpressionfor the two-point function in terms of a threefold integralover realvariables.This expressioncan be easilyevaluatednumerically.

Statistical theory of precompound reactions: The multistep compound process

Abstract Using a quantum-statistical framework, the method of the generating function involving both commuting and anticommuting variables, and the saddle-point approximation followed by the loop expansion, we derive a theoretical framework for multistep-compound reactions. Our statistical input distinguishes between several classes of states of increasing complexity; this distinction is possible only at the expense of relinquishing the orthogonal invariance of the distribution of Hamiltonian matrix elements usually required in compound-nucleus theories. Our result contains both the compound-nucleus scattering cross section and the theory of Agassi Weidenmuller, and Montzouranis ( Phys. Lett. C 22 (1975), 145.) developed earlier as special cases. It goes beyond this theory, and extends the framework of precompound theories in general, by allowing the couplings between classes, and to the channels, to be reasonably strong. A self-consistency condition embodied in the saddle-point equation implies in this case that the level densities used in precompound calculations must be modified. We investigate the modification in simple model cases. Our results suggest that the modification may be relevant for the high-energy tail of the spectrum of precompound particles.

Critique of the replica trick

It is shown that the replica trick fails to give the correct non-perturbative result for the two-point function S2 of the Gaussian unitary ensemble of N*N random matrices. The failure arises from an incorrect description of the symmetries of the random-matrix system in the limit N to infinity . The correct description, which involves integration over both non-compact and compact degrees of freedom, is obtained by using the method of superfields. Some implications for the localisation transition in disordered electronic systems and the theory of the quantised Hall effect are suggested.

Replica variables, loop expansion, and spectral rigidity of random-matrix ensembles

Abstract The replica trick of statistical mechanics is used to derive integral representations of n-point Greens functions both for the GOE and the EGOE. These integral representations are particularly suited for perturbative evaluation (loop expansion). Using the one-loop correction to the GOE one-point function, it is found that the density of states at the edge of the semicircle scales is ∼N −1 3 ϱ (N 2 3 δ ) where N is the dimension of the matrix ensemble. For the n-point functions with n ≥ 2, the existence of the microscopic limit to all orders in N−1 is proved by decomposing the integration variables into massive (i.e., macroscopic) and massless (microscopic) components. Evaluation of the EGOE two-point function to leading order in the inverse local distance variable yields the first analytic evidence that the long-range correlations of EGOE spectra are similar to the GOE but not-stationary.

Investigation of the formula for the average of two S-matrix elements in compound nucleus reactions

Recently the problem of calculating the average of two S-italic-matrix elements describing compound nucleus reactions was solved by using the technique of generating functions and grassmann integration. In this paper we analyze the solution, a three-dimensional integral, both analytically and numerically. We study the expansion of the integral in powers of the transmission coefficients and the expansion in inverse powers of the transmission coefficients. The former one, which also contains logarithmic terms, is shown to agree with previous work. The latter one is identical to the expansion obtained with the help of the replica trick. After a suitable change of integration variables the integrand becomes finite everywhere in its domain and numerical integration is without any problem. Earlier Monte Carlo calculations are confirmed. The average of two S-italic-matrix elements at the same energy obtained by a Mexican group using general properties of the S-italic-matrix and the maximum entropy principle is shown to coincide with the microscopic result.

Spectral fluctuation properties of Hamiltonian systems: the transition region between order and chaos

The authors study numerically the classical dynamical behaviour, and the spectral fluctuation properties, of a class of Hamiltonian systems with two degrees of freedom. The quantum mechanical properties of these systems are monotonic but non-universal functions of the fraction of classical phase space filled by chaotic trajectories. It is found that the observed spectral fluctuation measures can be reproduced by a random-matrix model which depends on one parameter only.

Destruction of order in nuclear spectra by a residual GOE interaction

Abstract We consider hamiltonians of the type H = H 0 + λV GOE where H 0 is a fixed N × N matrix and V GOE represents a gaussian orthogonal ensemble. The change as a function of λ of the average level density and of the eigenvector correlations is studied, and related to the distribution of branch points of H . It is shown that the GOE interaction completely dominates the spectral properties of H when its spectrum covers the spectrum of H 0 .

Quantum spectra of classically chaotic systems without time reversal invariance

Abstract A hamiltonian with two degrees of freedom displays a transition from ordered to strongly chaotic classical motion under either of two perturbations, one symmetric, the other antisymmetric under time reversal. We compute numerically the eigenvalues of the quantum hamiltonians and find that each perturbation alone leads to the level statistics of the gaussian orthogonal ensemble. Only if both perturbations individually produce strongly chaotic classical behaviour, do we find the level statistics of the gaussian unitary ensemble.

Evaluation of ensemble averages for simple Hamiltonians perturbed by a GOE interaction

Abstract Using an expansion in powers of N −1 , where N is the dimension of the Hamiltonian matrix, we evaluate ensemble averages of the resolvent, of products involving several resolvents, and of the moments of the Hamiltonian H 0 + λV . Here, H 0 is arbitrary but fixed, and V is a GOE ensemble. The nature of the N −1 expansion is also discussed.

A statistical study of shell-model eigenvectors

The components of shell-model eigenvectors show a non-gaussian distribution that can be derived by the use of the Porter-Thomas proposition.