It is argued that spectral features of quantal systems with random interactions can be given a geometric interpretation. This conjecture is investigated in the context of two simple models: a system of randomly interacting d bosons and one of randomly interacting fermions in a $j=\frac{7}{2}$ shell. In both examples the probability for a given state to become the ground state is shown to be related to a geometric property of a polygon or polyhedron which is entirely determined by particle number, shell size, and symmetry character of the states. The extensions to more general situations are discussed.It is argued that spectral features of quantal systems with random interactions can be given a geometric interpretation. This conjecture is investigated in the context of two simple models: a system of randomly interacting d bosons and one of randomly interacting fermions in a $j=\frac{7}{2}$ shell. In both examples the probability for a given state to become the ground state is shown to be related to a geometric property of a polygon or polyhedron which is entirely determined by particle number, shell size, and symmetry character of the states. The extensions to more general situations are discussed.

arXiv:nucl-th/0301061Physical Review CDec 23, 200226 citations

How random are random nuclei? Shapes, triangles and kites

R. Bijker, A. Frank

The energy systematics of medium and heavy even-even nuclei shows very regular features, such as for example the tripartite classification of nuclear structure into seniority, anharmonic vibrator and rotor regions [1,2]. Traditionally, this regular behavior has been interpreted as a consequence of particular interactions, such as an attractive pairing force in semimagic nuclei and an attractive neutron-proton quadrupole-quadrupole interaction for deformed nuclei. It came as a surprise, therefore, that recent studies of even-even nuclei in the nuclear shell model [3–5] and in the interacting boson model (IBM) [6–8] with random interactions also displayed a high degree of order. Both models showed a statistical preference for L = 0 ground states, despite the random nature of the interactions. In addition, in the shell model evidence was found for the occurrence of pairing properties [5], and in the IBM for vibrational and rotational bands [6,7]. These unexpected results have sparked a large number of investigations to try to understand their origin [9]. In this contribution, we discuss the phenomenom of emerging regular spectral features from the IBM with random interactions, and itsThe energy systematics of medium and heavy even-even nuclei shows very regular features, such as for example the tripartite classification of nuclear structure into seniority, anharmonic vibrator and rotor regions [1,2]. Traditionally, this regular behavior has been interpreted as a consequence of particular interactions, such as an attractive pairing force in semimagic nuclei and an attractive neutron-proton quadrupole-quadrupole interaction for deformed nuclei. It came as a surprise, therefore, that recent studies of even-even nuclei in the nuclear shell model [3–5] and in the interacting boson model (IBM) [6–8] with random interactions also displayed a high degree of order. Both models showed a statistical preference for L = 0 ground states, despite the random nature of the interactions. In addition, in the shell model evidence was found for the occurrence of pairing properties [5], and in the IBM for vibrational and rotational bands [6,7]. These unexpected results have sparked a large number of investigations to try to understand their origin [9]. In this contribution, we discuss the phenomenom of emerging regular spectral features from the IBM with random interactions, and its

Jan 1, 20020 citations

Nuclear structure from random interactions

R. Bijker, Alejandro Frank, Stuart Pittel

Low-lying states in nuclei are investigated using an ensemble of random interactions. Both in the nuclear shell model and in the interacting boson model we find a dominance of $J^P=0^+$ ground states. It is shown that this feature is not due to time reversal symmetry. In the shell model, evidence is found for the occurrence of pairing properties, and in the interacting boson model for both vibrational and rotational band structures. Our results suggest that these features represent general and robust properties of the model space, and do not depend on details of the interactions.Low-lying states in nuclei are investigated using an ensemble of random interactions. Both in the nuclear shell model and in the interacting boson model we find a dominance of $J^P=0^+$ ground states. It is shown that this feature is not due to time reversal symmetry. In the shell model, evidence is found for the occurrence of pairing properties, and in the interacting boson model for both vibrational and rotational band structures. Our results suggest that these features represent general and robust properties of the model space, and do not depend on details of the interactions.

arXiv:nucl-th/0004032Revista Mexicana de FísicaApr 12, 20002 citations

Ground State Properties with a Random Two-Body Interaction

Noritaka Shimizu, Takaharu Otsuka

We apply a two-body random ensemble to a hypothetical system of spinless fermions occupying a one-dimensional rotationally invariant orbit in order to study the statistical tendencies of spins of ground states for a random two-body interaction. We also investigate the origin of the dominance of spin-zero ground states of a single-j shell in the nuclear shell model, which can be interpreted geometrically as a particular case of the simple hypothetical model we consider. Assuming a specific symmetry, or the doublet structure due to time-reversal symmetry, it is found that the ensemble has a preference for a ground state whose spin is zero. We demonstrate how the time-reversal symmetry and its doublet structure affect this ground state tendency.We apply a two-body random ensemble to a hypothetical system of spinless fermions occupying a one-dimensional rotationally invariant orbit in order to study the statistical tendencies of spins of ground states for a random two-body interaction. We also investigate the origin of the dominance of spin-zero ground states of a single-j shell in the nuclear shell model, which can be interpreted geometrically as a particular case of the simple hypothetical model we consider. Assuming a specific symmetry, or the doublet structure due to time-reversal symmetry, it is found that the ensemble has a preference for a ground state whose spin is zero. We demonstrate how the time-reversal symmetry and its doublet structure affect this ground state tendency.

Progress of Theoretical PhysicsSep 1, 20072 citations

Random Interaction Matrix Ensembles in Mesoscopic Physics

Manan Vyas

We analyze several ground state related properties of mesoscopic systems using the random interaction matrix model EGOE(1+2)-$\cs$ (or RIMM) for many fermion systems with spin degree of freedom and the Hamiltonian containing pairing and exchange interactions in addition to the mean-field one-body and random two-body parts. RIMM reproduces the essential features of various properties: odd-even staggering in ground state energies as a function of particle number, delay in ground state magnetization and conductance peak spacing distributions. The analytical formula, we have derived, for the ensemble averaged spectral variances provides a simple understanding of some of these properties.We analyze several ground state related properties of mesoscopic systems using the random interaction matrix model EGOE(1+2)-$\cs$ (or RIMM) for many fermion systems with spin degree of freedom and the Hamiltonian containing pairing and exchange interactions in addition to the mean-field one-body and random two-body parts. RIMM reproduces the essential features of various properties: odd-even staggering in ground state energies as a function of particle number, delay in ground state magnetization and conductance peak spacing distributions. The analytical formula, we have derived, for the ensemble averaged spectral variances provides a simple understanding of some of these properties.

arXiv:1004.2761Apr 16, 20102 citations

Eigenvalue correlations and the distribution of ground state angular momenta for random many-body quantum systems

José Barea, R. Bijker, Alejandro Frank

The observed preponderance of ground states with angular momentum $L=0$ in many-body quantum systems with random two-body interactions is analyzed in terms of correlation coefficients (covariances) among different eigenstates. It is shown that the geometric analysis of Chau et al. can be interpreted in terms of correlations (covariances) between energy eigenvalues, thus providing an entirely statistical explanation of the distribution of ground state angular momenta of randomly interacting quantum systems that, in principle, is valid for both fermionic and bosonic systems. The method is illustrated for the interacting boson model.The observed preponderance of ground states with angular momentum $L=0$ in many-body quantum systems with random two-body interactions is analyzed in terms of correlation coefficients (covariances) among different eigenstates. It is shown that the geometric analysis of Chau et al. can be interpreted in terms of correlations (covariances) between energy eigenvalues, thus providing an entirely statistical explanation of the distribution of ground state angular momenta of randomly interacting quantum systems that, in principle, is valid for both fermionic and bosonic systems. The method is illustrated for the interacting boson model.

arXiv:0811.1240Physical Review CMay 4, 20093 citations

j=\frac{7}{2}

shell. In both examples the probability for a given state to become the ground state is shown to be related to a geometric property of a polygon or polyhedron which is entirely determined by particle number, shell size, and symmetry character of the states. The extensions to more general situations are discussed.

Low-lying states in nuclei are investigated using an ensemble of random interactions. Both in the nuclear shell model and in the interacting boson model we find a dominance of

J^P=0^+

ground states. It is shown that this feature is not due to time reversal symmetry. In the shell model, evidence is found for the occurrence of pairing properties, and in the interacting boson model for both vibrational and rotational band structures. Our results suggest that these features represent general and robust properties of the model space, and do not depend on details of the interactions.

The observed preponderance of ground states with angular momentum

L=0

in many-body quantum systems with random two-body interactions is analyzed in terms of correlation coefficients (covariances) among different eigenstates. It is shown that the geometric analysis of Chau et al. can be interpreted in terms of correlations (covariances) between energy eigenvalues, thus providing an entirely statistical explanation of the distribution of ground state angular momenta of randomly interacting quantum systems that, in principle, is valid for both fermionic and bosonic systems. The method is illustrated for the interacting boson model.