Numerical results for ground states of spin glasses on Bethe lattices

The average ground state energy and entropy for ±J spin glasses on Bethe lattices of connectivities k + 1 = 3..., 26 at T = 0 are approximated numerically. To obtain sufficient accuracy for large system sizes (up to n = 2¹²), the Extremal Optimization heuristic is employed which provides high-quality results not only for the ground state energies per spin e_k+1 but also for their entropies s_k+1. The results indicate sizable differences between lattices of even and odd connectivities. The extrapolated ground state energies compare very well with recent one-step replica symmetry breaking calculations. These energies can be scaled for all even connectivities k + 1 to within a fraction of a percent onto a simple functional form, e _{k + 1} = E _SK - (2E _SK + )/, where E _SK = - 0.7633 is the ground state energy for the broken replica symmetry in the Sherrington-Kirkpatrick model. But this form is in conflict with perturbative calculations at large k + 1, which do not distinguish between even and odd connectivities. We also find non-zero entropies per spin s_k+1 at small connectivities. While s_k+1 seems to vanish asymptotically with 1/(k + 1) for even connectivities, it is numerically indistinguishable from zero already for odd k + 1 ≥ 9. Received 9 August 2002 Published online 27 January 2003 RID="a" ID="a"e-mail: sboettc@emory.edu www.physics.emory.edu/faculty/boettcher     

Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

 By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick model, and the Derrida p-spin model. Here we extend this argument in order to compare the limiting free energy with the expression given by the Parisi Ansatz, and including full spontaneous replica symmetry breaking. Our main result is that the quenched average of the free energy is bounded from below by the value given in the Parisi Ansatz, uniformly in the size of the system. Moreover, the difference between the two expressions is given in the form of a sum rule, extending our previous work on the comparison between the true free energy and its replica symmetric Sherrington-Kirkpatrick approximation. We give also a variational bound for the infinite volume limit of the ground state energy per site. Received: 6 May 2002 / Accepted: 6 September 2002 Published online: 13 January 2003     

Finite-Size Scaling in the Energy-Entropy Plane for the 2D ± Ising Spin Glass

For L × L square lattices with L ≤ 20 the 2D Ising spin glass with +1 and −1 bonds is found to have a strong correlation between the energy and the entropy of its ground states. A fit to the data gives the result that each additional broken bond in the ground state of a particular sample of random bonds increases the ground state degeneracy by approximately a factor of 10/3. For x=0.5 (where x is the fraction of negative bonds), over this range of L, the characteristic entropy defined by the energy-entropy correlation scales with size as L ^1.78(2). Anomalous scaling is not found for the characteristic energy, which essentially scales as L ². When x=0.25, a crossover to L ² scaling of the entropy is seen near L=12. The results found here suggest a natural mechanism for the unusual behavior of the low temperature specific heat of this model, and illustrate the dangers of extrapolating from small L. PACS numbers: 75.10.Nr, 75.40.Mg, 75.50.Lk     

Metastable states of spin glasses on random thin graphs

In this paper we calculate the mean number of metastable states for spin glasses on so called random thin graphs with couplings taken from a symmetric binary distribution . Thin graphs are graphs where the local connectivity of each site is fixed to some value c. As in totally connected mean field models we find that the number of metastable states increases exponentially with the system size. Furthermore we find that the average number of metastable states decreases as c in agreement with previous studies showing that finite connectivity corrections of order 1/c increase the number of metastable states with respect to the totally connected mean field limit. We also prove that the average number of metastable states in the limit is finite and converges to the average number of metastable states in the Sherrington-Kirkpatrick model. An annealed calculation for the number of metastable states of energy E is also carried out giving a lower bound on the ground state energy of these spin glasses. For small c one may obtain analytic expressions for . Received 14 October 1999 and Received in final form 14 December 1999     

Classification and completeness of Bethe states and excitation spectrum for the spin 4 Fateev-Zamolodchikov model

The classification and completeness of Bethe states is undertaken for the spin 4 Fateev-Zamolodchikov model. This special case is of particular interest as the energy can be derived in a simple closed form even for lattices of arbitrary finite size. The excitation spectrum over the ferromagnetic ground state is also derived.     

Accurate Mass Determination of Neutron-Deficient Nuclides Close to Z=82 with ISOLTRAP

The recent implementation of gas-filled radiofrequency traps for efficient ion beam bunching extended the applicability of the Penning trap mass spectrometer ISOLTRAP/CERN to non-surface ionizable species. In a first series of successful runs the masses of ^182–197Hg, ^196,198Pb, ¹⁹⁷Bi, ¹⁹⁸Po and ²⁰³At have been determined with an accuracy of 1⋅10⁻⁷. In order to unambiguously determine the ground state mass the ground and isomeric states of ^{185,187,191,193,197}Hg were separated applying a resolving power of up to 3.7⋅10⁶. First experimental values for the isomeric excitation energy of ^187,191Hg were obtained. This revised version was published online in July 2006 with corrections to the Cover Date.     

μSR on an induced‐moment spin glass system: Hg1-xFexSe

Zero and longitudinal field μSR have been used to examine the singlet ground state magnetism of the diluted magnetic semiconductor Hg_1-xFe_xSe. For x\geq 0.05\ μSR experiments indicate spin glass behaviour which is limited to sample regions with locally enhanced Fe²⁺ content. Longitudinal field spectra in the glassy state reveal a static local field \sim 50 mT due to induced moments of Fe²⁺. These are proposed to originate from a distribution of bound magnetic polaron energies. This revised version was published online in July 2006 with corrections to the Cover Date.     

Transition energies of atomic lawrencium

Transition energies of the superheavy element lawrencium, including the ionization potential, excitation energies and electron affinities, are calculated by the intermediate Hamiltonian coupled cluster method. A large basis set (37s31p26d21f16g11h6i) is used, as well as an extensive P space (6s5p4d2f1g). The outer 43 electrons are correlated. Accuracy is monitored by applying the same approach to lutetium, the lighter homologue of Lr, and comparing with experimentally known energies. QED corrections are included. The main goal is to predict excitation energies, in anticipation of planned spectroscopy of Lr. The ground state of Lr is , unlike the of Lu. Predicted Lr excitations with large transition moments in the prime range for the planned experiment, 20 000–30 000 cm^-1, are 7p→8s at 20 100 cm^-1 and 7p→7d at 28 100 cm^-1. The average absolute error of 20 excitation energies of Lu is 423 cm ^-1, and the error limits for Lr are put at 700 cm^-1. The two electron affinities measured recently for Lu are reproduced within 55 cm^-1, and a third bound state of Lu^- is predicted.     

二氢化钚分子激发态结构的外场效应

采用密度泛函(DFT)方法B3LYP/Gen，在Pu为SDD基组、H为6-311++G^**基组水平上优化得到了分子轴方向不同电偶极场(-0.005—0.005a.u.)作用下，二氢化钚的基态电子状态、几何结构、电偶极矩和分子总能量.在优化构型下用同样的基组采用含时密度泛函(TDDFT)方法(TD-B3LYP)研究了同样外电场条件下对二氢化钚的激发能和振子强度的影响.计算结果表明，分子几何构型与电场大小和方向呈现较强的依赖，电场强度增加基态偶极矩随电场强度线性增加，H-Pu-H的角度线性减小，分子总能量线性减小；激发能随电场强度增加而减小，且对电场方向的依赖呈现近似对称性，满足Grozema关系.电场对振子强度的影响比较复杂，但仍满足跃迁选择定则. 关键词： 二氢化钚 激发态 电偶极场 TD-DFT     

Free-energy fluctuations and chaos in the Sherrington-Kirkpatrick model

The sample-to-sample fluctuations Delta FN of the free-energy in the Sherrington-Kirkpatrick model are shown rigorously to be related to bond chaos. Via this connection, the fluctuations become analytically accessible by replica methods. The replica calculation for bond chaos shows that the exponent mu governing the growth of the fluctuations with system size N, Delta FN approximately Nmu, is bounded by mu< or =1/4.     

Gaussian Mean Field Lattice Gas

We study rigorously a lattice gas version of the Sherrington–Kirckpatrick spin glass model. In discrete optimization literature this problem is known as unconstrained binary quadratic programming and it belongs to the class NP-hard. We prove that the fluctuations of the ground state energy tend to vanish in the thermodynamic limit, and we give a lower bound of such ground state energy. Then we present a heuristic algorithm, based on a probabilistic cellular automaton, which seems to be able to find configurations with energy very close to the minimum, even for quite large instances.     

The classical limit of quantum spin systems

We derive a classical integral representation for the partition function,Z ^Q , of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentumJ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality isZ ^C (J)Z ^Q (J)Z ^C (J+1).On leave from the Department of Mathematics, M.I.T., Cambridge, Mass. 02139, USA. Work partially supported by National Science Foundation Grant GP-31674X and by a Guggenheim Memorial Foundation Fellowship.     

Numerical study of the directed polymer in a 1 + 3 dimensional random medium

The directed polymer in a 1+3 dimensional random medium is known to present a disorder-induced phase transition. For a polymer of length L, the high temperature phase is characterized by a diffusive behavior for the end-point displacement R² ∼L and by free-energy fluctuations of order ΔF(L) ∼O(1). The low-temperature phase is characterized by an anomalous wandering exponent R²/L ∼L^ω and by free-energy fluctuations of order ΔF(L) ∼L^ω where ω∼0.18. In this paper, we first study the scaling behavior of various properties to localize the critical temperature T_c. Our results concerning R²/L and ΔF(L) point towards 0.76 < T_c ≤T₂=0.79, so our conclusion is that T_c is equal or very close to the upper bound T₂ derived by Derrida and coworkers (T₂ corresponds to the temperature above which the ratio remains finite as L ↦ ∞). We then present histograms for the free-energy, energy and entropy over disorder samples. For T ≫T_c, the free-energy distribution is found to be Gaussian. For T ≪T_c, the free-energy distribution coincides with the ground state energy distribution, in agreement with the zero-temperature fixed point picture. Moreover the entropy fluctuations are of order ΔS ∼L^1/2 and follow a Gaussian distribution, in agreement with the droplet predictions, where the free-energy term ΔF ∼L^ω is a near cancellation of energy and entropy contributions of order L^1/2.     

Solutions of a fourth order Bethe Salpeter equation

A Bethe Salpeter equation with interaction up to the fourth order is solved numerically for a bound state with energy equal to zero. It is found that the fourth order term is important even for very heavy exchanged mesons, and for angular momentum equal to 1. The correction found is to be considered the highest limit for the corrections with energy different from zero.     

Relativistic and QED corrections for the beryllium atom

Complete relativistic and quantum electrodynamics corrections of order alpha(2) Ry and alpha(3) Ry are calculated for the ground state of the beryllium atom and its positive ion. A basis set of correlated Gaussian functions is used, with exponents optimized against nonrelativistic binding energies. The results for Bethe logarithms ln(k(0)(Be)=5.750 34(3) and ln(k(0)(Be+)=5.751 67(3) demonstrate the availability of high precision theoretical predictions for energy levels of the beryllium atom and light ions. Our recommended value of the ionization potential 75 192.514(80) cm(-1) agrees with equally accurate available experimental values.     

The nature of binding in the ground state of the scandium dimer

For the study of the nature of binding in the Sc₂ dimer, the ground state, X⁵Σ _u ^–, was calculated by the valence multireference configuration interaction method with single and double excitations plus Davidson correction, MRCISD (+Q), at the complete basis set (CBS) limit. The employment of the C_2v symmetry group, allowed us to obtain the Sc atoms in different states at the dissociation limit. From the Mulliken population analysis and comparison with atomic energies follows that in the ground state Sc₂ dissociates on one Sc in the ground state and the other in the second excited quartet state, ⁴F _u . The spectroscopic parameters of the ground potential curve, obtained at the valence MRCISD (+Q)/CBS level, are: R _e  = 5.20 bohr, D _e  = 50.37 kcal/mol, and ω _e  = 234.5 cm^-1. The obtained value for the harmonic frequency agrees very well with the experimental one, ω _e  = 239.9 cm^-1. The dissociation energy with reference to the dissociation on two Sc in the ground states was estimated as D _e  = 9.98 kcal/mol. In contrast with many other studied transition-metal dimers, which are attributed to the van der Waals bonded molecules, the Sc₂ dimer is stabilized by the covalent bonding on the hybrid atomic orbitals.     

Orbital ordering and frustration of p-band Mott insulators

We investigate the general structure of orbital exchange physics in Mott-insulating states of p-orbital systems in optical lattices. Orbital orders occur in both the triangular and kagome lattices. In contrast, orbital exchange in the honeycomb lattice is frustrated as described by a novel quantum 120 degrees model. Its classical ground states are mapped into configurations of the fully packed loop model with an extra U(1) rotation degree of freedom. Quantum orbital fluctuations select a six-site plaquette ground state ordering pattern in the semiclassical limit from the "order from disorder" mechanism. This effect arises from the appearance of a zero energy flat band of orbital excitations.     

Adiabatic and non-adiabatic corrections to properties of the hydrogen molecular cation and its isotopomers: dissociation energies and bond lengths

A study is presented of adiabatic and non-adiabatic corrections to the dissociation energies and bond lengths of H⁺ ₂, D⁺ ₂ and HD⁺ in vibration-rotation levels of their ground electronic states, with particular attention to isotopic scaling. In previous work (MOSS, R. E., 1999, Molec. Phys., 97, 1) on rotationless levels, an anomalous non-adiabatic correction to the bond length was found for v = 20, N = 0 of HD⁺. Other levels close to dissociation are identified that display anomalous non-adiabatic corrections to the dissociation energies and to the bond lengths. The source of these anomalies is discussed.     

Extremal optimization for Sherrington-Kirkpatrick spin glasses

Extremal Optimization (EO), a new local search heuristic, is used to approximate ground states of the mean-field spin glass model introduced by Sherrington and Kirkpatrick. The implementation extends the applicability of EO to systems with highly connected variables. Approximate ground states of sufficient accuracy and with statistical significance are obtained for systems with more than N=1000 variables using ±J bonds. The data reproduces the well-known Parisi solution for the average ground state energy of the model to about 0.01%, providing a high degree of confidence in the heuristic. The results support to less than 1% accuracy rational values of ω=2/3 for the finite-size correction exponent, and of ρ=3/4 for the fluctuation exponent of the ground state energies, neither one of which has been obtained analytically yet. The probability density function for ground state energies is highly skewed and identical within numerical error to the one found for Gaussian bonds. But comparison with infinite-range models of finite connectivity shows that the skewness is connectivity-dependent.     

Excitation spectra of one-dimensional spin-1/2 Fermi gas with an attraction

Using an exact Bethe ansatz solution, we rigorously study excitation spectra of the spin-1/2 Fermi gas (called Yang–Gaudin model) with an attractive interaction. Elementary excitations of this model involve particle-hole excitation, hole excitation and adding particles in the Fermi seas of pairs and unpaired fermions. The gapped magnon excitations in the spin sector show a ferromagnetic coupling to the Fermi sea of the single fermions. By numerically and analytically solving the Bethe ansatz equations and the thermodynamic Bethe ansatz equations of this model, we obtain excitation energies for various polarizations in the phase of the Fulde–Ferrell–Larkin–Ovchinnikov-like state. For a small momentum (long-wavelength limit) and in the strong interaction regime, we analytically obtained their linear dispersions with curvature corrections, effective masses as well as velocities in particle-hole excitations of pairs and unpaired fermions. Such a type of particle-hole excitations display a novel separation of collective motions of bosonic modes within paired and unpaired fermions. Finally, we also discuss magnon excitations in the spin sector and the application of Bragg spectroscopy for testing such separated charge excitation modes of pairs and single fermions.