Large-N-limit of mixed valence: Diagrammatic expansion and rigorous results

For an intermediate valence model, containing one simply and oneN-fold degeneratef-configuration at a specific lattice site, the large-N-limit is studied within a concise diagrammatic expansion and also rigorously. At zero temperatureT rigorous upper and lower bounds for the ground state energy are found, while at finiteT the large-N-expansion is shown to be equivalent to the principle of classifying the diagrams according the maximum number of spin flips, whereupon various physical quantities can be calculated.On leave of absence from Department of Physics, Beijing University, Beijing, People's Republic of China with support from DAAD     

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     

Topology Trivialization and Large Deviations for the Minimum in the Simplest Random Optimization

Finding the global minimum of a cost function given by the sum of a quadratic and a linear form in N real variables over (N?1)-dimensional sphere is one of the simplest, yet paradigmatic problems in Optimization Theory known as the “trust region subproblem” or “constraint least square problem”. When both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field. We first identify and study two distinct large-N scaling regimes in which the linear term (magnetic field) leads to a gradual topology trivialization, i.e. reduction in the total number $\mathcal{N}_{tot}$ of critical (stationary) points in the cost function landscape. In the first regime $\mathcal{N}_{tot}$ remains of the order N and the cost function (energy) has generically two almost degenerate minima with the Tracy-Widom (TW) statistics. In the second regime the number of critical points is of the order of unity with a finite probability for a single minimum. In that case the mean total number of extrema (minima and maxima) of the cost function is given by the Laplace transform of the TW density, and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the TW law. Though the full form of that distribution is not yet known to us, one of its far tails can be inferred from the large deviation theory for the global minimum. In the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large-deviation approximation by finding both the rate function and the leading pre-exponential factor.     

One-dimensional Ising chain with competing interactions: Exact results and connection with other statistical models

We study the ground state properties of a one-dimensional Ising chain with a nearest-neighbor ferromagnetic interactionJ ₁, and akth neighboranti-ferromagnetic interactionJ _k . WhenJ _k/J₁=–1/k, there exists a highly degenerate ground state with a residual entropy per spin. For the finite chain with free boundary conditions, we calculate the degeneracy of this state exactly, and find that it is proportional to the (N+k–1)th term in a generalized Fibonacci sequence defined by,F _N ^(k) =F _N–1 ^(k) +F _N–k ^(k) . In addition, we show that this one-dimensional model is closely related to the following problems: (a) a fully frustrated two-dimensional Ising system with a periodic arrangement of nearest-neighbor ferro- and antiferromagnetic bonds, (b) close-packing of dimers on a ladder, a 2× strip of the square lattice, and (c) directed self-avoiding walks on finite lattice strips.Work partially supported by grants from AFOSR and ARO.     

Two point functions in theO(N+1) supersymmetric σ model beyond leading order

We calculate two point functions involving the Lagrange multiplier fields of theO(N+1) supersymmetric σ model beyond leading order in the largeN expansion, at zero momentum. We show they are not divergent and determine their finite part exactly ind dimensions. There is mixing between the bosonic Lagrange multiplier fields, similar to the leading order case. Further, the corrections to the bound state propagators are found to be functions ofN?1 and notN+1. This is consistent with the next to leading order modifications to the β-function.     

Atomic N00N state generation in distant cavities by virtual excitations

A general scheme of generating N00N states of virtually-excited 2N atoms is proposed. The two cavities are fibre-connected with N atoms in each cavity. Although we focus on the case of N=2, the system can be extended to a few atoms with N>2. It is found that all 2N atoms can be entangled in the form of N00N states if the atoms in the first cavity are initially in the excited states and atoms in the second cavity are all in the ground states. The feasibility of the scheme is carefully discussed, it shows that the N00N state with a few atoms can be generated with good fidelity and the scheme is feasible in experiment.     

Spin Operator Matrix Elements in the Superintegrable Chiral Potts Quantum Chain

We derive spin operator matrix elements between general eigenstates of the superintegrable ℤ _N -symmetric chiral Potts quantum chain of finite length. Our starting point is the extended Onsager algebra recently proposed by Baxter. For each pair of spaces (Onsager sectors) of the irreducible representations of the Onsager algebra, we calculate the spin matrix elements between the eigenstates of the Hamiltonian of the quantum chain in factorized form, up to an overall scalar factor. This factor is known for the ground state Onsager sectors. For the matrix elements between the ground states of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters. For the Ising quantum chain in a transverse field (N=2 case) the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the Separation of Variables method.     

Strongly coupled massive QCD3 and frustrated Heisenberg antiferromagnets: fractional statistics and chiral-spin order

Strongly coupled massive SU(N_C) and U(N_C) QCD₃ on a lattice is studied using the 1/N_C expansion. The quark mass terms have a definite sign in the present model, and therefore the system explicitly breaks the parity symmetry. The continuum counterpart generates the Maxwell + Chern-Simons theory by integrating out the quark field. In the present paper, we shall integrate out the gauge fields using the strong-coupling expansion and obtain a frustrated quantum Heisenberg model as an effective model. The ground state of the above effective quantum spin model is studied using the large-N_C approximation. There are two phases; one is a Neel-ordered state and the other is a state with a chiral-spin order. It is explicitly shown that the chiral-spin ordered state corresponds to a state with spontaneous generation of color magnetic flux in the original theory and fractional statistics appears in that phase. This result strongly suggests that there are (at least) two phases in the massive QCD₃ and Maxwell-CS theory. One is the confinement phase and the other is the perturbative deconfinement phase with fractional-statistics excitations.     

Free Energy and Complexity of Spherical Bipartite Models

We investigate both free energy and complexity of the spherical bipartite spin glass model. We first prove a variational formula in high temperature for the limiting free energy based on the well-known Crisanti–Sommers representation of the mixed \(p\) -spin spherical model. Next, we show that the mean number of local minima at low levels of energy is exponentially large in the size of the system and we derive a bound on the location of the ground state energy.     

First excitations of the spin 1/2 Heisenberg antiferromagnet on the kagomé lattice

We study the exact low energy spectra of the spin 1/2 Heisenberg antiferromagnet on small samples of the kagomé lattice of up to N=36 sites. In agreement with the conclusions of previous authors, we find that these low energy spectra contradict the hypothesis of Néel type long range order. Certainly, the ground state of this system is a spin liquid, but its properties are rather unusual. The magnetic () excitations are separated from the ground state by a gap. However, this gap is filled with nonmagnetic () excitations. In the thermodynamic limit the spectrum of these nonmagnetic excitations will presumably develop into a gapless continuum adjacent to the ground state. Surprisingly, the eigenstates of samples with an odd number of sites, i.e. samples with an unsaturated spin, exhibit symmetries which could support long range chiral order. We do not know if these states will be true thermodynamic states or only metastable ones. In any case, the low energy properties of the spin 1/2 Heisenberg antiferromagnet on the kagomé lattice clearly distinguish this system from either a short range RVB spin liquid or a standard chiral spin liquid. Presumably they are facets of a generically new state of frustrated two-dimensional quantum antiferromagnets. Received: 27 November 1997 / Accepted: 29 January 1998     

Renormalized trajectory for non-linear sigma model and improved scaling behavior

We apply the block-spin renormalization group method to the O(N) Heisenberg spin model. Extending a previous work of Hirsch and Shenker, we find the renormalized trajectory for O(∞) in two dimensions. For finite N models, we choose a four-parameter action near the large-N renormalized trajectory and demonstrate a remarkable improvement in the approach to continuum limit by performing Monte Carlo simulation of O(3) and O(4) models.     

Investigation of excitation energies and Hund's rule in open shell quantum dots by diffusion Monte Carlo

We use diffusion Monte Carlo to study the ground state, the low-lying excitation spectrum and the spin densities of circular quantum dots with parabolic radial potentials containing N = 16 and N = 24 electrons, each having four open-shell electrons and compare the results to those obtained from Hartree-Fock (HF) and density functional local spin density approximation (LSDA) calculations. We find that Hund's first rule is obeyed in both cases and that neither HF nor LSDA correctly predict the ordering of the energy levels. Received 20 November 2001 and Received in final form 20 February 2002 Published online 6 June 2002     

Large-N universality of variant actions

Corrections to large-N universality in mixed action lattice gauge theories imply constraints on the cut-off dependence. We state the form of these constraints and show that they are satisfied up to and including two loops (weak coupling). This implies that the second coefficient of the β function on the lattice is indeed universal as expected. We then evaluate the complete form of the corrections to large-N universality up to and including two loops. They give a sizeable contribution and determine qualitative aspects of equal string tension data like the bending seen for SU(2).     

Out-of-equilibrium quantum field dynamics in external fields

The quantum dynamics of the symmetry-broken λ(Φ ²)² scalar-field theory in the presence of an homogeneous external field is investigated in the large-N limit. We consider an initial thermal state of temperature T for a constant external field J. A subsequent sign flip of the external field, J→ - J, gives rise to an out-of-equilibrium nonperturbative quantum field dynamics. We review here the dynamics for the symmetry-broken λ(Φ ²)² scalar N component field theory in the large-N limit, with particular stress in the comparison between the results when the initial temperature is zero and when it is finite. The presence of a finite temperature modifies the dynamical effective potential for the expectation value, and also makes that the transition between the two regimes of the early dynamics occurs for lower values of the external field. The two regimes are characterized by the presence or absence of a temporal trapping close to the metastable equilibrium position of the potential. In the cases when the trapping occurs it is shorter for larger initial temperatures.     

Exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions

The exact solutions of a one-dimensional mixture of spinor bosons and spinor fermions with δ-function interactions are studied. Some new sets of Bethe ansatz equations are obtained by using the graded nest quantum inverse scattering method. Many interesting features appear in the system. For example, the wave function has the SU(2|2) supersymmetry. It is also found that the ground state of the system is partial polarized, where the fermions form a spin singlet state and the bosons are totally polarized. From the solution of Bethe ansatz equations, it is shown that all the momentum, spin and isospin rapidities at the ground state are real if the interactions between the particles are repulsive; while the fermions form two-particle bounded states and the bosons form one large bound state, which means the bosons condensed at the zero momentum point, if the interactions are attractive. The charge, spin and isospin excitations are discussed in detail. The thermodynamic Bethe ansatz equations are also derived and their solutions at some special cases are obtained analytically.     

Benzoic acid and phenol derivatives of nitronyl nitroxide biradical as building blocks of organic molecule-based ferrimagnets

A crystal-engineering approach to organic ferrimagnets is reported. Coulombic energy between an anionic biradical withS = 1 and a cationic monoradical withS = 1/2 can be utilized as a driving force of cocrystallization of open-shell molecules with different spin quantum numbers, leading to organic salt ferrimagnets. In this study, 3,5-substituted phenol and benzoic acid derivatives of nitronyl nitroxide biradicals were synthesized as an ionizableS = 1 component of organic salt ferrimagnets. The molecular ground states of the biradicals in the neutral state were examined by continuous wave electron spin resonance (ESR) spectroscopy and static paramagnetic susceptibility measurements in the solid state. The molecular ground state of the phenol derivative was found to be triplet (S = 1) with the singlet-triplet energy gap of ΔE/k_B ≈ 25 K, indicating that the biradical can be a building block of organic salt ferrimagnetics. The benzoic acid derivative was found to have a singlet (S = 0) ground state (ΔE/k_B ≧ −5 K), exemplifying thatmeta-(3,5)-linkage of unpaired electrons in π-aromatic rings does not necessarily give a triplet ground state for heteroatomic-substituted π conjugation. The molecular ground states of the biradicals determined in the ESR experiments were confirmed by the susceptibility in the solid state.     

Superconducting State in the Lagrangian Formalism of the Generalized Hubbard Model

The nonperturbative large-N expansion applied to the generalized Hubbard model describing N-fold-degenerate correlated bands is considered. Our previous results, obtained in the framework of the Lagrangian formalism for the normal-state case, are extended to the superconducting state. The standard Feynman diagrammatics is obtained and the renormalized physical quantities are computed and analyzed. Our purpose is to obtain the 1/N corrections to the renormalized boson and fermion propagators when a state with Cooper-pair condensation (i.e., the superconducting state) is considered.     

Dynamic generation of entangling wave packets in XY spin system with decaying long-range couplings

The dynamic generation of spin entanglement between two distant sites in a XY model with 1/r ² decay long-range couplings was studied. Due to the linear dispersion relation ε(k)∼|k| of magnons in such a model, a well-located spin state can be dynamically split into two moving entangled local wave packets without changing their shapes. Interestingly, when such two wave packets meet at the diametrically opposite site after the fast period τ = Nπ/J, the initial well-located state is completely recurrent. Numerical calculation was performed to confirm the analytical result even if the ring system of sizes N up to several thousands is considered. The truncation approximation for the coupling strengths was also studied. Numerical simulation shows that the above conclusions still hold even if the range of the coupling strength is truncated to a relatively short scale compared with the size of the spin system. Supported by the National Natural Science Foundation of China (Grant Nos. 90203018, 10474104 and 60433050) and the National Fundamental Research Program of China (Grant Nos. 2001CB309310 and 2005CB724508)     

Effective actions and large-N limits

The saddlepoint action for a large-N theory is given as an effective action for composite operators. This effective action is computed explicitly forO(N) models and as a series in large-N invariants for matrix models. In the latter case, the use of the first term of the series is found to give good numerical agreement with the exact solutions of the solvable models.     

轴对称晶场中d3离子激发态对4A2基态自旋哈密顿参量的影响

采用完全对角化方法，讨论了三角对称和四角对称下d³离子自旋二重态和自旋四重态对基态⁴A₂(⁴F)自旋哈密顿(SH)参量(包括零场分裂(ZFS)和g因子)的影响机理. 并对影响基态SH参量的四种机理(SO机理，SS机理，SOO机理和SO-SS-SOO联合机理)进行了分析. 结果表明，自旋二重态与四重态对d³离子基态零场分裂都具有重要贡献；而基态g因子主要由四重态决定，二重态对g因子贡献很小. 此外，发现SS机理和SOO机理对基态EPR参量的贡献主要由四重态决定，二重态的影响很小.