Periodic Striped Ground States in Ising Models with Competing Interactions

We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent p of the long range interaction is larger than d + 1, with d the space dimension, this happens for all values of J smaller than a critical value J_c(p), beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for p > 2d and J in a left neighborhood of J_c(p). In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes (d = 2) or slabs (d = 3), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity.     

On the Absence of Ferromagnetism in Typical 2D Ferromagnets

We consider the Ising systems in d dimensions with nearest-neighbor ferromagnetic interactions and long-range repulsive (antiferromagnetic) interactions that decay with power s of the distance. The physical context of such models is discussed; primarily this is d = 2 and s = 3 where, at long distances, genuine magnetic interactions between genuine magnetic dipoles are of this form. We prove that when the power of decay lies above d and does not exceed d + 1, then for all temperatures the spontaneous magnetization is zero. In contrast, we also show that for powers exceeding d + 1 (with d ≥ 2) magnetic order can occur.     

Boundary Conditions for Topological Quantum Field Theories,Anomalies and Projective Modular Functors

We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level m, and describe extended anomalous theories as natural transformations of invertible field theories of this type. We show how in such a framework anomalous theories give rise naturally to homotopy fixed points for n-characters on ∞-groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for n + 1-dimensional TQFTs produce n-dimensional anomalous field theories. Finally, we analyse the case of fully extended TQFTs, and show that any fully extended anomalous theory produces a suitable boundary condition for the anomaly field theory.     

Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems

Gapped ground states of quantum spin systems have been referred to in the physics literature as being ‘in the same phase’ if there exists a family of Hamiltonians H(s), with finite range interactions depending continuously on \({s\in [0,1]}\), such that for each s, H(s) has a non-vanishing gap above its ground state and with the two initial states being the ground states of H(0) and H(1), respectively. In this work, we give precise conditions under which any two gapped ground states of a given quantum spin system that ’belong to the same phase’ are automorphically equivalent and show that this equivalence can be implemented as a flow generated by an s-dependent interaction which decays faster than any power law (in fact, almost exponentially). The flow is constructed using Hastings’ ‘quasi-adiabatic evolution’ technique, of which we give a proof extended to infinite-dimensional Hilbert spaces. In addition, we derive a general result about the locality properties of the effect of perturbations of the dynamics for quantum systems with a quasi-local structure and prove that the flow, which we call the spectral flow, connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a co-cycle of automorphisms of the algebra of quasi-local observables of the infinite spin system. This proves that the ground state phase structure is preserved along the curve of models H(s), 0 ≤ s ≤ 1.     

On the Dimension of the Singular Set of Solutions to the Navier–Stokes Equations

In this paper we prove that if a suitable weak solution u of the Navier–Stokes equations is an element of \({L^w(0,T;L^s(\mathbb{R}^3))}\), where 1 ≤ 2/w + 3/s ≤ 3/2 and 3 < w, s < ∞, then the box-counting dimension of the set of space-time singularities is no greater than max{w, s}(2/w + 3/s ? 1). We also show that if \({\nabla u \in L^w(0,T;L^s(\Omega))}\) with 2 < s ≤ w < ∞, then the Hausdorff dimension of the singular set is bounded by w(2/w + 3/s ? 2). In this way we link continuously the bounds on the dimension of the singular set that follow from the partial regularity theory of Caffarelli, Kohn, &; Nirenberg (Commun. Pure Appl. Math. 35:771–831, 1982) to the regularity conditions of Serrin (Arch. Ration. Mech. Anal. 9:187–191, 1962) and Beirão da Veiga (Chin. Ann. Math. Ser. B 16(4):407–412, 1995).     

A multidimensional global monopole and nonsingular cosmology

We consider a spherically symmetric global monopole in general relativity in (D=d+2)-dimensional space-time. For γ<d?1, where γ is a parameter characterizing the gravitational field strength, the monopole is shown to be asymptotically flat up to a solid angle defect. In the range d?1< γ<2d(d+1)/(d+2), the monopole space-time contains a cosmological horizon. Outside the horizon, the metric corresponds to a cosmological model of the Kantowski-Sachs type, where spatial sections have the topology ? × S ^d . In the important case where the horizon is far from the monopole core, the temporal evolution of the Kantowski-Sachs metric is described analytically. The Kantowski-Sachs space-time contains a subspace with a (d+1)-dimensional Friedmann-Robertson-Walker metric, whose possible cosmological application is discussed. Some estimates in the d=3 case show that this class of nonsingular cosmologies can be viable. In particular, the symmetry-breaking potential at late times can give rise to both dark matter and dark energy. Other results, generalizing those known in 4-dimensional space-time, are derived, in particular, the existence of a large class of singular solutions with multiple zeros of the Higgs field magnitude.     

Analysis of the spectrum of the Zn-like Kr VII ion: Highly excited 4<Emphasis Type="Italic">p</Emphasis>4<Emphasis Type="Italic">d</Emphasis> and 4<Emphasis Type="Italic">p</Emphasis>5<Emphasis Type="Italic">s</Emphasis> configurations

The spectrum of the Zn-like Kr VII ion, excited in a capillary discharge and recorded with a high resolution in the wavelength range of 300–1000 Å, was studied. Previously performed identification of the transitions from the levels of the 4s4f, 4s5s, 4s5p, and 4s5d configurations is confirmed and extended, and the energies of these levels are specified. The (4p ²+4s4d)?4p4d and (4p ²+4s5s)?4p5s transitions are identified for the first time, and the energies of all the levels of the 4p4d and 4p5s configurations are determined. The results of the analysis performed are confirmed by semiempirical calculations in terms of the Hartree-Fock method. These results are also shown to conform to the experimental data obtained for lighter ions of the Zn I isoelectronic sequence.     

Dependence of structure factor and correlation energy on the width of electron wires

The structure factor and correlation energy of a quantum wire of thickness b ? a _B are studied in random phase approximation (RPA) and for the less investigated region r _s < 1. Using the single-loop approximation, analytical expressions of the structure factor are obtained. The exact expressions for the exchange energy are also derived for a cylindrical and harmonic wire. The correlation energy in RPA is found to be represented by ? _c (b, r _s ) = α(r _s )/b + β(r _s ) ln(b) + η(r _s ), for small b and high densities. For a pragmatic width of the wire, the correlation energy is in agreement with the quantum Monte Carlo simulation data.     

Vanishing Critical Magnetization in the Quantum Ising Model

Adapting the recent argument of Aizenman, Duminil-Copin and Sidoravicius for the classical Ising model, it is shown here that the magnetization in the transverse-field Ising model vanishes at the critical point. The proof applies to the ground state in dimension d ≥ 2 and to positive-temperature states in dimension d ≥ 3, and relies on graphical representations as well as an infrared bound.     

Inverse Scattering at Fixed Energy on Surfaces with Euclidean Ends

On a fixed Riemann surface (M ₀, g ₀) with N Euclidean ends and genus g, we show that, under a topological condition, the scattering matrix S _V (λ) at frequency λ > 0 for the operator Δ+V determines the potential V if \({V\in C^{1,\alpha}(M_0)\cap e^{-\gamma d(\cdot,z_0)^j}L^\infty(M_0)}\) for all γ > 0 and for some \({j\in\{1,2\}}\) , where d(z, z ₀) denotes the distance from z to a fixed point \({z_0\in M_0}\) . The topological condition is given by \({N\geq \max(2g+1,2)}\) for j = 1 and by N ≥ g + 1 if j = 2. In \({\mathbb {R}^2}\) this implies that the operator S _V (λ) determines any C ^{1, α} potential V such that \({V(z)=O(e^{-\gamma|z|^2})}\) for all γ > 0.     

A Liouville-type Theorem for Schrödinger Operators

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P ₁, such that a nonzero subsolution of a symmetric nonnegative operator P ₀ is a ground state. Particularly, if P _j : = ?Δ + V _j , for j = 0,1, are two nonnegative Schrödinger operators defined on \(\Omega\subseteq \mathbb{R}^d\) such that P ₁ is critical in Ω with a ground state φ, the function \(\psi\nleq 0\) is a subsolution of the equation P ₀ u = 0 in Ω and satisfies \(\psi_+\leq C\varphi\) in Ω, then P ₀ is critical in Ω and \(\psi\) is its ground state. In particular, \(\psi\) is (up to a multiplicative constant) the unique positive supersolution of the equation P ₀ u = 0 in Ω. Similar results hold for general symmetric operators, and also on Riemannian manifolds.     

Spontaneous Raman scattering upon resonant three-photon ionization of a samarium atom

The manifestation of spontaneous Raman scattering upon the three-photon ionization of a samarium atom has been investigated. The dependence of the Sm⁺ yield on the laser frequency, along with the peaks due to the two-photon excitation of bound states from different levels of the 4f ⁶6s ^{2 7} F ground term, exhibits a strong peak due to single-photon excitation of the 4f ⁵5d6s ^{2 7} D°₃ state from the 4f ⁶5d6s ⁹ H ₂ excited state. The 4f ⁶5d6s ⁹ H ₂ state was populated as a result of spontaneous Raman scattering. The large amplitude of the observed peak indicates a high efficiency of this process. The estimations performed indicate that the probability of resonant ionization through an excited state populated due to spontaneous Raman scattering can be comparable with the probability of three-photon ionization through the two-photon intermediate resonance. The necessary conditions for implementing this channel of resonant three-photon ionization are formulated.     

Hyperfeinstruktur-Untersuchungen im Rhenium-I-Spektrum

By optical means (i.e. hollow cathode, Fabry-Perot-interferometer and spectrograph) the hyperfine structure of 15 transitions in the Re-I spectrum was investigated. This permits to state the hfs coupling constantsA, B for 15 levels belonging to the configurations (5d ⁵) 6s ², (5d ⁵) 6s 6p, (5d ⁵) 6s 7s, (5d ⁶) 6s, (5d ⁶) 6p and (5d ⁴) 6s ² 6p. It was found that the (5d ⁵) 6s ^{2 6} S _5/2 ground state, too, shows a small hfs. Of its magnetic splitting both size and sign could be determined (A _0,185,187=?2,2±0,4 mK); for a possible quadrupole splitting upper limits were obtained. Estimates of the quadrupole moments resulted inQ _185,187?2,9b. — In the case of two levels (x ⁶ P _7/2 andy ⁶ F _11/2) the hfs analysis raises doubts as to their classification.     

Analyse von Hyperfeinstrukturen des Eu151, Am241 und einiger 3 d-Elemente mittels „Exchange Polarization“

It is shown how several discrepancies in the optical hfs of the Eu can be understood as consequences of the exchange polarization of the inner and outers-electrons by the spin of the half filled (4f ⁷)-subshell, an effect which should produce additional magnetic fields at the nucleus. Thus from the two different values of the electronic splitting constanta _6s in the two Eu-II ground states the polarization field from the 6s-shell (Δ H _6s ) is determined to be ca. +260 KG, and the formal splitting constantσ (??3 mK) of the (4f ⁷)-subshell yields ca. ?350 KG for the fieldΔ H _(1?5) from the five innern s-shells (n=1?5) in good agreement with the strength of the inner field obtained from recent Mössbauer effect studies.Δ H _(1?5) is deduced to be approximately equal in all sufficiently analysed ground and excited configurations of the neutral and ionised Eu atom ((4f ⁷) 6s, 6p, 5d, 6s ² and 6s 6p). Other elements with half filled subshells (Am, Mn) show similar features in their optical hfs. For Am⁺ ((5f ⁷) 7s) ca. ?2200 KG are found for the inner field (Δ H _(1?6)). For several 3d-elements it was found that the agreement between the calculated polarization fields and those following from experimental results is better than assumed so far.     

Fluctuations for the Ginzburg-Landau $${\nabla \phi}$$ Interface Model on a Bounded Domain

We study the massless field on \({D_n = D \cap \tfrac{1}{n} \mathbf{Z}^2}\), where \({D \subseteq \mathbf{R}^2}\) is a bounded domain with smooth boundary, with Hamiltonian \({\mathcal {H}(h) = \sum_{x \sim y} \mathcal {V}(h(x) - h(y))}\). The interaction \({\mathcal {V}}\) is assumed to be symmetric and uniformly convex. This is a general model for a (2 + 1)-dimensional effective interface where h represents the height. We take our boundary conditions to be a continuous perturbation of a macroscopic tilt: h(x) = n x · u + f(x) for \({x \in \partial D_n,\,u \in \mathbf{R}^2}\), and f : R ² → R continuous. We prove that the fluctuations of linear functionals of h(x) about the tilt converge in the limit to a Gaussian free field on D, the standard Gaussian with respect to the weighted Dirichlet inner product \({(f,g)_\nabla^\beta = \int_D \sum_i \beta_i \partial_i f_i \partial_i g_i}\) for some explicit β = β(u). In a subsequent article, we will employ the tools developed here to resolve a conjecture of Sheffield that the zero contour lines of h are asymptotically described by SLE(4), a conformally invariant random curve.     

Quenched and Annealed Critical Points in Polymer Pinning Models

We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential u + V _n which the chain encounters when it visits a special state 0 at time n. The disorder (V _n ) is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when u exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length n has the form \({n^{-c}\varphi(n)}\) with c ≥  1 and φ slowly varying. Comparing to the corresponding annealed system, in which the V _n are effectively replaced by a constant, it was shown in [1,4,13] that the quenched and annealed critical points differ at all temperatures for 3/2 < c < 2 and c > 2, but only at low temperatures for c < 3/2. For high temperatures and 3/2 < c < 2 we establish the exact order of the gap between critical points, as a function of temperature. For the borderline case c = 3/2 we show that the gap is positive provided \({\varphi(n) \to 0}\) as n → ∞, and for c > 3/2 with arbitrary temperature we provide an alternate proof of the result in [4] that the gap is positive, and extend it to c = 2.     

Superconducting pairing symmetry in frustrating t-J model: Electronic Raman absorption

Motivated by the controversy on the pairing symmetry of layered organic superconductors,we study electronic Raman scattering spectra on a frustrating lattice. A two-dimensionalt-t′-J-J′model and the Gutzwiller projectional variational method is used. The pairing symmetry isobtained self-consistently. Basing on this, we perform a systematic investigation of thedensity of states and electronic Raman spectra as a function oft′/t: ranging fromt′ = 0, the square lattice model, tot′ = t, the isotropic triangular latticemodel. We discuss the polarization dependence of the Raman spectra, which could be used toidentify the relevant superconducting pairing symmetry of frustrating systems such aslayered organic superconductors.     

Observation of high efficiency of Raman scattering in the process of the three-photon ionization of samarium atom

Very high efficiency of Raman scattering on samarium atoms in the process of three-photon ionization has been detected. An intense maximum attributed to the one-photon transition 4f ⁶5d6s ⁹ H ₂ → 4f ⁵5d6s ^{2 7} D ₃° between two excited states is observed in the dependence of the yield of Sm⁺ ions on the laser-radiation frequency. The 4f ⁶5d6s ⁹ H ₂ state is populated owing to the spontaneous Raman scattering 4f ⁶6s ^{2 7} F ₃ + ω ? ν → 4f ⁶5d6s ⁹ H ₂.