From Atiyah Classes to Homotopy Leibniz Algebras

A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold X makes T _X [?1] into a Lie algebra object in D ⁺ (X), the bounded below derived category of coherent sheaves on X. Furthermore, Kapranov proved that, for a Kähler manifold X, the Dolbeault resolution \({\Omega^{\bullet-1}(T_X^{1, 0})}\) of T _X [?1] is an L _∞ algebra. In this paper, we prove that Kapranov’s theorem holds in much wider generality for vector bundles over Lie pairs. Given a Lie pair (L, A), i.e. a Lie algebroid L together with a Lie subalgebroid A, we define the Atiyah class α _E of an A-module E as the obstruction to the existence of an A-compatible L-connection on E. We prove that the Atiyah classes α _L/A and α _E respectively make L/A[?1] and E[?1] into a Lie algebra and a Lie algebra module in the bounded below derived category \({D^+(\mathcal{A})}\) , where \({\mathcal{A}}\) is the abelian category of left \({\mathcal{U}(A)}\) -modules and \({\mathcal{U}(A)}\) is the universal enveloping algebra of A. Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of L/A and E, and inducing the aforesaid Lie structures in \({D^+(\mathcal{A})}\) .     

Decay channels of the standard Higgs boson

In this paper, we review the results of studies on the decay channels of the standard Higgs boson: H → f + f?, H → Z + f + f?, H → W + f + f?′, H → γ + γ, H → γ + Z and H → g + g. Here ff? or ff?′are the fundamental fermions pair (leptons, quarks). Within the framework of the Standard Model analytical expressions for the partial widths of the indicated decays were obtained and their dependence on the mass of the Higgs boson was studied.     

<Emphasis Type="Italic">Q</Emphasis>-systems as Cluster Algebras II: Cartan Matrix of Finite Type and the Polynomial Property

We define the cluster algebra associated with the Q-system for the Kirillov–Reshetikhin characters of the quantum affine algebra \({U_q(\widehat{\mathfrak {g}})}\) for any simple Lie algebra \({\mathfrak {g}}\), generalizing the simply-laced case treated in (Kedem in Q-systems as cluster algebras. arXiv:0712.2695 [math.RT], 2007). We describe some special properties of this cluster algebra, and explain its relation to the deformed Q-systems which appeared on our proof of the combinatorial-KR conjecture. We prove that the polynomiality of the cluster variables in terms of the “initial cluster seeds”, including solutions of the Q-system, is a consequence of the Laurent phenomenon and the boundary conditions. We also define the cluster algebra associated with T-systems, or general systems which take the form of T-systems in the bipartite case. Such systems describe the recursion relations satisfied by the q-characters of Kirillov–Reshetikhin modules and also appear in the categorification picture in terms of preprojective algebras of Geiss, Leclerc and Schröer. We give a formulation of both Q-systems and generalized T-systems as cluster algebras with coefficients. This provides a proof of the polynomiality of solutions of all such “generalized T-systems” with appropriate boundary conditions.     

A Formal Model of Berezin-Toeplitz Quantization

We give a new construction of symbols of the differential operators on the sections of a quantum line bundle L over a Kähler manifold M using the natural contravariant connection on L. These symbols are the functions on the tangent bundle TM polynomial on fibres. For high tensor powers of L, the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on TM corresponding to some pseudo-Kähler structure on TM. Surprisingly, this star product is intimately related to the formal symplectic groupoid with separation of variables over M. We extend the star product on TM to generalized functions supported on the zero section of TM. The resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the Bergman projection operator. Using this idempotent, we define an algebra of Toeplitz elements and show that it is naturally isomorphic to the algebra of Berezin-Toeplitz deformation quantization on M.     

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.     

Thermally activated flux flow,vortex-glass phase transition and the mixed-state Hall effect in 112-type iron pnictide superconductors

The transport properties in the mixed state of high-quality Ca_(0.8)La_(0.2)Fe_(0.98)Co_(0.02)As_2single crystal,a newly discovered 112-type iron pnictide superconductor,are comprehensively studied by magneto-resistivity measurement.The field-dependent activation energy,U_0,is derived in the framework of thermally activated flux flow(TAFF)theory,yielding a power law dependence U_0～H~αwith a crossover at a magnetic field around 2 T in both H⊥ab and H//ab,which is ascribed to the different pinning mechanisms.Moreover,we have clearly observed the vortex phase transition from vortex-glass to vortex-liquid according to the vortex-glass model,and vortex phase diagrams are constructed for both H⊥ab and H//ab.Finally,the results of mixed-state Hall effect show that no sign reversal of transverse resistivityρ_(xy)(H)is detected,indicating that the Hall component arising from the vortex flow is in theories or experiments previously reported on some high-T_ccuprates.     

Symmetry energy of the nucleus in the relativistic Thomas–Fermi approach with density-dependent parameters

We introduce an inhomogeneous term, f(t,x), into the right-hand side of the usual Burgers equation and examine the resulting equation for those functions which admit at least one Lie point symmetry. For those functions f(t,x) which depend nontrivially on both t and x, we find that there is just one symmetry. If f is a function of only x, there are three symmetries with the algebra s l(2,R). When f is a function of only t, there are five symmetries with the algebra s l(2,R) ⊕ _s 2A ₁. In all the cases, the Burgers equation is reduced to the equation for a linear oscillator with nonconstant coefficient.     

Temperature Dependence of the Upper Critical Field in Disordered Hubbard Model with Attraction

We study disorder effects upon the temperature behavior of the upper critical magnetic field in an attractive Hubbard model within the generalized DMFT+Σ approach. We consider the wide range of attraction potentials U—from the weak coupling limit, where superconductivity is described by BCS model, up to the strong coupling limit, where superconducting transition is related to Bose–Einstein condensation (BEC) of compact Cooper pairs, formed at temperatures significantly higher than superconducting transition temperature, as well as the wide range of disorder—from weak to strong, when the system is in the vicinity of Anderson transition. The growth of coupling strength leads to the rapid growth of H_c2(T), especially at low temperatures. In BEC limit and in the region of BCS–BEC crossover H_c2(T), dependence becomes practically linear. Disordering also leads to the general growth of H_c2(T). In BCS limit of weak coupling increasing disorder lead both to the growth of the slope of the upper critical field in the vicinity of the transition point and to the increase of H_c2(T) in the low temperature region. In the limit of strong disorder in the vicinity of the Anderson transition localization corrections lead to the additional growth of H_c2(T) at low temperatures, so that the H_c2(T) dependence becomes concave. In BCS–BEC crossover region and in BEC limit disorder only slightly influences the slope of the upper critical field close to T _c . However, in the low temperature region H_c2 (T may significantly grow with disorder in the vicinity of the Anderson transition, where localization corrections notably increase H_c2 (T = 0) also making H_c2(T) dependence concave.     

Hysteresis of the magnetoresistance of granular HTSC YBa<Subscript>2</Subscript>Cu<Subscript>3</Subscript>O<Subscript>7 − δ</Subscript> in weak fields

Hysteresis of the magnetoresistance of ceramic YBa₂Cu₃O_～6.95 HTSC samples is studied at T = 77.3 K in an external magnetic field H _ext changing in 0 → H _max → 0 cycles, where H _max is the maximum magnitude of H _ext. Information is obtained about the dependences of the critical fields of Josephson weak links H _c2J , the lower critical fields of superconducting grains H _c1A , and the critical fields H _BG-VG of the Bragg glass-vortex glass phase transition in the vortex matter on transport current I, magnetic field, and the mutual orientation of I and H _ext. It is found that the magnetoresistance δρ⁺/ρ_{273 K} measured with increasing H _ext is significantly higher than Δρ^?/ρ_{273 K} and that H _c2J ⁺ < H _c2J ^? , H _c1A ⁺ < H _c1A ^? , and H _BG-VG ⁺ < H _BG-VG ^? .     

Resolvent Expansion and Time Decay of the Wave Functions for Two-Dimensional Magnetic Schrödinger Operators

We consider two-dimensional Schrödinger operators H(B, V) given by Eq. (1.1) below. We prove that, under certain regularity and decay assumptions on B and V, the character of the expansion for the resolvent (H(B, V) ? λ)^?1 as λ → 0 is determined by the flux of the magnetic field B through \({\mathbb{R}^2}\) . Subsequently, we derive the leading term of the asymptotic expansion of the unitary group e ^{?i t H(B, V)} as t → ∞ and show how the magnetic field improves its decay in t with respect to the decay of the unitary group e ^{?i t H(0, V)}.     

<Emphasis Type="Italic">C</Emphasis><Subscript>2</Subscript>-Cofiniteness of the 2-Cycle Permutation Orbifold Models of Minimal Virasoro Vertex Operator Algebras

In this article, we give a sufficient and necessary condition for the C ₂-cofiniteness of \({\widetilde{V} = (V\otimes V)^\sigma}\) for a C ₂-cofinite vertex operator algebra V and the 2-cycle permutation σ of \({V\otimes V}\) . As an application, we show that the 2-cycle permutation orbifold model of the simple Virasoro vertex operator algebra L(c, 0) of minimal central charge c is C ₂-cofinite.     

The structure of a vortex line and the lower critical field in superconducting alloys

The structure of an isolated vortex line, and the lower critical fieldH _c 1, is calculated by means of the generalized Ginzburg-Landau (GL) theory for arbitrary values of the GL-parameterk(≧1/√2) and the mean free pathl at temperaturesT in the vicinity ofT _c . The free energy functional including the corrections of order [1?(T/T _c )] to the GL-functional is derived exactly. The corresponding Euler-Lagrange equations determining the zero-order (GL) contributions and the corrections of order [1?(T/T _c )] to the order parameter,f(r), and the superfluid velocity,v(r), have been solved numerically. The shapes of the first-order corrections off(r), v(r), and the magnetic field,h(r) are found to depend markedly, for a given value ofκ, on a second parameter,α=0.882(ξ ₀ /l) (whereξ ₀ is theBCS-coherence-distance). The deviations from the GL-solutions become largest forh(r) at parameter valuesk≈ 1 andα ≈ 0(the deviation ofh(0) is about 6% atT=0.9T _c forκ=1 andα=0). The ratioH _c1/H _c (where the thermodynamic criticalH _c has the BCS-temperature-dependence) is found to increase slightly in the “clean” limit (α=0), and to decrease slightly in the “dirty” limit (α=∞) asT decreases (the variation ofH _c 1/H _c is always less than 3% for arbitrary values ofκ andα asT decreases fromT _c to 0.9T _c ).     

Peculiarities of magnetic,galvanomagnetic, elastic,and magnetoelastic properties of Eu<Subscript>0.55</Subscript>Sr<Subscript>0.45</Subscript>MnO<Subscript>3</Subscript>

Magnetic, elastic, magnetoelastic, transport, and magnetotransport properties of the Eu_0.55Sr_0.45MnO₃ ceramics have been studied. A break was detected in the temperature dependence of electrical resistivity ρ(T) near the temperature of the magnetic phase transformation (41 K), with the material remaining an insulator down to the lowest measurement temperature reached (ρ=10⁶ Ω cm at 4.2 K). In the interval 4.2≤T≤50 K, the isotherms of the magnetization, volume magnetostriction, and ρ were observed to undergo jumps at the critical field H_C1, which decreases with increasing T. For 50≤T≤120 K, the jumps in the above curves persist, but the pattern of the curves changes and H_C1 grows with increasing T. The magnetoresistance Δρ/ρ = (ρ _H ?ρ_H=0)/ρ _H is positive for H<H_C1 and passes through a maximum at 41 K, where Δρ/ρ = 6%. For H>H_C1, the magnetoresistance is negative, passes through a minimum near 41 K, and reaches a colossal value of 3×10⁵ % at H=45 kOe. The volume magnetostriction is negative and attains a giant value of 4.5×10^?4atH=45 kOe. The observed properties are assigned to the existence of three phases in Eu_0.55Sr_0.45MnO₃, namely, a ferromagnetic (FM) phase, in which carriers are concentrated because of the gain in s-d exchange energy, and two antiferromagnetic (AFM) phases of the A and CE types. Their fractional volumes at low temperatures were estimated to be as follows: ～3% of the sample volume is occupied by the FM phase; ～67%, by the CE-type AFM phase; and ～30%, by the A-type AFM phase.     

A Note on Quantum Coherence

In this paper, we discuss the coherence of the reduced state in system H _A ?H _B under taking different quantum operations acting on subsystem H _B . Firstly, we show that for a pure bipartite state, the coherence of the final subsystem H _A under the sum of two orthonormal rank 1 projections acting on H _B is less than or equal to the sum of the coherence of the state after two orthonormal projections acting on H _B , respectively. Secondly, we obtain that the coherence of reduced state in subsystem H _A under random unitary channel \({\Phi }(\rho )={\sum }_{s}\lambda _{s}U_{s}\rho U_{s}^{\ast }\) acting on H _B , is equal to the coherence of the state after each operation \({\Phi }_{s}(\rho )=\lambda _{s}U_{s}\rho U_{s}^{\ast }\) acting on H _B for every s. In addition, for general quantum operation \({\Phi }(\rho )={\sum }_{s}F_{s}\rho F_{s}^{\ast }\) on H _B , we get the relation

$$ C\left (\left ((I\otimes {\Phi })\rho ^{AB}\right )^{A}\right )\leq \sum \limits _{s}C\left (\left ((I\otimes {\Phi }_{s})\rho ^{AB}\right )^{A}\right ). $$

     

Scattering the Geometry of Weighted Graphs

Given two weighted graphs (X, b_k, m_k), k =?1,2 with b₁ ～ b₂ and m₁ ～ m₂, we prove a weighted L¹-criterion for the existence and completeness of the wave operators W_±(H₂, H₁, I_1,2), where H_k denotes the natural Laplacian in ?²(X, m_k) w.r.t. (X, b_k, m_k) and I_1,2 the trivial identification of ?²(X, m₁) with ?²(X, m₂). In particular, this entails a general criterion for the absolutely continuous spectra of H₁ and H₂ to be equal.     

Generality of spontaneous and stimulated magnetization reversal in MnSb clusters embedded in GaMnSb thin films

Generality of the spontaneous and stimulated magnetization reversal in MnSb clusters embedded in GaMnSb thin films is established. In experiments, the similarity of the thermoactivation and field magnetization reversal processes can be observed as the coincidence of the maximum in the field dependences of magnetic viscosity S(H) with the sample coercivity H _C . Analysis of this experimental fact yields the relation between H _C and parameters of the model describing the S(H) dependences. The obtained formula is identical to the well-known Kneller law determining the H _C (T) dependence of noninteracting superparamagnetic nanoparticles.     

Enhancement of superconductivity in disordered films by parallel magnetic field

We show that the superconducting transition temperature T _c (H) of a very thin highly disordered film with strong spin-orbital scattering can be increased by a parallel magnetic field H. This effect is due to the polarization of magnetic impurity spins, which reduces the full exchange scattering rate of electrons; the largest effect is predicted for spin-1/2 impurities. Moreover, for some range of magnetic impurity concentrations, the phenomenon of superconductivity induced by magnetic field is predicted: the superconducting transition temperature T _c (H) is found to be nonzero in the range of magnetic fields 0 < H* ≤ H ≤ H _c .     

Magnetization behavior and metamagnetic transitions in electron-doped manganites induced by a strong magnetic field

The behavior of magnetization M of the R_xA_1?xMnO₃ manganites (R=La, Pr, Nd, Sm, etc., A=Ca, Sr, Ba) in the electron doping region (x<0.4) is studied as a function of external magnetic field H. The M(H) relations for homogeneous magnetic structures are obtained by performing band calculations in the double-exchange model. Three different types of magnetization behavior corresponding to three electron concentration ranges (x<0.14, 0.14<x<0.27, x>0.27) are revealed. The M(H) relations are interpreted in terms of the phase diagram for the homogeneous ground state of the manganites calculated for H=0, and the results agree qualitatively with experimental data on the magnetization of Sm_xCa_1?xMnO₃.     

Constructal Design Applied to the Geometric Evaluation of an Oscillating Water Column Wave Energy Converter Considering Different Real Scale Wave Periods

The present work presents numerical study of the influence of geometry on the performance of an oscillating water column (OWC) wave energy converter by means of a constructal design. The main purpose is to maximize the root mean square hydrodynamic power of device, (P_hyd)_RMS, subject to several real scale waves with different periods. The problem has two constraints: hydropneumatic chamber volume (V _HC ) and total OWC volume (V _T ), and two degrees of freedom: H₁/L (ratio of height to length of the hydropneumatic chamber) and H₃ (OWC submergence). For the numerical solution it was used a computational fluid dynamic (CFD) code, based on the finite volume method (FVM). The multiphasic volume of fluid (VOF) model is applied to tackle with the water–air interaction. The results led to important theoretical recommendations about the design of OWC device. For instance, the best shape for OWC chamber, which maximizes the (P_hyd)_RMS, was achieved when the ratio (H₁/L) was four times higher than the ratio of height to length of incident wave (H/λ), (H₁/L) _o = 4(H/λ). Moreover, the optimal submergence (H₃) was achieved as a function of wave height (H) and water depth (h), more precisely given by the following relation: h ? (3H/4) ≤ (H₃) _o ≤ h.     

Noncompact <Emphasis Type="Italic">u</Emphasis><Subscript>q</Subscript>(2, 1) quantum algebra: Discrete series of highest weight irreducible representations

The structure of unitary irreducible representations of the noncompact u_q(2, 1) quantum algebra that are related to a negative discrete series is examined. With the aid of projection operators for the su_q(2) subalgebra, a q analog of the Gelfand-Graev formulas is derived in the basis corresponding to the reduction u_q(2, 1) → su_q(2)×u(1). Projection operators for the su_q(1, 1) subalgebra are employed to study the same representations for the reduction u_q(2, 1) → u(1)×su_q(1, 1). The matrix elements of the generators of the u_q(2, 1) algebra are computed in this new basis. A general analytic expression for an element of the transformation brackets <U∣T>_q between the bases associated with the above two reductions (the elements of this matrix are referred to as q Weyl coefficients) is obtained for a general case where the deformation parameter q is not equal to a root of unity. It is shown explicitly that, apart from a phase, the q Weyl coefficients coincide with the q Racah coefficients for the su_q(2) quantum algebra.