199 research outputs found

    Local Lagrangian Formalism and Discretization of the Heisenberg Magnet Model

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    In this paper we develop the Lagrangian and multisymplectic structures of the Heisenberg magnet (HM) model which are then used as the basis for geometric discretizations of HM. Despite a topological obstruction to the existence of a global Lagrangian density, a local variational formulation allows one to derive local conservation laws using a version of N\"other's theorem from the formal variational calculus of Gelfand-Dikii. Using the local Lagrangian form we extend the method of Marsden, Patrick and Schkoller to derive local multisymplectic discretizations directly from the variational principle. We employ a version of the finite element method to discretize the space of sections of the trivial magnetic spin bundle N=M×S2N = M\times S^2 over an appropriate space-time MM. Since sections do not form a vector space, the usual FEM bases can be used only locally with coordinate transformations intervening on element boundaries, and conservation properties are guaranteed only within an element. We discuss possible ways of circumventing this problem, including the use of a local version of the method of characteristics, non-polynomial FEM bases and Lie-group discretization methods.Comment: 12 pages, accepted Math. and Comp. Simul., May 200

    Integrability in QCD and beyond

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    Yang--Mills theories in four space-time dimensions possess a hidden symmetry which does not exhibit itself as a symmetry of classical Lagrangians but is only revealed on the quantum level. It turns out that the effective Yang--Mills dynamics in several important limits is described by completely integrable systems that prove to be related to the celebrated Heisenberg spin chain and its generalizations. In this review we explain the general phenomenon of complete integrability and its realization in several different situations. As a prime example, we consider in some detail the scale dependence of composite (Wilson) operators in QCD and super-Yang--Mills (SYM) theories. High-energy (Regge) behavior of scattering amplitudes in QCD is also discussed and provides one with another realization of the same phenomenon that differs, however, from the first example in essential details. As the third example, we address the low-energy effective action in a N=2 SYM theory which, contrary to the previous two cases, corresponds to a classical integrable model. Finally, we include a short overview of recent attempts to use gauge/string duality in order to relate integrability of Yang--Mills dynamics with the hidden symmetry of a string theory on a curved background.Comment: 87 pages, 4 figures; minor stylistic changes, references added. To be published in the memorial volume 'From Fields to Strings: Circumnavigating Theoretical Phyiscs', World Scientific, 2004. Dedicated to the memory of Ian Koga

    Gauge Theories as String Theories: the First Results

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    The brief review of the duality between gauge theories and closed strings propagating in the curved space is based on the lectures given at ITEP Winter School - 2005Comment: Latex, 35 pages, Lectures given at ITEP Winter School, March 200

    B\"acklund Transformation for the BC-Type Toda Lattice

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    We study an integrable case of n-particle Toda lattice: open chain with boundary terms containing 4 parameters. For this model we construct a B\"acklund transformation and prove its basic properties: canonicity, commutativity and spectrality. The B\"acklund transformation can be also viewed as a discretized time dynamics. Two Lax matrices are used: of order 2 and of order 2n+2, which are mutually dual, sharing the same spectral curve.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

    Integrable Matrix Models in Discrete Space-Time

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    We introduce a class of integrable dynamical systems of interacting classical matrix-valued fields propagating on a discrete space-time lattice, realized as many-body circuits built from elementary symplectic two-body maps. The models provide an efficient integrable Trotterization of non-relativistic σ\sigma-models with complex Grassmannian manifolds as target spaces, including, as special cases, the higher-rank analogues of the Landau-Lifshitz field theory on complex projective spaces. As an application, we study transport of Noether charges in canonical local equilibrium states. We find a clear signature of superdiffusive behavior in the Kardar-Parisi-Zhang universality class, irrespectively of the chosen underlying global unitary symmetry group and the quotient structure of the compact phase space, providing a strong indication of superuniversal physics.Comment: v2, 60 pages, 10 figures, 1 tabl

    SU(N) Antiferromagnets and Strongly Coupled QED: Effective Field Theory for Josephson Junctions Arrays

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    We review our analysis of the strong coupling of compact QED on a lattice with staggered Fermions. We show that, for infinite coupling, compact QED is exactly mapped in a quantum antiferromagnet. We discuss some aspects of this correspondence relevant for effective field theories of Josephson junctions arrays.Comment: 33 pages,latex,Proceedings of "Common Trends in Condensed Matter and High Energy Physics",DFUPG 1/9

    Alternative Symmetries in Quantum Field Theory and Gravity

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    A general, incomplete and partisan overview of various areas of the theoretical investigation is presented. Most of this activity stems from the search for physics beyond quantum field theory and general relativity, a titanic struggle that, in my opinion, empowered the symmetry principles to a dangerous level of speculation. In the works (that are my own) commented upon here the attempt has been to proceed by holding to certain epistemological pillars (usually absent from the too speculative theories) such as, e.g., four or less dimensions, proposals for experimental tests of radical ideas, wide cross-fertilization, etc.. As for the latter, the enterprise is undertaken within a theoretical perspective that pushes till condensed matter and even biology the cross-fertilization between ``branches of physics''.Comment: 42 pages, 1 figure, Habilitation (associate professorship) dissertation at Charles University in Prague, the papers of Section 5 are not included but easy to fin

    Spin liquid nature in the Heisenberg J1J_{1}-J2J_{2} triangular antiferromagnet

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    We investigate the spin-12\frac{1}{2} Heisenberg model on the triangular lattice in the presence of nearest-neighbor J1J_1 and next-nearest-neighbor J2J_2 antiferromagnetic couplings. Motivated by recent findings from density-matrix renormalization group (DMRG) claiming the existence of a gapped spin liquid with signatures of spontaneously broken lattice point group symmetry [Zhu and White, Phys. Rev. B 92, 041105 (2015); Hu, Gong, Zhu, and Sheng, Phys. Rev. B 92, 140403 (2015)], we employ the variational Monte Carlo (VMC) approach to analyze the model from an alternative perspective that considers both magnetically ordered and paramagnetic trial states. We find a quantum paramagnet in the regime 0.08J2/J10.160.08\lesssim J_2/J_1\lesssim 0.16, framed by 120120^{\circ} coplanar (stripe collinear) antiferromagnetic order for smaller (larger) J2/J1J_2/J_1. By considering the optimization of spin-liquid wave functions of a different gauge group and lattice point group content as derived from Abrikosov mean-field theory, we obtain the gapless U(1)U(1) Dirac spin liquid as the energetically most preferable state in comparison to all symmetric or nematic gapped Z2\mathbb{Z}_{2} spin liquids so far advocated by DMRG. Moreover, by the application of few Lanczos iterations, we find the energy to be the same as the DMRG result within error-bars. To further resolve the intriguing disagreement between VMC and DMRG, we complement our methodological approach by the pseudofermion functional renormalization group (PFFRG) to compare the spin structure factors for the paramagnetic regime calculated by VMC, DMRG, and PFFRG. This model promises to be an ideal test-bed for future numerical refinements in tracking the long-range correlations in frustrated magnets.Comment: Editors' Suggestion. 16 pages, 13 figures, 4 table

    Two-particle decay and quantum criticality in dimerized antiferromagnets

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    In certain Mott-insulating dimerized antiferromagnets, triplet excitations of the paramagnetic phase can decay into the two-particle continuum. When such a magnet undergoes a quantum phase transition into a magnetically ordered state, this coupling becomes part of the critical theory provided that the lattice ordering wavevector is zero. One microscopic example is the staggered-dimer antiferromagnet on the square lattice, for which deviations from O(3) universality have been reported in numerical studies. Using both symmetry arguments and microscopic calculations, we show that a non-trivial cubic term arises in the relevant order-parameter quantum field theory, and assess its consequences using a combination of analytical and numerical methods. We also present finite-temperature quantum Monte Carlo data for the staggered-dimer antiferromagnet which complement recently published results. The data can be consistently interpreted in terms of critical exponents identical to that of the standard O(3) universality class, but with anomalously large corrections to scaling. We argue that the two-particle decay of critical triplons, although irrelevant in two spatial dimensions, is responsible for the leading corrections to scaling due to its small scaling dimension.Comment: 14 pages, 7 fig

    Subsystem Rényi Entropy of Thermal Ensembles for SYK-like models

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    The Sachdev-Ye-Kitaev model is an N-modes fermionic model with infinite range random interactions. In this work, we study the thermal Rényi entropy for a subsystem of the SYK model using the path-integral formalism in the large-N limit. The results are consistent with exact diagonalization [1] and can be well approximated by thermal entropy with an effective temperature [2] when subsystem size M ≤ N/2. We also consider generalizations of the SYK model with quadratic random hopping term or U(1) charge conservation
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