199 research outputs found
Local Lagrangian Formalism and Discretization of the Heisenberg Magnet Model
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 over an
appropriate space-time . 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
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
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
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
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
-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
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
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 - triangular antiferromagnet
We investigate the spin- Heisenberg model on the triangular
lattice in the presence of nearest-neighbor and next-nearest-neighbor
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 , framed by
coplanar (stripe collinear) antiferromagnetic order for smaller
(larger) . 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 Dirac spin
liquid as the energetically most preferable state in comparison to all
symmetric or nematic gapped 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
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
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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