iopscience.org/jphysa journal of Physics A Mathematical and Theoretical

Highlights 2010 Cover image: An artistic representation of the time evolution of a soliton solution of the KP equation. This solution might explain a large amplification of two-dimensional waves, such as a tsunami Yuji Kodama 2010 J. Phys. A: Math. Theor. 43 (2010) 434004. Journal of Physics A: Mathematical and Theoretical

Dear colleagues, This has been another successful year for Journal of Physics A: Mathematical and Theoretical. More people than ever before are reading the journal and enjoying issues of high-quality research throughout theoretical and mathematical physics. This collection of highlights displays some of the most highly rated articles published in the journal over the course of 2010. All of these were chosen primarily for their high-quality science. This selection also displays the broad scope of the journal and shows that it is a meeting place for researchers to share mathematically rich work across different disciplines. Some of our best articles have been published as fast track communications (FTCs). FTCs are short articles presenting important new developments in mathematical and theoretical physics. FTCs benefit from accelerated publication, are free-to-read Impact Factor and benefit from our recommended reader service, where we will notify researchers 1.641* of the author’s choice when the article is published. * As listed in ISI®’s 2010 Science Citation Index Journal citation reports This collection also shows details of the topical reviews and special issues that we have published throughout 2010.

AVERAGE DOWNLOADS Topical reviews are commissioned by our Board Members to provide timely PER ARTICLE overviews of the current state of research in areas of great interest and activity, while special issues are collections of articles focused on contemporary areas of research, with contributions from world-class authors, and guest edited by leading 139 researchers. 2011 will be another strong year for the journal, with special issues planned on Full-text downloads quantum integrable models and gauge-string duality and scattering amplitudes in gauge theories. Also look out for our new Insights, where we feature author- 1000000 written news items highlighting the key achievements of recent work published in 800000 the journal. We have awarded up to three Journal of Physics A Best Paper prizes 600000 to reward work that has excelled in novelty, achievement, potential impact and 400000 presentation. Please contact the journal office to make your nominations. 200000

0 We hope that you enjoy this collection. We would like to thank all of our authors 2008 2009 2010 for choosing to submit high-quality work to the journal as well as our referees for providing constructive peer review and for maintaining the quality standards of the journal. We look forward to working with you during 2011.

Editor-in-Chief Murray Batchelor

Highlights 2010 3 Journal of Physics A: Mathematical and Theoretical

Editorial Board

Editor-in-Chief M T Batchelor Australian National University, Canberra, Australia

Fast Track Communications Editor Mathematical Physics Editor P E Dorey University of Durham, UK P Forrester University of Melbourne, Australia

Reviews Editor Quantum Mechanics and Quantum Information Theory Editor J Feinberg Technion and University of Haifa, Israel V Scarani National University of Singapore, Singapore

Statistical Physics Editor Field Theory and String Theory Editor M R Evans University of Edinburgh, UK A Tseytlin Imperial College, London, UK

Chaotic and Complex Systems Editor Fluid and Plasma Theory Editor A Politi CNR-Institute of Complex Systems, Florence, Italy G Falkovich Weizmann Institute of Science, Israel

Editorial Board N Beisert Max-Planck-Institut fuer Gravitationsphysik, Germany G Korchemsky Université de Paris-Sud, France E Ben-Naim Los Alamos National Laboratory, USA R G Littlejohn University of California, Berkeley, USA M V Berry University of Bristol, UK S Majumdar Université de Paris-Sud, France K Binder Johannes Gutenberg-Universität, Mainz, Germany R Metzler Technische Universität, München, Germany D Bruß Heinrich-Heine-Universität Dusseldorf, Germany J A Minahan Uppsala Universitet, Sweden S Coppersmith University of Wisconsin, Madison, USA J H H Perk Oklahoma State University, Stillwater, USA M A del Olmo Universidad de Valladolid, Spain M Saraceno Comision Nacional de Energia Atomica—Departamento de B Derrida École Normale Supérieure, France Fisica, Buenos Aires, Argentina D Dhar Tata Institute of Fundamental Research, India P Schmelcher Universität Hamburg, Germany J Eisert University of Potsdam, Germany P Sollich King’s College London, UK E Elizalde CSIC, Barcelona, Spain Y Takei Kyoto University, Japan F Essler University of Oxford, UK R Tumulka Rutgers University, USA P Fendley University of Virginia, USA C Viallet Université Pierre et Marie Curie, Paris, France U Günther Forschungszentrum Rossendorf, Dresden, Germany M Visser Victoria University of Wellington, Wellington, New Zealand D D Holm Imperial College London, UK and Los Alamos National Laboratory, A Vulpiani Universita di Roma “La Sapienza”, Italy NM, USA P Wiegmann University of Chicago, USA M Horodecki Gdan´sk University, Poland H-Q Zhou Chongqing University, People’s Republic of China A V Kitaev Steklov Mathematical Institute, St Petersburg, Russia

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4 Highlights 2010 Journal of Physics A: Mathematical and Theoretical

Journal scope

Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.

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Research papers published in Journal of Physics A: Mathematical and Theoretical are catergorized into one of six subject sections:

•Statistical physics •Chaotic and complex systems •Mathematical physics •Quantum mechanics and quantum information theory •Field theory and string theory •Fluid and plasma theory

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Highlights 2010 5 Journal of Physics A: Mathematical and Theoretical

Contents

Statistical physics

Anomalous diffusion in correlated continuous time random walks 9 Vincent Tejedor and Ralf Metzler

Distribution of partition function zeros of the ±J model on the Bethe lattice 9 Yoshiki Matsuda, Markus Müller, Hidetoshi Nishimori, Tomoyuki Obuchi and Antonello Scardicchio

Exact and simple results for the XYZ and strongly interacting fermion chains 9 Paul Fendley and Christian Hagendorf

Exact results for an asymmetric annihilation process with open boundaries 10 Arvind Ayyer and Kirone Mallick

First-passage exponents of multiple random walks 10 E Ben-Naim and P L Krapivsky

Large deviations of the free energy in diluted mean-field spin-glass 10 Giorgio Parisi and Tommaso Rizzo

Phase diagram and spectral properties of a new exactly integrable spin-1 quantum chain 11 Francisco C Alcaraz and Gilberto M Nakamura

Spontaneous magnetization of the superintegrable chiral Potts model: calculation of the determinant DPQ 11 R J Baxter

The unusual asymptotics of three-sided prudent polygons 11 Nicholas R Beaton, Philippe Flajolet and Anthony J Guttmann

Time at which the maximum of a random acceleration process is reached 12 Satya N Majumdar, Alberto Rosso and Andrea Zoia

Ultra diffusions 12 Iddo Eliazar and Joseph Klafter

Universal amplitude ratios of two-dimensional percolation from field theory 12 Gesualdo Delfino, Jacopo Viti and John Cardy

Universality classes of polymer melts and conformal sigma models 13 C Candu, J L Jacobsen, N Read and H Saleur

Chaotic and complex systems Chaotic maps, Hamiltonian flows and holographic methods 13 Thomas L Curtright and Cosmas K Zachos

High-order classical adiabatic reaction forces: slow manifold for a spin model 14 M V Berry and P Shukla

Nonequilibrium dynamics of a stochastic model of anomalous heat transport 14 Stefano Lepri, Carlos Mejía-Monasterio and Antonio Politi

Weyl law for fat fractals 14 María E Spina, Ignacio García-Mata and Marcos Saraceno

6 Highlights 2010 Journal of Physics A: Mathematical and Theoretical

Mathematical physics Difference system for Selberg correlation integrals 15 Peter J Forrester and Masahiko Ito

Exercises with the universal R-matrix 15 Herman Boos, Frank Göhmann, Andreas Klümper, Khazret S Nirov and Alexander V Razumov

Families of classical subgroup separable superintegrable systems 15 E G Kalnins, J M Kress and W Miller Jr

Higher order decompositions of ordered operator exponentials 15 Nathan Wiebe, Dominic Berry, Peter Høyer and Barry C Sanders

High-order Fuchsian equations for the square lattice Ising model: χ(6) 16 S Boukraa, S Hassani, I Jensen, J-M Maillard and N Zenine

Identities in the superintegrable chiral Potts model 16 Helen Au-Yang and Jacques H H Perk

Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples 16 D D Holm and R I Ivano

Periodic orbits for an infinite family of classical superintegrable systems 17 Frédérick Tremblay, Alexander V Turbiner and Pavel Winternitz

Rogue waves, rational solutions, the patterns of their zeros and integral relations 17 Adrian Ankiewicz, Peter A Clarkson and Nail Akhmediev

Superintegrability of the Tremblay–Turbiner–Winternitz quantum Hamiltonians on a plane for odd k 17 C Quesne

The chiral Gaussian two-matrix ensemble of real asymmetric matrices 18 G Akemann, M J Phillips and H-J Sommers

Solving the ultradiscrete KdV equation 18 Ralph Willox, Yoichi Nakata, Junkichi Satsuma, Alfred Ramani and Basile Grammaticos

Derivation of determinantal structures for random matrix ensembles in a new way 18 Mario Kieburg and Thomas Guhr

The lowest eigenvalue of Jacobi random matrix ensembles and Painlevé VI 19 E Dueñez, D K Huynh, J P Keating, S J Miller and N C Snaith

Quantum mechanics and quantum information theory

PT-symmetry, Cartan decompositions, Lie triple systems and Krein space-related Clifford algebras 19 Uwe Günther and Sergii Kuzhel

Classical particle in a complex elliptic potential 20 Carl M Bender, Daniel W Hook and Karta Singh Kooner

Constructive counterexamples to the additivity of the minimum output Rényi entropy of quantum channels for all p > 2 20 Andrzej Grudka, Michał Horodecki and Łukasz Pankowski

Eigenphase preserving two-channel SUSY transformations 20 Andrey M Pupasov, Boris F Samsonov, Jean-Marc Sparenberg and Daniel Baye

Entanglement cost of two-qubit orthogonal measurements 20 Somshubhro Bandyopadhyay, Ramij Rahaman and William K Wootters

Highlights 2010 7 Journal of Physics A: Mathematical and Theoretical

Entanglement in valence-bond-solid states on symmetric graphs 21 Hosho Katsura, Naoki Kawashima, Anatol N Kirillov, Vladimir E Korepin and Shu Tanaka

Generalized minimal output entropy conjecture for one-mode Gaussian channels: definitions and some exact results 21 V Giovannetti, A S Holevo, S Lloyd and L Maccone

Ground-state fidelity and entanglement entropy for the quantum three-state Potts model in one spatial dimension 21 Yan-Wei Dai, Bing-Quan Hu, Jian-Hui Zhao and Huan-Qiang Zhou

Hilbert spaces from path integrals 22 Fay Dowker, Steven Johnston and Rafael D Sorkin

Imaginary cubic perturbation: numerical and analytic study 22 Jean Zinn-Justin and Ulrich D Jentschura

Looking for symmetric Bell inequalities 22 Jean-Daniel Bancal, Nicolas Gisin and Stefano Pironio

Quantum sign permutation polytopes 22 Colin Wilmott, Hermann Kampermann and Dagmar Bruß

The non-locality of n noisy Popescu–Rohrlich boxes 23 Matthias Fitzi, Esther Hänggi, Valerio Scarani and Stefan Wolf

Anyons emerging from fermions with conventional two-body interactions 23 Yue Yu and Yi Li

Field theory and string theory

Chiral correlators of the Ising conformal field theory 23 Eddy Ardonne and Germán Sierra

5 Exact computation of one-loop correction to the energy of folded spinning string in AdS5 × S 24 M Beccaria, G V Dunne, V Forini, M Pawellek and A A Tseytlin

Four-dimensional Yang–Mills theory, gauge invariant mass and fluctuating three-branes 24 Antti J Niemi and Sergey Slizovskiy

Magnon dispersion to four loops in the ABJM and ABJ models 24 J A Minahan, O Ohlsson Sax and C Sieg

Semionic supersymmetric solitons 25 Luca Mezincescu and Paul K Townsend

Y-system for scattering amplitudes 25 Luis F Alday, Juan Maldacena, Amit Sever and Pedro Vieira

Asymptotic Bethe equations for open boundaries in planar AdS/CFT 25 D H Correa and C A S Young

Open string pair creation from worldsheet instantons 25 Christian Schubert and Alessandro Torrielli

Fluid and plasma theory

Longitudinal wave-breaking limits in a unified geometric model of relativistic warm plasmas 26 D A Burton and A Noble

Unstable waves on a curved non-Newtonian liquid jet 26 V L Hawkins, C J Gurney, S P Decent, M J H Simmons and J Uddin

Kinematic magnetic dynamo in a random flow with strong average shear 26 V R Kogan, I V Kolokolov and V V Lebedev

8 Highlights 2010 Journal of Physics A: Mathematical and Theoretical

zeros directly in the thermodynamical limit. We clarify how the spin-glass q Statistical physics transition is characterized by the zeros of the partition function. It is also shown that in the spin-glass phase a continuous distribution of singularities touches the axes of real field and temperature. Anomalous diffusion in correlated continuous time random walks

Vincent Tejedor and Ralf Metzler Physics Department T30 g, Technical University of Munich, 85747 Garching, Germany

2010 J. Phys. A: Math. Theor. 43 082002

Fast Track Communications

We demonstrate that continuous time random walks in which successive waiting times are correlated by Gaussian statistics lead to anomalous Distribution of zeros on the complex T plane with p = 0.5 and H = 0 (left) and H = 0.5 (right). Zeros diffusion with the mean squared displacement r2(t) b t2/3. Long-ranged b 〈 〉 touch the real axis below the critical temperature TSG 1.13 (left) and 0.5 (right). The apparent correlations of the waiting times with a power-law exponent α absence of zeros near the origin may be due to numerical rounding errors of tanh β. (0 < α ≤ 2) give rise to subdiffusion of the form 〈r2(t)〉 b tα/(1 + α). In contrast, correlations in the jump lengths are shown to produce superdiffusion. We show that in both cases weak ergodicity breaking occurs. Our results are in excellent agreement with simulations. Exact and simple results for the XYZ and strongly interacting fermion chains 〈(r2(t)〉 for a waiting time correlated 3D Gaussian walk. The δψ follow an α-stable Paul Fendley and Christian Hagendorf law with scale factor c = 1; Department of Physics, University of Virginia, Charlottesville, VA 22904-4714, USA α decreases from top to bottom. Simulations (—) and power-laws (· · ·) with fitted 2010 J. Phys. A: Math. Theor. 43 402004 exponents 0.35, 0.50, 0.60, 0.66. Theoretical values α/ (α + 1): 0.33, 0.50, 0.60, Fast Track Communications 0.66. We conjecture exact and simple formulas for some physical quantities in two quantum chains. A classic result of this type is Onsager, Kaufman and Yang's formula for the spontaneous magnetization in the Ising model, subsequently generalized to the chiral Potts models. We conjecture that

analogous results occur in the XYZ chain when the couplings obey JxJy +

JyJz + JxJz = 0, and in a related fermion chain with strong interactions and supersymmetry. We find exact formulas for the magnetization and gap in Distribution of partition function zeros of the the former, and the staggered density in the latter, by exploiting the fact that certain quantities are independent of finite-size effects. ±J model on the Bethe lattice

Yoshiki Matsuda1, Markus Müller2, Hidetoshi Nishimori1, Tomoyuki Obuchi1 and Antonello Scardicchio2,3 1 Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan 2 International Center for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy 3 INFN, Sezione di Trieste, Strada Costiera 11, 34014, Trieste, Italy

2010 J. Phys. A: Math. Theor. 43 285002

The distribution of partition function zeros is studied for the ±J model of spin glasses on the Bethe lattice. We find a relation between the distribution of complex cavity fields and the density of zeros, which enables us to obtain ML(s) for s ≤ 1; the solid red curve is the conjecture for M∞(s), while the dashed curves are for the density of zeros for the infinite system size by using the cavity method. L = 5, 9, 13, 17. The phase boundaries thus derived from the location of the zeros are consistent with the results of direct analytical calculations. This is the first example in which the spin-glass transition is related to the distribution of

Highlights 2010 9 Journal of Physics A: Mathematical and Theoretical

Exact results for an asymmetric annihilation process with open boundaries

Arvind Ayyer and Kirone Mallick Institut de Physique Théorique, C E A Saclay, 91191 Gif-sur-Yvette Cedex, France

2010 J. Phys. A: Math. Theor. 43 045003

The compound two-dimensional random walk. The line indicates the random walk trajectory and We consider a nonequilibrium reaction–diffusion model on a finite one- the bullets show intermediate locations along this trajectory. The thick lines mark the absorbing dimensional lattice with bulk and boundary dynamics inspired by the boundary. (a) For n = 1, the random walk is confined to the exterior of the third (negative) Glauber dynamics of the Ising model. We show that the model has a rich quadrant. (b) For n = 2, the random walk is confined to the interior of the first (positive) quadrant. algebraic structure that we use to calculate its properties. In particular, we show that the Markov dynamics for a system of a given size can be embedded into the dynamics of systems of higher sizes. This remark leads us to devise a technique which we call the transfer matrix Ansatz Large deviations of the free energy in diluted that allows us to determine the steady-state distribution and correlation functions. Furthermore, we show that the disorder variables satisfy very mean-field spin-glass simple properties and we give a conjecture for the characteristic polynomial of Markov matrices. Finally, we compare the transfer matrix Ansatz used Giorgio Parisi1,2 and Tommaso Rizzo1 here with the matrix product representation of the steady state of one- 1 Dipartimento di Fisica, Università di Roma 'La Sapienza', P.le Aldo Moro 2, 00185 dimensional stochastic models. Roma, Italy 2 Statistical Mechanics and Complexity Center (SMC)-INFM-CNR, Italy

First-passage exponents of multiple random 2010 J. Phys. A: Math. Theor. 43 045001 walks Sample-to-sample free-energy fluctuations in spin-glasses display a 1 2 E Ben-Naim and P L Krapivsky markedly different behaviour in finite-dimensional and fully connected 1 Theoretical Division and Center for Nonlinear Studies, Los Alamos National models, namely Gaussian versus non-Gaussian. Spin-glass models defined Laboratory, Los Alamos, NM 87545, USA on various types of random graphs are in an intermediate situation between 2 Department of Physics, Boston University, Boston, MA 02215, USA these two classes of models and we investigate whether the nature of their free-energy fluctuations is Gaussian or not. It has been argued that 2010 J. Phys. A: Math. Theor. 43 495008 Gaussian behaviour is present whenever the interactions are locally non- homogeneous, i.e. in most cases with the notable exception of models ~ with fixed connectivity and random couplings Jij = ±J . We confirm these We investigate first-passage statistics of an ensemble of N noninteracting expectations by means of various analytical results concerning the large random walks on a line. Starting from a configuration in which all particles deviations of the free energy. In particular we unveil the connection between the spatial fluctuations of the populations of fields defined at different sites are located in the positive half-line, we study Sn(t), the probability that the nth rightmost particle remains in the positive half-line up to time t. of the lattice and the Gaussian nature of the free-energy fluctuations. In ßn contrast, on locally homogeneous lattices the populations do not fluctuate This quantity decays algebraically, Sn(t) ~ t– , in the long-time limit. over the sites and as a consequence the small deviations of the free energy Interestingly, there is a family of nontrivial first-passage exponents, β1 < are non-Gaussian and scale as in the Sherrington–Kirkpatrick model. β2 < … < βN − 1; the only exception is the two-particle case where β1 = 1/3.

In the N → ∞ limit, however, the exponents attain a scaling form, βn(N) → β(z) with z = (n – N/2)/√N. We also demonstrate that the smallest exponent decays exponentially with N. We deduce these results from first- passage kinetics of a random walk in an N-dimensional cone and confirm them using numerical simulations. Additionally, we investigate the family of exponents that characterizes leadership statistics of multiple random walks and find that in this case, the cone provides an excellent approximation.

Fast Track Communications FTCs are free to publish and free to read

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10 Highlights 2010 Journal of Physics A: Mathematical and Theoretical

Phase diagram and spectral properties of a Spontaneous magnetization of the new exactly integrable spin-1 quantum chain superintegrable chiral Potts model: calculation of the determinant D Francisco C Alcaraz and Gilberto M Nakamura PQ Instituto de Física de São Carlos, Universidade de São Paulo, CP 369, 13560-970, R J Baxter São Carlos, São Paulo, Brazil Mathematical Sciences Institute, The Australian National University, Canberra, ACT 0200, Australia 2010 J. Phys. A: Math. Theor. 43 155002 2010 J. Phys. A: Math. Theor. 43 145002 The spectral properties and phase diagram of the exactly integrable spin-1 quantum chain introduced by Alcaraz and Bariev are presented. The model has a U(1) symmetry and its integrability is associated with an unknown For the Ising model, the calculation of the spontaneous magnetization leads R-matrix whose dependence on the spectral parameters is not of a different to the problem of evaluating a determinant. Yang did this by calculating the form. The associated Bethe ansatz equations that fix the eigenspectra are eigenvalues in the large-lattice limit. Montroll, Potts and Ward expressed it distinct from those associated with other known integrable spin models. The as a Toeplitz determinant and used Szego´´’s theorem: this is almost certainly the route originally travelled by Onsager. For the corresponding problem model has a free parameter tp. We show that at the special point tp = 1, the model acquires an extra U(1) symmetry and reduces to the deformed SU(3) in the superintegrable chiral Potts model, neither approach appears to work: here we show that the determinant D can be expressed as that of Perk–Schultz model at a special value of its anisotropy q = exp (i2π/3) PQ and in the presence of an external magnetic field. Our analysis is carried a product of two Cauchy-like matrices. One can then use the elementary out either by solving the associated Bethe ansatz equations or by direct exact formula for the Cauchy determinant. One of course regains the known diagonalization of the quantum Hamiltonian for small lattice sizes. The result, originally conjectured in 1989. phase diagram is calculated by exploring the consequences of conformal invariance on the finite-size corrections of the Hamiltonian eigenspectrum. The model exhibits a critical phase ruled by the c = 1 conformal field theory separated from a massive phase by first-order phase transitions. The unusual asymptotics of three-sided prudent polygons

Nicholas R Beaton1, Philippe Flajolet2 and Anthony J Guttmann1 1 ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia 2 Algorithms Project, INRIA—Rocquencourt, 78153 Le Chesnay, France

2010 J. Phys. A: Math. Theor. 43 342001

Fast Track Communications

We have studied the area-generating function of prudent polygons on the square lattice. Exact solutions are obtained for the generating function of two-sided and three-sided prudent polygons, and a functional equation is found for four-sided prudent polygons. This is used to generate series coefficients in polynomial time, and these are analysed to determine the asymptotics numerically. A careful asymptotic analysis of the three-sided polygons produces a most surprising result. A transcendental critical exponent is found, and the leading amplitude is not quite a constant, but is a constant plus a small oscillatory component with an amplitude The values of x (t , ) as a function of for several values of t in regions 1 and 2. The curves with −8 p p ρ ρ p approximately 10 times that of the leading amplitude. This effect cannot tp < 1/2 and tp ≥ 1/2 belong to regions 1 and 2 of figure 1, respectively. The endpoints of the curves are the densities separating regions 1 and 2 from 3 and 5, respectively. For t ≥ 1/2 the be seen by any standard numerical analysis, but it may be present in other p models. If so, it changes our whole view of the asymptotic behaviour of endpoints of the curves are xp = 1/6 ∼ 0.166 66. lattice models.

Examples of (a) a two-sided prudent SAW; (b) a three-sided prudent SAW; (c) an (unrestricted) prudent SAW; and (d) a prudent SAW leading to a prudent SAP.

Highlights 2010 11 Journal of Physics A: Mathematical and Theoretical

motions driven by symmetric stable Lévy motions and M/G/∞ processes. Time at which the maximum of a random A methodological framework of ultra diffusions is established— acceleration process is reached accommodating transport processes which display, simultaneously, both ‘anomalous-diffusion’ temporal behavior and ‘fat-tailed’ amplitudinal Lévy fluctuations. Ultra diffusions with power-law temporal and amplitudinal 1 1 2 Satya N Majumdar , Alberto Rosso and Andrea Zoia statistics are shown to emerge universally from a general superposition 1 CNRS, Université Paris-Sud, LPTMS, UMR8626, Bât. 100, 91405 Orsay Cedex, model of stochastic processes. France 2 CEA/Saclay, DEN/DM2S/SERMA/LTSD, Bât. 454, 91191 Gif-sur-Yvette Cedex, France Universal amplitude ratios of two- 2010 J. Phys. A: Math. Theor. 43 115001 dimensional percolation from field theory

Gesualdo Delfino1,2, Jacopo Viti1,2 and John Cardy3,4 We study the random acceleration model, which is perhaps one of the 1 simplest, yet nontrivial, non-Markov stochastic processes, and is key International School for Advanced Studies (SISSA), via Beirut 2-4, 34151 Trieste, Italy to many applications. For this non-Markov process, we present exact 2 Istituto Nazionale di Fisica Nucleare, sezione di Trieste, Italy analytical results for the probability density p(tm|T) of the time tm at which the process reaches its maximum, within a fixed time interval [0, T]. We 3 Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK study two different boundary conditions, which correspond to the process 4 All Souls College, Oxford, UK representing respectively (i) the integral of a Brownian bridge and (ii) the integral of a free Brownian motion. Our analytical results are also verified by 2010 J. Phys. A: Math. Theor. 43 152001 numerical simulations. Fast Track Communications

We complete the determination of the universal amplitude ratios of two- dimensional percolation within the two-kink approximation of the form factor approach. For the cluster size ratio, which has for a long time been elusive both theoretically and numerically, we obtain the value 160.2, in good agreement with the lattice estimate 162.5 ± 2 of Jensen and Ziff.

The website that connects Simulation results for the cumulative distribution P(z) = ∫z p(zʹ) dzʹ (circles) as compared to the 0 physicists and engineers to a wide analytical formula in equation (6) (solid line), for the integral of a Brownian Bridge. range of employers

Ultra diffusions

Iddo Eliazar1 and Joseph Klafter2,3 1 Department of Technology Management, Holon Institute of Technology, PO Box 305, Holon 58102, Israel defence aerospace manufacturing 2 School of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel 3 Freiburg Institute for Advanced Studies (FRIAS), University of Freiburg, 79104, Freiburg, Germany

2010 J. Phys. A: Math. Theor. 43 132002 telecommunications engineering energy and utilities Fast Track Communications Register today for our jobswire This communication presents and explores ultra diffusions—a class of random transport processes which generalizes the class of ‘classic’ brightrecruits.com/register diffusions. Examples of ultra diffusions include Lévy motions, fractional Brownian motions, fractional stable Lévy motions, Ornstein–Uhlenbeck

12 Highlights 2010 Journal of Physics A: Mathematical and Theoretical

Universality classes of polymer melts and q Chaotic and complex systems conformal sigma models

C Candu1, J L Jacobsen1,2, N Read3 and H Saleur1,4 Chaotic maps, Hamiltonian flows and 1 Institut de Physique Théorique CEA, IPhT, CNRS, URA 2306, F-91191 Gif-sur-Yvette, holographic methods France

2 Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 rue Lhomond, Thomas L Curtright1 and Cosmas K Zachos2 75005 Paris, France 1 Department of Physics, University of Miami, Coral Gables, FL 33124-8046, USA 3 Department of Physics, Yale University, PO Box 208120, New Haven, CT 06520- 2 High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439- 8120, USA 4815, USA 4 Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA 2010 J. Phys. A: Math. Theor. 43 445101 2010 J. Phys. A: Math. Theor. 43 142001 Holographic functional methods are introduced as probes of discrete Fast Track Communications time-stepped maps that lead to chaotic behavior. The methods provide continuous time interpolation between the time steps, thereby revealing the In the usual statistical model of a dense polymer (a single space-filling loop maps to be quasi-Hamiltonian systems underlain by novel potentials that on a lattice) in two dimensions the loop does not cross itself. We modify govern the motion of a perceived point particle. Between turning points, the this by including intersections in which three lines can cross at the same particle is strictly driven by Hamiltonian dynamics, but at each encounter point, with some statistical weight w per crossing. We show that our model with a turning point the potential changes abruptly, loosely analogous to the describes a line of critical theories with continuously varying exponents switchbacks on a mountain road. A sequence of successively deepening depending on w, described by a conformally invariant nonlinear sigma switchback potentials explains, in physical terms, the frequency cascade model with varying coupling constant g2 ≥ 0. For the boundary critical σ and trajectory folding that occur on the particular route to chaos revealed by behavior, or the model defined in a strip, we propose an exact formula for 2 the logistic map. the l-leg exponents, hl = g σ l(l − 2)/8, which is shown numerically to hold very well.

E(t) for initial x = 1/2, using potentials VP, with P = 0, 1, 2, 3 and 4.

Vertices, weights and sample configuration for dense polymers on a square lattice of width L = 3. Boundary conditions are free in the horizontal (space) direction and periodic in the vertical (imaginary time) direction. The alternating , ¯ representations correspond to a lattice orientation, conserved along each loop.

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Highlights 2010 13 Journal of Physics A: Mathematical and Theoretical

baths. The noise amounts to random collisions between nearest-neighbour High-order classical adiabatic reaction oscillators that exchange their momenta. In a recent paper (Lepri et al 2009 J. Phys. A: Math. Theor. 42 025001), we have studied the forces: slow manifold for a spin model stationary state of this system with fixed boundary conditions, finding analytical exact expressions for the temperature profile and the heat 1 2 M V Berry and P Shukla current in the thermodynamic (continuum) limit. In this paper, we extend 1 H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL, UK the analysis to the evolution of the covariance matrix and to generic 2 Department of Physics, Indian Institute of Technology, Kharagpur, India boundary conditions. Our main purpose is to construct a hydrodynamic description of the relaxation to the stationary state, starting from the exact 2010 J. Phys. A: Math. Theor. 43 045102 equations governing the evolution of the correlation matrix. We identify and adiabatically eliminate the fast variables, arriving at a continuity equation for the temperature profile T(y, t), complemented by an ordinary equation The influence of a fast system on the hamiltonian dynamics of a slow that accounts for the evolution in the bulk. Altogether, we find that the system coupled to it is explored by calculating, in a model, high-order evolution of T(y, t) is the result of fractional diffusion. smooth (nonoscillating) adiabatic reaction forces (i.e. beyond Born– Spectrum { } of the linear Oppenheimer and geometric magnetism). The model is a spin (fast) driven λl by, and reacting on, the position vector (slow) of a particle coupled to it. The equation. The eigenvalues are expressed in ε3ω2/γ search for smooth solutions is equivalent to determining the slow manifold time units. The straight line in the full phase space, on which, in the model system, the spin would not corresponds to a power law precess. The series of reactions for the nonlinear coupled system diverges with a rate 3/2. factorially, as in the simpler linear case of a spin being driven passively by a position vector changing in a prescribed manner. When the particle is closest to the origin, all terms in the divergent series have the same sign, indicating a Stokes phenomenon and suggesting that a solution of the slow manifold equation exists but contains exponentially weak precession oscillations. The predicted oscillations are observed numerically, and shown to be inevitable for the exactly solvable linearized slow manifold which is equivalent to the Landau–Majorana–Zener model of quantum mechanics. Weyl law for fat fractals

María E Spina1, Ignacio García-Mata1,2 and Marcos Saraceno1,3 1 Departamento de Física, CNEA, Libertador 8250, C1429BNP Buenos Aires, Argentina 2 CONICET, Avda. Rivadavia 1917, CP C1033AAJ, Buenos Aires, Argentina 3 Escuela de Ciencia y Tecnología, UNSAM, Alem 3901, B1653HIM Villa Ballester, Argentina

2010 J. Phys. A: Math. Theor. 43 392003

Fast Track Communications

It has been conjectured that for a class of piecewise linear maps the closure of the set of images of the discontinuity has the structure of a fat fractal, that is, a fractal with positive measure. An example of such maps is the sawtooth map in the elliptic regime. In this work we analyze this problem quantum Spin S precessing about the instantaneous position vector R. mechanically in the semiclassical regime. We find that the fraction of states localized on the unstable set satisfies a modified fractal Weyl law, where the exponent is given by the exterior dimension of the fat fractal. Nonequilibrium dynamics of a stochastic Phase-space portrait of model of anomalous heat transport the map (equation (1)) for (left) K = −0.5, (right) K = 2 cos –2. The same number Stefano Lepri, Carlos Mejía-Monasterio1 and Antonio Politi of points was used to draw both maps to provide a Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, via Madonna del ‘visual’ representation of the piano 10, I-50019 Sesto Fiorentino, Italy measurer of the set. 1 Present address: Department of Mathematics and Statistics, University of Helsinki, 4 PO Box 68, FIN-00014, Helsinki, Finland

2010 J. Phys. A: Math. Theor. 43 065002

We study the dynamics of covariances in a chain of harmonic oscillators with conservative noise in contact with two stochastic Langevin heat

14 Highlights 2010 Journal of Physics A: Mathematical and Theoretical

in the momenta, for some families of generalized oscillator and Kepler– q Mathematical physics Coulomb systems, hence demonstrating their superintegrability. The latter generalizes recent results of Verrier and Evans, and Rodríguez, Tempesta and Winternitz. Another example is given of a superintegrable system on a Difference system for Selberg correlation non-conformally flat space. integrals

Peter J Forrester1 and Masahiko Ito2 Higher order decompositions of ordered 1 Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia operator exponentials 2 School of Science and Technology for future life, Tokyo Denki University, Tokyo 101-8457, Japan Nathan Wiebe1, Dominic Berry2,3, Peter Høyer1,4 and Barry C Sanders1 1 Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, 2010 J. Phys. A: Math. Theor. 43 175202 Canada 2 Institute for Quantum Computing, University of Waterloo, Ontario N2 L 3G1, Canada 3 Centre for Quantum Computer Technology, Macquarie University, Sydney, The Selberg correlation integrals are averages of the products with respect NSW 2109, Australia to the Selberg density. Our interest is in the case m = 1, μ1 = μ, when 4 Department of Computer Science, University of Calgary, Alberta T2N 1N4, Canada this corresponds to the µth moment of the corresponding characteristic polynomial. We give the explicit form of an (n + 1) × (n + 1) matrix linear 2010 J. Phys. A: Math. Theor. 43 065203 difference system in the variable μ which determines the average, and we give the Gauss decomposition of the corresponding (n + 1) × (n + 1) matrix. For μ a positive integer the difference system can be used to efficiently We present a decomposition scheme based on Lie–Trotter–Suzuki product compute the power series defined by this average. formulae to approximate an ordered operator exponential with a product of ordinary operator exponentials. We show, using a counterexample, that Lie–Trotter–Suzuki approximations may be of a lower order than expected when applied to problems that have singularities or discontinuous Exercises with the universal R-matrix derivatives of appropriate order. To address this problem, we present a set of criteria that is sufficient for the validity of these approximations, prove Herman Boos1, Frank Göhmann1, Andreas Klümper1, Khazret S Nirov1,2 convergence and provide upper bounds on the approximation error. This and Alexander V Razumov3 work may shed light on why related product formulae fail to be as accurate 1 Fachbereich C—Physik, Bergische Universität Wuppertal, 42097 Wuppertal, as expected when applied to Coulomb potentials. Germany 2 Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Ave. 7a, 117312 Moscow, Russia 3 Institute for High Energy Physics, 142281 Protvino, Moscow region, Russia

2010 J. Phys. A: Math. Theor. 43 415208

Using the formula for the universal R-matrix proposed by Khoroshkin and Tolstoy, we give a detailed derivation of L-operators for the quantum groups (1) (1) associated with the generalized Cartan matrices A 1 and A 2.

|| || 5 3 This is a plot of ζ = U(Δλ, 0)−U2(Δλ, 0) 2/ Δλ for Aa = λ sin(1/λ) 1 in (a) and Ab = cos(λ)1 in (b). The error in (a) is proportional to Δλ4 as opposed to the O(Δλ)5 scaling predicted for Suzuki’s Families of classical subgroup separable corresponding decomposition. The error in (b) is proportional to Δλ5 as expected for that Suzuki decomposition. superintegrable systems

E G Kalnins1, J M Kress2 and W Miller Jr3 1 Department of Mathematics, University of Waikato, Hamilton, New Zealand 2 School of Mathematics and Statistics, University if New South Wales, Sydney, Australia 3 School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

2010 J. Phys. A: Math. Theor. 43 092001

Fast Track Communications

We describe a method for determining a complete set of integrals for a classical Hamiltonian that separates in orthogonal subgroup coordinates. As examples, we use it to determine complete sets of integrals, polynomial

Highlights 2010 15 Journal of Physics A: Mathematical and Theoretical

High-order Fuchsian equations for the square Multi-component generalizations of the CH lattice Ising model: χ(6) equation: geometrical aspects, peakons and numerical examples S Boukraa1, S Hassani2, I Jensen3, J-M Maillard4 and N Zenine2 1 LPTHIRM and Département d'Aéronautique, Université de Blida, Algeria D D Holm1 and R I Ivanov1,2 2 Centre de Recherche Nucléaire d'Alger, 2 Bd. Frantz Fanon, BP 399, 16000 Alger, 1 Department of Mathematics, Imperial College London, London SW7 2AZ, UK Algeria 2 School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, 3 ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Dublin 8, Ireland Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia 4 LPTMC, Université de Paris 6, Tour 24, 4ème étage, case 121, 4 Place Jussieu, 2010 J. Phys. A: Math. Theor. 43 492001 75252 Paris Cedex 05, France Fast Track Communications 2010 J. Phys. A: Math. Theor. 43 115201 The Lax pair formulation of the two-component Camassa–Holm equation (CH2) is generalized to produce an integrable multi-component family,

(6) CH(n, k), of equations with n components and 1 ≤ |k| ≤ n velocities. All This paper deals with χ- , the six-particle contribution to the magnetic of the members of the CH(n, k) family show fluid-dynamics properties with susceptibility of the square lattice Ising model. We have generated, modulo (6) coherent solitons following particle characteristics. We determine their a prime, series coefficients for χ- . The length of the series is sufficient to Lie–Poisson Hamiltonian structures and give numerical examples of their produce the corresponding Fuchsian linear differential equation (modulo a soliton solution behaviour. We concentrate on the CH(2, k) family with one prime). We obtain the Fuchsian linear differential equation that annihilates (6) (6) (4) (2) or two velocities, including the CH(2, −1) equation in the Dym position of the ‘depleted’ series φ = χ- – 2/3χ- + 2/45 χ- . The factorization the CH2 hierarchy. A brief discussion of the CH(3, 1) system reveals the of the corresponding differential operator is performed using a method underlying graded Lie-algebraic structure of the Hamiltonian formulation for of factorization modulo a prime, introduced in a previous paper. The CH(n, k) when n ≥ 3. ‘depleted’ differential operator is shown to have a structure similar to the corresponding operator for χ-(5). It splits into factors of smaller orders, with the left-most factor of order 6 being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral E. The right-most factor has a direct sum structure, and using series calculated modulo several primes, all the factors in the direct sum have been reconstructed in exact arithmetics.

Identities in the superintegrable chiral Potts model Dam-break results for the CH(2,1) system in equations (3.1)–(3.2) show evolution of the density ρ (a) and velocity u (b), arising from initial conditions (3.4) in a periodic domain. The colour bars show positive density on the left and both positive and negative velocity on the right. The soliton Helen Au-Yang1,2 and Jacques H H Perk1,2 solutions are seen to emerge symmetrically leftward and rightward after a finite time, and the evolution of both variables generates more and more solitons propagating in both directions as 1 Department of Physics, Oklahoma State University, 145 Physical Sciences, time progresses. Figures are courtesy of L O´ Na´raigh. Stillwater, OK 74078-3072, USA 2 Centre for Mathematics and its Applications & Department of Theoretical Physics, Australian National University, Canberra, ACT 2600, Australia

2010 J. Phys. A: Math. Theor. 43 025203

We present proofs for a number of identities that are needed to study the superintegrable chiral Potts model in the Q ≠ 0 sector. DID YOU KNOW? Following publication all articles are free to read for 30 days

16 Highlights 2010 Journal of Physics A: Mathematical and Theoretical

Periodic orbits for an infinite family of Superintegrability of the Tremblay–Turbiner– classical superintegrable systems Winternitz quantum Hamiltonians on a plane for odd k Frédérick Tremblay1, Alexander V Turbiner2 and Pavel Winternitz1 1 Centre de recherches mathématiques and Département de mathématiques et de C Quesne statistique, Université de Montreal, C.P. 6128, Succ. Centre-Ville, Montréal (QC) H3C 3J7, Canada Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium 2 Instituto de Ciencias Nucleares, UNAM, A.P. 70-543, 04510 México, Mexico

2010 J. Phys. A: Math. Theor. 43 082001 2010 J. Phys. A: Math. Theor. 43 015202 Fast Track Communications We show that all bounded trajectories in the two-dimensional classical In a recent communication paper by Tremblay et al (2009 J. Phys. A: Math. system with the potential are closed for all integer and rational values Theor. 42 205206), it has been conjectured that for any integer value of of k. The period is and does not depend on k. This agrees with our k, some novel exactly solvable and integrable quantum Hamiltonian Hk on earlier conjecture suggesting that the quantum version of this system is a plane is superintegrable and that the additional integral of motion is a superintegrable. 2kth-order differential operator Y2k. Here we demonstrate the conjecture for the infinite family of Hamiltonians Hk with odd k ≥ 3, whose first member corresponds to the three-body Calogero–Marchioro–Wolfes model after Rogue waves, rational solutions, the patterns elimination of the centre-of-mass motion. Our approach is based on the construction of some D2k-extended and invariant Hamiltonian, which can of their zeros and integral relations be interpreted as a modified boson oscillator Hamiltonian. The latter is then shown to possess a D2k-invariant integral of motion, from which Y2k can be obtained by projection in the D identity representation space. Adrian Ankiewicz1, Peter A Clarkson2 and Nail Akhmediev1 2k 1 Optical Sciences Group, Research School of Physics and Engineering, Institute of Advanced Studies, The Australian National University, Canberra ACT 0200, Australia 2 School of Mathematics, Statistics and Actuarial Science, University of Kent, Journal of Physics A: Mathematical and Theoretical Canterbury CT2 7NF, UK 2010 Topical Reviews

2010 J. Phys. A: Math. Theor. 43 122002 Properties of multi-particle Green’s and vertex functions within Keldysh formalism Fast Track Communications Severin G Jakobs, Mikhail Pletyukhov and Herbert Schoeller 2010 J. Phys. A: Math. Theor. 43 103001 The focusing nonlinear Schrödinger equation, which describes generic Deterministic thermostats, theories of nonequilibrium systems and nonlinear phenomena, including waves in the deep ocean and light pulses parallels with the ergodic condition in optical fibres, supports a whole hierarchy of recently discovered rational Owen G Jepps and Lamberto Rondoni solutions. We present recurrence relations for the hierarchy, the pattern of 2010 J. Phys. A: Math. Theor. 43 133001 zeros for each solution and a set of integral relations which characterizes them. Discrete breathers and the anomalous decay of luminescence E Mihóková and L S Schulman 2010 J. Phys. A: Math. Theor. 43 183001 Finite tight frames and some applications Nicolae Cotfas and Jean Pierre Gazeau 2010 J. Phys. A: Math. Theor. 43 193001 Dark solitons in atomic Bose–Einstein condensates: from theory to experiments D J Frantzeskakis 2010 J. Phys. A: Math. Theor. 43 213001 n-ary algebras: a review with applications J A de Azcárraga and J M Izquierdo 2010 J. Phys. A: Math. Theor. 43 293001 A first course on twistors, integrability and gluon scattering amplitudes Martin Wolf 2010 J. Phys. A: Math. Theor. 43 393001

The 15 upper half plane zeros of F5(x, 0) from the fifth-order rational solution. Poles of G5/F5 with A pedestrian’s view on interacting particle systems, KPZ universality and residue +2i are marked with red dots, while those with residue −2i are marked with blue dots. This random matrices resembles a scaled version of figure 2 with an additional upper blue arc. Thomas Kriecherbauer and Joachim Krug 2010 J. Phys. A: Math. Theor. 43 403001 Lectures on nonlinear sigma-models in projective superspace Sergei M Kuzenko 2010 J. Phys. A: Math. Theor. 43 443001

Highlights 2010 17 Journal of Physics A: Mathematical and Theoretical

The chiral Gaussian two-matrix ensemble of Solving the ultradiscrete KdV equation real asymmetric matrices Ralph Willox1, Yoichi Nakata1, Junkichi Satsuma2, Alfred Ramani3 and Basile Grammaticos4 G Akemann1, M J Phillips1 and H-J Sommers2 1 Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, 1 Department of Mathematical Sciences and BURSt Research Centre, Brunel Meguro-ku, 153-8914 Tokyo, Japan University West London, Uxbridge UB8 3PH, UK 2 Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1 2 Fakultät für Physik, Universität Duisburg-Essen, 47048 Duisburg, Germany Fuchinobe, Sagamihara-shi, Japan 3 Centre de Physique Théorique, Ecole Polytechnique, CNRS, 91128 Palaiseau, 2010 J. Phys. A: Math. Theor. 43 085211 France 4 IMNC, Université Paris VII & XI, CNRS, UMR 8165, Bât. 440, 91406 Orsay, France

We solve a family of Gaussian two-matrix models with rectangular 2010 J. Phys. A: Math. Theor. 43 482003 N × (N + ) matrices, having real asymmetric matrix elements and ν depending on a non-Hermiticity parameter μ. Our model can be thought Fast Track Communications of as the chiral extension of the real Ginibre ensemble, relevant for Dirac operators in the same symmetry class. It has the property that its We show that a generalized cellular automaton, exhibiting solitonic eigenvalues are either real, purely imaginary or come in complex conjugate interactions, can be explicitly solved by means of techniques first eigenvalue pairs. The eigenvalue joint probability distribution for our model introduced in the context of the scattering problem for the KdV equation. is explicitly computed, leading to a non-Gaussian distribution including We apply this method to calculate the phase-shifts caused by interactions K-Bessel functions. All n-point density correlation functions are expressed between the solitonic and non-solitonic parts into which arbitrary initial for finite N in terms of a Pfaffian form. This contains a kernel involving states separate in time. Laguerre polynomials in the complex plane as a building block which was previously computed by the authors. This kernel can be expressed in terms of the kernel for complex non-Hermitian matrices, generalizing the known relation among ensembles of Hermitian random matrices. Compact expressions are given for the density at finite N as an example, as well as Derivation of determinantal structures for its microscopic large-N limits at the origin for fixed ν at strong and weak random matrix ensembles in a new way non-Hermiticity. Mario Kieburg and Thomas Guhr Universität Duisburg-Essen, Lotharstraße 1, 47048 Duisburg, Germany

2010 J. Phys. A: Math. Theor. 43 075201

There are several methods to treat ensembles of random matrices in symmetric spaces, circular matrices, chiral matrices and others. Orthogonal polynomials and the supersymmetry method are particular powerful techniques. Here, we present a new approach to calculate averages over ratios of characteristic polynomials. At first sight paradoxically, one can The complex spectral density RC (z) for finite N = 10 at maximal non-Hermiticity μ2 = 1 (left) 1Dirac call our approach ‘supersymmetry without supersymmetry’ because we use and intermediate μ2 = 0.5 (right), both for ν = 0. We show only the first quadrant for symmetry reasons. For μ = 1, we see a circular ‘support’ growing with √N, apart from the repulsion from the structures from supersymmetry without actually mapping onto superspaces. axes. For decreasing μ, the ‘support’ becomes an ellipse, with the eigenvalues moving towards, as We address two kinds of integrals which cover a wide range of applications well as onto, the real axis. Note the increased height in the right plot. for random matrix ensembles. For probability densities factorizing in the eigenvalues we find determinantal structures in a unifying way. As a new application we derive an expression for the k-point correlation function of an arbitrary rotation invariant probability density over the Hermitian matrices in the presence of an external field.

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18 Highlights 2010 Journal of Physics A: Mathematical and Theoretical

The lowest eigenvalue of Jacobi random q Quantum mechanics and quantum matrix ensembles and Painlevé VI information theory

E Dueñez1, D K Huynh2, J P Keating3, S J Miller4 and N C Snaith5 1 Department of Mathematics, University of Texas at San Antonio, San Antonio, PT -symmetry, Cartan decompositions, TX 78249, USA Lie triple systems and Krein space-related 2 Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada Clifford algebras 3 School of Mathematics, University of Bristol, Bristol BS8 1TW, UK 4 Department of Mathematics and Statistics, Williams College, Williamstown, Uwe Günther1 and Sergii Kuzhel2 MA 01267, USA 1 Research Center Dresden-Rossendorf, PO Box 510119, D-01314 Dresden, 5 School of Mathematics, University of Bristol, Bristol BS8 1TW, UK Germany 2 Institute of Mathematics of the NAS of Ukraine, 01601 Kyiv, Ukraine 2010 J. Phys. A: Math. Theor. 43 405204 2010 J. Phys. A: Math. Theor. 43 392002

We present two complementary methods, each applicable in a different Fast Track Communications range, to evaluate the distribution of the lowest eigenvalue of random matrices in a Jacobi ensemble. The first method solves an associated Gauged PT quantum mechanics (PTQM) and corresponding Krein space Painlevé VI nonlinear differential equation numerically, with suitable setups are studied. For models with constant non-Abelian gauge potentials initial conditions that we determine. The second method proceeds via and extended parity inversions compact and noncompact Lie group constructing the power-series expansion of the Painlevé VI function. components are analyzed via Cartan decompositions. A Lie-triple structure Our results are applied in a forthcoming paper in which we model the is found and an interpretation as PT -symmetrically generalized Jaynes– distribution of the first zero above the central point of elliptic curve Cummings model is possible with close relation to recently studied cavity L-function families of finite conductor and of conjecturally orthogonal QED setups with transmon states in multilevel artificial atoms. For models symmetry. with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space-related J-self- adjoint extensions for PTQM setups with ultra-localized potentials.

a+1/2,b+1/2 π Plot of ν N ( N θ), the scaled distribution of the first eigenvalue. On the left N = 2, a = 0 and b = 0. The numerical solver was given the initial conditions (dotted line) and produced the solution shown with the dot-dashed line. This is indistinguishable from the series expansion (solid line) using 100 terms. On the right N = 5, a = −0.5 and b = 0.5. The numerical solver was given the initial conditions (dotted line) and produced the solution shown with the dot-dashed line. The tail agrees with the series expansion (solid line) using 99 terms. In both figures the numerical solver −7 −5 −6 was run with values of t0 = 1–10 , reltol = 10 and abstol = 10 .

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Highlights 2010 19 Journal of Physics A: Mathematical and Theoretical

Classical particle in a complex elliptic We present a constructive example of the violation of the additivity of minimum output Rényi entropy for each p > 2. The example is provided by an potential antisymmetric subspace of a suitable dimension. We discuss the possibility of extension of the result to go beyond p > 2 and obtain additivity for p = 0 for a Carl M Bender1, Daniel W Hook2 and Karta Singh Kooner2 class of entanglement breaking channels. 1 Department of Physics, Washington University, St. Louis, MO 63130, USA 2 Theoretical Physics, Imperial College, London SW7 2AZ, UK

2010 J. Phys. A: Math. Theor. 43 165201 Eigenphase preserving two-channel SUSY transformations

This paper reports a numerical study of complex classical trajectories of Andrey M Pupasov1,2, Boris F Samsonov1, Jean-Marc Sparenberg2 a particle in an elliptic potential. This study of doubly periodic potentials and Daniel Baye2 is a natural sequel to earlier work on complex classical trajectories in 1 Physics Department, Tomsk State University, 36 Lenin Avenue, 634050 Tomsk, trigonometric potentials. For elliptic potentials, there is a two-dimensional Russia array of identical cells in the complex plane, and each cell contains a pair of 2 Physique Quantique, C.P. 165/82, Université Libre de Bruxelles, B 1050 Bruxelles, turning points. The particle can travel both horizontally and vertically as it visits Belgium these cells, and sometimes the particle is captured temporarily by a pair of turning points. If the particle's energy lies in a conduction band, the particle 2010 J. Phys. A: Math. Theor. 43 155201 drifts through the lattice of cells and is never captured by the same pair of turning points more than once. However, if the energy of the particle is not in a conduction band, the particle can return to previously visited cells. We propose a new kind of supersymmetric (SUSY) transformation in the case of the two-channel scattering problem with equal thresholds for partial waves of the same parity. This two-fold transformation is based on two imaginary factorization energies with opposite signs and with mutually conjugated factorization solutions. We call it an eigenphase preserving SUSY transformation as it relates two Hamiltonians, the scattering matrices of which have identical eigenphase shifts. In contrast to known phase-equivalent transformations, the mixing parameter is modified by the eigenphase preserving transformation.

Entanglement cost of two-qubit orthogonal measurements

Somshubhro Bandyopadhyay1, Ramij Rahaman2 and William K Wootters3,4 Trajectory of a particle in the complex cnoidal potential Cn(x, 1/10 000). The particle is initially 1 Department of Physics and Center for Astroparticle Physics and Space Science, at x(0) = i and its path x(t) is plotted in the complex-x plane for 0 ≤ t ≤ 124.4. The energy of the Bose Institute, Block EN, Sector V, Kolkata 700 091, India particle is E = 1/2+i/10. In the left panel, the turning points are indicated by small squares and the pole singularities of the cnoidal function are indicated by small circles. In the right panel, the 2 Department of Informatics, University of Bergen, PB-7803, Bergen-5020, Norway particle trajectory is superimposed on a three-dimensional relief plot of the real part of the cnoidal 3 Department of Physics, Williams College, Williamstown, MA 01267, USA function. 4 Department of Applied Physics, Kigali Institute of Science and Technology, B.P. 3900, Kigali, Rwanda

2010 J. Phys. A: Math. Theor. 43 455303 Constructive counterexamples to the additivity of the minimum output Rényi The ‘entanglement cost’ of a bipartite measurement is the amount of shared entropy of quantum channels for all p > 2 entanglement two participants need to use up in order to carry out the given measurement by means of local operations and classical communication. Andrzej Grudka1, Michał Horodecki2 and Łukasz Pankowski2,3 We numerically investigate the entanglement cost of generic orthogonal measurements on two qubits. Our results strongly suggest that for almost all 1 Faculty of Physics, Adam Mickiewicz University, 61-614 Poznan´, Poland measurements of this kind, the entanglement cost is strictly greater than the 2 Institute of Theoretical Physics and Astrophysics, University of Gdan´sk, 80-952 average entanglement of the eigenstates associated with the measurements, Gdan´sk, Poland implying that the nonseparability of a two-qubit orthogonal measurement is 3 Institute of Informatics, University of Gdan´sk, 80-952 Gdan´sk, Poland generically distinct from the nonseparability of its eigenstates.

2010 J. Phys. A: Math. Theor. 43 425304

20 Highlights 2010 Journal of Physics A: Mathematical and Theoretical

Blue dots: plot of the lower bound of the entanglement Generalized minimal output entropy cost CL(M) with respect to the entropy bound for general two-qubit orthogonal conjecture for one-mode Gaussian channels: measurements. Red line: plot of the entropy bound. definitions and some exact results

V Giovannetti1, A S Holevo2, S Lloyd3 and L Maccone3 1 NEST, Scuola Normale Superiore & CNR-INFM, Piazza dei Cavalieri 7, I-56126 Pisa, Italy 2 Steklov Mathematical Institute, Gubkina 8, 119991 Moscow, Russia 3 Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Entanglement in valence-bond-solid states on symmetric graphs 2010 J. Phys. A: Math. Theor. 43 415305

Hosho Katsura1, Naoki Kawashima2, Anatol N Kirillov3, Vladimir E Korepin4 A formulation of the generalized minimal output entropy conjecture for and Shu Tanaka4 Gaussian channels is presented. It asserts that, for states with fixed input 1 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, entropy, the minimal value of the output entropy of the channel (i.e. the CA 93106, USA minimal output entropy increment for fixed input entropy) is achieved by 2 Institute for Solid State Physics, The University of Tokyo, 5-1-5 Kashiwanoha, Gaussian states. In the case of centered channels (i.e. channels which do not Kashiwa, Chiba 277-8581, Japan add squeezing to the input state) this implies that the minimum is obtained by 3 Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto thermal (Gibbs) inputs. The conjecture is proved to be valid in some special 606-8502, Japan cases. 4 C N Yang Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11794-3840, USA Ground-state fidelity and entanglement 2010 J. Phys. A: Math. Theor. 43 255303 entropy for the quantum three-state Potts

We study quantum entanglement in the ground state of the Affleck–Kennedy– model in one spatial dimension Lieb–Tasaki model defined on two-dimensional graphs with reflection and/ or inversion symmetry. The ground state of this spin model is known as the Yan-Wei Dai, Bing-Quan Hu, Jian-Hui Zhao and Huan-Qiang Zhou valence-bond-solid state. We investigate the properties of a reduced density Centre for Modern Physics and Department of Physics, Chongqing University, matrix of a subsystem which is a mirror image of the other one. Thanks to Chongqing 400044, People's Republic of China reflection symmetry, the eigenvalues of the reduced density matrix can be obtained by numerically diagonalizing a real symmetric matrix whose 2010 J. Phys. A: Math. Theor. 43 372001 elements are calculated by Monte Carlo integration. We calculate the von Neumann entropy of the reduced density matrix. The obtained results indicate Fast Track Communications that there is some deviation from the naive expectation that the von Neumann entropy per valence bond on the boundary between the subsystems is ln The ground-state fidelity per lattice site is computed for the quantum three- 2. This deviation is interpreted in terms of the hidden spin chain along the state Potts model in a transverse magnetic field on an infinite-size lattice in boundary between the subsystems. In some cases where graphs are on one spatial dimension in terms of the infinite matrix product state algorithm. ladders, the numerical results are analytically or algebraically confirmed. It is found that, on the one hand, a pinch point is identified on the fidelity surface around the critical point, and on the other hand, the ground-state fidelity per lattice site exhibits bifurcations at pseudo critical points for different values of the truncation dimension, which in turn approach the critical point as the truncation dimension becomes large. This implies that the ground-state fidelity per lattice site enables us to capture spontaneous symmetry breaking when the control parameter crosses the critical value. In addition, a finite-entanglement scaling of the von Neumann entropy is performed with respect to the truncation dimension, resulting in a precise determination of the central charge at the critical point. Finally, we compute the transverse magnetization, from which the critical exponent β is extracted VBS state on a 2-leg hexagonal ladder. It is cut by the reflection line indicated by the broken line. from the numerical data. The ground-state fidelity surface defined by the

ground-state fidelity per site, d(λ1, λ2), as a function

of λ1 and λ2. The critical point λc = 1 occurs as a pinch point in the ground-state fidelity surface. Note that the black line denotes the normalization condition: d(λ, λ) = 1.

Highlights 2010 21 Journal of Physics A: Mathematical and Theoretical

Hilbert spaces from path integrals Looking for symmetric Bell inequalities

Fay Dowker1, Steven Johnston1 and Rafael D Sorkin2,3 Jean-Daniel Bancal1, Nicolas Gisin1 and Stefano Pironio2 1 Theoretical Physics, Blackett Laboratory, Imperial College London, London, 1 Group of Applied Physics, University of Geneva, 20 rue de l’Ecole-de Médecine, SW7 2AZ, UK CH-1211 Geneva 4, Switzerland 2 Perimeter Institute, 31 Caroline Street North, Waterloo ON, N2L 2Y5 Canada 2 Laboratoire d'Information Quantique, Université Libre de Bruxelles, Belgium 3 Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA 2010 J. Phys. A: Math. Theor. 43 385303 2010 J. Phys. A: Math. Theor. 43 275302

Finding all Bell inequalities for a given number of parties, measurement It is shown that a Hilbert space can be constructed for a quantum system settings and measurement outcomes is in general a computationally hard starting from a framework in which histories are fundamental. The task. We show that all Bell inequalities which are symmetric under the decoherence functional provides the inner product on this ‘history Hilbert exchange of parties can be found by examining a symmetrized polytope space’. It is also shown that the history Hilbert space is the standard Hilbert which is simpler than the full Bell polytope. As an illustration of our method, space in the case of non-relativistic quantum mechanics. we generate 238 885 new Bell inequalities and 1085 new Svetlichny inequalities. We find, in particular, facet inequalities for Bell experiments involving two parties and two measurement settings that are not of the Collins–Gisin–Linden–Massar–Popescu type. Imaginary cubic perturbation: numerical and analytic study

Jean Zinn-Justin1 and Ulrich D Jentschura2 1 CEA/IRFU, Centre de Saclay, 91191 Gif-sur-Yvette Cedex, France 2 Missouri University of Science and Technology, Rolla, MO 65409-0640, USA

2010 J. Phys. A: Math. Theor. 43 425301

(a) Example of a polytope P in the vector space R3 The analytic properties of the ground-state resonance energy E(g) of the . (b) Subspace S symmetric under the exchange of coordinates e1 and e2. Ps (grey) is the projection of the polytope onto this subspace. cubic potential are investigated as a function of the complex coupling 3 (c) fs and gs are two facets of Ps, and f and g are their symmetric extensions to the whole space R . parameter g. We explicitly show that it is possible to analytically continue f is a symmetric facet of the original polytope P, whereas g is just a valid inequality for P. E(g) by means of a resummed strong coupling expansion, to the second sheet of the Riemann surface, and we observe a merging of resonance and antiresonance eigenvalues at a critical point along the line arg(g) = 5π/4. In addition, we investigate the convergence of the resummed weak-coupling Quantum sign permutation polytopes expansion in the strong-coupling regime, by means of various modifications of order-dependent mappings (ODMs), that take special properties of the Colin Wilmott, Hermann Kampermann and Dagmar Bruß cubic potential into account. Various ODMs are adapted to different regimes Institut für Theortische Physik III, Heinrich-Heine-Universität, Düsseldorf, Germany of the coupling constant. We also determine a large number of terms of the strong-coupling expansion by resumming the weak-coupling expansion using the ODMs, demonstrating the interpolation between the two regimes made 2010 J. Phys. A: Math. Theor. 43 505306 possible by this summation method.

Convex polytopes are convex hulls of point sets in the n-dimensional space Image in the complex λ n plane of the negative real E that generalize two-dimensional convex polygons and three-dimensional g axis under the mapping convex polyhedra. We concentrate on the class of n-dimensional polytopes g = ρ λ/(1 − λ)5/2, for ρ = in En called sign permutation polytopes. We characterize sign permutation 1/2. A point infinitesimally displaced above the polytopes before relating their construction to constructions over the space negative real axis is mapped of quantum density matrices. Finally, we consider the problem of state onto a point in the upper identification and show how sign permutation polytopes may be useful in complex λ plane. addressing issues of robustness.

Truncated octahedron: sign permutation polytope with the basis vector a = (0, 1, 2) centered at the origin. The truncated octahedron is an Archimedean solid consisting of 8 regular hexagonal facets, 6 square facets, 36 edges and 24 vertices.

22 Highlights 2010 Journal of Physics A: Mathematical and Theoretical

The non-locality of n noisy Popescu–Rohrlich boxes

Matthias Fitzi1, Esther Hänggi1, Valerio Scarani2 and Stefan Wolf1 1 Computer Science Department, ETH Zurich, Switzerland 2 Centre for Quantum Technologies and Department of Physics, National University of Singapore, Singapore

2010 J. Phys. A: Math. Theor. 43 465305

We quantify the amount of non-locality contained in n noisy versions of the so-called Popescu–Rohrlich boxes (PRBs), i.e. bipartite systems violating The ground state (taking |G↑〉 as an example) and the low-lying excitations in a set of decoupled

Ising spin chains which form a square lattice. From left to right and up to down, they are |G↑〉, |Hia〉, the CHSH Bell inequality maximally. Following the approach by Elitzur, d h i i |D a〉, |F a〉, |Wp〉, |WP,Pʹ〉, |WP,Pʹ〉 and |Wpʹ〉. Up and down arrows label the fermion with spin-up and Popescu and Rohrlich, we measure the amount of non-locality of a system by spin-down. The empty circle is unoccupied site and up–down arrow is double occupied. The white s ¯s representing it as a convex combination of a local behaviour, with maximal plaquette P'' has (G P'' ,G P'' ) = (1, 1), the yellow has (−1, 1), the red has (1,−1) and the grey has possible weight, and a non-signalling system. We show that the local part of n (−1,−1). systems, each of which approximates a PRB with probability 1 − ε, is of order Θ(ε [n/2] ) in the isotropic, and equal to (3ε)n in the maximally biased case.

q Field theory and string theory

Anyons emerging from fermions with Chiral correlators of the Ising conformal field conventional two-body interactions theory Yue Yu1 and Yi Li2,3 Eddy Ardonne1 and Germán Sierra2 1 Institute of Theoretical Physics, Chinese Academy of Sciences, PO Box 2735, Beijing 100190, People's Republic of China 1 Nordita, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden 2 Department of Physics, Fudan University, Shanghai 200433, People’s Republic of 2 Instituto de Física Téorica, UAM-CSIC, Madrid, Spain China 3 Department of Physics, University of California, San Diego, CA 92093, USA 2010 J. Phys. A: Math. Theor. 43 505402

2010 J. Phys. A: Math. Theor. 43 105306 We derive explicit expressions for the conformal blocks of the Ising conformal field theory for the correlators of an arbitrary number of primary fields. These Emergent anyons are the key elements of the topological quantum results are obtained from the bosonized description of the Ising model. computation and topological quantum memory. We study a two-component Interestingly, correlators involving Majorana fermions can be obtained in two fermion model with conventional two-body interaction in a fine-tuned external different ways, giving rise to identities between the ‘bosonic’ and ‘fermionic’ field and show that several subsets in the low-lying excitations obey the same description of these correlators. These identities are generalizations of the fusion rules as those of the toric code model. Those string-like (or domain famous Cauchy identity. The conformal blocks of the Ising model are used to wall) excitations whose energy congregates in a small spatial region (a wall) derive the expression for the conformal blocks of the su(2)2 WZW conformal may be thought of as quasiparticles which, in a given subset, obey mutual field theory. semionic statistics. We show how to peel off one of such subsets from other degenerate subsets and manipulate anyons in cold dipolar Fermi atoms or cold dipolar fermionic heteronuclear molecules in optical lattices by means of the established techniques.

Illustration of the macrogroups, corresponding to (1, 3, 6, 8, 9, 12, 14) (2, 4, 5, 7, 10, 11, 13). The value for q = 22 or in binary digits q = 010110.

Highlights 2010 23 Journal of Physics A: Mathematical and Theoretical

Exact computation of one-loop correction to Four-dimensional Yang–Mills theory, gauge the energy of folded spinning string in invariant mass and fluctuating three-branes AdS × S5 5 Antti J Niemi1 and Sergey Slizovskiy2 1 Department of Physics and Astronomy, Uppsala University, PO Box 803, S-75108, M Beccaria1, G V Dunne2, V Forini3, M Pawellek4 and A A Tseytlin5,6,6 Uppsala, Sweden 1 Physics Department, Salento University and INFN, 73100 Lecce, Italy 2 Laboratoire de Mathematiques et Physique Theorique CNRS UMR 6083, Fédération 2 Department of Physics, University of Connecticut, Storrs CT 06269-3046, USA Denis Poisson, Université de Tours, Parc de Grandmont, F37200, Tours, France 3 Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut Am Mühlenberg 1, D-14476 Potsdam, Germany 2010 J. Phys. A: Math. Theor. 43 425402 4 Institut für Theoretische Physik III, Universität Erlangen-Nürnberg, Staudtstr.7, D-91058 Erlangen, Germany 5 The Blackett Laboratory, Imperial College, London SW7 2AZ, UK We are interested in a gauge invariant coupling between four-dimensional 6 Also at Lebedev Institute, Moscow Yang–Mills field and a three-brane that can fluctuate into higher dimensions. For this we interpret the Yang–Mills theory as a higher dimensional bulk 2010 J. Phys. A: Math. Theor. 43 165402 gravity theory with dynamics that is governed by the Einstein action, and with a metric tensor constructed from the gauge field in a manner that displays the original gauge symmetry as an isometry. The brane moves in this higher We consider the one-loop correction to the energy of the folded spinning dimensional spacetime under the influence of its bulk gravity, with dynamics 5 determined by the Nambu action. This introduces the desired interaction string solution in the AdS3 part of AdS5 × S . The classical string solution is expressed in terms of elliptic functions so an explicit computation of between the brane and the gauge field in a way that preserves the original the corresponding fluctuation determinants for generic values of the spin gauge invariance as an isometry of the induced metric. After a prudent change appears to be a non-trivial problem. We show how it can be solved exactly by of variables the result can be interpreted as a gauge invariant and massive 4 using the static-gauge expression for the string partition function (which we vector field that propagates in the original spacetime R . The presence of the demonstrate to be equivalent to the conformal gauge one) and observing that brane becomes entirely invisible, expect for the mass. all the corresponding second-order fluctuation operators can be put into the standard (single-gap) Lamé form. We systematically derive the small-spin and large-spin expansions of the resulting expression for the string energy and comment on some of their applications. Magnon dispersion to four loops in the ABJM and ABJ models

J A Minahan1, O Ohlsson Sax1 and C Sieg2 1 Department of Physics and Astronomy, Uppsala University, SE-751 08 Uppsala, Sweden 2 The Niels Bohr International Academy, The Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen, Denmark

2010 J. Phys. A: Math. Theor. 43 275402

The ABJM model is a superconformal Chern–Simons theory with N = 6 supersymmetry which is believed to be integrable in the planar limit. However, there is a coupling-dependent function that appears in the magnon dispersion relation and the asymptotic Bethe ansatz that is only known to leading order at strong and weak coupling. We compute this function to four Potentials V (red) and V (blue) in (3.31), for k = 0.5, 0.9, 0.99 and 0.999, from bottom to ψ+ ψ− loops in perturbation theory by an explicit Feynman diagram calculation for top. Note that V±(σ ) are identical in from, but are displaced from one another by a half-period π; they are self-isospectral. both the ABJM model and the ABJ extension. The ABJM four-loop correction has mixed transcendentality, while the ABJ extension adds a term to the ABJM correction with highest transcendentality. We then compute the four- loop wrapping correction for a scalar operator in the 20 representation of SU(4) and find that it agrees with a recent prediction of the ABJM Y-system by Gromov, Kazakov and Vieira. We also propose a limit of the ABJ model that might be perturbatively integrable at all loop orders but has a short range Hamiltonian.

24 Highlights 2010 Journal of Physics A: Mathematical and Theoretical

Semionic supersymmetric solitons Asymptotic Bethe equations for open boundaries in planar AdS/CFT Luca Mezincescu1 and Paul K Townsend2 1 Department of Physics, University of Miami, Coral Gables, FL 33124, USA D H Correa1 and C A S Young2 2 Department of Applied Mathematics and Theoretical Physics, Centre for 1 DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge Road, Cambridge CB3 0WA, UK CB3 0WA, UK 2 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

2010 J. Phys. A: Math. Theor. 43 465401 2010 J. Phys. A: Math. Theor. 43 145401

The Bogomolnyi vortex of the N = 2 supersymmetric Abelian–Higgs model in 2 + 1 dimensions is shown to be a ‘semion’ of spin 1/4. Specifically, We solve, by means of a nested coordinate Bethe ansatz, the open- the effective superparticle action for one vortex is shown to describe, upon boundaries scattering theory describing the excitations of a free open string 5 quantization, a parity self-dual centrally charged ‘short’ supermultiplet of propagating in AdS5 × S , carrying large angular momentum J = J56, and ‘relativistic helicities’ (–¼, –¼, ¼, ¼). ending on a maximal giant graviton whose angular momentum is in the same plane. We thus obtain the all-loop Bethe equations describing the spectrum, for J, finite but large, of the energies of such strings, or equivalently, on the gauge side of the AdS/CFT correspondence, the anomalous dimensions of Y-system for scattering amplitudes certain operators built using the ε tensor of SU(N). We also give the Bethe equations for strings ending on a probe D7-brane, corresponding to meson- N Luis F Alday1, Juan Maldacena1, Amit Sever2 and Pedro Vieira2 like operators in an = 2 gauge theory with fundamental matter. 1 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA 2 Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J 2W9, Canada Open string pair creation from worldsheet 2010 J. Phys. A: Math. Theor. 43 485401 instantons

We compute N = 4 super Yang–Mills planar amplitudes at strong coupling Christian Schubert1,2 and Alessandro Torrielli3 1 by considering minimal surfaces in AdS5 space. The surfaces end on a null Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Mühlenberg 1, polygonal contour at the boundary of AdS. We show how to compute the area D-14476 Potsdam, Germany of the surfaces as a function of the conformal cross ratios characterizing the 2 Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de polygon at the boundary. We reduce the problem to a simple set of functional Hidalgo, Edificio C-3, Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacán, México equations for the cross ratios as functions of the spectral parameter. These 3 Institute for Theoretical Physics and Spinoza Institute, Utrecht University, equations have the form of thermodynamic Bethe ansatz (TBA) equations. Leuvenlaan 4, 3584 CE Utrecht, The Netherlands The area is the free energy of the TBA system. We consider any number of gluons and in any kinematic configuration. 2010 J. Phys. A: Math. Theor. 43 402003

Fast Track Communications

Worldline instantons provide a particularly elegant way to derive Schwinger's well-known formula for the pair creation rate due to a constant electric field in quantum electrodynamics. In this communication, we show how to extend this method to the corresponding problem of open string

The polygon is specified at the AdS boundary by the positions of the cusps xi. These positions are related to an ordered sequence of momenta ki by ki = xi−xi−1. The two-dimensional minimal surface stretches in the AdS bulk and ends on the polygonal contour at the boundary. 821,772 Journal of Physics A: Mathematical and Theoretical full-text article downloads in 2010

Highlights 2010 25 Journal of Physics A: Mathematical and Theoretical

q Fluid and plasma theory A spiralling slender non-Newtonian liquid jet emerging from a rapidly rotating orifice is examined. The effect of the non-Newtonian rheology on the trajectory of the jet and its linear instability are determined using a mixture of computational and asymptotic methods. The sizes of the droplets produced Longitudinal wave-breaking limits in a unified by this instability are determined by considering the most unstable wave mode. This enables a quantitative comparison between theoretical and geometric model of relativistic warm plasmas experimental results to be made, by comparing droplet sizes predicted from the theory with experimental measurements. At lower Weber numbers some D A Burton1,2 and A Noble1,3 good points of agreement have been obtained. At higher Weber numbers jet 1 Department of Physics, Lancaster University, Lancaster LA1 4YB, UK break-up becomes increasingly complex and the possibility of break-up being due to absolute, rather than convective, instability is discussed. 2 The Cockcroft Institute, Daresbury WA4 4AD, UK 3 Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK Experimental images of the jet (a) We = 5.37, Rb = 0.552, Oh = 0.0229, Re = 101.00, 2010 J. Phys. A: Math. Theor. 43 075502 ρ = 1025 kgm−3, k = 0.012 Pa sn, n = 0.920; (b) We = 23.50, Rb = 0.588, Oh = 0.0356, Re = 136.11, ρ = 1026 kgm−3, n The covariant Vlasov–Maxwell system is used to study the breaking of k = 0.061 Pa s , n = 0.748 and (c) We = 65.99, Rb = 0.473, Oh = 0.0519, relativistic warm plasma waves. The well-known theory of relativistic warm Re = 156.31, ρ = 1027 kgm−3, plasmas due to Katsouleas and Mori (KM) is subsumed within a unified k = 0.199 Pa sn, n = 0.654. geometric formulation of the ‘waterbag‘ paradigm over spacetime. We calculate the maximum amplitude Emax of nonlinear longitudinal electric waves for a particular class of waterbags whose geometry is a simple three- dimensional generalization (in velocity) of the one-dimensional KM waterbag

(in velocity). It has been shown previously that the value of limv → cEmax (with the effective temperature of the plasma electrons held fixed) diverges for the KM model; however, we show that a certain class of simple three-dimensional waterbags exhibit a finite value for limv → cEmax, where v is the phase velocity of the wave and c is the speed of light. Kinematic magnetic dynamo in a random flow with strong average shear

V R Kogan1,2, I V Kolokolov1,2 and V V Lebedev1,2 1 Landau Institute for Theoretical Physics RAS, 119334 Kosygina 2, Moscow, Russia 2 Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Moscow, Russia

2010 J. Phys. A: Math. Theor. 43 182001

Fast Track Communications

We analyze the kinematic dynamo in a conducting fluid where the stationary shear flow is accompanied by relatively weak random velocity fluctuations. The diffusionless and diffusion regimes are described. The growth rates of the magnetic field moments are related to the statistical characteristics of the flow Two illustrations of the 4-velocity dependence of a particular ‘bowl’ waterbag. The axis of describing the divergence of Lagrangian trajectories. A degree of anisotropy of . symmetry is aligned along χ3, and f equals the positive constant α within the ‘wall’ of the bowl the magnetic field is estimated. We demonstrate that Zeldovich’s ‘antidynamo and vanishes outside. The maximum electric field amplitude is achieved during the oscillation in which the top of the waterbag (a circle) grazes the phase speed of the wave. theorem’ is wrong.

Unstable waves on a curved non-Newtonian liquid jet

V L Hawkins1, C J Gurney2, S P Decent2, M J H Simmons1 and J Uddin2 1 School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK 2 School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

2010 J. Phys. A: Math. Theor. 43 055501 Sketch of typical magnetic blobs during the diffusive kinematic stage.

26 Highlights 2010 Journal of Physics A: Mathematical and Theoretical

2010 special issues

Quantum Phases: 50 years of the Aharonov–Bohm effect and 25 years of the Berry phase

Guest Editors: Mark Dennis, Sandu Popescu and Lev Vaidman 2010 J. Phys. A: Math. Theor. 43 issue 35

This special issue celebrates the discovery of two of the most important aspects of quantum mechanics: the Aharonov–Bohm (AB) effect and the Berry phase. The AB effect and the Berry phase are ubiquitous in modern physics and have implications ranging from the most fundamental to the very applied. This special issue is a snapshot of the state-of-the-art 50 and 25 years, respectively, from their discoveries.

Current trends in integrability and nonlinear phenomena

Guest Editors: David Gómez-Ullate, Sara Lombardo, Manuel Mañas, Marta Mazzocco, Frank Nijhoff and Matteo Sommacal 2010 J. Phys. A: Math. Theor. 43 issue 43

Nonlinear phenomena appear everywhere in nature, their description and understanding is therefore of great interest both from the theoretical and applicative point of view. If a nonlinear phenomenon can be represented by an integrable system then we have at our disposal a variety of tools to achieve a better mathematical description of the phenomenon. This special issue is largely dedicated to investigations of nonlinear phenomena which are related to the concept of integrability, either involving integrable systems themselves or because they use techniques from the theory of integrability.

Spectral and transport properties of quantum systems: in memory of Pierre Duclos (1948–2010)

Guest Editors: Jean-Michel Combes, Pavel Exner and Valentin A Zagrebnov 2010 J. Phys. A: Math. Theor. 43 issue 47

This issue is devoted to Pierre Duclos who passed away suddenly and prematurely in Prague on 12 January 2010. We want to honour his memory in the way he would have liked, by collecting fresh and original work from his area of interest.

Forthcoming special issues for 2011

Quantum integrable models and gauge-string duality Guest Editors: Patrick Dorey, Joseph Minahan and Arkady Tseytlin

Scattering amplitudes in gauge theories: progress and outlook Guest Editors: Radu Roiban, Marcus Spradlin and Anatasia Volovich

Highlights 2010 27 We would like to thank all of our authors, referees, board members and supporters across the world for their vital contribution to the work and progress of Journal of Physics A.

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