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September 2009

Volume 50, Issue 9,  Articles (09xxxx)

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SPECIAL ISSUE: INTEGRABLE QUANTUM SYSTEMS AND SOLVABLE STATISTICAL MECHANICS MODELS

Introduction to Special Issue: Integrable Quantum Systems and Solvable Statistical Mechanics Models

Alexander Its, Michio Jimbo, and Jean-Michel Maillet

J. Math. Phys. 50, 095101 (2009) (1 page)

Online Publication Date: 27 August 2009

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05.40.-a, 02.50.Ga, 02.50.Cw

The noncommutative sine-Gordon breather

André Fischer and Olaf Lechtenfeld

J. Math. Phys. 50, 095201 (2009) (10 pages)

Online Publication Date: 9 April 2009

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As shown by Lechtenfeld et al. [Nucl. Phys. B 705, 447 (2005)], there exists a noncommutative deformation of the sine-Gordon model which remains (classically) integrable but features a second scalar field. We employ the dressing method (adapted to the Moyal-deformed situation) for constructing the deformed kink-antikink and breather configurations. Explicit results and plots are presented for the leading noncommutativity correction to the breather. Its temporal periodicity is unchanged.
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11.10.Lm, 11.10.Nx, 11.15.-q

Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II. Painlevé transcendent potentials

Ian Marquette

J. Math. Phys. 50, 095202 (2009) (18 pages)

Online Publication Date: 9 April 2009

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We consider a superintegrable quantum potential in two-dimensional Euclidean space with a second and a third order integral of motion. The potential is written in terms of the fourth Painlevé transcendent. We construct for this system a cubic algebra of integrals of motion. The algebra is realized in terms of parafermionic operators and we present Fock-type representations which yield the corresponding energy spectra. We also discuss this potential from the point of view of higher order supersymmetric quantum mechanics and obtain ground state wave functions.
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03.65.Fd, 11.30.Pb

Restricted partition functions of the two-dimensional Ising model on a half-infinite cylinder

Louis-Pierre Arguin and Yvan Saint-Aubin

J. Math. Phys. 50, 095203 (2009) (8 pages)

Online Publication Date: 20 April 2009

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We study the Ising model on a half-infinite cylinder at the critical temperature. On the boundary circle, we fix four intervals of constant signs. Let theta1, theta2, theta3, and theta4 be the positions where the flips occur, labeled counterclockwise in that order. From each theta starts a contour between clusters of opposite signs. The contour leaving theta1 may end only at theta2 or theta4. Using an argument based on conformal field theory, we give the probability distribution that the contour leaving theta1 ends at theta2. The behavior of this function when theta2−theta1-->0 is described by a power law with an exponent ((5/3)) that belongs to the Kac table but that corresponds to a nonunitarizable highest-weight representation. We check that this prediction agrees with a Monte Carlo simulation.
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05.50.+q
FREE

Total current fluctuations in the asymmetric simple exclusion process

Craig A. Tracy and Harold Widom

J. Math. Phys. 50, 095204 (2009) (4 pages)

Online Publication Date: 8 June 2009

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A limit theorem for the total current in the asymmetric simple exclusion process (ASEP) with step initial condition is proven. This extends the result of Johansson on TASEP to ASEP.
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05.40.-a, 02.50.Ga, 02.50.Cw

Fine structure of the asymptotic expansion of cyclic integrals

K. K. Kozlowski

J. Math. Phys. 50, 095205 (2009) (11 pages)

Online Publication Date: 11 June 2009

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The asymptotic expansion of n-dimensional cyclic integrals was expressed as a series of functionals acting on the symmetric function involved in the cyclic integral. In this article, we give an explicit formula for the action of these functionals on a specific class of symmetric functions. These results are necessary for the computation of the O(1) part in the long-distance asymptotic behavior of correlation functions in integrable models.
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02.30.Sa, 02.30.Rz

Completeness of a fermionic basis in the homogeneous XXZ model

H. Boos, M. Jimbo, T. Miwa, and F. Smirnov

J. Math. Phys. 50, 095206 (2009) (10 pages)

Online Publication Date: 11 June 2009

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With the aid of the creation operators introduced in our previous works, we show how to construct a basis of the space of quasilocal operators for the homogeneous XXZ chain.
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05.30.Fk, 02.30.Tb

Light-cone matrix product

M. B. Hastings

J. Math. Phys. 50, 095207 (2009) (17 pages)

Online Publication Date: 11 June 2009

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We show how to combine the light-cone and matrix product algorithms to simulate quantum systems far from equilibrium for long times. For the case of the XXZ spin chain at Delta=0.5, we simulate to a time of [approximate]22.5. While part of the long simulation time is due to the use of the light-cone method, we also describe a modification of the infinite time-evolving bond decimation algorithm with improved numerical stability, and we describe how to incorporate symmetry into this algorithm. While statistical sampling error means that we are not yet able to make a definite statement, the behavior of the simulation at long times indicates the appearance of either “revivals” in the order parameter as predicted by Hastings and Levitov (e-print arXiv:0806.4283) or of a distinct shoulder in the decay of the order parameter.
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75.10.Jm, 75.10.Pq, 02.60.-x, 02.10.Yn, 03.65.Fd, 03.67.Ac, 03.65.Ud

Thermodynamic limit for the Mallows model on Sn

Shannon Starr

J. Math. Phys. 50, 095208 (2009) (15 pages)

Online Publication Date: 10 July 2009

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The Mallows model on Sn is a probability distribution on permutations, qd(pi,e)/Pn(q), where d(pi,e) is the distance between pi and the identity element, relative to the Coxeter generators. Equivalently, it is the number of inversions: pairs (i,j) where 1<=i<j<=n, but pii>pij. Analyzing the normalization Pn(q), Diaconis and Ram calculated the mean and variance of d(pi,e) in the Mallows model, which suggests that the appropriate n-->[infinity] limit has qn scaling as 1−beta/n. We calculate the distribution of the empirical measure in this limit, u(x,y)dxdy=limn-->[infinity](1/n)[summation]<sub>i = 1</sub><sup>n</sup>delta(i,pii). Treating it as a mean-field problem, analogous to the Curie–Weiss model, the self-consistent mean-field equations are ([partial-derivative]2/[partial-derivative]x[partial-derivative]y)ln u(x,y)=2betau(x,y), which is an integrable partial differential equation, known as the hyperbolic Liouville equation. The explicit solution also gives a new proof of formulas for the blocking measures in the weakly asymmetric exclusion process and the ground state of the [script U]q([fraktur s][fraktur l]2)-symmetric XXZ ferromagnet.
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05.70.-a, 02.60.Nm, 02.60.Lj, 02.50.Cw, 02.30.Jr

On the thermodynamic limit of form factors in the massless XXZ Heisenberg chain

N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, and V. Terras

J. Math. Phys. 50, 095209 (2009) (24 pages)

Online Publication Date: 14 July 2009

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We consider the problem of computing form factors of the massless XXZ Heisenberg spin-1/2 chain in a magnetic field in the (thermodynamic) limit where the size M of the chain becomes large. For that purpose, we take the particular example of the matrix element of the operator sigmaz between the ground state and an excited state with one particle and one hole located at the opposite ends of the Fermi interval (umklapp-type term). We exhibit its power-law decrease in terms of the size of the chain M and compute the corresponding exponent and amplitude. As a consequence, we show that this form factor is directly related to the amplitude of the leading oscillating term in the long-distance asymptotic expansion of the correlation function <sigma<sub>1</sub><sup>z</sup>sigma<sub>m + 1</sub><sup>z</sup>>.
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75.10.Jm, 75.10.Pq

Paths and partitions: Combinatorial descriptions of the parafermionic states

Pierre Mathieu

J. Math. Phys. 50, 095210 (2009) (43 pages)

Online Publication Date: 22 July 2009

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The [script Z]k parafermionic conformal field theories, despite the relative complexity of their modes algebra, offer the simplest context for the study of the bases of states and their different combinatorial representations. Three bases are known. The classic one is given by strings of the fundamental parafermionic operators whose sequences of modes are in correspondence with restricted partitions with parts at distance k−1 differing at least by 2. Another basis is expressed in terms of the ordered modes of the k−1 different parafermionic fields, which are in correspondence with the so-called multiple partitions. Both types of partitions have a natural (Bressoud) path representation. Finally, a third basis, formulated in terms of different paths, is inherited from the solution of the restricted solid-on-solid model of Andrews–Baxter–Forrester. The aim of this work is to review, in a unified and pedagogical exposition, these four different combinatorial representations of the states of the [script Z]k parafermionic models. The first part of this article presents the different paths and partitions and their bijective relations; it is purely combinatorial, self-contained, and elementary; it can be read independently of the conformal-field-theory applications. The second part links this combinatorial analysis with the bases of states of the [script Z]k parafermionic theories. With the prototypical example of the parafermionic models worked out in detail, this analysis contributes to fix some foundations for the combinatorial study of more complicated theories. Indeed, as we briefly indicate in ending, generalized versions of both the Bressoud and the Andrews–Baxter–Forrester paths emerge naturally in the description of the minimal models.
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05.30.Fk

Transport and control in one-dimensional systems

Lea F. Santos

J. Math. Phys. 50, 095211 (2009) (8 pages)

Online Publication Date: 28 July 2009

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We study transport of local magnetization in a Heisenberg spin-1/2 chain at zero temperature. The system is initially prepared in a highly excited pure state far from equilibrium and its evolution is analyzed via exact diagonalization. Integrable and nonintegrable regimes are obtained by adjusting the parameters of the Hamiltonian, which allows for the comparison of transport behaviors in both limits. In the presence of nearest-neighbor interactions only, the transport behavior in the integrable clean system contrasts with the chaotic chain with on-site defects, oscillations in the first suggesting ballistic transport, and a fast decay in the latter indicating diffusive transport. The results for a nonintegrable system with frustration are less conclusive, similarities with the integrable chain being verified. We also show how methods of quantum control may be applied to chaotic systems to induce a desired transport behavior, such as that of an integrable system.
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73.23.Ad, 75.60.Ej, 75.10.Jm

Bethe ansatz approach to quench dynamics in the Richardson model

Alexandre Faribault, Pasquale Calabrese, and Jean-Sébastien Caux

J. Math. Phys. 50, 095212 (2009) (14 pages)

Online Publication Date: 13 August 2009

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By instantaneously changing a global parameter in an extended quantum system, an initially equilibrated state will afterwards undergo a complex nonequilibrium unitary evolution whose description is extremely challenging. A nonperturbative method giving a controlled error in the long time limit remained highly desirable to understand general features of the quench induced quantum dynamics. In this paper we show how integrability (via the algebraic Bethe ansatz) gives one numerical access, in a nearly exact manner, to the dynamics resulting from a global interaction quench of an ensemble of fermions with pairing interactions (Richardson's model). This possibility is deeply linked to the specific structure of this particular integrable model which gives simple expressions for the scalar product of eigenstates of two different Hamiltonians. We show how, despite the fact that a sudden quench can create excitations at any frequency, a drastic truncation of the Hilbert space can be carried out therefore allowing access to large systems. The small truncation error which results does not change with time and consequently the method grants access to a controlled description of the long time behavior which is a hard to reach limit with other numerical approaches.
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03.65.Ge, 02.10.Ud, 05.30.Fk, 02.30.Rz

Approximating the ground state of gapped quantum spin systems

Eman Hamza, Spyridon Michalakis, Bruno Nachtergaele, and Robert Sims

J. Math. Phys. 50, 095213 (2009) (16 pages)

Online Publication Date: 9 September 2009

Full Text: PDF (175 kB)

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We consider quantum spin systems defined on finite sets V equipped with a metric. In typical examples, V is a large, but finite subset of [openface Z]d. For finite range Hamiltonians with uniformly bounded interaction terms and a unique, gapped ground state, we demonstrate a locality property of the corresponding ground state projector. In such systems, this ground state projector can be approximated by the product of observables with quantifiable supports. In fact, given any subset [script X][subset or is implied by]V the ground state projector can be approximated by the product of two projections, one supported on [script X] and one supported on [script X]c, and a bounded observable supported on a boundary region in such a way that as the boundary region increases, the approximation becomes better. This result generalizes to multidimensional models, a result of Hastings that was an important part of his proof of an area law in one dimension [“An area law for one dimensional quantum systems,” J. Stat. Mech.: Theory Exp. 2007, 08024].
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03.65.Ud, 75.10.Dg, 03.67.Ac

Correlation functions of integrable models: A description of the ABACUS algorithm

Jean-Sébastien Caux

J. Math. Phys. 50, 095214 (2009) (18 pages)

Online Publication Date: 24 September 2009

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Recent developments in the theory of integrable models have provided the means of calculating dynamical correlation functions of some important observables in systems such as Heisenberg spin chains and one-dimensional atomic gases. This article explicitly describes how such calculations are generally implemented in the ABACUS C++ library, emphasizing the universality in treatment of different cases coming as a consequence of unifying features within the Bethe ansatz.
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03.65.Ta, 02.30.Sa, 02.30.Rz

A commutative algebra on degenerate [openface C][openface P]1 and Macdonald polynomials

B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, and S. Yanagida

J. Math. Phys. 50, 095215 (2009) (42 pages)

Online Publication Date: 30 September 2009

Full Text: PDF (507 kB)

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We introduce a unital associative algebra [script A] associated with degenerate [openface C][openface P]1. We show that [script A] is a commutative algebra and whose Poincaré series is given by the number of partitions. Thereby, we can regard [script A] as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices 11, 531 (1997)]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using [script A] and the Heisenberg representation of the commutative family studied by Shiraishi [ Commun. Math. Phys. 263, 439 (2006)]. It is found that the Ding–Iohara algebra [Lett. Math. Phys. 41, 183 (1997)] provides us with an algebraic framework for the free field construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting [Leningrad Math. J. 1, 1419 (1990)] in the sence of Babelon-Bernard–Billey [Phys. Lett. B. 375, 89 (1996)], the Ruijsenaars difference operator [Commun. Math. Phys. 110, 191 (1987)], and the operator M(q,t1,t2) of Okounkov–Pandharipande [e-print arXiv:math-ph/0411210].
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02.10.Ud, 02.10.De, 02.30.Tb

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