Random matrix averages and the impenetrable Bose gas in Dirichlet and Neumann boundary conditions

Forrester, Peter J.; Frankel, N. E.; Garoni, Timothy M.

doi:10.1063/1.1599954

Cited by 21 publications

(48 citation statements)

References 28 publications

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“…We point out that the same class of averages over the groups O + (2N + 1) or O − (2N + 1) result from considering the density matrix for the impenetrable Bose gas in the case of mixed Dirichlet and Neumann boundary conditions. In [9] the sought asymptotics were calculated on the basis of a combination of analytic and log-gas arguments, and a Fisher-Hartwig type generalization (with b r = 0) conjectured. The conjecture of [9] can used to predict the asymptotic form in the case of mixed Dirichlet and Neumann boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

“…In [9] the sought asymptotics were calculated on the basis of a combination of analytic and log-gas arguments, and a Fisher-Hartwig type generalization (with b r = 0) conjectured. The conjecture of [9] can used to predict the asymptotic form in the case of mixed Dirichlet and Neumann boundary conditions. Moreover we show that this asymptotic form can be proved by making use of asymptotic formulas recently obtained [10] for Toeplitz + Hankel determinants det[a j−k + a j+k+1 ] j,k=0,...,n−1 (1.11) in the case of singular generating functions (1.6).…”

Section: Introductionmentioning

confidence: 99%

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Applications and generalizations of Fisher–Hartwig asymptotics

Forrester

Frankel

2004

Journal of Mathematical Physics

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Fisher-Hartwig asymptotics refers to the large n form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin-spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher-Hartwig formula to the asymptotic decay of the Ising correlations above T c , while the study of the Bose gas density matrix leads us to generalize the Fisher-Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the FisherHartwig asymptotic form known for Toeplitz determinants.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Applications and generalizations of Fisher–Hartwig asymptotics

Forrester

Frankel

2004

Journal of Mathematical Physics

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“…we were able to compute the asymptotic form of ρ θ N using the Fisher-Hartwig conjecture [38,39]. For N ≫ 1 the one-particle anyon density matrix reads:…”

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confidence: 99%

Entanglement properties and momentum distributions of hard-core anyons on a ring

Santachiara¹,

Stauffer²,

Cabra³

2007

J. Stat. Mech.

119

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We study the one-particle von Neumann entropy of a system of N hard-core anyons on a ring. The entropy is found to have a clear dependence on the anyonic parameter which characterizes the generalized fractional statistics described by the anyons. This confirms the entanglement is a valuable quantity to investigate topological properties of quantum states. We derive the generalization to anyonic statistics of the Lenard formula for the one-particle density matrix of N hard-core bosons in the large N limit and extend our results by a numerical analysis of the entanglement entropy, providing additional insight into the problem under consideration.In recent years an intense research activity has been devoted to the study of entanglement in many-body states. Initially, this effort has been mostly motivated by the fact that quantum correlated many-body states, which appear in various solid-state models, can be valuable resources for information processing and quantum computation [1,2]. The theory of entanglement is now attracting even more attention because of its fundamental implication for the development of new efficient numerical methods for quantum systems [3,4,5] and for the characterization of quantum critical phases [6,7,8].Generally speaking, entanglement measures nonlocal properties of composite quantum systems and it can provide additional information to that obtained by investigating local observables or traditional correlation functions. In this respect entanglement might be a sensitive probe into the topological properties of quantum states. A particularly significant quantity is the entanglement entropy S A , which is defined in a bipartite system A ∪ B and quantified as the von Neumann entropy S A = −Trρ A ln ρ A associated to the reduced density matrix ρ A of a subsystem A. In two-dimensional systems a firm connection between topological order and entanglement entropy has been established in [9,10], where the entanglement entropy was defined by spatial partitioning. Recent studies on Laughlin states [11,12] have considered the entanglement entropy associated to particle partitioning [11,12]. Also in this case, the entanglement entropy turns out to reveal important aspects of the topological order in Laughlin States.The two-dimensional case is of particular interest due to the existence of models whose elementary excitations exhibit generalized fractional statistics. Anyons, the particles obeying such statistics, play a fundamental role in the description of the fractional quantum Hall effect [13]. Although this concept is essentially twodimensional, anyons can also occur in one-dimensional (1D) systems [14,15,16,17,18,19,20], where statistics and interactions are inextricable, leading to strong shortrange correlations. The 1D anyonic models have proven useful to study persistent charge and magnetic currents in 1D mesoscopic rings [15]. This possibility and their own pure theoretical interest lead us to investigate the effects of the anyonic statistics on the entanglement entropy in the present Letter....

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“…Rewriting (2.5) in the form 10) recalling the explicit form of T from (2.1), and equating successive powers of λ on both sides gives…”

Section: Calculation Of a Jacobian For Tridiagonal Matricesmentioning

confidence: 99%

“…[8]). Like their counterparts from U (N ), such random matrices are of fundamental importance in applications of random matrix theory to combinatorial models [20,3], analytic number theory [16] and the quantum many body problem [10].…”

Section: Introductionmentioning

confidence: 99%

Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices

Forrester

Rains

2006

International Mathematics Research Notices

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In a recent work Killip and Nenciu gave random recurrences for the characteristic polynomials of certain unitary and real orthogonal upper Hessenberg matrices. The corresponding eigenvalue p.d.f.'s are β-generalizations of the classical groups. Left open was the direct calculation of certain Jacobians. We provide the sought direct calculation. Furthermore, we show how a multiplicative rank 1 perturbation of the unitary Hessenberg matrices provides a joint eigenvalue p.d.f. generalizing the circular β-ensemble, and we show how this joint density is related to known inter-relations between circular ensembles. Projecting the joint density onto the real line leads to the derivation of a random three-term recurrence for polynomials with zeros distributed according to the circular Jacobi β-ensemble.

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Random matrix averages and the impenetrable Bose gas in Dirichlet and Neumann boundary conditions

Cited by 21 publications

References 28 publications

Applications and generalizations of Fisher–Hartwig asymptotics

Applications and generalizations of Fisher–Hartwig asymptotics

Entanglement properties and momentum distributions of hard-core anyons on a ring

Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices

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