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Koprucki, Thomas; Wagner, Heinz-Jürgen

doi:10.1023/a:1018683727464

Cited by 8 publications

(7 citation statements)

References 11 publications

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“…The present system of interacting Brownian bridges is temporally inhomogeneous, however, and the Kolmogorov equation is mapped into a Schrödinger-type equation with time-dependent Hamiltonian and the time-dependent ground energy [17]. The obtained quantum system is elliptic, but different from the elliptic Calogero-Moser-Sutherland model extensively studied as a quantum integrable system [7,25,31,39,40]. We found at the same time that the interaction among particles vanishes when the parameter is chosen to be a special value (β = 2 in our case) as found in the usual Calogero-Moser-Sutherland models.…”

Section: Introduction and Main Resultsmentioning

confidence: 98%

Elliptic Determinantal Processes and Elliptic Dyson Models

Katori¹

2017

SIGMA

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Abstract. We introduce seven families of stochastic systems of interacting particles in onedimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families A N −1 , B N , C N and D N , we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Here we use the notion of martingales in probability theory and the elliptic determinant evaluations of the Macdonald denominators of irreducible reduced affine root systems given by Rosengren and Schlosser.

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Section: Introduction and Main Resultsmentioning

confidence: 98%

Elliptic Determinantal Processes and Elliptic Dyson Models

Katori¹

2017

SIGMA

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“…The study of such models was initiated by Sutherland and Calogero [68,70,116] and extended in Refs. [117][118][119]. We recall how models with this property are constructed using a slightly more general approach, and give a list of corresponding Hamiltonians in the Appendix A.…”

Section: Tablementioning

confidence: 99%

“…The next extension generalizes the models on the fourth line of the Table 1 which are related to the JβE. The interaction potential W (x, y) is the same as in Table 1, but the potential V (x) contains additional cos x and cos 2x terms, see formula (119). The ground state wave function has the form (5) with v(x) = c 1 log sin x 2 + c 2 log cos x 2 + c 3 cos x.…”

Section: Tablementioning

confidence: 99%

Full counting statistics for interacting trapped fermions

et al. 2021

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We study NN spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index \betaβ. In the fermion model \betaβ controls the strength of the interaction, \beta=2β=2 corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions N_DND in a domain DD of macroscopic size in the bulk of the Fermi gas. We predict that for general \betaβ the variance of N_DND grows as A_{\beta} \log N + B_{\beta}AβlogN+Bβ for N \gg 1N≫1 and we obtain a formula for A_\betaAβ and B_\betaBβ. This is based on an explicit calculation for \beta\in\left\{ 1,2,4\right\}β∈{1,2,4} and on a conjecture that we formulate for general \betaβ. This conjecture further allows us to obtain a universal formula for the higher cumulants of N_DND. Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter K = 2/\betaK=2/β, and allow to go beyond. In addition we present families of interacting fermion models in one dimension which, in their ground states, can be mapped onto random matrix models. We obtain the mean fermion density for these models for general interaction parameter \betaβ. In some cases the fermion density exhibits interesting transitions, for example we obtain a noninteracting fermion formulation of the Gross-Witten-Wadia model.

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“…[29]. The latter classification was finally completed several years later by Koprucki and Wagner [31], who obtained Forrester's model as a particular case.…”

Section: Introductionmentioning

confidence: 95%

“…In the classification of long-range interaction potentials with two-body interactions and Jastrow-type ground state performed in Refs. [28,29,31], the rational, trigonometric and hyperbolic solutions are obtained precisely from the three choices of the function ϕ used in the previous examples. In fact, in this case there is an additional solution given 13…”

Section: Examplesmentioning

confidence: 99%

Jastrow-like ground states for quantum many-body potentials with near-neighbors interactions

et al. 2018

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We completely solve the problem of classifying all one-dimensional quantum potentials with nearest-and next-to-nearest-neighbors interactions whose ground state is Jastrowlike, i.e., of Jastrow type but depending only on differences of consecutive particles. In particular, we show that these models must necessarily contain a three-body interaction term, as was the case with all previously known examples. We discuss several particular instances of the general solution, including a new hyperbolic potential and a model with elliptic interactions which reduces to the known rational and trigonometric ones in appropriate limits.Here x ≡ (x 1 , . . . , x N ) andfor the (harmonic) Calogero model, whilefor the Sutherland model (with ω > 0 and a > −1/2). As first noted by Sutherland [2, 23], this property makes it possible to compute in closed form certain correlation functions of the latter models by exploiting their connection with random matrix theory. Indeed, for Calogero's model ψ 2 coincides with the joint probability density of the eigenvalues of the Gaussian orthogonal, unitary and symplectic ensembles respectively for 2ω = 2a = 1, 2, 4, while the same relation holds for the ground state of the Sutherland model and Dyson's circular unitary ensembles (with eigenvalues parametrized as e 2ix k ) [24]. Later on, Dyson [25] showed how to construct analogues of the Gaussian ensembles with eigenvalues distributed according to ψ 2 in Eq. (1.1), with essentially arbitrary ρ and χ(x) = |x| β/2 (as usual, β will be assumed to take the values 1, 2, 4 for the orthogonal, unitary and symplectic ensembles, respectively). In view of these results, it is natural to look for the most general quantum Hamiltonian of Calogero-Sutherland (CS) type (i.e, with one-and two-body long-range interactions) whose ground state is of the form (1.1). A restricted version of this problem (with ρ = 1) was already formulated by Sutherland himself [2], who later found a solution thereof with an elliptic two-body interaction potential [26,27]. Shortly afterwards, Calogero [28] showed that this is in fact the most general solution of this restricted problem. The general problem (with ρ not necessarily equal to 1) was tackled by Inozemtsev and Meshcheryakov [29], who claimed to have found a complete solution. A decade later, however, Forrester [30] found a model of CS type whose ground state, which exhibits longrange crystalline order in the thermodynamic limit, is of the factorized form (1.1) and yet did not appear in the classification of Ref. [29]. The latter classification was finally completed several years later by Koprucki and Wagner [31], who obtained Forrester's model as a particular case.The probability distribution p β (s) of the (normalized) spacing s between two consecutive eigenvalues of the Gaussian β-ensembles is approximately given by Wigner's surmise p β (s) = A β s β e −c β s 2 , where the positive parameters A β , c β are fixed by normalization and the condition that the mean spacing be equal to 1 (see, e.g., Refs. [32,33]). By contra...

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Untitled

Cited by 8 publications

References 11 publications

Elliptic Determinantal Processes and Elliptic Dyson Models

Elliptic Determinantal Processes and Elliptic Dyson Models

Full counting statistics for interacting trapped fermions

Jastrow-like ground states for quantum many-body potentials with near-neighbors interactions

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