Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2104.08749 · Journal of Physics A Mathematical and Theoretical

Exact matrix elements of the field operator in the thermodynamic limit of the Lieb–Liniger model

Eldad Bettelheim

Published Aug 13, 2021• 0 citations
arXivHTMLPDF

Abstract

Abstract We study a matrix element of the field operator in the Lieb–Liniger model using the Bethe ansatz technique coupled with a functional approach to compute Slavnov determinants. We obtain the matrix element exactly in the thermodynamic limit for any coupling constant c , and compare our results to known semiclassics at the limit c → 0

DOI
10.1088/1751-8121/ac1d8c

Related papers

Exact formulas for the form factors of local operators in the Lieb–Liniger model

Lorenzo Piroli, Pasquale Calabrese

We present exact formulas for the form factors of local operators in the repulsive Lieb–Liniger model at finite size. These are essential ingredients for both numerical and analytical calculations. From the theory of algebraic Bethe ansatz, it is known that the form factors of local operators satisfy a particular type of recursive relations. We show that in some cases these relations can be used directly to derive practical expressions in terms of the determinant of a matrix whose dimension scales linearly with the system size. Our main results are determinant formulas for the form factors of the operators and Ψ R (0), for arbitrary integer R , where Ψ, are the usual field operators. From these expressions, we also derive the infinite size limit of the form factors of these local operators in the attractive regime.

arXiv:1506.06539•Journal of Physics A Mathematical and Theoretical•Oct 20, 2015•44 citations

Determinant formula for the field form factor in the anyonic Lieb–Liniger model

Lorenzo Piroli, Stefano Scopa, Pasquale Calabrese

Abstract We derive an exact formula for the field form factor in the anyonic Lieb–Liniger model, valid for arbitrary values of the interaction c, anyonic parameter κ, and number of particles N. Analogously to the bosonic case, the form factor is expressed in terms of the determinant of an N × N matrix, whose elements are rational functions of the Bethe quasimomenta but explicitly depend on κ. The formula is efficient to evaluate, and provide an essential ingredient for several numerical and analytical calculations. Its derivation consists of three steps. First, we show that the anyonic form factor is equal to the bosonic one between two special off-shell Bethe states, in the standard Lieb–Liniger model. Second, we characterize its analytic properties and provide a set of conditions that uniquely specify it. Finally, we show that our determinant formula satisfies these conditions.

arXiv:2002.12629•Journal of Physics A Mathematical and Theoretical•May 20, 2020•5 citations

One-dimensional Lieb-Liniger Bose gas as nonrelativistic limit of the sinh-Gordon model

Márton Kormos, Giuseppe Mussardo, Andrea Trombettoni

The repulsive Lieb-Liniger model can be obtained as the nonrelativistic limit of the sinh-Gordon model: all physical quantities of the latter model ($S$-matrix, Lagrangian, and operators) can be put in correspondence with those of the former. We use this mapping, together with the thermodynamical Bethe ansatz equations and the exact form factors of the sinh-Gordon model, to set up a compact and general formalism for computing the expectation values of the Lieb-Liniger model both at zero and finite temperatures. The computation of one-point correlators is thoroughly detailed and when possible compared with known results in the literature.

arXiv:0912.3502•Physical Review A•Apr 12, 2010•99 citations

Determinant formula for the field form factor in the anyonic Lieb-Liniger model

Piroli, L., Scopa, S., Pasquale Calabrese

We derive an exact formula for the field form factor in the anyonic Lieb-Liniger model, valid for arbitrary values of the interaction $c$, anyonic parameter $\kappa$, and number of particles $N$. Analogously to the bosonic case, the form factor is expressed in terms of the determinant of a $N\times N$ matrix, whose elements are rational functions of the Bethe quasimomenta but explicitly depend on $\kappa$. The formula is efficient to evaluate, and provide an essential ingredient for several numerical and analytical calculations. Its derivation consists of three steps. First, we show that the anyonic form factor is equal to the bosonic one between two special off-shell Bethe states, in the standard Lieb-Liniger model. Second, we characterize its analytic properties and provide a set of conditions that uniquely specify it. Finally, we show that our determinant formula satisfies these conditions.

•Oct 1, 2020•9 citations

Density form factors of the 1D Bose gas for finite entropy states

Jacopo De Nardis, Miłosz Panfil

We consider the Lieb–Liniger model for a gas of bosonic δ -interacting particles. Using Algebraic Bethe Ansatz results we compute the thermodynamic limit of the form factors of the density operator between finite entropy eigenstates such as finite temperature states or generic non-equilibrium highly excited states. These form factors are crucial building blocks to obtain the thermodynamic exact dynamic correlation functions of such physically relevant states. As a proof of principle we compute an approximated dynamic structure factor by including only the simplest types of particle-hole excitations and show the agreement with known results.

arXiv:1411.4537•Journal of Statistical Mechanics Theory and Experiment•Feb 25, 2015•67 citations

Dynamics of the attractive 1D Bose gas: analytical treatment from integrability

Pasquale Calabrese, Jean-Sébastien Caux

The physics of the attractive one-dimensional Bose gas (Lieb–Liniger model) is investigated with techniques based on the integrability of the system. Combining a knowledge of particle quasi-momenta to exponential precision in the system size with determinant representations of matrix elements of local operators coming from the algebraic Bethe ansatz, we obtain rather general analytical results for the zero-temperature dynamical correlation functions of the density and field operators. Our results thus provide quantitative predictions for possible future experiments in atomic gases or optical waveguides.

arXiv:0707.4115•Journal of Statistical Mechanics Theory and Experiment•Aug 28, 2007•51 citations
Abstract We study a matrix element of the field operator in the Lieb–Liniger model using the Bethe ansatz technique coupled with a functional approach to compute Slavnov determinants. We obtain the matrix element exactly in the thermodynamic limit for any coupling constant c , and compare our results to known semiclassics at the limit c → 0
We present exact formulas for the form factors of local operators in the repulsive Lieb–Liniger model at finite size. These are essential ingredients for both numerical and analytical calculations. From the theory of algebraic Bethe ansatz, it is known that the form factors of local operators satisfy a particular type of recursive relations. We show that in some cases these relations can be used directly to derive practical expressions in terms of the determinant of a matrix whose dimension scales linearly with the system size. Our main results are determinant formulas for the form factors of the operators and Ψ R (0), for arbitrary integer R , where Ψ, are the usual field operators. From these expressions, we also derive the infinite size limit of the form factors of these local operators in the attractive regime.
Abstract We derive an exact formula for the field form factor in the anyonic Lieb–Liniger model, valid for arbitrary values of the interaction c, anyonic parameter κ, and number of particles N. Analogously to the bosonic case, the form factor is expressed in terms of the determinant of an N × N matrix, whose elements are rational functions of the Bethe quasimomenta but explicitly depend on κ. The formula is efficient to evaluate, and provide an essential ingredient for several numerical and analytical calculations. Its derivation consists of three steps. First, we show that the anyonic form factor is equal to the bosonic one between two special off-shell Bethe states, in the standard Lieb–Liniger model. Second, we characterize its analytic properties and provide a set of conditions that uniquely specify it. Finally, we show that our determinant formula satisfies these conditions.
The repulsive Lieb-Liniger model can be obtained as the nonrelativistic limit of the sinh-Gordon model: all physical quantities of the latter model (SSS-matrix, Lagrangian, and operators) can be put in correspondence with those of the former. We use this mapping, together with the thermodynamical Bethe ansatz equations and the exact form factors of the sinh-Gordon model, to set up a compact and general formalism for computing the expectation values of the Lieb-Liniger model both at zero and finite temperatures. The computation of one-point correlators is thoroughly detailed and when possible compared with known results in the literature.
We derive an exact formula for the field form factor in the anyonic Lieb-Liniger model, valid for arbitrary values of the interaction ccc, anyonic parameter κ\kappaκ, and number of particles NNN. Analogously to the bosonic case, the form factor is expressed in terms of the determinant of a N×NN\times NN×N matrix, whose elements are rational functions of the Bethe quasimomenta but explicitly depend on κ\kappaκ. The formula is efficient to evaluate, and provide an essential ingredient for several numerical and analytical calculations. Its derivation consists of three steps. First, we show that the anyonic form factor is equal to the bosonic one between two special off-shell Bethe states, in the standard Lieb-Liniger model. Second, we characterize its analytic properties and provide a set of conditions that uniquely specify it. Finally, we show that our determinant formula satisfies these conditions.
We consider the Lieb–Liniger model for a gas of bosonic δ -interacting particles. Using Algebraic Bethe Ansatz results we compute the thermodynamic limit of the form factors of the density operator between finite entropy eigenstates such as finite temperature states or generic non-equilibrium highly excited states. These form factors are crucial building blocks to obtain the thermodynamic exact dynamic correlation functions of such physically relevant states. As a proof of principle we compute an approximated dynamic structure factor by including only the simplest types of particle-hole excitations and show the agreement with known results.
The physics of the attractive one-dimensional Bose gas (Lieb–Liniger model) is investigated with techniques based on the integrability of the system. Combining a knowledge of particle quasi-momenta to exponential precision in the system size with determinant representations of matrix elements of local operators coming from the algebraic Bethe ansatz, we obtain rather general analytical results for the zero-temperature dynamical correlation functions of the density and field operators. Our results thus provide quantitative predictions for possible future experiments in atomic gases or optical waveguides.