Duality between Weak and Strong Interactions in Quantum Gases

Granet, Etienne; Bertini, Bruno; Eßler, Fabian H. L.

doi:10.1103/physrevlett.128.021604

Cited by 11 publications

(19 citation statements)

References 65 publications

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“…the spin-1/2 Heisenberg XXZ chain. Third, our framework is readily generalized to quench dynamics [115] by combining it with the quench action approach [89,90]. Here the novel feature is that the spectral sum involves "overlaps" that multiply the form factors.…”

Section: Discussionmentioning

confidence: 99%

A systematic $1/c$-expansion of form factor sums for dynamical correlations in the Lieb-Liniger model

Granet,

Essler

2020

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We introduce a framework for calculating dynamical correlations in the Lieb-Liniger model in arbitrary energy eigenstates and for all space and time, that combines a Lehmann representation with a 1/c expansion. The n th term of the expansion is of order 1/c n and takes into account all n 2 +1 particle-hole excitations over the averaging eigenstate. Importantly, in contrast to a "bare" 1/c expansion it is uniform in space and time. The framework is based on a method for taking the thermodynamic limit of sums of form factors that exhibit non integrable singularities. We expect our framework to be applicable to any local operator.We determine the first three terms of this expansion and obtain an explicit expression for the density-density dynamical correlations and the dynamical structure factor at order 1/c 2 . We apply these to finite-temperature equilibrium states and non-equilibrium steady states after quantum quenches. We recover predictions of (nonlinear) Luttinger liquid theory and generalized hydrodynamics in the appropriate limits, and are able to compute sub-leading corrections to these.

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Section: Discussionmentioning

confidence: 99%

A systematic $1/c$-expansion of form factor sums for dynamical correlations in the Lieb-Liniger model

Granet,

Essler

2020

Preprint

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“…We will call η this pointlike interaction. Depending on the regularization, this η potential can act like an infinite wall for bosons [14,25] or be transparent [28].…”

Section: The Potentialmentioning

confidence: 99%

“…In higher dimensions the definition and regularization of the δ potential is more involved [9][10][11]. But in 1D there also exist other pointlike interactions that find applications and that require a more involved functional form and scaling [12][13][14].…”

Section: Introductionmentioning

confidence: 99%

“…However their definition therein is only formal and it is known that the connection conditions obtained by integrating the Schrödinger equation with higher derivatives δ functions are often incorrect, see e.g. [16] and the appendix of [14]. To the best of our knowledge, our paper is the first to define energy-dependent potentials constructed from usual, regular and hermitian potentials in non-relativistic quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%

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Regularization of energy-dependent pointlike interactions in 1D quantum mechanics

Granet

2022

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We construct a family of hermitian potentials in 1D quantum mechanics that converges in the zero-range limit to a δ interaction with an energy-dependent coupling. It falls out of the standard four-parameter family of pointlike interactions in 1D. Such classification was made by requiring the pointlike interaction to be hermitian. But we show that although our Hamiltonian is hermitian for the standard inner product when the range of the potential is finite, it becomes hermitian for a different inner product in the zero-range limit. This inner product attributes a finite probability (and not probability density) for the particle to be exactly located at the position of the potential. Such pointlike interactions can then be used to construct potentials with a finite support with an energydependent coupling.

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“…)) : ∆(k 1 , k 2 , k 3 , k 4 ) > 0}, (sm-35) where we introduced s ± (x, y, z, t) := 1 2x + y + z − t ∓ ∆(x, y, x, t)∆(x, y, z, t) := x 2 − 2xy − 2xz + 2xt + y 2 − 2yz + 2yt + z 2 + 2zt − 3t 2 . (sm-36)Now we proceed in two steps.…”

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BBGKY Hierarchy and Generalised Hydrodynamics

Bertini,

Essler,

Granet

2022

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We consider fermions defined on a continuous one-dimensional interval and subject to weak repulsive two-body interactions. We show that it is possible to perturbatively construct an extensive number of mutually compatible conserved charges for any interaction potential. However, the contributions to the densities of these charges at second order and higher are generally non-local and become spatially localized only if the potential fulfils certain compatibility conditions. We prove that the only solution to the first of these conditions with strictly local densities of charges is the Cheon-Shigehara potential (fermionic dual to the Lieb-Liniger model), and observe numerically that the only other solutions appear to be the Calogero-Sutherland potentials. We use our construction to show how Generalized Hydrodynamics (GHD) emerges from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, and argue that GHD in the weak interaction regime is robust under non-integrable perturbations.

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Duality between Weak and Strong Interactions in Quantum Gases

Cited by 11 publications

References 65 publications

A systematic $1/c$-expansion of form factor sums for dynamical correlations in the Lieb-Liniger model

A systematic $1/c$-expansion of form factor sums for dynamical correlations in the Lieb-Liniger model

Regularization of energy-dependent pointlike interactions in 1D quantum mechanics

BBGKY Hierarchy and Generalised Hydrodynamics

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