An Introduction to Integrable Techniques for One-Dimensional Quantum Systems

Franchini, Fabio

doi:10.1007/978-3-319-48487-7

Cited by 223 publications

(253 citation statements)

References 143 publications

(248 reference statements)

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“…Working directly in the limit L → ∞, this is a well-known problem in terms of the thermodynamic Bethe Ansatz (TBA) [46,47]. Indeed, the eigenstates of the XXZ chain are parametrized by rapidities λ, corresponding to magnon excitations created upon the fully polarized state.…”

Section: Magnetization Profilesmentioning

confidence: 99%

“…In particular, the ground state has a simple Fermi sea character, with real rapidities fully occupying an interval λ ∈ [−Λ, Λ]. While for finite L the rapidities satisfy some appropriate quantization conditions, in the thermodynamic limit the roots become continuous, and their density ρ(λ) follows from the TBA equation [46,47] …”

Section: Magnetization Profilesmentioning

confidence: 99%

“…the initial density is a step function) by setting ξ = 0 in (46). This yields ϕ = π/2 and the initial correlations (47) have to be evaluated with an argument t. For the domain-wall case the density and current are well-known scaling functions of the variable v = j/t and read [22] …”

Section: A Exact Solution For the XX Chainmentioning

confidence: 99%

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Front dynamics and entanglement in the XXZ chain with a gradient

Eisler

Bauernfeind

2017

Phys. Rev. B

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We consider the XXZ spin chain with a magnetic field gradient and study the profiles of the magnetization as well as the entanglement entropy. For a slowly varying field it is shown that, by means of a local density approximation, the ground-state magnetization profile can be obtained with standard Bethe ansatz techniques. Furthermore, it is argued that the low-energy description of the theory is given by a Luttinger liquid with slowly varying parameters. This allows us to obtain a very good approximation of the entanglement profile using a recently introduced technique of conformal field theory in curved spacetime. Finally, the front dynamics is also studied after the gradient field has been switched off, following arguments of generalized hydrodynamics for integrable systems. While for the XX chain the hydrodynamic solution can be found analytically, the XXZ case appears to be more complicated and the magnetization profiles are recovered only around the edge of the front via an approximate numerical solution.

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Section: Magnetization Profilesmentioning

confidence: 99%

Section: Magnetization Profilesmentioning

confidence: 99%

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Front dynamics and entanglement in the XXZ chain with a gradient

Eisler

Bauernfeind

2017

Phys. Rev. B

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“…The effect of frustration induced by the boundary conditions has been already considered in integrable systems with a continuous U(1) symmetry at vanishing external field as the XXZ chain obtained by setting γ=0 and h=0 in (1) [21][22][23][24], where the eigenstates can be constructed in terms of individual excitations. Thus, while for even lengths N=2M the ground-state can achieve zero total magnetization S 0 T Z = and be characterized as a spinon vacuum [25], in the frustrated case N=2M+1 there are two equivalent ground-states with S T Z 1 2 =  (whose degeneracy is immediately lifted for a nonzero h), which can be interpreted as due to the presence of a traveling single spinon excitation. The goal of the present paper is to analyze the case of systems with discrete global symmetries 2  , which, thus, do not conserve particle number.…”

Section: Introductionmentioning

confidence: 99%

The frustration of being odd: universal area law violation in local systems

2019

Self Cite

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At the core of every frustrated system, one can identify the existence of frustrated rings that are usually interpreted in terms of single-particle physics. We check this point of view through a careful analysis of the entanglement entropy of both models that admit an exact single-particle decomposition of their Hilbert space due to integrability and those for which the latter is supposed to hold only as a low energy approximation. In particular, we study generic spin chains made by an odd number of sites with short-range antiferromagnetic interactions and periodic boundary conditions, thus characterized by a weak, i.e. nonextensive, frustration. While for distances of the order of the correlation length the phenomenology of these chains is similar to that of the non-frustrated cases, we find that correlation functions involving a number of sites scaling like the system size follow different rules. We quantify the long-range correlations through the von Neumann entanglement entropy, finding that indeed it violates the area law, while not diverging with the system size. This behavior is well fitted by a universal law that we derive from the conjectured single-particle picture.

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“…, 1 where the two-body interaction between the particles is described by a zero-range delta-function potential of strength g [44]. While exact solutions for this Hamiltonian exist in the absence of an external potential [45,46], only limiting cases can be solved in other situations. One regime in which the Hamiltonian (1) can be solved exactly is the strongly interacting TG limit.…”

Section: Basic Modelmentioning

confidence: 99%

Static and dynamic phases of a Tonks–Girardeau gas in an optical lattice

2018

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We investigate the properties of a Tonks-Girardeau gas in the presence of a one-dimensional lattice potential. Such a system is known to exhibit a pinning transition when the lattice is commensurate with the particle density, leading to the formation of an insulating state even at infinitesimally small lattice depths. Here we examine the properties of the gas at all lattices depths and, in addition to the static properties, also consider the non-adiabatic dynamics induced by the sudden motion of the lattice potential with a constant speed. Our work provides a continuum counterpart to the work done in discrete lattice models.

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An Introduction to Integrable Techniques for One-Dimensional Quantum Systems

Cited by 223 publications

References 143 publications

Front dynamics and entanglement in the XXZ chain with a gradient

Front dynamics and entanglement in the XXZ chain with a gradient

The frustration of being odd: universal area law violation in local systems

Static and dynamic phases of a Tonks–Girardeau gas in an optical lattice

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