Self-consistent harmonic approximation in presence of non-local couplings(a)

Giachetti, Guido; Defenu, Nicoló; Ruffo, Stefano; Trombettoni, Andrea

doi:10.1209/0295-5075/133/57004

Cited by 11 publications

(7 citation statements)

References 58 publications

(94 reference statements)

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“…However, these results, have been recently challenged (Cescatti et al, 2019). Moreover, self-consistent harmonic approximation results give an upper bound for σ * equal to 2 (Giachetti et al, 2021b). No MC results for the 2d XY with (nondisordered) power-law long-range couplngs around σ = 2 are avaialble, to the best of our knowledge.…”

Section: Berezinskii-kosterlitz-thouless Scalingmentioning

confidence: 92%

Long-range interacting quantum systems

Defenu,

Donner,

Macrì

et al. 2021

Preprint

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108

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The presence of non-local and long-range interactions in quantum systems induces several peculiar features in their equilibrium and out-of-equilibrium behavior. In current experimental platforms control parameters such as interaction range, temperature, density and dimension can be changed. The existence of universal scaling regimes, where diverse physical systems and observables display quantitative agreement, generates a common framework, where the efforts of different research communities can be -in some cases rigorously -connected. Still, the application of this general framework to particular experimental realisations requires the identification of the regimes where the universality phenomenon is expected to appear. In the present review we summarise the recent investigations of many-body quantum systems with long-range interactions, which are currently realised in Rydberg atom arrays, dipolar systems, trapped ion setups and cold atoms in cavity experiments. Our main aim is to present and identify the common and (mostly) universal features induced by long-range interactions in the behaviour of quantum many-body systems. We will discuss both the case of very strong non-local couplings, i.e. the non-additive regime, and the one in which energy is extensive, but nevertheless low-energy, long wavelength properties are altered with respect to the short-range limit. Cases of competition with other local effects in the above mentioned setups are also reviewed.

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Section: Berezinskii-kosterlitz-thouless Scalingmentioning

confidence: 92%

Long-range interacting quantum systems

Defenu,

Donner,

Macrì

et al. 2021

Preprint

Self Cite

108

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“…[22,38,39], in this case J jumps discontinuously to zero as the temperature increases, implying that the exponent of the correlations becomes infinite in the high-temperature region. This path has been pursued for the case of a power-law interaction as well: in this case both the coupling J and the exponent of the interaction σ are replaced by variational parameters J and s. The results are unambiguous as long as σ > 2, predicting a phenomenology analogous to the SR case [22]. Our aim is to generalize such an analysis to a general functional equation, in order to correctly analyze to the σ ∈ (0, 2] regime.…”

Section: Self-consistent Approachmentioning

confidence: 88%

“…In the recent paper [21], the fate of the BKT line of critical points has been discussed. For σ > 2 simple arguments (not directly applicable to σ < 2) show the existence of the BKT transition [22], while what happens for σ < 2 is considerably more subtle. In Ref.…”

Section: Introductionmentioning

confidence: 99%

“…In the present paper, the scenario presented in Ref. [21] for the 2d XY model is analyzed thoughtfully and re-produced using different methods: the low-temperature expansion, an extension of the traditional self-consistent harmonic approximation (SCHA) [22,38,39] to σ < 2 and renormalization group (RG) field theoretical arguments. The difficulties inherent in the use of the Villain approximation for σ < 2 are pointed out and contrasted with the SR case, where, at variance, the 2d Villain and XY model are mappable thorough controllable approximations and lie in the same universality class.…”

Section: Introductionmentioning

confidence: 99%

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$BKT$ transitions in classical and quantum long-range systems

Giachetti¹,

Trombettoni²,

Ruffo³

et al. 2022

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In the past decades considerable efforts have been made in order to understand the critical features of both classical and quantum long-range interacting models. The case of the Berezinskii -Kosterlitz -Thouless (BKT) universality class, as in the 2d classical XY model, is considerably complicated by the presence, for short-range interactions, of a line of renormalization group fixed points. In this paper we discuss a field theoretical treatment of the 2d XY model with long-range couplings and we compare it with results from the self-consistent harmonic approximation. Both these methods lead to a rich phase diagram, where both power-law BKT scaling and spontaneous symmetry breaking appear for the same (intermediate) decay rates of long-range interactions. The methods is also applied to the long-range quantum XXZ spin chain at zero temperature. We discuss the relation between the phase diagrams of the two models and we give predictions about the scaling of the order parameter of the quantum chain close to the transition. Finally, we discuss the Villain approximation for the 2d XY model and we show how, in the long-range regime, it fails to reproduce the correct critical behavior.

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“…Paradigmatic examples of such interactions are the gravitational and the electrostatic one, but also dipolar forces in three dimensions or effective interactions between atoms in an optical cavity mediated by the electromagnetic field [3] are long-ranged. The behavior of long-range systems is peculiar both in equilibrium and non-equilibrium, for additive and non-additive systems as well [4][5][6][7][8]. Here we shall mainly be concerned with non-equilibrium aspects of non-additive long-range systems.…”

Section: Introductionmentioning

confidence: 99%

Violent relaxation in the Hamiltonian mean field model: II. Non-equilibrium phase diagrams

Santini,

Giachetti,

Casetti

2021

Preprint

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A classical long-range-interacting N -particle system relaxes to thermal equilibrium on time scales growing with N ; in the limit N → ∞ such a relaxation time diverges. However, a completely non-collisional relaxation process, known as violent relaxation, takes place on a much shorter time scale independent of N and brings the system towards a non-thermal quasi-stationary state. A finite system will eventually reach thermal equilibrium, while an infinite system will remain trapped in the quasi-stationary state forever. For times smaller than the relaxation time the distribution function of the system obeys the collisionless Boltzmann equation, also known as the Vlasov equation. The Vlasov dynamics is invariant under time reversal so that it does not "naturally" describe a relaxational dynamics. However, as time grows the dynamics affects smaller and smaller scales in phase space, so that observables not depending upon small-scale details appear as relaxed after a short time. Herewith we present an approximation scheme able to describe violent relaxation in a one-dimensional toy-model, the Hamiltonian Mean Field (HMF). The approach described here generalizes the one proposed in [1], that was limited to "cold" initial conditions, to generic initial conditions, allowing us to to predict non-equilibrium phase diagrams that turn out to be in good agreement with those obtained from the numerical integration of the Vlasov equation.

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Self-consistent harmonic approximation in presence of non-local couplings(a)

Cited by 11 publications

References 58 publications

Long-range interacting quantum systems

Long-range interacting quantum systems

$BKT$ transitions in classical and quantum long-range systems

Violent relaxation in the Hamiltonian mean field model: II. Non-equilibrium phase diagrams

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