# Thermal and Quantum Fluctuation Effects in Quasiperiodic Systems in External Potentials

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## Abstract

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## 1. Introduction

## 2. Model Hamiltonians and Methodology

## 3. Trapped Quasicrystal: Thermal Fluctuations

## 4. Trapped Quasicrystal: Quantum Fluctuations

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J.W. Metallic Phase with Long-Range Orientational Order and No Translational Symmetry. Phys. Rev. Lett.
**1984**, 53, 1951–1953. [Google Scholar] [CrossRef] [Green Version] - Suck, J.B.; Schreiber, M.; Häussler, P. Quasicrystals; Springer: Berlin/Heidelberg, Germany, 2002. [Google Scholar] [CrossRef]
- Likos, C.N.; Lang, A.; Watzlawek, M.; Löwen, H. Criterion for determining clustering versus reentrant melting behavior for bounded interaction potentials. Phys. Rev. E
**2001**, 63, 031206. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shin, H.; Grason, G.M.; Santangelo, C.D. Mesophases of soft-sphere aggregates. Soft Matter
**2009**, 5, 3629–3638. [Google Scholar] [CrossRef] [Green Version] - Fetter, A.L.; Walecka, J.D. Quantum Theory of Many-Particle Systems; McGraw-Hill: Boston, MA, USA, 1971. [Google Scholar]
- Lahaye, T.; Menotti, C.; Santos, L.; Lewenstein, M.; Pfau, T. The physics of dipolar bosonic quantum gases. Rep. Prog. Phys.
**2009**, 72, 126401. [Google Scholar] [CrossRef] - Wächtler, F.; Santos, L. Quantum filaments in dipolar Bose-Einstein condensates. Phys. Rev. A
**2016**, 93, 061603. [Google Scholar] [CrossRef] [Green Version] - Henkel, N.; Cinti, F.; Jain, P.; Pupillo, G.; Pohl, T. Supersolid Vortex Crystals in Rydberg-Dressed Bose-Einstein Condensates. Phys. Rev. Lett.
**2012**, 108, 265301. [Google Scholar] [CrossRef] - Macrì, T.; Maucher, F.; Cinti, F.; Pohl, T. Elementary excitations of ultracold soft-core bosons across the superfluid-supersolid phase transition. Phys. Rev. A
**2013**, 87, 061602. [Google Scholar] [CrossRef] [Green Version] - Cinti, F.; Macrì, T.; Lechner, W.; Pupillo, G.; Pohl, T. Defect-induced supersolidity with soft-core bosons. Nat. Commun.
**2014**, 5, 3235. [Google Scholar] [CrossRef] [Green Version] - Macrì, T.; Pohl, T. Rydberg dressing of atoms in optical lattices. Phys. Rev. A
**2014**, 89, 011402. [Google Scholar] [CrossRef] [Green Version] - Macrì, T.; Saccani, S.; Cinti, F. Ground State and Excitation Properties of Soft-Core Bosons. J. Low Temp. Phys.
**2014**, 177, 59–71. [Google Scholar] [CrossRef] [Green Version] - Butenko, S.; Chaovalitwongse, W.A.; Pardalos, P.M. Clustering Challenges in Biological Networks; World Scientific: Singapore, 2009. [Google Scholar] [CrossRef]
- Likos, C.N. Effective interactions in soft condensed matter physics. Phys. Rep.
**2001**, 348, 267–439. [Google Scholar] [CrossRef] - Díaz-Méndez, R.; Mezzacapo, F.; Lechner, W.; Cinti, F.; Babaev, E.; Pupillo, G. Glass Transitions in Monodisperse Cluster-Forming Ensembles: Vortex Matter in Type-1.5 Superconductors. Phys. Rev. Lett.
**2017**, 118, 067001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Díaz-Méndez, R.; Mezzacapo, F.; Cinti, F.; Lechner, W.; Pupillo, G. Monodisperse cluster crystals: Classical and quantum dynamics. Phys. Rev. E
**2015**, 92, 052307. [Google Scholar] [CrossRef] [Green Version] - Cinti, F.; Cappellaro, A.; Salasnich, L.; Macrì, T. Superfluid Filaments of Dipolar Bosons in Free Space. Phys. Rev. Lett.
**2017**, 119, 215302. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cinti, F.; Boninsegni, M. Classical and quantum filaments in the ground state of trapped dipolar Bose gases. Phys. Rev. A
**2017**, 96, 013627. [Google Scholar] [CrossRef] [Green Version] - Cinti, F.; Boninsegni, M. Absence of Superfluidity in 2D Dipolar Bose Striped Crystals. J. Low Temp. Phys.
**2019**, 196, 413–422. [Google Scholar] [CrossRef] [Green Version] - Macrì, T.; Cinti, F. Many-Body Physics of Low-Density Dipolar Bosons in Box Potentials. Condens. Matter
**2019**, 4, 17. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y.C.; Maucher, F.; Pohl, T. Supersolidity around a Critical Point in Dipolar Bose-Einstein Condensates. Phys. Rev. Lett.
**2019**, 123, 015301. [Google Scholar] [CrossRef] [Green Version] - Cinti, F.; Boninsegni, M.; Pohl, T. Exchange-induced crystallization of soft-core bosons. New J. Phys.
**2014**, 16, 033038. [Google Scholar] [CrossRef] [Green Version] - Saccani, S.; Moroni, S.; Boninsegni, M. Excitation Spectrum of a Supersolid. Phys. Rev. Lett.
**2012**, 108, 175301. [Google Scholar] [CrossRef] [Green Version] - Cappellaro, A.; Macrì, T.; Salasnich, L. Collective modes across the soliton-droplet crossover in binary Bose mixtures. Phys. Rev. A
**2018**, 97, 053623. [Google Scholar] [CrossRef] [Green Version] - Cidrim, A.; dos Santos, F.E.A.; Henn, E.A.L.; Macrì, T. Vortices in self-bound dipolar droplets. Phys. Rev. A
**2018**, 98, 023618. [Google Scholar] [CrossRef] [Green Version] - Cinti, F.; Wang, D.W.; Boninsegni, M. Phases of dipolar bosons in a bilayer geometry. Phys. Rev. A
**2017**, 95, 023622. [Google Scholar] [CrossRef] [Green Version] - Kadau, H.; Schmitt, M.; Wenzel, M.; Wink, C.; Maier, T.; Ferrier-Barbut, I.; Pfau, T. Observing the Rosensweig instability of a quantum ferrofluid. Nature
**2016**, 530, 194–197. [Google Scholar] [CrossRef] [PubMed] - Chomaz, L.; Baier, S.; Petter, D.; Mark, M.J.; Wächtler, F.; Santos, L.; Ferlaino, F. Quantum-Fluctuation- Driven Crossover from a Dilute Bose-Einstein Condensate to a Macrodroplet in a Dipolar Quantum Fluid. Phys. Rev. X
**2016**, 6, 041039. [Google Scholar] [CrossRef] - Tanzi, L.; Lucioni, E.; Famà, F.; Catani, J.; Fioretti, A.; Gabbanini, C.; Bisset, R.N.; Santos, L.; Modugno, G. Observation of a Dipolar Quantum Gas with Metastable Supersolid Properties. Phys. Rev. Lett.
**2019**, 122, 130405. [Google Scholar] [CrossRef] [Green Version] - Léonard, J.; Morales, A.; Zupancic, P.; Esslinger, T.; Donner, T. Supersolid formation in a quantum gas breaking a continuous translational symmetry. Nature
**2017**, 543, 87–90. [Google Scholar] [CrossRef] [Green Version] - Li, J.R.; Lee, J.; Huang, W.; Burchesky, S.; Shteynas, B.; Top, F.Ç.; Jamison, A.O.; Ketterle, W. A stripe phase with supersolid properties in spin–orbit-coupled Bose–Einstein condensates. Nature
**2017**, 543, 91–94. [Google Scholar] [CrossRef] - Barkan, K.; Engel, M.; Lifshitz, R. Controlled Self-Assembly of Periodic and Aperiodic Cluster Crystals. Phys. Rev. Lett.
**2014**, 113, 098304. [Google Scholar] [CrossRef] [Green Version] - Dotera, T.; Oshiro, T.; Ziherl, P. Mosaic two-lengthscale quasicrystals. Nature
**2014**, 506, 208–211. [Google Scholar] [CrossRef] - Pupillo, G.; Ziherl, P.; Cinti, F. Quantum Cluster Quasicrystals. arXiv
**2019**, arXiv:1905.12073. [Google Scholar] - Ceperley, D.M. Path integrals in the theory of condensed helium. Rev. Mod. Phys.
**1995**, 67, 279–355. [Google Scholar] [CrossRef] - Krauth, W. Statistical Mechanics: Algorithms and Computations; Oxford Master Series in Physics; Oxford University Press: Oxford, UK, 2006. [Google Scholar]
- Boninsegni, M. Permutation Sampling in Path Integral Monte Carlo. J. Low Temp. Phys.
**2005**, 141, 27–46. [Google Scholar] [CrossRef] [Green Version] - Chin, S.A. Symplectic integrators from composite operator factorizations. Phys. Lett. A
**1997**, 226, 344–348. [Google Scholar] [CrossRef] - Jain, P.; Cinti, F.; Boninsegni, M. Structure, Bose-Einstein condensation, and superfluidity of two-dimensional confined dipolar assemblies. Phys. Rev. B
**2011**, 84, 014534. [Google Scholar] [CrossRef] [Green Version] - Boninsegni, M.; Prokof’ev, N.V. Colloquium: Supersolids: What and where are they? Rev. Mod. Phys.
**2012**, 84, 759–776. [Google Scholar] [CrossRef] [Green Version] - Cinti, F.; Jain, P.; Boninsegni, M.; Micheli, A.; Zoller, P.; Pupillo, G. Supersolid Droplet Crystal in a Dipole-Blockaded Gas. Phys. Rev. Lett.
**2010**, 105, 135301. [Google Scholar] [CrossRef] - Chiacchiera, S.; Macrì, T.; Trombettoni, A. Dipole oscillations in fermionic mixtures. Phys. Rev. A
**2010**, 81, 033624. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(

**a**) Pair potential in real space identifying a cluster quasicrystal with a 10-fold symmetry. (

**b**) Fourier of the pair potential proposed in (

**a**) around the minima that mark the quasicrystal. (

**c**) Typical configuration of a classical simulation obtained at $t=0.05$ using 2048 particles (canonical ensemble). The parameters of the interparticle potential in Equation (2) used for the simulations were $\sigma =0.69$, ${c}_{0}=1$, ${c}_{2}=-0.79$, ${c}_{4}=0.25$, ${c}_{6}=-0.02$, and ${c}_{8}=6.0\times {10}^{-4}$, as reported in Ref. [32]. These parameters lead to two degenerate minima of the Fourier transform of the potential. Length scales and parameters were chosen to fix the first minimum at ${k}_{min}^{1}=1$.

**Figure 2.**Instantaneous configuration for a trapped system made of $N=2048$ classical particles at temperature (

**a**) $t=0.05$ and (

**b**) $t=1.0$. (

**c**) Grey points: density profiles for panel (

**a**); Orange points: density profiles for panel (

**b**).

**Figure 3.**Snapshot of $N=2048$ boltzmannons simulated at temperature $t=0.05$ and confined into an harmonic trap varying the kinetic term of (1): (

**a**) $\lambda =0.05$ and (

**b**) $\lambda =0.5$. Different colors identify different world lines. (

**c**) Grey points: density profiles for panel (

**a**); Orange points: density profiles for panel (

**b**)—error bars lie within point size.

**Figure 4.**Fourier transform (logarithmic scale) of the density for the classical and quantum configurations. (

**a**) Fourier transform of Figure 2a (thermal fluctuations), (

**b**) Fourier transform of Figure 3a (quantum fluctuations). Whereas in (

**b**) the peak structure clearly indicates the presence of a hexagonal lattice, the Fourier transform of the classical configuration in (

**a**) displays a central peak surrounded by four pairs of closely spaced peaks, compatible with an irregular structure of the clusters.

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**MDPI and ACS Style**

Cinti, F.; Macrì, T.
Thermal and Quantum Fluctuation Effects in Quasiperiodic Systems in External Potentials. *Condens. Matter* **2019**, *4*, 93.
https://doi.org/10.3390/condmat4040093

**AMA Style**

Cinti F, Macrì T.
Thermal and Quantum Fluctuation Effects in Quasiperiodic Systems in External Potentials. *Condensed Matter*. 2019; 4(4):93.
https://doi.org/10.3390/condmat4040093

**Chicago/Turabian Style**

Cinti, Fabio, and Tommaso Macrì.
2019. "Thermal and Quantum Fluctuation Effects in Quasiperiodic Systems in External Potentials" *Condensed Matter* 4, no. 4: 93.
https://doi.org/10.3390/condmat4040093