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The phase transtions of a dilute Bose gas in two and three
dimensions
Markus HolzmannLPMMC, CNRS UGA Grenoble
Gordon Baym, Jean-Paul Blaizot, Franck Laloë, Dominique Vautherin
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Dilute Bose gas
● Finite temperature T>0 (β=1/T)
● Homogeneous systems
● Thermodynamic limit, constant density n=N/V
● Short range interaction described by scattering length a (g=4πa/m)
● Dilute Bosons: na³→0
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Questions & Results (3D):
● Critical temperature shift (with respect to ideal gas TC
0)?
Leading order: linear in a, constant c>0 is “universal”
Next-to-leading order: a^2 log arestricts linear region drastically
G. Baym, J.-P. Blaizot, M. H., F. Laloë, D. Vautherin,Phys. Rev. Lett. 83, 1703 (1999)
M. H., G. Baym, J.-P. Blaizot, F. Laloë,Phys. Rev. Lett. 87, 120403 (2001)
Best value for c from classical field calculations on a lattice: c≈1.3
P. Arnold, G. Moore, Phys. Rev. Lett. 87, 120401 (2001);V. A. Kashurnikov, N.V. Prokof ’ev, B.V. Svistunov, Phys. Rev. Lett. 87, 120402 (2001).
Perturbative calculation of c:P. Arnold, G. Moore, B. Tomasik, Phys. Rev. A 65, 013606 (2001).
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● Critical region, scaling behavior?
condensate/ superfluid fraction inside critical region:
In particular:
M. H., G. Baym, Phys. Rev. Lett. 90, 040402 (2003)
● External, harmonic trap?
P. Arnold, B. TomášikPhys. Rev. A 64, 053609 (2001)
S. Giorgini, L. P. Pitaevskii, S. Stringari,Phys. Rev. A 54, R4633 (1996)
thermodynamic limit in trap
(perturbative)
Non-perturbative,uses universal “c” resultof the homogeneous system
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● Experiments (trap)?
F. Gerbier, J. H. Thywissen, S. Richard, M. Hugbart, P. Bouyer, A. Aspect, Phys. Rev. Lett. 92, 030405 (2004)
Mean-field shift
Beyond Mean-field shiftR. Smith, R. Campbell, N. Tammuz, Z. HadzibabicPhys. Rev. Lett. 106, 250403 (2011)
Condensate density shift:
R. Smith, N. Tammuz, R. Campbell, M. H., Z. HadzibabicPhys. Rev. Lett. 107, 190403
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Homogeneous Bose gas: linear shift
● Critical temperature shift vs critical density shift: Ideal gas BEC:
● Density expressed in finite temperature Green’s function
Self-energy (contains all interaction effects)
● BEC transition point (for chemical potential, T and a fixed):
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Self-energy: diagramatic analysis
● Diagrams linear in a => mean field
No shift in TC !
● a2: second order diagram
dominant contribution for small wavevectors:
If we have at critical point:
thenM. H., P. Grüter, F. Laloë,Eur. Phys. J. B10, 739 (1999)
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● Higher order diagrams:
Power counting of infrarouge:
at criticality we have
All orders are relevant!!!
● Classical phi^4 field theory is sufficient to get linearity, prefactor universal
use UV cutoff Λ (lattice) to regularizeand simple 2nd order counter-term toseparate IR and UV (no mixing of logs!)
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Universal behavior in critical region
● Scaling behavior of self energy
● Scaling function u(x) “dominates” inside critical region
counter term needed to renormalize 2nd order UV divergence:
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Matching with “real world”
● Strategy:
➢ calculate non-perturbative contributions numerically (classical field, lattice)➢ tabulate scaled functions➢ add perturbative contributions with mean-field ➢ use local density approximation for smooth external potentials
density, condensate density…:
classical field contributioncalculate numerically f
add quantum/UV contributions perturbatively, e.g. mean-field
use local density approximation (LDA) to account for trap potential:
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2D: Berenzinskii-Kosterlitz-Thouless transition
● Perturbative analysis:close to transition IR divergencies, classical phi^4 theory dominates
● Classical phi^4 “easier”: mean-field terms are enough for renormalization
● transition point as in 3D, but structure below is different:no condensate density in the thermodynamic limit butfinite superfluid density (use Josephson’s relation to relate them with G(k)
● calculate classical \phi^4 contributions numerically match with mean-field
● Typically: classical field contributions dominate over broad region,matching relatively simple, works quantitatively well….
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ExperimentalMeasurements:
P.A. Murphy, I. Boettcher, L. Bayha,M.H., D. Kedar, M. Neidig, M.G. RiesA.N. Wenz, G. Zürn, S. Jochim,Phys. Rev. Lett. 115, 010401 (2015).
Algebraic Decayof correlations?
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Berezinskii-Kosterlitz-Thouless cross-over ina trapped exciton gas (GaAs)
Berezinskii-Kosterlitz-Thouless cross-over ina trapped exciton gas (GaAs)
S. Dang, R. Anankine, C. Gomez, A. Lemaîre, M. H., F. Dubin, Phys. Rev. Lett. 122, 17402 (2019)
GaAs Bilayer excitons:● Low mass M
X≈0.22 m
electron T=340mK...2K
● 4 almost degenerate states (dark & bright excitons)● Dipolar interaction => large effective coupling g=3...7