bkt transition and sine-gordon theory: from...
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BKT transition and Sine-Gordon theory:
from superconductors to cold atomic gases
T. Giamarchi
http://dqmp.unige.ch/giamarchi/
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n 1d quantum (clean)
L. Sanchez-Palencia (Polytechnique)G. Modugno, M. Inguscio (LENS)M. A. Cazalilla (Taiwan), A.F. Ho (Royal Holloway)
n Disorder
H.J. Schulz* (LPS), G. Roux (LPTMS), T. Barthel (Duke), G. Modugno, M. Inguscio (LENS)
n Superconducting films
L. Benfatto (Rome U.), C. Castellani (Rome U.)
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General referencesTG, arXiv/0605472 (Salerno lectures)
M. Cazalilla et al., Rev. Mod. Phys.83 1405 (2011)
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BKT transition
n BKT: remarkable transision going outside the paradigm of Landau’s phase transitions
n A transition wihout an order parameter
n Topological Vortex excitations
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Where to look for BKT
n Classical two dimensional systems (XY model)
n Two dimensional quantum problems: superfluid films or superconducting films
n Yes but 2+1 (time): needs finite temperature or dissipation to get BKT
n Alternative: look for 1d quantum problems: 1+1
n Yes but here temperature is the ennemy
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Mapping 2D Cl. to 1D quantum2 21 ( ) ( ) co s(2 )
2 xuH dx uK g dxK qp q f
pé ù= P + ¶ -ê úë ûò ò
n Sine-Gordon Hamiltonian
[ ( ), ( )] ( )x x i x xqq d¢ ¢P = - ( ) ( )xx xqp fP = ¶
2 21 ( ) ( ) cos(2 )2 xKS dxd u g dx
u tq q ftp
é ù= ¶ + ¶ -ê úë ûò ò
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Vortex operator
| |x+aiaPe xñ = ñ ( , ) ( , )x
x dx xqf t p t-¥
¢ ¢= Pò
x
t
02p 1 1cos(2 ( , ))xf t
n Vortex operator for q
21 ( ) ( ) cos(2 )2 xKS dxd u g dx
u tq q ftp
é ù= ¶ + ¶ -ê úë ûò ò
n K : inverse temperature
n g : vortex fugacity
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Why sine-Gordon is important
n Describes a very large number of quantum interacting 1D systems
n Example: 1d interacting bosons
n Bosonization: use collective variables
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Bosonization
Quantum fluctuations
Superfluid phase
K,u: depend on the interactions
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Mott transition in 1D
• Commensutate: Q = 2 p r0
0 cos( ) ( )H dxV Qx xr= ò
0(2 2 ( ))0 0cos( ) i x xH dxV Qx e pr fr -= ò
0 0 cos(2 ( ))LS V dxd xr t f= - òn BKT transition at K=2
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Test in cold atomic gasesE. Haller et al. Nature 466 597 (2010)
RenormalizedSine-Gordon
G. Boeris et al. PRA 93 011601® (2016)
Shows:
K* = 2
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Dirty interacting 1D bosonsTG + H. J. Schulz EPL 3 1287 (1987); PRB 37 325 (1988)
n BKT-like transition
n Vortex have long range itneractions in time only
n K*= 3/2
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Bose glass phase
U [interactions]
Db
0
Uc
Bose GlassSuperfluid
TG + H. J. Schulz EPL 3 1287 (87); PRB 37 325 (1988);
M.P.A. Fisher et al. PRB 40 546 (1989)
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Cold atomic gases (bosons + QP)C. D’Errico et al. PRL 113, 095301 (2014)L. Gori et al. PRA 93, 033650 (2016)
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Other systems
n Quantum Spin chains
n Spin 1/2
n BKT transitions in a various spin chains and ladders
TG, Int J. Mod. Phys. B 26 1244004 (2012)
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Superconducting filmsL. Benfatto, C. Castellani, TG in
n Thin (d < x) superconducting film
n 2D dependence of the superconducting phase
n Should see BKT physics
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Amplitude-Phase representation
n Superfluid stiffness J
n Vortices will try to reduce J
n Fugacity of the vortices
n Other excitations (single particle) affect J
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Typical quantities measuredn Superfluid density (via penetration length)
I.Hetel, T.R.Lemberger and M.Randeria, Nat. Phys. 3, 700 (2007)
n Transport
2
1~vnrx
µ
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BKT Signatures/parameters
n Parameters
n Exponential growth of x
n Universal superfluid density at the transition
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Does it work ?
M. Mondal et al. PRL 106, 047001 (2011)
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Coupling between layers
n Lawrence-Doniach model2
1cos( ( ) ( ))j j jj
H H J d r r rq q^ += - -å ò
n Bi-layer system
n Many such coupled cells
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How to treat
n Mapping to sine-Gordon0 01 2 1 2 1 2cos(2 ) cos(2 ) cos( )H H H g g Jf f q q= + - - - -
n Double sine-Gordon model
n Difficult !!
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Bilayer
I.Hetel, T.R.Lemberger and M.Randeria, Nat. Phys. 3, 700 (2007)
L. Benfatto, C. Castellani, TGPRB 77, 100506 ® (2008)
n Strange dependence in Tc of the fugacity
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Many layers (High Tc bulk)
L. Benfatto, C. Castellani, TG PRL 98, 117008 (2007)
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Related problemsn Coupled 1d
quantum tubesn Coupled 2d
superfluid pancakes
M. A. Cazalilla, A.F. Ho, TG, New J. Physics 8 158 (2006)
M. A. Cazalilla, A.F. Ho, TG, PRA 75, 051603 ® (2007)
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Conclusionsn BKT: many consequences in 2d superfluids, 2d
superconductors, 1d interacting quantum systems
n Very convenient mapping between quantum and classical problems
n Experimental signatures of BKT in 1d quantum systems and superconducting films
n Competition vortices – Josephson coupling for layered systems
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Open problems
n Effects of the competition Mott-Superfluidity, vortices-Josephson coupling
n Effects of disorder in 1d quantum problems
n Effects of disorder on 2d problems
n Dynamics vs thermodynamics