BKT transition and Sine-Gordon theory:

from superconductors to cold atomic gases

T. Giamarchi

http://dqmp.unige.ch/giamarchi/

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.)

General referencesTG, arXiv/0605472 (Salerno lectures)

M. Cazalilla et al., Rev. Mod. Phys.83 1405 (2011)

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

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

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

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

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

Bosonization

Quantum fluctuations

Superfluid phase

K,u: depend on the interactions

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

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

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

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)

Cold atomic gases (bosons + QP)C. D’Errico et al. PRL 113, 095301 (2014)L. Gori et al. PRA 93, 033650 (2016)

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)

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

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

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

BKT Signatures/parameters

n Parameters

n Exponential growth of x

n Universal superfluid density at the transition

Does it work ?

M. Mondal et al. PRL 106, 047001 (2011)

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

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 !!

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

Many layers (High Tc bulk)

L. Benfatto, C. Castellani, TG PRL 98, 117008 (2007)

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)

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

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