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Higgs boson in a 2D superfluid
2F
m
To be, or not to be in d=2What’s the drama?
N. Prokof’ev
ICTP, Trieste, July 18, 2012
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WIBG: ~ /CT n m
: collective excitations are overdamped (classical criticality)
~ CT T In a Galilean system phase and density are canonical variables and the spectrum is exhausted by Bogoliubov quasiparticles
0T
Strongly interacting superlfuids: ~ (0)CT nU
At we have and the amplitude mode energy is comparable to overlap with other modes.
CT T 2~ n
(0)nU
Suppressing by interactions:( ) 0CT U
is the necessary condition for emergence of the new soft mode (Higgs), but …Liquid-Solid first order transition may happen instead
Why not to be in a generic superfluid?
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†
,
( 1) ( )2i g i i i i i
i g i i
UH J b b n n v n Bose Hubbard model:
1n
Particle-hole symmetricLorentz-invariant QCP 4.8(2)c aJ ( ) 5.30(5)
JT a
T
Capogrosso-Sansone, Soyler et al. ‘08
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To be or not to be in d=3,2 ?
22 211 1
2 2dS d r r
g
Asymptotically exact mean-fieldHiggs mode is well-defined.1/2(1 / )CU U
Overdamped due to strong decay into two Goldstone modes.
No Higgs resonance at low energy in any correlation function in close vicinity to the QCP
Chubukov, Sachdev, Ye ’93Altman, Auerbach ’02Zwerger ‘04Podolsky, Auerbach, Arovas ’11
d=3 d=2
Does it help to move away from QCP towards Galilean system? [Yes --- mean-field/variational, 1/N, RPA] Huber, Buchler, Theiler, Altman, Blatter ’08,
’07Menotti, Trivedi ’08
???
Chubukov, Sachdev, Ye ’93Podolsky, Auerbach, Arovas ’11
Look at the right response function!Scalar susceptibility is a better candidate
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Not to be in d=2: 1/N predictions for scalar susceptibility
2 3
2 2 2 2 2( )
( ) 4
US
m
2 3
2 2 2 2( )
2 ( ( / 2 ) )
U xS
x m x
/ 2x
8 2 (1 / )
/ 8Cm J U U
U
Altman, Auerbach ’02Polkovnikov, Altman, Demler, Halperin, Lukin ‘05
Podolsky, Auerbach, Arovas (2011)
Peak width INCREASES as CU U Peak maximum > non-universal scale ,
no Higgs resonance in the relativistic limit. ~ 4J
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Universal scaling predictions
Chubukov, Sachdev, Ye ’93
Sachdev ’99
3 2( ) / (1 / )CS U U
3 2/( ) ( / )
( ) , 0.6717C
S F
U U
~
3 2/ 0.0225( )S
( )S
J
A
B
Podolsky et al.
MISSING SPECTRAL DENSITY
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Scalar response through lattice modulation
†
,
( ) , ( ) , i tBH i g i
i g
JH H t K t e K J b b
J
Linear response for small /J J
( ) (0) ImK K K Energy dissipation rate :
Total energy absorbed: : Im 2 / ImM
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Recent experiment @ Munich: The onset of quantum critical continuum.
Resonance can not be seen due to inhomogeneous broadening.
Onset frequency
/T U S
J
U
/ 14U J / 16.7424CU J
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TIME TO CALL WORMS!
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( ) (0)n n
i iK K K m
Quantum Monte Calro: BH model in path integral representation + WA
No systematic errors but
(ii) finite system size L=20: + explicit checks of no size dependence
(i) finite simulation time: for lowest frequencies (ii) imaginary time (Matsubara frequencies) analytic continuation
/ 1L 510
Ill-posed problem:
( )
0
( )MC m e e d
MaxEnt=“most likely” “< all good fits >” “most featureless”
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0
space
Lattice path-integral = expansion of in hopping transitions, or kinks
1i i
HTr e
Kinetic energy = sum of all hopping transitions
MCkinks
K
0
( ) ( )n n ki iMC n MC
k kinks
K i d e K e
2( ) (0) ( )
nMC ni
K K K i
kink-kink correlation function
Results are person, continent, and CPU indendent, and agree with accuracy for the lowest frequencies 510
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There is a resonance atlow frequency which
- emerges at
- softens as
- gets more narrow as
- preserves its amplitude (roughly)
14U
CU U
CU U
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Side-by-side comparison
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Higgs resonance is present only in close vicinity of QCP. Barely seen at U=14, impossible to disentangle from other modes at U=12
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Higgs resonance in the MI phase – where is the Mexican hat potential?
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/ 1T J
( 16) 0.45CT U J
/ 0.5T J
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14U J
( 14)CT U J
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Power-point attempt to compare signals (amplitude adjusted)
One (small ?) problem for direct comparison: experiment = ( , ( ( ))S T S
/ 14 / 1.2CU J j j
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Most recent 1/N calculation by Podolsky & Sachdev [arXiv:1205.2700]
Universal part of the scalar response has an oscillating component !
(S
Pade approximants
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( )S
/
Conclusions:
3
0.0225
Universal part QMC simulation
Higgs resonance
Possible to extract experimentally in traps and at finite temperature.H
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