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

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?

†

,

( 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

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

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

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

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

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

TIME TO CALL WORMS!

( ) (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”

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

There is a resonance atlow frequency which

- emerges at

- softens as

- gets more narrow as

- preserves its amplitude (roughly)

14U

CU U

CU U

Side-by-side comparison

Higgs resonance is present only in close vicinity of QCP. Barely seen at U=14, impossible to disentangle from other modes at U=12

Higgs resonance in the MI phase – where is the Mexican hat potential?

/ 1T J

( 16) 0.45CT U J

/ 0.5T J

14U J

( 14)CT U J

Power-point attempt to compare signals (amplitude adjusted)

One (small ?) problem for direct comparison: experiment = ( , ( ( ))S T S

/ 14 / 1.2CU J j j

Most recent 1/N calculation by Podolsky & Sachdev [arXiv:1205.2700]

Universal part of the scalar response has an oscillating component !

(S

Pade approximants

( )S

/

Conclusions:

3

0.0225

Universal part QMC simulation

Higgs resonance

Possible to extract experimentally in traps and at finite temperature.H