Lattice modulation of a strongly interacting 2d superfluid: could - - PowerPoint PPT Presentation

Nikolay Prokof’ev

in collaboration with:

Lode Pollet

Lattice modulation of a strongly interacting 2d superfluid: could it be...the Higgs particle?

PRL 2012, Editor’s suggestion

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Today’s goal:

• 1. cold gas experiment (shaking)
• 2. QMC
• 3. result

Overview

• 1. Physics of the Bose-Hubbard model
• 2. lattice modulation

...or how there can still be secrets in the simplest interacting lattice bose model

Physics of Bose-Hubbard in a nutshell

U(1) symmetry decoupling approximation (mean-field)

• Integer density
• zero compressibility
• gap
• insulating

Mott phase: Quantum phase transition:

• S. Sachdev, Quantum Phase Transitions, 1999
• M. P

. A. Fisher et al, PRB 1989

b†

ibj = ψ(b† i + bj) − ψ2

ψ = hbii = hb†

ii

global bi → bieiφ

SF

Mott Mott Mott

Bogoliubov approximation

After Fourier transform:

b0 = √n0 + δb0 b†

0 = √n0 + δb†

keep terms up to second order, and diagonalize by a Bogoliubov transformation

• No Mott transition can be found this way
• phase and density are canonically conjugate;

i.e. Bogoliubov sound modes exhaust the collective degrees of freedom weak interactions

Mott insulator

atomic limit: ground state (product wave function) particle excitation hole excitation strong interactions

3d Mott excitations

hole particle

relativistic

• B. Capogrosso-Sansone, B.
• V. Svistunov, and N.V. Prokof’ev, Phys. Rev. B 75, 134302 (2007).

Friday, August 6, 2010

strong interactions

sketch

particle condenses; hole remains gaped

• ne gapless mode (phase -- sound)

two gapless modes at QCP phase(sound) and amplitude(Higgs) particle and hole condense Bogoliubov regime

• ne

gapless mode (sound)

• M. P

. A. Fisher et al, PRB 1989

?

(3d quantum antiferromagnet)

Phase diagram of 2d Bose-Hubbard model

finite temperature, commensurate density Bogoliubov region

phase mode exhausts all collective excitations

critical

correlation length at Uc: 5.30(5) J/T speed of sound at Uc: c=4.8J Uc = 16.7424

superfluid normal

Mott insulator

(from quantum Monte Carlo simulations)

Soyler, Capogrosso-Sansone, Prokof’ev, Svistunov, PRB 2008

O(N) field theories

d=2, n=1,2

d>3 : u is irrelevant (Gaussian free field theory)

χ0(q) = u q2 + (rc + r)

mean-field pole at amplitude mass

Π0(q) ∼ Z 1 k2(k + q)2 dd+1k (2π)d+1

two dimensions

longitudinal susceptibility has branch cut no pole-like structure at a frequency of order ρs(0)

two dimensions

``The longitudinal fluctuations of the Neel order thus lead to a critical continuum above the spin wave pole at w~ cq, which decays only

• algebraically. The continuum results from the decay of a normally massive

amplitude mode with momentum p into a pair of spin waves with momenta q and p-q, which is possible for any w > cq, with a singular cross section because of the large phase space. The amplitude mode is thus completely

• verdamped in two dimensions.”

derived same formula’s, and used them in the dynamic structure factor:

Mexican hat -- radial fluctuations

O(2) relativistic field theory

Φ = (Φ0 + σ, π) Φ = Φ0(1 + √ Nρ)ˆ n

Scalar and longitudinal susceptibility

Chubukov, Sachdev, Ye ’93 Podolsky, Auerbach, Arovas ’11

What’s the drama?

Asymptotically exact mean-field Higgs mode is well-defined. 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 ’93 Altman, Auerbach ’02 Zwerger ’04

d=3 d=2

Does it help to move away from QCP towards Galilean system where, in the limit, Higgs definitely does not exist? [Yes --- mean-field/variational, 1/N, RPA]

Huber, Buchler, Theiler, Altman, Blatter ’08, ’07 Menotti, Trivedi ’08

Podolsky, Auerbach, Arovas, PRA 2011

U → Uc

Altman, Auerbach ’02 Polkovnikov, Altman, Demler, Halperin, Lukin ‘05

Peak width INCREASES as Peak maximum > non-universal scale , no Higgs resonance in the relativistic limit.

(see however Podolsky and Sachdev, PRB 2012, for a 1/N expansion in the scaling regime with results more in line with the Monte Carlo results that follow)

Universal scaling predictions

Chubukov, Sachdev, Ye ’93 Sachdev ’99

A B

Podolsky et al. MISSING SPECTRAL DENSITY

Overview

• 1. Physics of the Bose-Hubbard model
• 2. lattice modulation: simulation and experiment

quantum simulation of the 2d Higgs particle

Higgs boson well, a 2D version of LHC

• n your marks

get set go

not-so Large Boson Shaker

190(36) particles

Technique pioneered in Zurich (Stoeferle et al); see also Kollath et al, etc

Energy dissipation rate Total energy absorbed:

Nature 2012

The experimental results

The experimental results

softening of onset of spectral weight on approach to the critical point

Our response

5 10 15 0.02 0.04 0.06 0.08 0.1 0.12

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U=16

Long Monte Carlo simulations (LMC)

• ur main result

Long Monte Carlo simulations (LMC)

0.001 0.01 0.1 1 10 50 100 150 200 250 300 350 χ(iωn) iωn

Monte Carlo and data processing

0.1 1 10 10 20 30 40 50 χ(iωn) iωn β=2 β=10

Add tail from higher temperatures for ground state

data processing

1e-05 0.0001 0.001 0.01 0.1 1 0.2 0.4 0.6 0.8 1 1.2 1.4 χs(τ) τ 1/τ4

tail compatible with w3 behavior for small w

Analytic continuation

...an ill-posed problem, so everybody may judge...

• 0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 5 10 15 20 25 30 35 40 45 50 S(ω) ω/J heuristic maxent Mischchenko

U/t = 18

Our response

2 4 6 8 10 0.04 0.08 0.12

Temperature effects

5 10 15 0.02 0.04 0.06 0.08 0.1 0.12

T/J=0.1 T/J=0.2 T/J=0.5 T/J=1.0 Tc ~ 0.3

Trap effects

5 10 15 0.04 0.08

homogeneous

U=14

Attempt to compare signals (amplitude adjusted) One (small ?) problem for direct comparison: experiment =

universal scaling function

in collaboration with Kun Chen et al.

also investigated by Podolsky et al.

results by Podolsky et al

• S. Gazit et al, arXiv:1212.3759, PRL

ωH = 1.9(1)∆ ωH = 3.2(8)∆

• n SF side:

compare to ours:

ωH = 2.1(3)∆

Conclusions

• Observation of a identifiable Higgs mode,

which softens down to the quantum critical point

• Onset in experimental spectral weight is

compatible with the Higgs mode

• Experimental trap is too small

5 10 15 0.02 0.04 0.06 0.08 0.1 0.12

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Further work:

• finite momenta (sharper!) ?
• O(N) models?
• peak in epxeriment?
• tuning away from commensurate

densities

• 1d; 1d-2d crossover
• universal conductivity -AdS/CFT

link

Kun Chen, L. Yang, Y.J. Deng, N.

• V. Prokof’ev, L. Pollet, et al.

conclusion

do cold atomic gases live up to their promise? Maybe it’s good that there are still Monte Carlo simulations. But we are reaching the limits of what we can do numerically in a reliable way can such systems be used as a quantum simulator?

5 10 15 0.02 0.04 0.06 0.08 0.1 0.12

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towards magnetism for spinor bosons quantum Monte Carlo towards pseudo-gap phase in the Hubbard model towards simulations of systems with long-range interactions further developments in diagrammatic Monte Carlo

1 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 image courtesy of I. Bloch

polarons and impurities

image courtesy of I. Bloch

Anderson-Higgs mechanism

“...The purpose of the present note is to report that, as a consequence of this coupling, the spin-one quanta of some of the gauge fields acquire mass; the longitudinal degrees

• f freedom of these particles (which would be absent if their mass were zero) go over

into the Goldstone bosons when the coupling tends to zero. This phenomenon is just the relativistic analog of the plasmon phenomenon to which Anderson has drawn attention: that the scalar zero-mass excitations of a superconducting neutral Fermi gas become longitudinal plasmon modes of finite mass when the gas is charged.”

spontaneous symmetry breaking

Consider a relativistic quantum field theory with mass m, and a complex scalar field

L = ∂µφ∗∂µφ − m2φ∗φ − 1 2λ(φ∗φ)2

• r, for negative mass,

L = ∂µφ∗∂µφ + m2φ∗φ − 1 2λ(φ∗φ)2

The Lagrangian has a global U(1) symmetry

φ(x) → φ(x)eiθ

In terms of the Mexican hat potential,

V (φ) = −1 2λνφ∗φ + 1 2λ(φ∗φ)2 ν = −−2m2 λ

the minimum occurs for

|φ|2 = ν2 2

Spontaneous symmetry breaking

We pick one of the minima and expand around it,

φ = 1 √ 2(ν + ϕ1 + iϕ2)

The low-energy Lagrangian is then

L = 1 2 ⇥ (∂µϕ1)2 + (∂µϕ2)2⇤ − 1 2λν2ϕ2

1 + . . .

where we see a massless Goldstone mode and a massive Higgs mode.

Anderson-Higgs mechanism

Consider now the case of coupling to a gauge field and local gauge invariance,

θ → θ(x) Aµ → Aµ − 1 e∂µθ(x) Dµφ = ∂µφ + ieAµφ L = Dµφ∗Dµφ − 1 4FµνF µν − V (φ)

Breaking the symmetry now leads to

L = 1 2 ⇥ (∂µϕ1)2 + (∂µϕ2 + eνAµ)2⇤ − 1 4FµνF µν − 1 2λν2ϕ2

1 + . . .

So we see that the photons have become massive (Anderson 63). The same can be applied to the non-Abelian U(1) x SU(2) so that W,Z bosons acquire mass (Englert, Brout, Higgs, Hagen, Guralnik, Kibble, Weinberg, Salam,...