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

Lattice modulation of a strongly interacting 2d superfluid: could it be...the Higgs particle? Lode Pollet in collaboration with: PRL 2012, Nikolay Prokof’ev Editor’s suggestion

Today’s goal: 1. cold gas experiment (shaking) 2. QMC 3. result H 0.12 0.1 0.08 0.06 0.04 0.02 0 0 5 10 15

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 M. P . A. Fisher et al, PRB 1989 global b i → b i e i φ Mott U(1) symmetry decoupling approximation (mean-field) SF b † i b j = ψ ( b † i + b j ) − ψ 2 ψ = h b i i = h b † i i Mott Mott phase: • Integer density • zero compressibility • gap • insulating Mott Quantum phase transition: S. Sachdev, Quantum Phase Transitions, 1999

Bogoliubov approximation weak interactions After Fourier transform: b 0 = √ n 0 + δ b 0 b † 0 = √ n 0 + δ b † 0 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

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

3d Mott excitations hole particle relativistic strong interactions B. Capogrosso-Sansone, B. V. Svistunov, and N.V. Prokof’ev, Phys. Rev. B 75 , 134302 (2007). Friday, August 6, 2010

sketch Bogoliubov regime one gapless mode (phase -- sound) one particle condenses; two gapless gapless hole remains gaped modes at QCP mode phase(sound) and (sound) ? amplitude(Higgs) particle and hole condense M. P . A. Fisher et al, PRB 1989

(3d quantum antiferromagnet)

Phase diagram of 2d Bose-Hubbard model finite temperature, commensurate density (from quantum Monte Carlo simulations) correlation length normal at Uc: 5.30(5) J/T speed of sound superfluid at Uc: c=4.8J Mott insulator critical Bogoliubov region Uc = 16.7424 phase mode exhausts all collective excitations Soyler, Capogrosso-Sansone, Prokof’ev, Svistunov, PRB 2008

O(N) field theories d>3 : u is irrelevant (Gaussian free field theory) u χ 0 ( q ) = q 2 + ( r c + r ) mean-field pole at amplitude mass d d +1 k 1 Z Π 0 ( q ) ∼ (2 π ) d +1 k 2 ( k + q ) 2 d=2, n=1,2

two dimensions longitudinal susceptibility has branch cut no pole-like structure at a frequency of order ρ s (0)

two dimensions derived same formula’s, and used them in the dynamic structure factor: ``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 overdamped in two dimensions.”

Mexican hat -- radial fluctuations O(2) relativistic field theory Φ = ( Φ 0 + σ , π ) √ Φ = Φ 0 (1 + N ρ )ˆ n

Scalar and longitudinal susceptibility Chubukov, Sachdev, Ye ’93 Podolsky, Auerbach, Arovas ’11

What’s the drama? d=3 d=2 Overdamped due to strong decay Asymptotically exact mean-field into two Goldstone modes. Higgs mode is well-defined. Chubukov, Sachdev, Ye ’93 Altman, Auerbach ’02 Zwerger ’04 No Higgs resonance at low energy in any correlation function in close vicinity to the QCP 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 Altman, Auerbach ’02 Polkovnikov, Altman, Demler, Halperin, Lukin ‘05 U → U c 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 B A MISSING SPECTRAL DENSITY Podolsky et al.

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 on your marks get set go

not-so Large Boson Shaker Technique pioneered in Zurich (Stoeferle et al); see also Kollath et al, etc 190(36) particles Nature 2012 Energy dissipation rate Total energy absorbed:

The experimental results

The experimental results softening of onset of spectral weight on approach to the critical point

Our response U=16 H 0.12 0.1 0.08 0.06 0.04 0.02 0 0 5 10 15

Long Monte Carlo simulations (LMC) our main result

Long Monte Carlo simulations (LMC) 10 1 χ (i ω n ) 0.1 0.01 0.001 0 50 100 150 200 250 300 350 i ω n

Monte Carlo and data processing 10 β =2 β =10 χ (i ω n ) 1 0.1 0 10 20 30 40 50 i ω n Add tail from higher temperatures for ground state

data processing 1/ τ 4 1 0.1 0.01 χ s ( τ ) 0.001 0.0001 1e-05 0 0.2 0.4 0.6 0.8 1 1.2 1.4 τ tail compatible with w 3 behavior for small w

Analytic continuation 0.08 heuristic maxent 0.07 Mischchenko 0.06 0.05 0.04 S( ω ) 0.03 0.02 0.01 0 ...an ill-posed -0.01 0 5 10 15 20 25 30 35 40 45 50 ω /J problem, so everybody may U/t = 18 judge...

Our response 0.12 0.08 0.04 0 0 2 4 6 8 10

Temperature effects 0.12 Tc ~ 0.3 0.1 T/J=0.1 T/J=0.5 T/J=0.2 T/J=1.0 0.08 0.06 0.04 0.02 0 0 5 10 15

Trap effects U=14 homogeneous 0.08 0.04 0 0 5 10 15

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 on SF side: ω H = 1 . 9(1) ∆ ω H = 2 . 1(3) ∆ compare to ours: ω H = 3 . 2(8) ∆

Conclusions - Observation of a identifiable Higgs mode, which softens down to the quantum critical point - Onset in experimental spectral weight is H 0.12 compatible with the Higgs mode - Experimental trap is too small 0.1 Further work: 0.08 Kun Chen, L. Yang, Y.J. Deng, N. V. Prokof’ev, L. Pollet, et al. - finite momenta (sharper!) ? - O(N) models? 0.06 - peak in epxeriment? - tuning away from commensurate 0.04 densities - 1d; 1d-2d crossover - universal conductivity -AdS/CFT 0.02 link 0 0 5 10 15

conclusion do cold atomic gases live up to their promise? can such systems be used as a quantum simulator? H 0.12 0.1 0.08 0.06 0.04 0.02 0 0 5 10 15 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

image courtesy of I. Bloch image courtesy of I. Bloch quantum 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 towards pseudo-gap phase in the Hubbard model towards magnetism for spinor bosons polarons and impurities further developments in diagrammatic towards simulations of systems with Monte Carlo long-range interactions

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 of 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 = ∂ µ φ ∗ ∂ µ φ − m 2 φ ∗ φ − 1 2 λ ( φ ∗ φ ) 2 or, for negative mass, L = ∂ µ φ ∗ ∂ µ φ + m 2 φ ∗ φ − 1 2 λ ( φ ∗ φ ) 2 The Lagrangian has a global U(1) symmetry φ ( x ) → φ ( x ) e i θ In terms of the Mexican hat potential, V ( φ ) = − 1 2 λνφ ∗ φ + 1 ν = −− 2 m 2 2 λ ( φ ∗ φ ) 2 λ the minimum occurs for | φ | 2 = ν 2 2

Spontaneous symmetry breaking We pick one of the minima and expand around it, 1 φ = 2( ν + ϕ 1 + i ϕ 2 ) √ The low-energy Lagrangian is then L = 1 − 1 ( ∂ µ ϕ 1 ) 2 + ( ∂ µ ϕ 2 ) 2 ⇤ 2 λν 2 ϕ 2 ⇥ 1 + . . . 2 where we see a massless Goldstone mode and a massive Higgs mode.