Collective modes at a disordered quantum phase transition Thomas - PowerPoint PPT Presentation

Collective modes at a disordered quantum phase transition Thomas Vojta Department of Physics, Missouri University of Science and Technology Los Alamos, January 27, 2020

Outline • Collective modes: Goldstone and amplitude (Higgs) • Superfluid-Mott glass quantum phase transition • Fate of the collective modes at the Martin Puschmann superfluid-Mott glass transition Jose Hoyos • Conclusions Jack Crewse Cameron Lerch

Spontaneous symmetry breaking Does a symmetric Hamiltonian imply a symmetric equilibrium state? • world of this pencil is completely isotropic, all directions are equal • symmetry is lost when pencil falls over, now only one direction holds • state of lowest energy has lower symmetry than system Rotational symmetry has been broken spontaneously!

Broken symmetries and collective modes • systems with broken continuous symmetry : − planar magnet breaks O(2) rotation symmetry − superfluid wave function breaks U(1) symmetry • Higgs (amplitude) mode : corresponds to fluctuations of order parameter amplitude • Goldstone (phase) mode : corresponds to fluctuations of order parameter phase • Amplitude mode is condensed matter analogue of famous Higgs boson Goldstone theorem: ”Mexican hat” potential for order When a continuous symmetry is spontaneously parameter in symmetry-broken phase broken, massless Goldstone modes appear. F = t m 2 + u m 4

Higgs (amplitude) mode in condensed matter? • Is the Higgs mode a sharp, particle-like excitation or is it overdamped because it decays into other modes? Raman scattering data for NbSe 2 [from Measson et.al., Phys. Rev. B 89 , 060503 (2014)]

What is the fate of the Goldstone and Higgs modes near a disordered quantum phase transition?

• Collective modes: Goldstone and Higgs • Superfluid-Mott glass quantum phase transition • Fate of the collective modes at the superfluid-Mott glass transition • Conclusions

Disordered interacting bosons Ultracold atoms in optical potentials: • disorder: speckle laser field • interactions: tuned by Feshbach resonance and/or density F. Jendrzejewski et al., Nature Physics 8, 398 (2012) 1.0 Disordered superconducting films: 3 0.8 G(V)/G(4mV) 0.6 • energy gap in insulating as well as (V) 0.4 H=11T 0.2 2 superconducting phase 0.0 (V)/G -4 -3 -2 -1 0 1 2 3 4 V(mV) • preformed Cooper pairs ⇒ superconducting H=0 1 transition is bosonic G -2 -1 0 1 2 3 4 V(mV) Sherman et al., Phys. Rev. Lett. 108, 177006 (2012)

Disordered interacting bosons Bosonic quasiparticles in doped quantum magnets: Yu et al., Nature 489, 379 (2012) • bromine-doped dichloro-tetrakis-thiourea-nickel (DTN) • coupled antiferromagnetic chains of S = 1 Ni 2+ ions • S = 1 spin states can be mapped onto bosonic states with n = m s + 1

Bose-Hubbard model Bose-Hubbard Hamiltonian in two dimensions: H = U n i ) 2 − J ij ( a † � � (ˆ n i − ¯ i a j + h.c. ) 2 i � i,j � • superfluid ground state if Josephson couplings J ij dominate • insulating ground state if charging energy U dominates • chemical potential µ i = U ¯ n i Particle-hole symmetry: • large integer filling ¯ n i = k with integer k ≫ 1 ⇒ Hamiltonian invariant under ( ˆ n i − ¯ n i ) → − (ˆ n i − ¯ n i )

Phase diagrams (b) (c) (a) ~ µ ~ ~ µ /U µ /U /U 0 0 0 3 _ _ 3 3 _ SF SF SF 2 2 2 MI MI <n>=1 <n>=1 MI <n>=1 1 n=1 1 1 n=1 +δ n=1 1 _ ~ ~ (µ) µ ( ) 2 J J 0 1 _ ______ 1 _ 1 _ 0,c − δ + ______ BG BG MG − + 2 U 2 2 U 0 0 −δ 1 _ MI MI MI <n>=0 2 <n>=0 <n>=0 0 0 0 J /U 0 J J /U 0 n=0 /U n=0 0 0 0 n=0 0 ~ (µ) J 0,c −1 _ 1 _ −1 _ ________ − SF BG BG SF (1+δ ) 2 2 2 U + 0 −1 MI MI <n>=−1 MI <n>=−1 <n>=−1 −1 −1 n=−1 n=−1 n=−1 SF 3 3 −3 _ _ _ − −2 2 2 clean random potentials random couplings Weichman et al., Phys. Rev. B 7, 214516 (2008)

Stability of clean quantum critical point against dilution Site dilution: • randomly remove a fraction p of lattice sites • superfluid phase possible for 0 ≤ p ≤ p c (percolation threshold) Harris criterion: • for dilution p = 0 , quantum critical point is in 3D XY universality class • correlation length critical exponent ν ≈ 0 . 6717 • clean ν violates Harris criterion dν > 2 with d = 2 ⇒ clean critical behavior unstable against disorder (dilution) Critical behavior of superfluid-Mott glass transition must be in new universality class

Monte Carlo simulations • large-scale Monte Carlo simulations in 2d and 3d quantum fluctuations (~U/J) • conventional power-law critical behavior • universal critical exponents for dilutions 0 < p < p c • Griffiths singularities exponentially weak ( see classification in J. Phys. A 39 , R143 (2006), PRL 112 , 075702 (2014) ) (3+1)D exponents (2+1)D exponents exponent clean disordered exponent clean disordered z 1 1.52 z 1 1.67 ν 0.5 0.90 ν 0.6717 1.16 β/ν 1 1.09 β/ν 0.518 0.48 γ/ν 2 2.50 γ/ν 1.96 2.52

• Collective modes: Goldstone and amplitude (Higgs) • Superfluid-Mott glass quantum phase transition • Fate of the collective modes at the superfluid-Mott glass transition • Conclusions

Amplitude mode: scalar susceptibility • parameterize order parameter fluctuations into amplitude and direction � φ = φ 0 (1 + ρ )ˆ n • Amplitude mode is associated with scalar susceptibility χ ρρ ( � x, t ) = i Θ( t ) � [ ρ ( � x, t ) , ρ (0 , 0)] � • Monte-Carlo simulations compute imaginary time correlation function χ ρρ ( � x, τ ) = � ρ ( � x, τ ) ρ (0 , 0) � • Wick rotation required: analytical continuation from imaginary to real times/frequencies ⇒ maximum entropy method

Analytic continuation - maximum entropy method • Matsubara susceptibility χ ρρ ( iω m ) vs. spectral function A ( ω ) = χ ′′ ρρ ( ω ) /π � ∞ 2 ω χ ρρ ( iω m ) = dωA ( ω ) . ω 2 m + ω 2 0 Maximum entropy method: • inversion is ill-posed problem, highly sensitive to noise • fit A ( ω ) to χ ρρ ( iω m ) MC data by minimizing 2 σ 2 − αS Q = 1 • parameter α balances between fit error σ 2 L-curve 10 5 and entropy S of A ( ω ) , i.e., between fitting 10 4 information and noise σ 2 10 3 • best α value chosen by L-curve method [see 10 2 Bergeron et al., PRE 94, 023303 (2016)] 10 1 4 6 8 10 12 14 16 ln α

Amplitude mode in clean undiluted system χ ρρ ( ω ) = | r | 3 ν − 2 X ( ω | r | − ν ) Scaling form of the scalar susceptibility: [Podolsky + Sachdev, PRB 86, 054508 (2012)] 0.14 1 ν=0.664 0.12 0.5 0.1 ω Η 0.08 A( ω ) = |r | 0.0909 0.2 0.06 0.0682 0.0545 0.0474 0.04 0.0363 0.01 0.03 0.1 0.0272 r 0.0181 0.02 0.0090 0.0045 0 0 0.5 1 1.5 2 2.5 3 ω • sharp Higgs peak in spectral function • Higgs energy (mass) ω H scales as expected with distance from criticality r

Dispersion of the clean amplitude mode 0.125 q= 0 0.05 0.1 0.1 0.25 0.49 0.98 1.96 0.075 A( ω ) 0.05 0.025 0 0 1 2 3 4 ω spectral density at different q for Higgs mode dispersion r = − 0 . 0045

Amplitude mode in disordered system dilution p=1/3 0.05 0.04 | r |= 0.2391 0.03 A( ω ) 0.1756 0.1439 0.1122 0.02 0.0805 0.0488 0.0330 0.01 0.0171 0.0013 0 0 0.5 1 1.5 2 2.5 3 ω • spectral function shows broad peak near ω = 1 • peak is noncritical: does not move as quantum critical point is approached • amplitude fluctuations not soft at criticality • violates expected scaling form χ ρρ ( ω ) = | r | ( d + z ) ν − 2 X ( ω | r | − zν )

What is the reason for the absence of a sharp amplitude mode at the superfluid-Mott glass transition?

Quantum mean-field theory H = U n i ) 2 − J � � ǫ i ǫ j ( a † ǫ i (ˆ n i − ¯ i a j + h.c. ) 2 i � i,j � • truncate Hilbert space: keep only states | ¯ n − 1 � , | ¯ n � , and | ¯ n + 1 � on each site Variational wave function: � 1 � � θ i � � θ i �� � � e iφ i | ¯ n + 1 � i + e − iφ i | ¯ � | Ψ MF � = | g i � = cos | ¯ n � i + sin √ n − 1 � i 2 2 2 i i • locally interpolates between Mott insulator, θ = 0 , and superfluid limit, θ = π/ 2 Mean-field energy: � θ i � E 0 = � Ψ MF | H | Ψ MF � = U � ǫ i sin 2 � − J ǫ i ǫ j sin( θ i ) sin( θ j ) cos( φ i − φ j ) 2 2 i � ij � • solved by minimizing E 0 w.r.t. θ i ⇒ coupled nonlinear equations

m Diluted lattice: order parameter • local order parameter: m i = � a i � = sin( θ i ) e iφ i (dilution p = 1 / 3 ) 0.6 typical mean 0.4 U = 12 U = 14 U = 8 U = 10 0.0 0.2 0.4 0.6 0.8 1.0 mi 0.2 0.0 7 8 9 10 11 12 13 14 U