Strong-disorder ferromagnetic quantum phase transitions Thomas Vojta - PowerPoint PPT Presentation

Strong-disorder ferromagnetic quantum phase transitions Thomas Vojta Department of Physics, Missouri University of Science and Technology Dresden, May 6, 2019

Collective modes at a disordered quantum phase transition Thomas Vojta Department of Physics, Missouri University of Science and Technology, USA Dresden, May 6, 2019

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

Broken symmetries and collective modes • collective excitation in systems with broken continuous symmetry , e.g., − planar magnet breaks O(2) rotation symmetry − superfluid wave function breaks U(1) symmetry • Amplitude(Higgs) mode : corresponds to fluctuations of order parameter amplitude • Goldstone mode : corresponds to fluctuations of order parameter phase • Amplitude mode is condensed matter analogue of famous Higgs boson effective potential for order parameter in symmetry-broken phase

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

• Collective modes: Goldstone and amplitude (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 2 0.2 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 (a) (b) (c) ~ µ ~ ~ µ /U µ /U /U 0 0 0 3 _ 3 _ 3 _ SF SF SF 2 2 2 MI MI <n>=1 MI <n>=1 <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 /U n=0 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 Mote Carlo simulations in 6 p = 5 2d and 3d 1 / 8 1 / 5 4 2 / 7 • conventional critical behavior with 1 / 3 3 /L 9 / 25 universal exponents for dilutions L max 0 < p < p c 2 τ • Griffiths singularities exponentially weak ( see classification in J. Phys. A 39 , 1 R143 (2006), PRL 112 , 075702 (2014) ) 10 100 L (2+1)D exponents (3+1)D exponents exponent clean disordered exponent clean disordered z 1 1.52 z 1 1.67 ν 0.6717 1.16 ν 0.5 0.90 β/ν 0.518 0.48 β/ν 1 1.09 γ/ν 1.96 2.52 γ/ν 2 2.50

• 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

Amplitude mode in clean undiluted system Scaling form of the scalar susceptibility: [Podolsky + Sachdev, PRB 86, 054508 (2012)] χ ρρ ( ω ) = | r | 3 ν − 2 X ( ω | r | − ν ) 0.14 1 ν=0.664 0.12 0.5 0.1 ω Η 0.08 A( ω ) T= 2.002 0.2 0.06 2.052 2.082 2.102 0.04 0.01 0.03 0.1 2.122 T c -T 2.142 2.162 0.02 2.182 2.192 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

Amplitude mode in disordered system dilution p=1/3 0.05 T c =1.577 0.04 T= 1.200 0.03 A( ω ) 1.300 1.350 1.400 0.02 1.450 1.500 1.525 0.01 1.550 1.575 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

Mean-field theory: excitations • define local excitations (orthogonal to | g i � , OP phase fixed at 0) � 1 � θ i � � θ i | g i � = cos | ¯ n � i + sin √ 2 ( | ¯ n + 1 � i + | ¯ n − 1 � i ) 2 2 � 1 � θ i � � θ i √ | θ i � = sin | ¯ n � i − cos 2 ( | ¯ n + 1 � i + | ¯ n − 1 � i ) 2 2 1 √ | φ i � = 2 ( | ¯ n + 1 � i − | ¯ n − 1 � i ) • expand H to quadratic order in excitations: H = E 0 + H θ + H φ    U  ǫ i b † � � H θ = 2 + 2 J sin( θ i ) sin( θ j ) θi b θi j ′ i � cos( θ i ) cos( θ j ) ǫ i ǫ j ( b † θi + b θi )( b † − J θj + b θj ) � ij � H φ has similar structure but different coefficients

15 200 ..../ 0 5 10 computation) 20 Higgs ( m Goldstone (computation) ../ ..../ €?, 100 0 5 • 0 --- ' ' 6* ' -.� � ' ' 1,--� ω 02 25 20 Higgs 15 \ \ \ ' Goldstone 0.5 �--......./....../...../...../ • Computation -- Superfmuid - w = v�l-�( V�/1-6�)�2 Insulator - W = 0 o ..../....../ -......./...../ � ...../....../..../....../...../...../...../....../..../...../ �, Clean system: results • mean-field quantum phase transition at U = 16 J • all excitations are spatially extended (plane waves) Mott insulator • all excitations are gapped U Superfluid - -0/ - -0/ • Goldstone mode is gapless • amplitude (Higgs) modes is gapped, gap vanishes at QCP 10 U

Diluted lattice: Goldstone mode • Goldstone mode becomes massless in superfluid phase, as required by Goldstone’s theorem • wave function of lowest excitation for U = 8 to 15 • localized in insulator, delocalizes in superfluid phase