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Strong-disorder ferromagnetic quantum phase transitions
Thomas VojtaDepartment of Physics, Missouri University of Science and Technology
Dresden, May 6, 2019
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Collective modes at a disordered quantum phase transition
Thomas Vojta
Department of Physics, Missouri University of Science and Technology, USA
Dresden, May 6, 2019
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Outline
• Collective modes: Goldstone and
amplitude (Higgs)
• Superfluid-Mott glass quantum phase
transition
• Fate of the collective modes at the
superfluid-Mott glass transition
• Conclusions
Martin PuschmannJose Hoyos
Jack CrewseCameron Lerch
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Broken symmetries and collective modes
• collective excitation in systems with brokencontinuous symmetry, e.g.,− planar magnet breaks O(2) rotation symmetry− superfluid wave function breaks U(1) symmetry
• Amplitude(Higgs) mode: corresponds tofluctuations of order parameter amplitude
• Goldstone mode: corresponds to fluctuations oforder parameter phase
• Amplitude mode is condensed matter analogueof famous Higgs boson
effective potential for order parameter
in symmetry-broken phase
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What is the fate of the Goldstone and amplitude modes near adisordered quantum phase transition?
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• Collective modes: Goldstone and amplitude (Higgs)
• Superfluid-Mott glass quantum phase transition
• Fate of the collective modes at the superfluid-Mott glass transition
• Conclusions
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Disordered interacting bosons
Ultracold atoms in opticalpotentials:
• disorder: speckle laser field
• interactions: tuned byFeshbach resonance and/ordensity
F. Jendrzejewski et al., Nature Physics 8, 398 (2012)
-2 -1 0 1 2 3 4
1
2
3
-4 -3 -2 -1 0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
GH
=0(V
)/GH
=11T
(V)
V(mV)
G(V
)/G(4
mV
)
V(mV)
Sherman et al., Phys. Rev. Lett. 108, 177006 (2012)
Disordered superconducting films:
• energy gap in insulating as well assuperconducting phase
• preformed Cooper pairs ⇒ superconductingtransition is bosonic
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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 Ni2+ ions
• S = 1 spin states can be mapped onto bosonic states with n = ms + 1
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Bose-Hubbard model
Bose-Hubbard Hamiltonian in two dimensions:
H =U
2
∑i
(n̂i − n̄i)2 −∑〈i,j〉
Jij(a†iaj + h.c.)
• superfluid ground state if Josephson couplings Jij dominate
• insulating ground state if charging energy U dominates
• chemical potential µi = Un̄i
Particle-hole symmetry:
• large integer filling n̄i = k with integer k � 1⇒ Hamiltonian invariant under (n̂i − n̄i)→ −(n̂i − n̄i)
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Phase diagrams
/U0
(µ)~
0
0
0
~
<n>=1
<n>=−1
MI
n=0
SF
SF
J
/Uµ
1
0<n>=0
(a)
−1MI
MI
n=−1
n=1
1−
_23−
_2
2
23_
U0,cJ______
_
1
/U0
J0
U0
______~
+− +−δµ ( )
0~
0
<n>=−1
<n>=1
<n>=0
SF
MI
MI
n=1MI
n=0
n=−1
BG
BG
SF
(b)
1
0−δ
+δ
J
µ /U
−1
12_
12_
_32
_−23
−12_
_12
+(1+δ )0
(µ)~/U
~0
0 0
<n>=0
<n>=1
<n>=−1
MI
MI
________J0,cBG
BG
SF
MG
SF
J
/Uµ(c)
0
−1
1n=1
n=0
n=−1MI
U
12_
−12_
−32_
_32
clean random potentials random couplingsWeichman et al., Phys. Rev. B 7, 214516 (2008)
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Stability of clean quantum critical point against dilution
Site dilution:
• randomly remove a fraction p of lattice sites
• superfluid phase possible for 0 ≤ p ≤ pc (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 newuniversality class
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Monte Carlo simulations
10 100
1
2
3
4
56
L
Lm
ax
τ/L
1/81/52/71/3
9/25
p =
(2+1)D exponents
exponent clean disorderedz 1 1.52ν 0.6717 1.16β/ν 0.518 0.48γ/ν 1.96 2.52
• large-scale Mote Carlo simulations in2d and 3d
• conventional critical behavior withuniversal exponents for dilutions0 < p < pc
• Griffiths singularities exponentiallyweak (see classification in J. Phys. A 39,
R143 (2006), PRL 112, 075702 (2014))
(3+1)D exponents
exponent clean disorderedz 1 1.67ν 0.5 0.90β/ν 1 1.09γ/ν 2 2.50
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• Collective modes: Goldstone and amplitude (Higgs)
• Superfluid-Mott glass quantum phase transition
• Fate of the collective modes at the superfluid-Mott glass transition
• Conclusions
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Amplitude mode: scalar susceptibility
• parameterize order parameter fluctuations intoamplitude and direction
~φ = φ0(1 + ρ)n̂
• Amplitude mode is associated with scalarsusceptibility
χρρ(~x, t) = iΘ(t) 〈[ρ(~x, t), ρ(0, 0)]〉
• Monte-Carlo simulations compute imaginary time correlation function
χρρ(~x, τ) = 〈ρ(~x, τ)ρ(0, 0)〉
• Wick rotation required: analytical continuation from imaginary to realtimes/frequencies ⇒ maximum entropy method
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Amplitude mode in clean undiluted system
Scaling form of the scalar susceptibility: [Podolsky + Sachdev, PRB 86, 054508 (2012)]
χρρ(ω) = |r|3ν−2X(ω|r|−ν)
0 0.5 1 1.5 2 2.5 3ω
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
A(ω
)
0.01 0.03 0.1Tc-T
0.2
0.5
1
ωΗ
T=2.0022.0522.0822.1022.1222.1422.1622.1822.192
ν=0.664
• sharp Higgs peak in spectral function• Higgs energy (mass) ωH scales as expected with distance from criticality r
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Amplitude mode in disordered system
0 0.5 1 1.5 2 2.5 3ω
0
0.01
0.02
0.03
0.04
0.05
A(ω
)
T=1.2001.3001.3501.4001.4501.5001.5251.5501.575
dilution p=1/3Tc =1.577
• 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 χρρ(ω) = |r|(d+z)ν−2X(ω|r|−zν)
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What is the reason for the absence of a sharp amplitude mode at thesuperfluid-Mott glass transition?
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Quantum mean-field theory
H =U
2
∑i
εi(n̂i − n̄i)2 − J∑〈i,j〉
εiεj(a†iaj + h.c.)
• truncate Hilbert space: keep only states |n̄− 1〉, |n̄〉, and |n̄+ 1〉 on each site
Variational wave function:
|ΨMF 〉 =∏i
|gi〉 =∏i
[cos
(θi2
)|n̄〉i + sin
(θi2
)1√2
(eiφi|n̄+ 1〉i + e−iφi|n̄− 1〉i
)]
• locally interpolates between Mott insulator, θ = 0, and superfluid limit, θ = π/2
Mean-field energy:
E0 = 〈ΨMF |H|ΨMF 〉 =U
2
∑i
εi sin2
(θi2
)− J
∑〈ij〉
εiεj sin(θi) sin(θj) cos(φi − φj)
• solved by minimizing E0 w.r.t. θi ⇒ coupled nonlinear equations
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Diluted lattice: order parameter
• local order parameter: mi = 〈ai〉 = sin(θi)eiφi (dilution p = 1/3)
U = 8 U = 10
U = 12 U = 14
0.0 0.2 0.4 0.6 0.8 1.0mi
7 8 9 10 11 12 13 140.0
0.2
0.4
0.6
U
m
typicalmean
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Mean-field theory: excitations
• define local excitations (orthogonal to |gi〉, OP phase fixed at 0)
|gi〉 = cos
(θi2
)|n̄〉i + sin
(θi2
)1√2
(|n̄+ 1〉i + |n̄− 1〉i)
|θi〉 = sin
(θi2
)|n̄〉i − cos
(θi2
)1√2
(|n̄+ 1〉i + |n̄− 1〉i)
|φi〉 =1√2
(|n̄+ 1〉i − |n̄− 1〉i)
• expand H to quadratic order in excitations: H = E0 +Hθ +Hφ
Hθ =∑i
U2
+ 2J∑j′
sin(θi) sin(θj)
εib†θibθi−J
∑〈ij〉
cos(θi) cos(θj)εiεj(b†θi + bθi)(b
†θj + bθj)
Hφ has similar structure but different coefficients
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Clean system: results
• mean-field quantum phase transitionat U = 16J
• all excitations are spatially extended(plane waves)
Mott insulator
• all excitations are gapped
Superfluid
• Goldstone mode is gapless
• amplitude (Higgs) modes is gapped,gap vanishes at QCP
1,--�
0.5
• Computation-- Superfluid - w = v�l-�(V�/1-6�)�2
Insulator - W = 0
o ............ --............. � ............................................................. �---.......................... � ..... --� ..... 0 5 10 15 20
U
m
- -0.- -0.
200 ...
€>,
100
0 5
•
0
- - -
'
'6) '
10
' ' �,
Goldstone (computation) Higgs ( computation)
Goldstone
\
'
\
\
15
Higgs
20 25
ω02
U
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Diluted lattice: Goldstone mode
• Goldstone mode becomes massless insuperfluid phase, as required by Goldstone’stheorem
• wave function of lowestexcitation for U = 8 to15
• localized in insulator,delocalizes insuperfluid phase
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Goldstone mode: localization properties
• inverse participation ratio:
P−1 = N∑i |ψi|4
P → 1 for delocalized statesP → 0 for localized states
10° 10°
L top to bottom
Goldstone 32
32 64
10-1 • 64 128
128Higgs 10-1
-�- 32
� 10-2--o-- 64 /J
� --- 128
""'Cl
0-t 0 0-t p'cf ,If
,Jl'rz .cl'
'2 c,.oO
p" ,ti' 10-2 10-3 a ....
"' .� ....
a,Da
,ti'
• �,
5 10 15 0 2 4 6
u w
• wave function atU = 8 as function ofexcitation energy
• delocalized at ω = 0,localized for higherenergies
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Amplitude (Higgs) mode
• amplitude mode strongly localized for allU and all excitation energies
10°
Goldstone 32
10-1 • 64128
Higgs -�- 32
� 10-2--o-- 64 /J
--- 128 ""'Cl
0-t 0 p'cf ,If
,Jl'rz .cl'
'2 c,.oO
p" ,ti' 10-3a .. ..
"' .� ....
a,Da
,ti'
• �,
5 10 15 u
• wave function of lowestexcitation for U = 8 to15
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Longitudinal and transverse susceptibilities (q = 0)
U = 15
U = 14
U = 13
U = 12
U = 11
U = 10
0 2 4 60
2
4
6
ω
χ′′
U = 17
U = 16
U = 15
U = 14
U = 13
U = 12
0 2 4 60
2
4
6
ω
χ′′
diluted, p = 1/3 clean
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Conclusions
• disordered interacting bosons undergo quantum phase transition betweensuperfluid state and insulating Mott glass state
• conventional critical behavior with universal critical exponents
• Griffiths effects exponentially weak [see classification in T.V., J. Phys. A 39, R143 (2006)]
• collective modes in superfluid phase show striking localization behavior
• Goldstone mode is delocalized at ω = 0 but localizes with increasing energy
• amplitude (Higgs) mode is strongly localized for all energies
• broad incoherent scalar response at q = 0, violates naive scaling
Exotic collective mode dynamics even if critical behavior is conventional
T.V., Jack Crewse, Martin Puschmann, Daniel Arovas, and Yury Kiselev, PRB 94, 134501 (2016)
Jack Crewse, Cameron Lerch and T.V., PRB 98, 054514 (2018)
Cameron Lerch and T.V., Eur. Phys. J. ST 227, 22753 (2019)
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Analytic continuation - maximum entropy method
• Matsubara susceptibility χρρ(iωm) vs. spectral function A(ω) = χ′′ρρ(ω)/π
χρρ(iωm) =
∫ ∞0
dωA(ω)2ω
ω2m + ω2
.
Maximum entropy method:
• inversion is ill-posed problem, highly sensitiveto noise
• fit A(ω) to χρρ(iωm) MC data by minimizing
Q = 12σ
2 − αS
• parameter α balances between fit error σ2
and entropy S of A(ω), i.e., between fittinginformation and noise
• best α value chosen by L-curve method [see
Bergeron et al., PRE 94, 023303 (2016)]
4 6 8 10 12 14 16ln α
101
102
103
104
105
σ2
L-curve