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Nonequilibrium electron dynamicsnear Mott transition
Sharareh Sayyad
In collaboration with: Martin Eckstein
Max Planck Institute for the Structure and Dynamics of Matter,University of Hamburg-CFEL, Hamburg, Germany
September 27, 2016
1
Photo induced Mott transition in 1T − TaS2
• Mott insulator at T=30 K
• Preparation of an excited state in crossover region↪→ Induced insulator-to-metal transition
Creation of hot carriers
Collapse of the gap < 100 fs
Fast thermalization < 100 fs
Relaxation of the excited state > 500 fs
Electronic relxation timescale< electron-lattice relxation timescale
� L. Perfetti, et al., Phys. Rev. Lett. 97, 067402 (2006).
2
Mott transition in the half-filled Hubbard model
H = −J∑
〈ij〉,σ
(c†iσcjσ + h.c.) + U
N∑
i=1
ni↑ni↓
U/W
(βW )−1
Metal
Insulator
Badmetal
-3 0 3
A(ω
)ω
U=4
β=1β=5
β=7
3
Relaxation to the Fermi liquid?
H = −J∑
〈ij〉,σ
(c†iσcjσ + h.c.) + U
N∑
i=1
ni↑ni↓
Preparing the system in the crossover region:
Excitation (quench)
Phys. Rev. Lett. 117, 096403(2016).
Fast thermalizationPhys. Rev. B 84, 035122(2011).
Studying the relaxation dynamics of the excited state
+electrons coupled to the environment
Σeb = λ U/W
(βW )−1
tcut
tf
Metal
Insulator
Crossover
4
Slow-relaxing electronic characteristic timescale
Solved by DMFT:
-3 0 3
A(ω
,t)
ω
β=3U=4, , λ=0.5
t=20t=50
-3 0 3
A(ω
,t)
ω
U=4, λ=0.5 β=10
t=20t=40t=80
Hubbard Bands :: Fast thermalization
→ inverse One-body energy-scales [ 14
]
Quasiparticle peak :: Slow retrieval (for β > 5)
→ timescale>> inverse One-body energy-scales [ 1
4]
>> inverse quasiparticle bandwidth [ 10.8
]
Bottleneck of dynamics (large β):
temperature-independent evolution of hqp
0.3
0.4
0.5
0.6
0.7
0.8
0.9
30 40 50 60 70 80
hq
p
time
U=4, λ=0.5
β=5β=6.5β=7.5
β=8β=10
5
Dynamical mean field theory
JU
lattice problem → impurity-bath problem
Solving the impurity problem
∆(t, t′)
t t′
DMFT self-consistency on Bethe lattice
Gimp(t, t′) ≡ ∑
kGlatt
k (t, t′)
Σlattk (t, t′) ≡ Σimp(t, t′)
ImpurityGlattk = 1
i∂t+µ−ǫk−Σlattk
solver
∆(t, t′) = J(t)Gimp(t, t′)J(t′)
approximation
Σimp, Gimp
Review on NEDMFT: Rev. Mod. Phys. 86, 779 (2014).
6
Impurity solver: U(1) slave-rotor
c†σ︸︷︷︸electron
= eiθ︸︷︷︸X:rotor
f†σ︸︷︷︸
spinon
⇔ Ge = GX .Gf
Atomic model:{|0〉, | ↑〉, | ↓〉, | ↑↓〉}︸ ︷︷ ︸
electron
≡ {|l = −1〉, |l = 0〉, |l = 0〉, |l = 1〉}︸ ︷︷ ︸Charge conservation: l=nf−1
⊗{|0〉, | ↑〉, | ↓〉, | ↑↓〉}︸ ︷︷ ︸
Spinon
On the impurity site:
Solving Dyson equations for
GX with ΣX = ∆.Gf
Gf with Σf = ∆.GX
Imposing constraints
Charge conservation
|X|2 = 1
Why slave-rotor impurity solver:
Accessibility of long time-evolution (Similar to NCA)
Accurate phase diagram near Mott transition (not the case in NCA)
� Equilibrium study: S. Florens and A. Georges, Phys. Rev. B. 66 165111 (2002).
� Nonequilibrium study: Sh. S and M. Eckstein, Phys. Rev. Lett. 117, 096403 (2016).
7
Equilibrium Physics: Slave-rotor Language
-3 0 3
A(ω
)
ω
U=4
β=1β=5
β=7
By decreasing the temperature:
Electron: height of the quasiparticle peak enhanced.
Rotor: density at zero frequency formed.
Spinon: nonmonotonous behavior as a function of β
-1
0
1
2
-4 -2 0 2 4
A(ω
)
ω
Af*0.2
β=1
a)
AfAx
-4 -2 0 2 4
ω
Af*0.04
β=5
b)
-4 -2 0 2 4
ω
Af*0.2
β=7
c)
Maximum in the spinon inverse bandwidth
0
60
120
180
240
0 2 4 6 8 10
τ eq
β
λ=0.5U=4
U=4.1U=4.2
U=4.25U=4.3
Footprint of this nonmonotonous response out of equilibrium ?
8
Spinon’s response: presence of a “⋂
” turn!
0
0.1
0.2
0.3
0.4
0.5
0 20 40 60 80 100 120
Gfre
t (t,t-s
)
s
U=4, λ=0.5, β=10
t=16
t=32
t=48
t=62
t=80
t=96t=112
t=128
9
Spinon’s response: presence of a “⋂
” turn!
0
0.1
0.2
0.3
0.4
0.5
0 20 40 60 80 100 120
Gfre
t (t,t-s
)
s
U=4, λ=0.5, β=10
t=16
t=32
t=48
t=62
t=80
t=96t=112
t=128
Time-evolution of Gretf :
1. Evolving against equilibrium value initially
2. “⋂
” turn: Reaching τmax at tmax
3. Start evolving towards equilibrium value
Nontrivial timescale:
“⋂
” turn: Bottleneck of dynamics at tmax
Quasiparticle retrieval only after t = tmax
τ−1ne = −∂sGret
f /Gretf
0
150
300
25 50 75
τn
e
t
s=16
U=4
U=4.1
U=4.2
U=4.25
U=4.3
10
Spinon “⋂
” turn: equilibrium and nonequilibrium
0
60
120
180
240
0 2 4 6 8 10
τ eq
β
λ=0.5
U=4U=4.1U=4.2
U=4.25U=4.3
Spinon nontrivial response:
Agreement between τmaxeq and τmax
neq
Spinon lifetime (tmax)∝ τmaxeq
Spinon lifetime reflected in electronic bottleneck of dynamics
11
Reflection of spinon lifetime in other observables?
Assume : =Gf (ω) ≈ δ(ω)/π
τ−1ne = −Σ
′′f (ω = 0) ≈ −πJ2 ∫
dωA(ω)A(ω)
cosh2(βω/2)
Emergence of a correlation timescale
0
60
120
180
240
300
0 2 4 6 8 10
τ eq
β
λ=0.5
U=4U=4.2U=4.3
Nontrivial spinon response in multi-orbital physics:
“Frozen” spin-spin correlation functionPh. Werner, et al., Phys. Rev. Lett. 101,166405 (2008).
Non-Fermi-liquid behavior of optical conductivity in perovskite ruthenates (∝ 1/√ω)
Y. S.Lee, et al., Phys. Rev. B 66, 041104 (2002).
12
Conclusion and Summary
• Study the Hubbard model near Mott transition
• Investigate the system under a quick ramp
• slave-rotor impurity solver + DMFT
Gimp(t, t′) ≡ ∑k G
lattk (t, t′)
Σlattk (t, t′) ≡ Σimp(t, t′)
ImpurityGlatt
k = 1i∂t+µ−ǫk−Σlatt
k solver
Slow retrieval of the quasiparticle density
Presence of a “⋂
” turn in the spinon retarded Green’s function
Presence of a nonmonotonous spinon behavior in equilibri1um
Agreement between equilibrium and nonequilibrium spinon response
0
60
120
180
240
300
0 2 4 6 8 10
τ eq
β
λ=0.5
U=4U=4.2U=4.3
Emergence of a correlation timescale
13
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