March meeting B45.00013

1. Transition from Quantum Many-body Localization to Quantum Spin
Glass
Zeyang Li1,2 Pai Peng3,2
1 Department of Physics, MIT
2 Research Laboratory of Electronics, MIT
3 Department of EECS, MIT
APS March Meeting, March 2-6, 2020
Zeyang Li (MIT) LIOMs for MBL to SG APS March Meeting, March 2-6, 2020 1 / 8

2. Outline
1 Introduction
Ergodicity Breaking: MBL and SG
Many-Body Localization, Spin Glass and Local Integrals Of Motion
2 Algorithm
3 Results on Transition
Phase transition for different N
4 Acknowledgement
Zeyang Li (MIT) LIOMs for MBL to SG APS March Meeting, March 2-6, 2020 2 / 8

3. Introduction
Many body non-ergodicity system:
Resist from relaxation to equilibrium by
(chaotic) interaction.
0.8
1 8 sites
12 sites
b
Uncorrelated Short-range correlated Long-range correlateda
c
1/L
Thermal
Critical
MBL
Δx Δx
d
e
=
=
=
Gc
(n)
Gtot
(n)
Gdis
(n)
= –
M
B
L
C
ritica
l
T
h
e
rm
a
l
0.15
0
Long-range correlated d –Gc
(n)
Gtot
(n)
=
–=
–
–
– – –
e
T
h
e
rm
a
l
––
=
=
– – –
T
h
e
rm
a
l
T
h
e
rm
a
l
–
T
h
e
rm
a
l
T
h
e
rm
a
l
–
–
– – ...
n=2n=3n=4
0.1 1 10 100
Time t ( )
10-1
100
101
Transportdistancex(sites)
Thermal (W = 1.0 J)
Critical (W = 4.8 J)
MBL (W = 8.9 J)
4 6 8
0
0.2
0.4
Disorder strength W (J)
Exponent
Time
A. Lukin, et al. Nature 573, 385-389 (2019)
Zeyang Li (MIT) LIOMs for MBL to SG APS March Meeting, March 2-6, 2020 2 / 8

4. Introduction
Many-body localization:
Not thermalizing due to an extensive set of local integrals of motion (LIOMs).
H =
L
i=1
hiσi
z + interaction hi ∈ [−W, W]; Wc ≈ 7
Zeyang Li (MIT) LIOMs for MBL to SG APS March Meeting, March 2-6, 2020 3 / 8

5. Introduction
Many-body localization:
Not thermalizing due to an extensive set of local integrals of motion (LIOMs).
H =
L
i=1
hiσi
z + interaction hi ∈ [−W, W]; Wc ≈ 7
=
L
i=1
ξiτi
z +
ij
Vijτi
zτj
z +
ijk
Vijkτi
zτj
z τk
z + · · ·
(1)
Spin Glass:
Metastable low-energy states separated by large energy barriers.
H = −
L
i=1
Jiσz
i σz
i+1 + h
L
i=1
σx
i + interaction Ji ∈ [−J, J]
= H({τz
i }) : Diagonal and “random”
(2)
For h |Ji|, the energy eigenstates are 2-fold degenerate. Many “domain-well” prevents connecting
near-resonant energy states.
Zeyang Li (MIT) LIOMs for MBL to SG APS March Meeting, March 2-6, 2020 3 / 8

6. LIOM
Non interaction: Wigner-Jordan transformation.
With interaction, how to understand? LIOM!
Numerically constructing LIOMs:
Infinite-time averaging (Chandran et. al. PRB 91, 085425 (2015), Geraedts et. al. PRB 95, 064204 (2017),
Goihl et. al. PRB 97, 134202 (2018))
Perturbation theory (Ros et. al. JNPB 841, 440 (2015), Rademaker M.Ortu˜no PRL 116, 010404 (2016))
Optimization (O’Brien et. al. PRB 94, 144208 (2016), Wahl et. al. PRX 7, 021018 (2017), Kulshreshtha
PRB 98, 184201 (2018))
Wegner-Wilson flow renormalization (Pekker et. al. PRL 119, 075701 (2017))
Zeyang Li (MIT) LIOMs for MBL to SG APS March Meeting, March 2-6, 2020 4 / 8

7. LIOM
Non interaction: Wigner-Jordan transformation.
With interaction, how to understand? LIOM!
Numerically constructing LIOMs:
Infinite-time averaging (Chandran et. al. PRB 91, 085425 (2015), Geraedts et. al. PRB 95, 064204 (2017),
Goihl et. al. PRB 97, 134202 (2018))
Perturbation theory (Ros et. al. JNPB 841, 440 (2015), Rademaker M.Ortu˜no PRL 116, 010404 (2016))
Optimization (O’Brien et. al. PRB 94, 144208 (2016), Wahl et. al. PRX 7, 021018 (2017), Kulshreshtha
PRB 98, 184201 (2018))
Wegner-Wilson flow renormalization (Pekker et. al. PRL 119, 075701 (2017))
Exact Diagonalization (P. Peng, ZL, H. Yan, K. X. Wei, P. Cappellaro, PRB 100, 214203 (2019))
Zeyang Li (MIT) LIOMs for MBL to SG APS March Meeting, March 2-6, 2020 4 / 8

8. Scheme of algorithm (“quick-sort” like)
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&' = 2
-1
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&' = 1
&' = 3
+1
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-1-1
+1 +1
-1 +1
-1 +1
Tr(σ2
zτ2
z ) > Tr(σ1
zτ1
z ), Tr(σ3
zτ3
z )
τj
z = 2L
k=1 aj
k|k k|, aj
k = ±1, k aj
k = 0; Tr(τi
zτj
z ) ∝ δi,j
Maximize Tr(σj
zτj
z ) = k k|σj
z|k aj
k
'
&
$
%'
&
$
%
Zeyang Li (MIT) LIOMs for MBL to SG APS March Meeting, March 2-6, 2020 5 / 8

9. Phase transition
H =
N
i=1
Si · Si+1 + JiSx
i Sx
i+1 + hiSz
i (3)
Overlap between LIOMs and local physical
operators: G = ∆ Tr(τjOj) i
0 10 20
0
10
20
∆LIOM i for N=4
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20
0
10
20
∆LIOM i for N=9
0.0
0.2
0.4
0.6
0.8
1.0
Spin Glass Order (for a chosen energy
eigenstate |n ): χSG = i,j n|Sx
i Sx
j |n 2/N
0 10 20
0
10
20
χSG/N for N = 4
0.0
0.2
0.4
0.6
0.8
1.0
0 10 20
0
10
20
χSG/N for N = 9
0.0
0.2
0.4
0.6
0.8
1.0
Zeyang Li (MIT) LIOMs for MBL to SG APS March Meeting, March 2-6, 2020 6 / 8

10. What’s the transition?
From ∆LIOM plot, we cannot observe any N-scaling from N = 4 to N = 9. However there is
a clear difference in the SG-order.
Zeyang Li (MIT) LIOMs for MBL to SG APS March Meeting, March 2-6, 2020 7 / 8

11. What’s the transition?
From ∆LIOM plot, we cannot observe any N-scaling from N = 4 to N = 9. However there is
a clear difference in the SG-order.
The reason is there are actually 3 phases: MBL, MBL SG, and thermal (where J ∼ h ).
The transition from MBL to MBL SG must go through the thermal phase. The SG parameter
doesn’t distinguish thermal with naive MBL, and there is only one transition whose sharpness
scales with N. For the LIOM’s difference, there are two transitions and the “transition width”
we observed is nothing more than the width of the thermal phase, and therefore should not
scale with N.
Zeyang Li (MIT) LIOMs for MBL to SG APS March Meeting, March 2-6, 2020 7 / 8

12. Acknowledgement
Based on extensive discussion with Pai Peng and Bing-Tian Ye.
Also thanks for motivation from Dmitry A. Abanin.
Mentally (and financially) supported by my family.
Zeyang Li (MIT) LIOMs for MBL to SG APS March Meeting, March 2-6, 2020 8 / 8