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Spectral statistics and many-body localization Jan Suntajs, Lev - - PowerPoint PPT Presentation

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Spectral statistics and many-body localization Jan Suntajs, Lev Vidmar, Janez Bon ca September 16, 2019 Jan Suntajs Spectral statistics and MBL September 16, 2019 1 / 17 Many-body localization (MBL) - what is it about? Occurring in

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SLIDE 1

Spectral statistics and many-body localization

Jan ˇ Suntajs, Lev Vidmar, Janez Bonˇ ca September 16, 2019

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 1 / 17

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SLIDE 2

Many-body localization (MBL) - what is it about?

1

Occurring in INTERACTING quantum systems with DISORDER

2

An IDEAL INSULATOR → at ANY temperature

3

Explains the FAILURE of some systems to THERMALIZE

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 2 / 17

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SLIDE 3

Many-body localization (MBL) - what is it about?

1

Occurring in INTERACTING quantum systems with DISORDER

2

An IDEAL INSULATOR → at ANY temperature

3

Explains the FAILURE of some systems to THERMALIZE

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 2 / 17

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SLIDE 4

Many-body localization (MBL) - what is it about?

1

Occurring in INTERACTING quantum systems with DISORDER

2

An IDEAL INSULATOR → at ANY temperature

3

Explains the FAILURE of some systems to THERMALIZE

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 2 / 17

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SLIDE 5

Many-body localization (MBL) - what is it about?

1

Occurring in INTERACTING quantum systems with DISORDER

2

An IDEAL INSULATOR → at ANY temperature

3

Explains the FAILURE of some systems to THERMALIZE

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 2 / 17

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SLIDE 6

The outline of the presentation

1

The properties of MBL systems

2

Introduction of the physical model

3

Numerical analysis of the spectral statistics

A brief introduction to spectral statistics Spectral form factor (SFF)

4

Our recent results and conclusion

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 3 / 17

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SLIDE 7

The outline of the presentation

1

The properties of MBL systems

2

Introduction of the physical model

3

Numerical analysis of the spectral statistics

A brief introduction to spectral statistics Spectral form factor (SFF)

4

Our recent results and conclusion

arXiv: 1905.06345 [cond-mat.str-el], 15. May 2019

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 3 / 17

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SLIDE 8

MBL - a quick recap

Closed quantum systems

Nandkishore, Huse, 2015

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 4 / 17

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SLIDE 9

MBL - a quick recap

Closed quantum systems

Nandkishore, Huse, 2015

Presence of interactions

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 4 / 17

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SLIDE 10

MBL - a quick recap

Closed quantum systems

Nandkishore, Huse, 2015

Presence of interactions Presence of disorder

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 4 / 17

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SLIDE 11

MBL - a quick recap

Closed quantum systems

Nandkishore, Huse, 2015

Presence of interactions Presence of disorder

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 4 / 17

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SLIDE 12

The properties of MBL systems

The absence of ergodicity

Abanin, Altman, Bloch, Serbyn, 2018

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 5 / 17

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SLIDE 13

The properties of MBL systems

The absence of ergodicity

Abanin, Altman, Bloch, Serbyn, 2018

THE ENTANGLEMENT ENTROPY:

Area law scaling for all eigenstates Logarithmic growth in time

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 5 / 17

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SLIDE 14

The properties of MBL systems

The absence of ergodicity

Abanin, Altman, Bloch, Serbyn, 2018

THE ENTANGLEMENT ENTROPY:

Area law scaling for all eigenstates Logarithmic growth in time

1 2 3 4 5 6 W 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ε

SA, L = 14, Nh = 0, Nu = 7, LA = L/2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

SA LA log(2)

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 5 / 17

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SLIDE 15

The properties of MBL systems

The absence of ergodicity

Abanin, Altman, Bloch, Serbyn, 2018

THE ENTANGLEMENT ENTROPY:

Area law scaling for all eigenstates Logarithmic growth in time

Special properties of the energy spectra

The subject of our numerical analysis

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 5 / 17

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SLIDE 16

The properties of MBL systems

The absence of ergodicity

Abanin, Altman, Bloch, Serbyn, 2018

THE ENTANGLEMENT ENTROPY:

Area law scaling for all eigenstates Logarithmic growth in time

Special properties of the energy spectra

The subject of our numerical analysis

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 5 / 17

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SLIDE 17

The model

The hamiltonian - paradigmatic ‘quantum chaotic’/ergodic: Heisenberg-like 1D ˆ H =

  • j∈{1,2}

Jj

L

  • ℓ
  • ˆ

sx

ℓ ˆ

sx

ℓ+j + ˆ

sy

ℓ ˆ

sy

ℓ+j + ∆jˆ

sz

ℓ ˆ

sz

ℓ+j

  • +

L

  • ℓ=1

wℓˆ sz

ℓ

l - site, L - chain length wℓ: randomly disordered potential

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 6 / 17

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SLIDE 18

The model

J1 = J2 = 1 ∆1 = ∆2 = 0.55 Disorder probability distribution:

−W W wl

1 2W

p(wl)

W - the disorder strength parameter

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 6 / 17

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Our (numerical) analysis of the MBL systems

We perform full or partial diagonalization of the Hamiltonians Spectrum: {E1 ≤ E2 ≤ · · · ≤ ED} Partial diagonalization: ≈ 500 eigenstates from the middle of the spectra Maximum Hilbert space dimensions: D = 48620 (full) D = 184756 (partial) Between 102 − 103 different disorder realizations for each model parameter

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 7 / 17

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Quantum chaos and energy spectra

Why do we study energy spectra? Quantum chaos conjecture (Bohigas, Giannoni, Schmidt, 1984): Quantum systems Spectral properties match the predictions of the random matrix theory (RMT). Corresponding classical systems The dynamics are completely chaotic.

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 8 / 17

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SLIDE 21

Quantum chaos and energy spectra

What about systems without a classical analogue?

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 8 / 17

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Quantum chaos and energy spectra

Many-body quantum chaos GENERIC systems → RMT-like spectral statistics

Montambaux et. al. (1993), Prosen (1999), Santos and Rigol (2008)

RMT statistics → hallmarks of ergodicity and thermalization in an isolated quantum system.

D’Alessio, Kafri, Polkovnikov, Rigol (2016)

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 8 / 17

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Statistical properties of the energy spectra

We analyse the statistical properties of the energy spectra We rely on the findings of the RMT:

Ergodic systems: spectral statistic match the Gaussian

  • rthogonal ensemble (GOE)

MBL systems: nearest levels distributed in accordance with the Poisson distribution

We compare our RESULTS with the above cases.

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 9 / 17

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SLIDE 24

Statistical properties of the energy spectra

We analyse the statistical properties of the energy spectra We rely on the findings of the RMT:

Ergodic systems: spectral statistic match the Gaussian

  • rthogonal ensemble (GOE)

MBL systems: nearest levels distributed in accordance with the Poisson distribution

We compare our RESULTS with the above cases.

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 9 / 17

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The mean ratio of the level spacings

The spacings between the nearest energy levels: δn = En+1 − En ≥ 0 We define the level spacing ratio: 0 ≤ ˜ rn = min{δn, δn−1}/ max{δn, δn−1} ≤ 1 KEYNOTE: the limiting values of ˜ r are well known: Ergodic: ˜ rGOE = 0.5307 MBL: ˜ rP = 2 ln 2 − 1 ≈ 0.3863

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 10 / 17

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The mean ratio of the level spacings

The spacings between the nearest energy levels: δn = En+1 − En ≥ 0 We define the level spacing ratio: 0 ≤ ˜ rn = min{δn, δn−1}/ max{δn, δn−1} ≤ 1 KEYNOTE: the limiting values of ˜ r are well known: Ergodic: ˜ rGOE = 0.5307 MBL: ˜ rP = 2 ln 2 − 1 ≈ 0.3863

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 10 / 17

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The mean ratio of the level spacings

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 10 / 17

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Pros and cons of ˜ r

Mean level spacings ratio: a commonly used indicator of a given system’s ergodicity + straightforward implementation

  • only considers correlation between the nearest energy levels

We would like to consider correlations between all the levels This is why we implement the spectral form factor (SFF) the implementation is more demanding

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 11 / 17

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SLIDE 29

Pros and cons of ˜ r

Mean level spacings ratio: a commonly used indicator of a given system’s ergodicity + straightforward implementation

  • only considers correlation between the nearest energy levels

We would like to consider correlations between all the levels This is why we implement the spectral form factor (SFF) the implementation is more demanding

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 11 / 17

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The spectral form factor (SFF)

Definition: K(τ) :=

  • 1

D

D

  • i,j

e−i(εi−εj)τ

  • ;

K(0) = D, K(τ → τH) = 1 D - Hilbert space dimension τ → an external parameter ... over disorder realizations Heisenberg time τH ∝ inverse mean level spacing (largest sensible timescale of a system) {εi} → energy levels after spectral unfolding

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 12 / 17

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SLIDE 31

The spectral form factor (SFF)

Definition: K(τ) :=

  • 1

D

D

  • i,j

e−i(εi−εj)τ

  • ;

K(0) = D, K(τ → τH) = 1 D - Hilbert space dimension τ → an external parameter ... over disorder realizations Heisenberg time τH ∝ inverse mean level spacing (largest sensible timescale of a system) {εi} → energy levels after spectral unfolding

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 12 / 17

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The spectral form factor (SFF)

A quick introduction to unfolding

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 12 / 17

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The spectral form factor (SFF)

Unfolding → mean level spacing = 1

RAW SPECTRA UNFOLDED SPECTRA

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 12 / 17

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The spectral form factor (SFF)

SFF - KEYNOTES: we again expect different behaviour for ergodic and uncorrelated spectra we investigated the behaviour of the Thouless time τTh Thouless time τTh → the onset of UNIVERSAL DYNAMICS

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 12 / 17

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The spectral form factor (SFF)

ERG MBL

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 12 / 17

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The spectral form factor (SFF)

ERG MBL DECAY

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 12 / 17

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The spectral form factor (SFF)

ERG MBL DECAY RAMP

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 12 / 17

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The spectral form factor (SFF)

We see universal behaviour after some transient time τTh K(τ) for uncorrelated spectra KP(τ) = 1 K(τ) in ergodic systems KGOE(τ) = 2 − τ log

2τ + 1

2τ − 1

  • Jan ˇ

Suntajs Spectral statistics and MBL September 16, 2019 12 / 17

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SFF - explaining the Thouless time τTh

Thouless time τTh: determines the energy scale at which the spectral correlations are universally determined by the GOE predictions (e.g., when the RAMP appears) large(r) τTh → small(er) spectral correlation length ETh

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 13 / 17

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SFF - explaining the Thouless time τTh

Thouless time τTh: determines the energy scale at which the spectral correlations are universally determined by the GOE predictions (e.g., when the RAMP appears) large(r) τTh → small(er) spectral correlation length ETh

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 13 / 17

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SFF - questions

We set out to find out different scalings

How does τTh scale with the system size L? How does τTh scale with disorder strength parameter W

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 14 / 17

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SLIDE 42

SFF - questions

We noticed some surprising results along the way

arXiv:1905.06345 [cond-mat.str-el], 15 May 2019

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 14 / 17

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SFF - results

We first checked for consistency of the K(τ) and ˜ r results

10−5 10−4 10−3 10−2 10−1 100 τ 10−5 10−3 10−1 101 103 105 K(τ) τTh τTh τH

W = 0.5 W = 4.0 W = 8.0 KGOE(τ)

4 8 W 0.4 0.5 ˜ r

Poisson GOE

L = 18

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 15 / 17

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SFF - results

We then numerically extracted τTh values and performed a scaling analysis w.r.t. both L and W In the subsequent scaling analysis, we introduce the PHYSICAL THOULESS TIME tTh, rescaling τTh by the mean level spacing δE of the RAW spectra: tTh = τTh/δE

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 15 / 17

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SFF - results

We then numerically extracted τTh values and performed a scaling analysis w.r.t. both L and W In the subsequent scaling analysis, we introduce the PHYSICAL THOULESS TIME tTh, rescaling τTh by the mean level spacing δE of the RAW spectra: tTh = τTh/δE

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 15 / 17

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SFF - results

The results of the scaling analysis

2 4 6 W 10−2 10−1 100

tTh/L2

a

L = 12 L = 14 L = 16 L = 18 t0 eW/Ω

We obtain the following double scaling: tTh = t0eW/ΩL2

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 15 / 17

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SFF - results

Verification of the scaling: data collapse

UNIVERSAL NONUNIVERSAL

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 15 / 17

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SFF - results

... Is there a MBL transition at all? a SUPPOSED MBL transition occurs for some CRITICAL disorder W ∗ when the energy spectrum becomes uncorrelated: tTh(W ∗) = tH tH scaling is given by: tH ∝ exp (L ln2) combining these results gives us W ∗ ≈ Ω ln(2) L ∝ L IMPLIES ABSENCE OF MBL IN THE THERMODYNAMIC LIMIT!

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 15 / 17

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SFF - results

... Is there a MBL transition at all? a SUPPOSED MBL transition occurs for some CRITICAL disorder W ∗ when the energy spectrum becomes uncorrelated: tTh(W ∗) = tH tH scaling is given by: tH ∝ exp (L ln2) combining these results gives us W ∗ ≈ Ω ln(2) L ∝ L IMPLIES ABSENCE OF MBL IN THE THERMODYNAMIC LIMIT!

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 15 / 17

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SFF - results

... Is there a MBL transition at all? a SUPPOSED MBL transition occurs for some CRITICAL disorder W ∗ when the energy spectrum becomes uncorrelated: tTh(W ∗) = tH tH scaling is given by: tH ∝ exp (L ln2) combining these results gives us W ∗ ≈ Ω ln(2) L ∝ L IMPLIES ABSENCE OF MBL IN THE THERMODYNAMIC LIMIT!

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 15 / 17

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SLIDE 51

SFF - results

... Is there a MBL transition at all? a SUPPOSED MBL transition occurs for some CRITICAL disorder W ∗ when the energy spectrum becomes uncorrelated: tTh(W ∗) = tH tH scaling is given by: tH ∝ exp (L ln2) combining these results gives us W ∗ ≈ Ω ln(2) L ∝ L IMPLIES ABSENCE OF MBL IN THE THERMODYNAMIC LIMIT!

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 15 / 17

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Verification of our results

We tested our conclusions against more commonly used statistics However, we interpreted our results DIFFERENTLY

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 16 / 17

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SLIDE 53

Verification of our results

We tested our conclusions against more commonly used statistics However, we interpreted our results DIFFERENTLY

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 16 / 17

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SLIDE 54

Verification of our results

We tested our conclusions against more commonly used statistics However, we interpreted our results DIFFERENTLY

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 16 / 17

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Verification of our results

BOTTOM LINE: results for different spectral statistics seem consistent

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 16 / 17

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Conclusions and further work

Our results for the PARADIGMATIC class of models expected to give MBL show NO INDICATIONS of the MBL transition The emergence of QUANTUM CHAOS for ANY disorder strength in the TD limit

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 17 / 17

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Conclusions and further work

We need to test our assumptions on other models in which MBL is predicted We need to examine and better understand the relationship between our results and the transport properties

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 17 / 17

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Conclusions and further work

Thank you for your attention!

Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 17 / 17