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
Many-body localization (MBL) - what is it about? Occurring in INTERACTING quantum systems with DISORDER 1 An IDEAL INSULATOR → at ANY temperature 2 Explains the FAILURE of some systems to THERMALIZE 3 Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 2 / 17
Many-body localization (MBL) - what is it about? Occurring in INTERACTING quantum systems with DISORDER 1 An IDEAL INSULATOR → at ANY temperature 2 Explains the FAILURE of some systems to THERMALIZE 3 Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 2 / 17
Many-body localization (MBL) - what is it about? Occurring in INTERACTING quantum systems with DISORDER 1 An IDEAL INSULATOR → at ANY temperature 2 Explains the FAILURE of some systems to THERMALIZE 3 Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 2 / 17
Many-body localization (MBL) - what is it about? Occurring in INTERACTING quantum systems with DISORDER 1 An IDEAL INSULATOR → at ANY temperature 2 Explains the FAILURE of some systems to THERMALIZE 3 Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 2 / 17
The outline of the presentation The properties of MBL systems 1 Introduction of the physical model 2 Numerical analysis of the spectral statistics 3 A brief introduction to spectral statistics Spectral form factor ( SFF ) Our recent results and conclusion 4 Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 3 / 17
The outline of the presentation The properties of MBL systems 1 Introduction of the physical model 2 Numerical analysis of the spectral statistics 3 A brief introduction to spectral statistics Spectral form factor ( SFF ) Our recent results and conclusion 4 arXiv: 1905.06345 [cond-mat.str-el], 15. May 2019 Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 3 / 17
MBL - a quick recap Closed quantum Nandkishore, Huse, 2015 systems Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 4 / 17
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
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
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
The properties of MBL systems The absence of Abanin, Altman, Bloch, Serbyn, 2018 ergodicity Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 5 / 17
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
The properties of MBL systems The absence of ergodicity Abanin, Altman, Bloch, Serbyn, 2018 S A , L = 14, N h = 0, N u = 7 , L A = L/ 2 1 . 0 0 . 9 THE ENTANGLEMENT ENTROPY: 0 . 8 0 . 7 Area law scaling for all eigenstates 0 . 6 0 . 5 ε Logarithmic growth in time 0 . 4 0 . 3 0 . 2 0 . 1 0 . 0 0 1 2 3 4 5 6 W 0 . 8 0 . 7 0 . 6 0 . 5 0 . 4 0 . 3 0 . 2 0 . 1 S A L A log(2) Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 5 / 17
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
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
The model The hamiltonian - paradigmatic ‘quantum chaotic’/ergodic : Heisenberg-like 1D L L � � ˆ s x s x s y s y s z s z s z � � � H = J j ˆ ℓ ˆ ℓ + j + ˆ ℓ ˆ ℓ + j + ∆ j ˆ ℓ ˆ + w ℓ ˆ ℓ + j ℓ ℓ ℓ =1 j ∈{ 1 , 2 } l - site, L - chain length w ℓ : randomly disordered potential Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 6 / 17
The model J 1 = J 2 = 1 ∆ 1 = ∆ 2 = 0 . 55 Disorder probability distribution: 1 2 W p ( w l ) 0 − W 0 W w l W - the disorder strength parameter Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 6 / 17
Our (numerical) analysis of the MBL systems We perform full or partial diagonalization of the Hamiltonians Spectrum: { E 1 ≤ E 2 ≤ · · · ≤ E D } Partial diagonalization: ≈ 500 eigenstates from the middle of the spectra Maximum Hilbert space dimensions: D = 48620 (full) D = 184756 (partial) Between 10 2 − 10 3 different disorder realizations for each model parameter Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 7 / 17
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
Quantum chaos and energy spectra What about systems without a classical analogue? Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 8 / 17
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
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 orthogonal 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
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 orthogonal 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
The mean ratio of the level spacings The spacings between the nearest energy levels: δ n = E n +1 − E n ≥ 0 We define the level spacing ratio : 0 ≤ ˜ r n = min { δ n , δ n − 1 } / max { δ n , δ n − 1 } ≤ 1 KEYNOTE: the limiting values of � ˜ r � are well known: Ergodic: � ˜ r � GOE = 0 . 5307 MBL: � ˜ r � P = 2 ln 2 − 1 ≈ 0 . 3863 Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 10 / 17
The mean ratio of the level spacings The spacings between the nearest energy levels: δ n = E n +1 − E n ≥ 0 We define the level spacing ratio : 0 ≤ ˜ r n = min { δ n , δ n − 1 } / max { δ n , δ n − 1 } ≤ 1 KEYNOTE: the limiting values of � ˜ r � are well known: Ergodic: � ˜ r � GOE = 0 . 5307 MBL: � ˜ r � P = 2 ln 2 − 1 ≈ 0 . 3863 Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 10 / 17
The mean ratio of the level spacings Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 10 / 17
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
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
The spectral form factor (SFF) Definition: D � � 1 � e − i ( ε i − ε j ) τ K ( τ ) := ; K (0) = D, K ( τ → τ H ) = 1 D i,j D - Hilbert space dimension τ → an external parameter � ... � over disorder realizations Heisenberg time τ H ∝ inverse mean level spacing (largest sensible timescale of a system) { ε i } → energy levels after spectral unfolding Jan ˇ Suntajs Spectral statistics and MBL September 16, 2019 12 / 17
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