Jeanne Colbois - Institut Néel CNRS - Grenoble - France

Nicolas Laflorencie

Fabien Alet

LPT Toulouse CNRS - France

Ashirbad Padhan

Phys. Rev. Lett. 136 (2026)

Long-Range Resonances in (finite-size) Quasiperiodic Many-Body Localization

Yesterday...

Ergodicity breaking transitions through entanglement entropy

Density-density correlation functions

Many-body localization & many-body resonances

Yesterday...

Density-density correlation functions

Many-body localization & many-body resonances

\(h\)

Ergodic

Finite-size crossover

Rare large correlations at long distances

Ergodicity breaking transitions through entanglement entropy

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{orange}h_i n_i}

Quasiperiodic potential: Aubry André model

\(h_i = h \cos(2\pi \beta i + \phi)\)

Harper Proc. Phys. Soc. (1955)

S. Aubry and G. Andre, Ann. Israel Phys. Soc(1980)

Hofstadter, PRB (1976)

S. Ya. Jitomirskaya, Ann. Math. 150, (1999)

Szabo & Schneider, PRB (2018)

Cookmeyer et al. PRB (2020)

Hopjan et al. PRB (2021)

Gottlob et al. , PRX quantum (2025)

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{orange}h_i n_i}

Quasiperiodic potential: Aubry André model

\(h_i = h \cos(2\pi \beta i + \phi)\)

Irrational

Harper Proc. Phys. Soc. (1955)

S. Aubry and G. Andre, Ann. Israel Phys. Soc(1980)

Hofstadter, PRB (1976)

S. Ya. Jitomirskaya, Ann. Math. 150, (1999)

Szabo & Schneider, PRB (2018)

Cookmeyer et al. PRB (2020)

Hopjan et al. PRB (2021)

Gottlob et al. , PRX quantum (2025)

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Quasiperiodic potential: Aubry André model

Irrational

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{orange}h_i n_i}

Harper Proc. Phys. Soc. (1955)

S. Aubry and G. Andre, Ann. Israel Phys. Soc(1980)

Hofstadter, PRB (1976)

S. Ya. Jitomirskaya, Ann. Math. 150, (1999)

Szabo & Schneider, PRB (2018)

Cookmeyer et al. PRB (2020)

Hopjan et al. PRB (2021)

Gottlob et al. , PRX quantum (2025)

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Quasiperiodic potential: Aubry André model

Irrational

random

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{orange}h_i n_i}

Harper Proc. Phys. Soc. (1955)

S. Aubry and G. Andre, Ann. Israel Phys. Soc(1980)

Hofstadter, PRB (1976)

S. Ya. Jitomirskaya, Ann. Math. 150, (1999)

Szabo & Schneider, PRB (2018)

Cookmeyer et al. PRB (2020)

Hopjan et al. PRB (2021)

Gottlob et al. , PRX quantum (2025)

Quasiperiodic potential: Aubry André model

Harper Proc. Phys. Soc. (1955)

S. Aubry and G. Andre, Ann. Israel Phys. Soc(1980)

Hofstadter, PRB (1976)

S. Ya. Jitomirskaya, Ann. Math. 150, (1999)

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Szabo & Schneider, PRB (2018)

Cookmeyer et al. PRB (2020)

Hopjan et al. PRB (2021)

Gottlob et al. , PRX quantum (2025)

Delocalized

\(h\)

\(h_c = 1\)

Localized

1. Example of delocalization-localization in 1D

2. SELF-DUAL

2. All localization lengths SAME

3. Universal properties determined by the continued fraction expansion of \(\beta\)

\(\xi_{\rm AA} = \frac{1}{\ln(h/J)}\)

Irrational

random

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{orange}h_i n_i}

Experimental relevance

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{orange}h_i n_i}

Experimental relevance

Roati et al. (Ignuscio), Nature (2008)

Localization in a quasiperiodic non-interacting BEC

\(\Delta\) is the amplitude

\(h/J\) grows

\(\Delta = 0\)

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{orange}h_i n_i}

Experimental relevance

Roati et al. (Ignuscio), Nature (2008)

Localization in a quasiperiodic non-interacting BEC

\(\Delta\) is the amplitude

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{cyan}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{orange}h_i n_i}

Attraction / repulsion

\(h/J\) grows

\(\Delta = 0\)

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

DOES LOCALIZATION "SURVIVE" IN THE PRESENCE OF INTERACTIONS AT HIGH ENERGY?

Fleischman, Anderson, (1980); Altschuler, et al (1997); Gornyi et al (2005); Basko et al (2006) ; Pal and Huse (2010); Luitz et al (2015) [...and a whole field!...]

2025

Experimental relevance

Roati et al. (Ignuscio), Nature (2008)

Localization in a quasiperiodic non-interacting BEC

\(\Delta\) is the amplitude

Attraction / repulsion

\(h/J\) grows

\(\Delta = 0\)

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Schreiber et al. (Bloch), Science (2015)

Imbalance in an interacting (spinful) Aubry-André model.

\(U\) is the onsite repulsion, \(J\) the hopping, \(\Delta\) the amplitude

\(h/J\) grows

\(\Delta \neq 0\)

\(h/J = 0\)

See also Lüschen et al., PRL (2017, 2018)

Attraction / repulsion

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

Ising interactions

1. QP vs Random-field Heisenberg chains at high energy

2. Spin-spin correlations

3. Large long-range correlations on the MBL side

Jordan-Wigner

\(L/2\) fermions

\(S^z_{\rm tot} = 0\)

scope

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

Iyer et al. PRB (2013), Naldesi et al. SciPost (2016), Doggen et al. PRB (2019), Sierant & Zakrewski PRB (2022), Faulend et al. (2026)...

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Quasiperiodic field heisenberg chain (QPHC) at high energy

\(J = 1, \Delta = 1\)

Deterministic, autocorrelated

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

Iyer et al. PRB (2013), Naldesi et al. SciPost (2016), Doggen et al. PRB (2019), Sierant & Zakrewski PRB (2022), Faulend et al. (2026)...

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Quasiperiodic field heisenberg chain (QPHC) at high energy

\(J = 1, \Delta = 1\)

Text

Ergodic

\(h\)

MBL

Volume law

Random matrix level stat.

Ergodic eigenstates

ETH

Area law

Poisson statistics

Multifractal eigenstates

Emergent integrability

In some aspects, similar to random-field:

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

Iyer et al. PRB (2013), Naldesi et al. SciPost (2016), Doggen et al. PRB (2019), Sierant & Zakrewski PRB (2022), Faulend et al. (2026)...

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Quasiperiodic field heisenberg chain (QPHC) at high energy

\(J = 1, \Delta = 1\)

Text

Ergodic

\(h\)

MBL

Volume law

Random matrix level stat.

Ergodic eigenstates

ETH

Area law

Poisson statistics

Multifractal eigenstates

Emergent integrability

In some aspects, similar to random-field:

Challenges:

Ultraslow dynamics

Strong finite-size drifts

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

Ergodic

\(h\)

MBL

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Orders of magnitude

\(J = 1, \Delta = 1\)

Iyer et al. PRB (2013), Naldesi et al. SciPost (2016), Doggen et al. PRB (2019), Sierant & Zakrewski PRB (2022)

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

Ergodic

\(h\)

MBL

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Orders of magnitude

\(J = 1, \Delta = 1\)

\(h_c^{{\rm finite}} \in [1.5, 5]\)

Iyer et al. PRB (2013), Naldesi et al. SciPost (2016), Doggen et al. PRB (2019), Sierant & Zakrewski PRB (2022)

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

Ergodic

\(h\)

MBL

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

\(S = -\mathrm{Tr} \rho_A \ln \rho_A\)

\(L\)

\(S/L\)

\(L\)

\(S/L\)

Padhan et al. PRL (2026)

\(J = 1, \Delta = 1\)

Khemani et al - PRL, PRX (2017), Aramthottil et al. PRB (2021), Falcao et al. PRB (2024), ....

RMT from Vidmar and M. Rigol, PRL (2017).

Orders of magnitude

\(h_c^{{\rm finite}} \in [1.5, 5]\)

High energy eigenstates:

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

Ergodic

\(h\)

MBL

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

ORDERS OF MAGNITUDE

Padhan et al. PRL (2026)

\(J = 1, \Delta = 1\)

Khemani et al - PRL, PRX (2017), Aramthottil et al. PRB (2021), Falcao et al. PRB (2024), ....

RMT from Vidmar and M. Rigol, PRL (2017).

\(h_c^{{\rm finite}} \in [1.5, 5]\)

High energy eigenstates:

\(S = -\mathrm{Tr} \rho_A \ln \rho_A\)

\(L\)

\(S/L\)

\(L\)

\(S/L\)

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

Ergodic

\(h\)

MBL

\(h_c^{{\rm finite}} \in [1.5, 5]\)

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Padhan et al. PRL (2026)

\(J = 1, \Delta = 1\)

Khemani et al - PRL, PRX (2017), Aramthottil et al. PRB (2021), Falcao et al. PRB (2024), ....

RMT from Vidmar and M. Rigol, PRL (2017).

Orders of magnitude

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

\(J = 1, \Delta = 1\)

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

1. Weaker \(h_c\) (finite-size)

Iyer et al. (2013), Khemani et al. PRL (2017)

Important differences with the RFHC

Autocorrelated, deterministic

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

\(J = 1, \Delta = 1\)

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Iyer et al. (2013), Khemani et al. PRL (2017)

2. Slower drift, sharper crossover

Khemani et al. PRL (2017), Aramthottil et al PRB (2021)

Important differences with the RFHC

Autocorrelated, deterministic

1. Weaker \(h_c\) (finite-size)

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

\(J = 1, \Delta = 1\)

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Iyer et al. (2013), Khemani et al. PRL (2017)

Important differences with the RFHC

Autocorrelated, deterministic

Khemani et al. PRX, PRL (2017) (with next-nearest neighbours)

3. Sample-to-sample fluctuations

Khemani et al. PRL (2017), Aramthottil et al PRB (2021)

Suntajs, et al . PRB (2020) (RHFC), Aramthottil et al PRB (2021)

1. Weaker \(h_c\) (finite-size)

2. Slower drift, sharper crossover

Khemani et al. PRL (2017), Aramthottil et al PRB (2021)

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

\(J = 1, \Delta = 1\)

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Iyer et al. (2013), Khemani et al. PRL (2017)

Important differences with the RFHC

Autocorrelated, deterministic

Khemani et al. PRX, PRL (2017) (with next-nearest neighbours)

3. Sample-to-sample fluctuations

Khemani et al. PRL (2017), Aramthottil et al PRB (2021)

Suntajs, et al . PRB (2020) (RHFC), Aramthottil et al PRB (2021)

Absence of rare regions in the potential

1. Weaker \(h_c\) (finite-size)

2. Slower drift, sharper crossover

Khemani et al. PRL (2017), Aramthottil et al PRB (2021)

avalanche instability

De Roeck, Huveneers, Luitz, Thierry,... (2017-2020)

Crowley, Chandran, Long, Vanoni (2020-now)

Morningstar et al (2022)

RFHC picture:

QPHC: more stable many-body localization?

avalanche instability

De Roeck, Huveneers, Luitz, Thierry,... (2017-2020)

Crowley, Chandran, Long, Vanoni (2020-now)

Morningstar et al (2022)

RFHC picture:

Yes...

But...

QPHC: more stable many-body localization?

Imbalance at strong amplitude

avalanche instability

De Roeck, Huveneers, Luitz, Thierry,... (2017-2020)

Crowley, Chandran, Long, Vanoni (2020-now)

Morningstar et al (2022)

RFHC picture:

Yes...

But...

Slower drift (log?) / sharper crossover (exp)

Khemani et al. PRL (2017), Aramthottil et al. PRB (2021), Falcao et al, PRB (2024), ....

QPHC: more stable many-body localization?

Agrawal et al. PRB (2022), Stralj et al. PRB (2022),

Crowley and Chandran (2022), Bordia et al. (2018), Hur et al. (2025),...

Sierant and Zakrewski PRB (2022)

Discussion around stability in 2D?

Agrawal et al. PRB (2022), Stralj et al. PRB (2022),

Crowley and Chandran (2022), Bordia et al. (2018), Hur et al. (2025),...

Imbalance at strong amplitude

Sierant and Zakrewski PRB (2022)

avalanche instability

De Roeck, Huveneers, Luitz, Thierry,... (2017-2020)

Crowley, Chandran, Long, Vanoni (2020-now)

Morningstar et al (2022)

RFHC picture:

Yes...

But...

Slower drift (log?) / sharper crossover (exp)

Khemani et al. PRL (2017), Aramthottil et al. PRB (2021), Falcao et al, PRB (2024), ....

Znidaric, Ljuobtina, PNAS (2018)

Sharper weak interaction instability

Slow "creep" - 2-point correl.

Weiner et al. PRB (2019)

QPHC: more stable many-body localization?

No proof of MBL in QP systems

Many-body resonances

many-body resonances

here - end to end QMI

\rangle_{\tau^z \, \mathrm{or}\, \sigma^z}

\ket{E', \pm} =

\(\pm\)

Gopalakrishnan et al (2015); Khemani et al. (2017); Kjäll (2018); Villalonga and Clark (2020); Garratt et al (2021); Crowley and Chandran (2022); Morningstar et al (2022); Colbois et al. (2024); Laflorencie et al (2025)

Morningstar et al PRB (2022)

RFHC picture:

Many-body resonances

many-body resonances

here - end to end QMI

\rangle_{\tau^z \, \mathrm{or}\, \sigma^z}

\ket{E', \pm} =

\(\pm\)

Morningstar et al PRB (2022)

RFHC picture:

many-body resonanceS in QP model

From Two point correlation functions

Two-point correlations

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

\alpha = z

\alpha = x, y

An Experimentally accessible...

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

\(C_{ij}^{zz} \rightarrow \langle n_i n_{j} \rangle - \langle n_i \rangle \langle n_{j} \rangle\)

Density-density correlations, QP Bose-Hubbard

An Experimentally accessible...

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

\(C_{ij}^{zz} \rightarrow \langle n_i n_{j} \rangle - \langle n_i \rangle \langle n_{j} \rangle\)

Density-density correlations, QP Bose-Hubbard

Correlation length

Lukin et al, Science (2019), 8 sites

An Experimentally accessible...

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

\(C_{ij}^{zz} \rightarrow \langle n_i n_{j} \rangle - \langle n_i \rangle \langle n_{j} \rangle\)

Density-density correlations, QP Bose-Hubbard

Correlation length

Correlation clusters

Lukin et al, Science (2019), 8 sites

Rispoli et al. et al, Nature (2019), 12 sites

An Experimentally accessible...

Lukin et al, Science (2019), 8 sites

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Rispoli et al. et al, Nature (2019), 12 sites

\(C_{ij}^{zz} \rightarrow \langle n_i n_{j} \rangle - \langle n_i \rangle \langle n_{j} \rangle\)

Density-density correlations, QP Bose-Hubbard

Léonard et al. et al, Nat. Phys. (2022), 6+6 sites

Clean

Correlation length

Correlation clusters

Avalanche spreading

(For a detailed numerical study on disordered systems up to 22 sites, see Szoldra et al, PRB (2024).

... And somewhat overlookeD Probe

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Localized

\(|C^{\alpha,\alpha}_{r} |= A e^{-r /\xi_{\alpha}}\)

... And somewhat overlookeD Probe

Localized

\(|C^{\alpha,\alpha}_{r} |= A e^{-r /\xi_{\alpha}}\)

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Delocalized

It depends

... And somewhat overlookeD Probe

A probe of the transition...

Weiner et al. (2019)

Localized

Delocalized

It depends

\(|C^{\alpha,\alpha}_{r} |= A e^{-r /\xi_{\alpha}}\)

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Pal and Huse, PRB (2010)

see also Znidaric et al PRB (2008) and Lim & Sheng, PRB (2016)

... And somewhat overlookeD Probe

A probe of the transition...

Weiner et al. (2019)

Localized

Delocalized

It depends

\(|C^{\alpha,\alpha}_{r} |= A e^{-r /\xi_{\alpha}}\)

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

...by now well studied in RFHC

Pal and Huse, PRB (2010)

see also Znidaric et al PRB (2008) and Lim & Sheng, PRB (2016)

...De Tomasi et al (2017), Bera et al. (2017), Colmenarez et al. (2019),

Varma et al., (2019), Weiner et al (2019), Villalonga & Clark (2020), Morningstar et al (2022),Szoldra et al. (2024), Colbois et al. (2024), Laflorencie et al. (2025),...

... And somewhat overlookeD Probe

A probe of the transition...

...but not so much in QPHC

Weiner et al. (2019)

LIOMS:

Localized

Delocalized

It depends

\(|C^{\alpha,\alpha}_{r} |= A e^{-r /\xi_{\alpha}}\)

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Singh et al (2021), Thomson et al. (2023), Jiang et al (2025)

...by now well studied in RFHC

Pal and Huse, PRB (2010)

see also Znidaric et al PRB (2008) and Lim & Sheng, PRB (2016)

...De Tomasi et al (2017), Bera et al. (2017), Colmenarez et al. (2019),

Varma et al., (2019), Weiner et al (2019), Villalonga & Clark (2020), Morningstar et al (2022),Szoldra et al. (2024), Colbois et al. (2024), Laflorencie et al. (2025),...

Weiner et al. PRB (2019)

Direct:

COrrelations at maximal distance

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

\(\exp(\overline{\ln|C^{\alpha \alpha}_{i, i+r}|})\)

Typical value:

COrrelations at maximal distance

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

\(\exp(\overline{\ln|C^{\alpha \alpha}_{i, i+r}|})\)

Typical value:

Distance-dependent \(|C^{\alpha\alpha}_r|\) :

see e.g. Varma et al., PRB (2019)

Villalonga and Clark (2020)

no \(r\)-dep. in the ergodic phase \(\epsilon = 0.5\)
inherent difficulties with PBC

COrrelations at maximal distance

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

\(\exp(\overline{\ln|C^{\alpha \alpha}_{i, i+r}|})\)

Typical value:

Distance-dependent \(|C^{\alpha\alpha}_r|\) :

see e.g. Varma et al., PRB (2019)

Villalonga and Clark (2020)

no \(r\)-dep. in the ergodic phase \(\epsilon = 0.5\)
inherent difficulties with PBC

Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L\)

used with QMI
risk of edge effects?

Morningstar et al (2022)

Laflorencie et al (2025)

COrrelations at maximal distance

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

\(\exp(\overline{\ln|C^{\alpha \alpha}_{i, i+r}|})\)

Typical value:

Distance-dependent \(|C^{\alpha\alpha}_r|\) :

see e.g. Varma et al., PRB (2019)

Villalonga and Clark (2020)

no \(r\)-dep. in the ergodic phase \(\epsilon = 0.5\)
inherent difficulties with PBC

Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L\)

used with QMI
risk of edge effects?

Morningstar et al (2022)

Laflorencie et al (2025)

Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L/2\)

COrrelations at maximal distance

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

\(\exp(\overline{\ln|C^{\alpha \alpha}_{i, i+r}|})\)

Typical value:

Distance-dependent \(|C^{\alpha\alpha}_r|\) :

see e.g. Varma et al., PRB (2019)

Villalonga and Clark (2020)

no \(r\)-dep. in the ergodic phase \(\epsilon = 0.5\)
inherent difficulties with PBC

Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L\)

used with QMI
risk of edge effects?

Morningstar et al (2022)

Laflorencie et al (2025)

Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L/2\)

COrrelations at maximal distance

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

\(\exp(\overline{\ln|C^{\alpha \alpha}_{i, i+r}|})\)

Typical value:

Distance-dependent \(|C^{\alpha\alpha}_r|\) :

see e.g. Varma et al., PRB (2019)

Villalonga and Clark (2020)

no \(r\)-dep. in the ergodic phase \(\epsilon = 0.5\)
inherent difficulties with PBC

Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L\)

used with QMI
risk of edge effects?

Morningstar et al (2022)

Laflorencie et al (2025)

Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L/2\)

COrrelations at maximal distance

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

\(\exp(\overline{\ln|C^{\alpha \alpha}_{i, i+r}|})\)

Typical value:

Distance-dependent \(|C^{\alpha\alpha}_r|\) :

see e.g. Varma et al., PRB (2019)

Villalonga and Clark (2020)

no \(r\)-dep. in the ergodic phase \(\epsilon = 0.5\)
inherent difficulties with PBC

Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L\)

used with QMI
risk of edge effects?

Morningstar et al (2022)

Laflorencie et al (2025)

Mid-chain correlations

Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L/2\)

Two SIMPLE limits

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Random vector

JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)

Ergodic typical eigenstate

Two SIMPLE limits

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Random vector

JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)

Ergodic typical eigenstate

No spatial dependence

Two SIMPLE limits

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Random vector

JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)

Ergodic typical eigenstate

Total spin conservation

|C^{zz}_{L/2}| \sim 1/4(L-1)

Two SIMPLE limits

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Random vector

JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)

Ergodic typical eigenstate

Total spin conservation

|C^{zz}_{L/2}| \sim 1/4(L-1)

Hilbert space dim.

|C^{xx}_{L/2}| \sim e^{-\ln 2 /L}

Two SIMPLE limits

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Random vector

JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)

Ergodic typical eigenstate

Total spin conservation

|C^{zz}_{L/2}| \sim 1/4(L-1)

Hilbert space dim.

|C^{xx}_{L/2}| \sim e^{-\ln 2 /L}

ZZ correlations dominate

Two SIMPLE limits

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Random vector

JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)

Ergodic typical eigenstate

Total spin conservation

|C^{zz}_{L/2}| \sim 1/4(L-1)

Hilbert space dim.

|C^{xx}_{L/2}| \sim e^{-\ln 2 /L}

Localized side of QP XX

ZZ correlations dominate

\(h = 1.4\)

Two SIMPLE limits

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Random vector

JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)

Ergodic typical eigenstate

Total spin conservation

|C^{zz}_{L/2}| \sim 1/4(L-1)

Hilbert space dim.

|C^{xx}_{L/2}| \sim e^{-\ln 2 /L}

ZZ correlations dominate

Localized side of QP XX

XX correlations dominate

\(h = 1.4\)

Two SIMPLE limits

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Random vector

JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)

Ergodic typical eigenstate

Total spin conservation

|C^{zz}_{L/2}| \sim 1/4(L-1)

Hilbert space dim.

|C^{xx}_{L/2}| \sim e^{-\ln 2 /L}

ZZ correlations dominate

Localized side of QP XX

XX correlations dominate

\(h = 1.4\)

Two SIMPLE limits

\alpha = z

\alpha = x, y

\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

Random vector

JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)

2\(\xi^{z}_{L/2}\sim \xi^{x}_{L/2} \sim \xi^{AA}\)

Ergodic typical eigenstate

Total spin conservation

|C^{zz}_{L/2}| \sim 1/4(L-1)

Hilbert space dim.

|C^{xx}_{L/2}| \sim e^{-\ln 2 /L}

ZZ correlations dominate

Localized side of QP XX

XX correlations dominate

4. MIDchain correlations in the QP Heisenberg chain

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

\(h_i = h \cos(2\pi \beta i + \phi) , \beta = \frac{\sqrt{5}-1}{2}, \phi \in [0,2\pi)\)

Periodic boundary conditions

\(J = 1, \Delta = 1\)

Entanglement entropy

Participation entropy

Gap ratio statistics

Extreme magnetization

\(\ln(C_{L/2,{\rm typ}}^{\alpha,\alpha}) =\overline{\ln|C_{L/2}^{\alpha, \alpha}|} =: -\frac{L}{2\xi_{\rm typ}^{\alpha}}+ \mathcal{O}(1) \)

typical mid-chain correlation length: QP Heisenberg

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

\(\ln(C_{L/2,{\rm typ}}^{\alpha,\alpha}) =\overline{\ln|C_{L/2}^{\alpha, \alpha}|} =: -\frac{L}{2\xi_{\rm typ}^{\alpha}}+ \mathcal{O}(1) \)

\(h = 3.6\)

typical mid-chain correlation length: QP Heisenberg

\(\ln(C_{L/2,{\rm typ}}^{\alpha,\alpha}) =\overline{\ln|C_{L/2}^{\alpha, \alpha}|} =: -\frac{L}{2\xi_{\rm typ}^{\alpha}}+ \mathcal{O}(1) \)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

typical mid-chain correlation length: QP Heisenberg

\(h = 3.6\)

\(\ln(C_{L/2,{\rm typ}}^{\alpha,\alpha}) =\overline{\ln|C_{L/2}^{\alpha, \alpha}|} =: -\frac{L}{2\xi_{\rm typ}^{\alpha}}+ \mathcal{O}(1) \)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

\(h = 3.6\)

typical mid-chain correlation length: QP Heisenberg

\(\ln(C_{L/2,{\rm typ}}^{\alpha,\alpha}) =\overline{\ln|C_{L/2}^{\alpha, \alpha}|} =: -\frac{L}{2\xi_{\rm typ}^{\alpha}}+ \mathcal{O}(1) \)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

\(h = 3.6\)

typical mid-chain correlation length: QP Heisenberg

\(\ln(C_{L/2,{\rm typ}}^{\alpha,\alpha}) =\overline{\ln|C_{L/2}^{\alpha, \alpha}|} =: -\frac{L}{2\xi_{\rm typ}^{\alpha}}+ \mathcal{O}(1) \)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

\(h = 3.6\)

typical mid-chain correlation length: QP Heisenberg

Heavy tails of large correlations

Heisenberg QP chain \(h = 3\)

Heavy tails of large correlations

Heisenberg QP chain \(h = 3\)

A. Padhan et al. (unpublished)

Heavy tails of large correlations

Heisenberg QP chain \(h = 3\)

A. Padhan et al. (unpublished)

Heavy tails of large correlations

Heisenberg QP chain \(h = 3\)

A. Padhan et al. (unpublished)

Heavy tails of large correlations

Heisenberg QP chain \(h = 3.6\)

A. Padhan et al. (unpublished)

Heavy tails of large correlations

Heisenberg QP chain \(h = 4\)

A. Padhan et al. (unpublished)

Heavy tails of large correlations

Heisenberg QP chain \(h = 5\)

A. Padhan et al. (unpublished)

Heavy tails of large correlations

Heisenberg QP chain \(h = 6\)

A. Padhan et al. (unpublished)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

Heavy tails of large correlations

JC, F. Alet, N. Laflorencie, PRB (2024)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

Heavy tails of large correlations

JC, F. Alet, N. Laflorencie, PRB (2024)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

Heavy tails of large correlations

JC, F. Alet, N. Laflorencie, PRB (2024)

RFHC \(h =6\)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

Heavy tails of large correlations

JC, F. Alet, N. Laflorencie, PRB (2024)

RFHC \(h =6\)

Can we say more about these events?

Strong correlations arise in pairs

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

Strong correlations arise in pairs

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

Quasiperiodic: \(h = 3.6,L = 22\)

Quasiperiodic: \(h = 3.6,L = 22\)

Strong correlations arise in pairs

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

RFHC, \(L = 16\)

QPHC \(h = 3.6,L = 22\)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

A toy model

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

QPHC \(h = 3.6,L = 22\)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

A toy model

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

QPHC \(h = 3.6,L = 22\)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

A toy model

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

RFHC \(h = 20,L = 12\)

QPHC \(h = 3.6,L = 22\)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

A toy model

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

RFHC \(h = 20,L = 12\)

QPHC \(h = 3.6,L = 22\)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

A toy model

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

RFHC \(h = 20,L = 12\)

QPHC \(h = 3.6,L = 22\)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

A toy model

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

RFHC \(h = 20,L = 12\)

\rangle_{\sigma^z}

\pm

QPHC \(h = 3.6,L = 22\)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

A toy model

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

RFHC \(h = 20,L = 12\)

\rangle_{\sigma^z}

\pm

\ket{\Phi_{2p}^{+}} =

\otimes \ket{\varphi_{L-2p}}

\ket{\Phi_{2p}^{-}} =

\otimes \ket{\varphi_{L-2p}}

\left[\cos\left(\frac{\theta}{2}\right) \ket{\varphi_{2p}} + \sin\left(\frac{\theta}{2}\right) \overline{\ket{\varphi_{2p}}} \right]

\left[\sin\left(\frac{\theta}{2}\right) \ket{\varphi_{2p}} - \cos\left(\frac{\theta}{2}\right) \overline{\ket{\varphi_{2p}}} \right]

QPHC \(h = 3.6,L = 22\)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

A toy model

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

RFHC \(h = 20,L = 12\)

\rangle_{\sigma^z}

\pm

\ket{\Phi_{2p}^{+}} =

\otimes \ket{\varphi_{L-2p}}

\ket{\Phi_{2p}^{-}} =

\otimes \ket{\varphi_{L-2p}}

Flipped

\left[\cos\left(\frac{\theta}{2}\right) \ket{\varphi_{2p}} + \sin\left(\frac{\theta}{2}\right) \overline{\ket{\varphi_{2p}}} \right]

\left[\sin\left(\frac{\theta}{2}\right) \ket{\varphi_{2p}} - \cos\left(\frac{\theta}{2}\right) \overline{\ket{\varphi_{2p}}} \right]

QPHC \(h = 3.6,L = 22\)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

A toy model

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

RFHC \(h = 20,L = 12\)

\rangle_{\sigma^z}

\pm

\ket{\Phi_{2p}^{+}} =

\otimes \ket{\varphi_{L-2p}}

\left[\cos\left(\frac{\theta}{2}\right) \ket{\varphi_{2p}} + \sin\left(\frac{\theta}{2}\right) \overline{\ket{\varphi_{2p}}} \right]

\ket{\Phi_{2p}^{-}} =

\left[\sin\left(\frac{\theta}{2}\right) \ket{\varphi_{2p}} - \cos\left(\frac{\theta}{2}\right) \overline{\ket{\varphi_{2p}}} \right]

\otimes \ket{\varphi_{L-2p}}

Quality of the cat state

Flipped

QPHC \(h = 3.6,L = 22\)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

A toy model

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

RFHC \(h = 20,L = 12\)

\rangle_{\sigma^z}

\pm

\ket{\Phi_{2p}^{+}} =

\otimes \ket{\varphi_{L-2p}}

\ket{\Phi_{2p}^{-}} =

\otimes \ket{\varphi_{L-2p}}

Quality of the cat state

Flipped

\left[\cos\left(\frac{\theta}{2}\right) \ket{\varphi_{2p}} + \sin\left(\frac{\theta}{2}\right) \overline{\ket{\varphi_{2p}}} \right]

\left[\sin\left(\frac{\theta}{2}\right) \ket{\varphi_{2p}} - \cos\left(\frac{\theta}{2}\right) \overline{\ket{\varphi_{2p}}} \right]

QPHC \(h = 3.6,L = 22\)

A. Padhan, JC, F. Alet, N. Laflorencie, PRL (2026)

A toy model

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

RFHC \(h = 20,L = 12\)

\rangle_{\sigma^z}

\pm

\ket{\Phi_{2p}^{+}} =

\otimes \ket{\varphi_{L-2p}}

\ket{\Phi_{2p}^{-}} =

\otimes \ket{\varphi_{L-2p}}

Simple analytical predictions

see also Falcao et al (2026) for further evidence

\left[\cos\left(\frac{\theta}{2}\right) \ket{\varphi_{2p}} + \sin\left(\frac{\theta}{2}\right) \overline{\ket{\varphi_{2p}}} \right]

\left[\sin\left(\frac{\theta}{2}\right) \ket{\varphi_{2p}} - \cos\left(\frac{\theta}{2}\right) \overline{\ket{\varphi_{2p}}} \right]

Resonance "counting" In the RFHC...

1. Number of rare events

Resonance "counting" In the RFHC...

1. Number of rare events

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

For accessible sizes:

Rare cat-like state events with large, \(\mathcal O(1)\) correlations

Resonance "counting" : random field Heisenberg chain

1. Number of rare events

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

For accessible sizes:

Rare cat-like state events with large, \(\mathcal O(1)\) correlations

2. Cat state anatomy:

\(\Delta\epsilon^{\mathrm{cat}}_{\mathrm{typ}} \propto A(h)e^{-L/\xi_{\mathrm{cat}}}\)

1.7

\(\Delta\epsilon_{\mathrm{typ}} \propto 2^{-L} = e^{-L/\xi_{\mathrm{ typ}}}\)

1.44

(... and many other observations)

perspective and outlook: QPHC

1. Resonance counting / understanding

Extent of this regime?
Scaling and fate of this regime? \(\rightarrow\) accessibility with other methods?

QP \(\rightarrow\) link between configuration and resonances? (see Faulend et al. 2026)

perspective and outlook: QPHC

2. Link with other observations:

Link with "creep dynamics" observed by Weiner et al. (2019)?
Link with failure to converge LIOMS below (relatively larger) disorder in Singh et al. (2021)?

1. Resonance counting / understanding

Extent of this regime?
Scaling and fate of this regime? \(\rightarrow\) accessibility with other methods?

QP \(\rightarrow\) link between configuration and resonances? (see Faulend et al. 2026)

Take-Home message

Spin chain in a quasiperiodic field

\(C_{L/2}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+L/2}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+L/2}^{\alpha} \rangle\)

Rare events:

Large longitudinal correlations at long distances

Additional slides

CREEP DYNAMICS

Weiner et al. PRB (2019)

Hunting cats

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

Large correlations counting

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

Anatomy of cat states : Spectral

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

Anatomy of cat states : Spatial

N. Laflorencie, JC, F. Alet, Phys. Rev. B 112, 224207 (2025)

\(L\) for ergodic states

\(L/2\) for typical (l-bits) MBL states

RFHC distributions

Khemani et al. PRX, PRL (2017) (with next-nearest neighbours)

Intra-sample

Inter sample

Entanglement entropy (J-J')

States

Cuts

Samples

Sample to sample fluctuations

Typical correlation lengths in AA for small sizes

Fits on 4 sizes

Fits on 5 sizes

Heisenberg

\(\xi_x > \xi_z\)

Inversion

Extrapolated \(h_c\)

\(\xi_z \rightarrow \infty\)

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

Heisenberg line: RHFC

Long-range correlations

N. Laflorencie et al. (PRB 2025)

Long-range correlations

JC, F. Alet, N. Laflorencie (PRB 2024)

Landmarks

D. Luitz, N. Laflorencie, F. Alet (2016)

Sierant and Zakrewski (2022)

Other probes

Some eigenstate

J. C., N. Laflorencie, PRB (2023)

\(|\langle S_i^{z}\rangle| < 1/2\)

\delta_i = 1/2 -| \langle S_i^z \rangle|

\delta_{\rm min} = 1/2 -\max_{i}| \langle S_i^z \rangle|

A simple many-body effect : Maximal magnetization?

Anderson chain / XX chain

Chain breaking

Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)

Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)

JC, Laflorencie, PRB 108, 144206 (2023)

Toy model:

\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}

\delta_{\rm min}^{\rm typ} = \exp(\overline{\ln \delta_{\min}})

\delta_{\rm min} = 1/2 -\max_{i}| \langle S_i^z \rangle|

SPIN FREEZING !

CHAIN BREAKING !

Participation entropy

Macé et al (2019)

Colbois, Alet, Laflorencie (2024)

Participation entropy

Phenomenology, theory and challenges

De Roeck & Huveneers 2017, Luitz, De Roeck & Huveneers 2017, Thiery et al 2018; Crowley and Chandran 2020

Condition for spin at \(r\) to relax thanks to the grain:

\Gamma \sim e^{-r/\zeta} \gg \delta_{\rm eff} \sim 2^{-(n_0+2r)}

\zeta > \zeta_{\rm av.}

Avalanche criterion:

Instabilities : Avalanches

Question:

Does the seed hybridize (absorb) the l-bits?

Answer: it depends on

(1) \(V_{ij}\) the matrix element coupling the seed to the l-bit

(2) \(1/ \rho\) the level spacing.

Typically \(V_{ij} \gg 1/\rho\).

The challenge is to quantify this, see Crowley and Chandran.

L-bits models

quasi-local integrals of motion
In spin models : dressed physical on-site Pauli spin operators
in fermionic models : Anderson orbitals dressed by local electron-hole excitations
"dressed" = quasilocal, finite-depth, unitary transformation (ideally, that diagonalizes H).
MODEL : instead of finding U, H', we define U (finite-depth circuit of 2-site gates) and H'

DEEP MBL :

strong overlap with physical dofs -> constants of motion

\tau_i^{z} := U \sigma_i^{z} U ^{\dagger}

H' = \sum_i h_i \tau_i^z + \sum_{i,j} J_{i,j} \tau_i^z \tau_j^z + \sum_{i,j,k} J_{i,j,k} \tau_i^{z} \tau_{j}^{z}\tau_{k}^{z}

investigate to what extent the growth of number entropy can be explained directly whithin the phenomenology of MBL
directly work with an effecting l-bits model:
- expontentially decaying support
- exponentially decaying interactions
is particle transport entirely absent in an l-bit model ?

ultra-slow growth, saturating in finite systems at a subextensive value increasing with system size

Other weak interactions results

Other spatial correlations results