REVIEW 4 major objections 4 minor 90 references
Sub-diffusion in the Anderson model on random regular graph
T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read An initially localized wave packet on a random regular graph spreads subdiffusively: both the wave front and the mean distance grow as $t^{\beta}$ with $\beta(W) \approx 1 - W/W_{\mathrm{AT}}$, evidence of a subdiffusive phase rather than…
desk verdict Plausible and well-organized numerical evidence for subdiffusive spreading on the RRG, but the phase claim leans on an inherited exponent that the collapses do not independently test. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the distance-resolved probability distribution $\Pi(x,t)$, the probability that the particle is at graph distance $x$ from its initial site at time $t$, restricted to a microcanonical energy shell around mid-spectrum. The wave front $X_{\mathrm{front}}(t)$, defined as the distance at which $\Pi(x,t)$ has its maximum, is the moving boundary that separates frozen, front, and relaxed regions in spacetime. The argument is carried by the relation between spatial and temporal relaxation: the stretched-exponential return probability $\Pi(0,t) \sim \exp(-\Gamma t^{\beta(W)})$ fixes the exponent $\beta(W) = 1 - W/W_{\mathrm{AT}}$, and the same exponent collapses both the wave-front motion and the growth of the first moment $X(t)$. The space-time factorization behind the front, $\Pi(x,t)-\Pi(x,\infty) = g(x)[\Pi(0,t)-\Pi(0,\infty)]$, is what ties local relaxation to the front motion.
What would settle it
A direct test would be to evolve wave packets on random regular graphs larger than $L = 2^{20}$ and longer than the current maximum times, extracting $\beta(W)$ separately from the wave-front position, the first moment $X(t)$, and the return probability. If the three fitted exponents disagree, or if $X(t)$ visibly crosses over to linear growth at any fixed $W$ in $8 \le W \le 14$, the subdiffusive-phase claim as stated would be falsified.
Extended reading notes
Core claim
On the Anderson model on a random regular graph, the full radial probability distribution $\Pi(x,t)$ of an initially localized wave packet is shown to organize into four spacetime regimes determined by a moving wave front. Before the front arrives, the distribution is frozen; near the front, $\Pi$ minus its infinite-time value collapses as a function of $X_{\mathrm{front}}(t) - x$; behind the front, it factorizes as $g(x)[\Pi(0,t)-\Pi(0,\infty)]$ and relaxes with the return probability; at very long times it saturates to the uniform-over-sites distribution. The central quantitative claim is that for disorder strengths $0.4W_{\mathrm{AT}} \lesssim W \lesssim 0.7W_{\mathrm{AT}}$, both the front $X_{\mathrm{front}}(t) \approx \Gamma(W) t^{\beta(W)}$ and the mean distance $X(t) \sim t^{\beta(W)}$ grow subdiffusively with $\beta(W) \approx 1 - W/W_{\mathrm{AT}}$. Because the same exponent also controls the stretched-exponential return probability from earlier work, the paper presents the spatial spreading and the temporal relaxation as two faces of one subdiffusive phase, in contrast to $\mathbb{Z}^d$ lattices with $d>2$ where subdiffusion appears only at the critical point.
Load-bearing premise
The paper's spatial exponents are not fitted independently: it assumes that the stretched-exponential return probability with $\beta(W) = 1 - W/W_{\mathrm{AT}}$, taken from its earlier study, is accurate and not a finite-time artifact, and pins the wave-front and mean-distance growth to this same $\beta$.
Editorial extensions
If this is right
- If the claim holds, the random regular graph hosts a genuine subdiffusive phase for a finite interval of disorder, not just a critical point, with the exponent decreasing linearly to zero at the Anderson transition.
- The Thouless time, when the front reaches the graph diameter, scales as $(\ln L / \ln K)^{1/\beta(W)}$, which diverges as a power of the logarithm near the transition; this is the same scaling found in subdiffusive many-body-localized systems.
- Slow spreading on the random regular graph implies slow relaxation of local observables in systems that map to it, offering a route to subdiffusion in many-body systems that does not invoke Griffiths effects.
- The observed slow dynamics rules out full ergodicity in the random-matrix sense for $W \ge 8$ even in parameter regions where eigenfunction-based measures such as the inverse participation ratio may look ergodic.
Reading between the lines
- If the stretched-exponential exponent survives at larger sizes, one could test whether the same $\beta$ controls entanglement growth in the corresponding many-body system, since the graph distance maps to the Hamming distance in Fock space.
- The four-regime structure is likely generic to locally tree-like graphs with exponential growth of the number of sites with distance; a possible extension is to derive $\beta(W)$ from statistics of rare resonances along the boundary rather than importing it from the return probability.
- A finite-time to diffusive crossover at larger graph sizes cannot be excluded from these data; the paper's own claim should be read as establishing the subdiffusive phase for the accessible time and size window, with the thermodynamic extrapolation as the open question.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This manuscript studies the finite-time dynamics of an initially localized wave packet in the Anderson model on a random regular graph (RRG), using the full probability distribution Π(x,t) as a function of distance from the initial site. The authors identify four space-time regimes controlled by a propagating wave front X_front(t) and present numerical evidence that both the front and the mean distance X(t) grow sub-diffusively as t^β with β(W) ≈ 1 − W/W_AT for 0.4 W_AT ≲ W ≲ 0.7 W_AT. They also demonstrate a space-time factorization Π(x,t) − Π(x,∞) = g(x)[Π(0,t) − Π(0,∞)] after the front passes, connect the return probability to the front position through an exponential relation, and argue that the RRG can serve as a proxy for many-body localization dynamics. The numerical results are for L up to 2^20, with the energy-shell fraction f = 1/8, and are supported by appendices on finite-size and energy-window dependence.
Significance. If the claims hold, this paper would establish that the RRG Anderson model hosts a genuine dynamical subdiffusive phase over a finite disorder interval, in contrast to Anderson models on Z^d with d>2 where subdiffusion is restricted to the critical point. The four-regime description of Π(x,t) and the factorization property are potentially useful organizing principles for slow dynamics near the MBL transition, and the connection to the Thouless time scaling is suggestive. The paper's strengths include the use of a well-defined observable (the radial distribution), consistency checks on the energy window and system size in Appendix A, and an explicit statement of the residual uncertainty about a thermodynamic-limit crossover to diffusion. However, the central exponent β(W) is taken from the same authors' previous return-probability work (Ref. [73]) rather than measured here, the numerical collapses use that value as an input, and the paper provides no error bars, realization counts, or code/data release. These issues make the quantitative phase claim weaker than the qualitative observation of sublinear spreading.
major comments (4)
- [§Model and methods, Eq. (5); §Results, Eqs. (10)–(11); Figs. 4(b), 5(b), 11] The exponent β(W) is not measured in this paper. It is imported from Ref. [73] as Eq. (5), and then used as the time-rescaling exponent in the collapses of Figs. 4(b), 5(b), and 11. Consequently, those collapses demonstrate consistency with the assumed β(W), not an independent extraction of β from X(t) or X_front(t). Since the central quantitative claim is X_front(t) ~ Γ(W)t^β and X(t) ~ t^β, the authors should fit β directly from the new data, report confidence intervals and collapse-quality metrics, and compare the fitted values with Eq. (5). Without this, the subdiffusive-phase exponent remains an inherited assumption rather than a result of the present analysis.
- [§Results, Figs. 2–5 and Appendix A] No statistical information is reported: the numbers of disorder realizations, graph samples, and initial states are not given, and no error bars appear on Π(x,t), X(t), or the collapse curves. For a numerical claim about a scaling exponent and a collapse, this is load-bearing: the apparent agreement with the proposed functional forms cannot be distinguished from finite-sampling scatter or systematic drift. Please provide uncertainties (for example, bootstrap over samples) and state the number of realizations for each W and L. Releasing the data-processing code would also allow the collapses to be checked.
- [Conclusions, final paragraph before Acknowledgments] The paper explicitly concedes that a crossover to β = 1 in the thermodynamic limit cannot be ruled out. Because the RRG diameter grows only as ln L and the pre-saturation time window is t < t_Th ~ (ln L)^{1/β}, all data shown are intrinsically finite-time. The abstract's claim of an "entire subdiffusive phase" is therefore stronger than the evidence presented. Either soften the claim to finite-time subdiffusive dynamics or add a scaling analysis in L showing that the exponent extracted in the available window is stable as L increases.
- [Abstract and Appendix D] The abstract states that the numerical results are supported by a "self-consistent semiclassical picture" relating β to the relaxation rate of the return probability, but no derivation of β is given. Appendix D shows only that the two collapse forms, Eqs. (7) and (8), are mutually consistent if the return probability decays exponentially with the front position, δR(x) ~ e^{−λx}; it does not determine β(W). The wording should be revised to describe this as a consistency condition or an ansatz, not a self-consistent derivation.
minor comments (4)
- [§Introduction, Fig. 1] The term "subdiffusive" is used with a nonstandard definition (X(t) ~ t^β with β < 1 on a hierarchical graph, where diffusive motion would be X(t) ~ t). This is explained in the text, but the abstract and Fig. 1 could benefit from an early clarification to avoid confusion with the standard mean-square-displacement terminology.
- [Appendix A, Fig. 6] The figure axis is labeled X^{−1}(0) while the text discusses X(0); please check the label or define the plotted quantity explicitly.
- [Footnote 79] The sentence "Xfront(t) it is given by the value of x for which Π(x,t) has a maximum" contains an extra "it is"; please correct to "Xfront(t) is given by...".
- [§Results, Fig. 4(a) inset] The inset is said to show all four stages of the evolution, but the caption does not explicitly mark the four stages; adding labels or arrows would make the claimed four-regime structure easier to verify.
Circularity Check
The subdiffusive phase claim inherits its exponent β(W)=1−W/W_AT from the same authors' prior return-probability fit; the collapses use that β as an input rather than extracting it.
-
self citation load bearing
[Model and methods, paragraph following Eq. (5)]
"In Ref. [73] we have shown that for small values of W (0<W<0.16WAT≃3) the return probability Π(0,t) is consistent with the result of Eq. (3), confirming that the system is in a fully ergodic phase. For larger disorder, W∈[0.4WAT,0.7WAT]≃[8,13], Π(0,t) decays as a stretched exponential ∼e^{−Γt^{β(W)}}, where the exponent is well approximated by β(W)≃1−W/WAT, 0.4WAT≲W≲0.7WAT (5)."
The central quantitative exponent β(W) is not derived in this paper; it is quoted from Ref. [73], a return-probability study by the same group (Bera, De Tomasi, Khaymovich, Scardicchio). That prior result is a fitted, numerical form, not an externally established theorem. The present paper then makes this β the load-bearing input for the claimed front propagation and X(t) growth, so the phase claim's exponent reduces to a self-citation of an earlier fit rather than being independently established here.
-
fitted input called prediction
[Results, Eqs. (10)-(11) and Fig. 5; also Fig. 4(b) and Appendix C]
"This collapse allows us to extract the following subdiffusive wave-front evolution Xfront(t)≃Γ(W)t^{β(W)}, β(W)<1. (10) ... Figure 5(a) shows the algebraic growth of X(t) in time for several W X(t)∼t^{β(W)}, (11) with the same subdiffusive exponent β(W), Eq. (5), as in the wave-front propagation Xfront(t)."
The collapse in Fig. 4(b) and the analogous collapses in Fig. 5(b) and Appendix C rescale the time axis with β(W) taken from Eq. (5); the data are therefore only tested for consistency with that assumed β, not used to extract it. Eq. (10) is presented as an 'extraction' even though β was fed into the rescaling, and Eq. (11) explicitly reuses Eq. (5) as the same exponent. Thus Eqs. (10) and (11) are, at the level of the exponent, restatements of the prior fitted input rather than independent predictions. The raw log-log growth of X(t) does provide some independent qualitative evidence of sublinear spreading, which prevents the entire claim from being purely circular.
full rationale
The paper contains substantial self-contained numerical content: the four-regime space-time structure, the space-time factorization for x<Xfront, the finite-size and energy-window checks, and the direct observation that X(t) grows sublinearly before saturation. These parts are not circular. However, the quantitative centerpiece, β(W)=1−W/WAT, is imported as Eq. (5) from Ref. [73] by the same authors and is then used to rescale time in all the collapses supporting Xfront(t)∼t^β and X(t)∼t^β. Consequently, the claimed exponent and the width of the claimed subdiffusive phase are inherited from a prior fitted result rather than independently extracted in this work; the collapses demonstrate self-consistency with the assumed β, not a free determination of it. The paper is also honest about the main limitation, stating: 'We do not report any crossover to diffusivity for our available system sizes and time scales. Although we cannot completely rule out this possibility in the thermodynamic limit.' That limitation weakens the phase claim but is not itself a circularity. The absence of error bars or collapse-quality metrics further limits the strength of the claimed confirmation. Overall, this is partial circularity: the central quantitative claim reduces, at the exponent level, to a self-cited fit, while the qualitative subdiffusive spreading has independent grounding. Hence a score of 6 is appropriate.
Assumptions & free parameters
free parameters (3)
- beta(W) = 1 - W/W_AT =
Range from about 0.56 (W=8) to about 0.23 (W=14), using W_AT about 18.1
- Gamma(W) =
not tabulated
- lambda =
not tabulated
assumptions (6)
- standard math Standard unitary time evolution and spectral decomposition of H with the projector P_DeltaE, Eqs. (1) and (2).
- domain assumption The RRG is locally tree-like with N(x) ~ K^x up to the diameter D = ln L / ln K.
- domain assumption The RRG is a valid proxy for the Fock-space structure of many-body localized systems.
- ad hoc to paper The stretched-exponential return probability Pi(0,t) ~ exp(-Gamma t^beta) with beta(W) approximately 1 - W/W_AT holds for 0.4 W_AT < W < 0.7 W_AT.
- ad hoc to paper Space-time factorization Pi(x,t) - Pi(x,infinity) = g(x)[Pi(0,t) - Pi(0,infinity)], Eq. (8), and the front profile Eq. (7) with f(z) = exp(-lambda z).
- domain assumption The energy window deltaE = f E_BW with f = 1/8 at band center is small enough to define a microcanonical wave packet and does not affect the dynamics.
Cite this review
Pith. "Pith review of Sub-diffusion in the Anderson model on random regular graph." pith.science (2026). https://pith.science/paper/IKLER7J5
@misc{pith2026190811388,
author = {Pith},
title = {Pith review of: Sub-diffusion in the Anderson model on random regular graph},
year = {2026},
howpublished = {\url{https://pith.science/paper/IKLER7J5}},
note = {Machine review of arXiv:1908.11388}
}
abstract
We study the finite-time dynamics of an initially localized wave-packet in the Anderson model on the random regular graph (RRG). Considering the full probability distribution $\Pi(x,t)$ of a particle to be at some distance $x$ from the initial state at time $t$, we give evidence that $\Pi(x,t)$ spreads sub-diffusively over a range of disorder strengths, wider than a putative non-ergodic phase. We provide a detailed analysis of the propagation of $\Pi(x,t)$ in space-time $(x,t)$ domain, identifying four different regimes. These regimes in $(x,t)$ are determined by the position of a wave-front $X_{\text{front}}(t)$, which moves sub-diffusively to the most distant sites $X_{\text{front}}(t) \sim t^{\beta}$ with an exponent $\beta < 1$. We support our numerical results by a self-consistent semiclassical picture of wavepacket propagation relating the exponent $\beta$ with the relaxation rate of the return probability $\Pi(0,t) \sim e^{-\Gamma t^\beta}$. Importantly, the Anderson model on the RRG can be considered as proxy of the many-body localization transition (MBL) on the Fock space of a generic interacting system. In the final discussion, we outline possible implications of our findings for MBL.
Figures
Figures from the paper (7 more)
Reference graph
Works this paper leans on
-
[73]
Return probability for the anderson model on the random regular graph,
S. Bera, G. De Tomasi, I. M. Khaymovich, and A. Scardicchio, “Return probability for the anderson model on the random regular graph,” Phys. Rev. B 98, 134205 (2018)
work page 2018
-
[1]
Metal- insulator transition in a weakly interacting many-electron system with localized single-particle states,
D.M. Basko, I.L. Aleiner, and B.L. Altshuler, “Metal- insulator transition in a weakly interacting many-electron system with localized single-particle states,” Annals of Physics 321, 1126 – 1205 (2006)
2006
-
[2]
Inter- acting electrons in disordered wires: Anderson localiza- tion and low-t transport,
I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, “Inter- acting electrons in disordered wires: Anderson localiza- tion and low-t transport,” Phys. Rev. Lett. 95, 206603 (2005)
2005
-
[3]
Many-Body Localiza- tion and Thermalization in Quantum Statistical Mechan- ics,
R. Nandkishore and D. A. Huse, “Many-Body Localiza- tion and Thermalization in Quantum Statistical Mechan- ics,” Annual Review of Condensed Matter Physics 6, 15– 38 (2015)
2015
-
[4]
Local integrals of motion in many-body localized systems,
J. Z Imbrie, V. Ros, and A. Scardicchio, “Local integrals of motion in many-body localized systems,” Annalen der Physik 529, 1600278 (2017)
2017
-
[5]
Recent progress in many- body localization,
D. A Abanin and Z. Papi´ c, “Recent progress in many- body localization,” Annalen der Physik 529, 1700169 (2017)
2017
-
[6]
Many-body localization: An introduction and selected topics,
F. Alet and N. Laflorencie, “Many-body localization: An introduction and selected topics,” Comptes Rendus Physique 19, 498 – 525 (2018), quantum simulation / Simulation quantique
2018
-
[7]
Col- loquium: Many-body localization, thermalization, and entanglement,
D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, “Col- loquium: Many-body localization, thermalization, and entanglement,” Rev. Mod. Phys. 91, 021001 (2019)
2019
Show all 90 references
-
[8]
Absence of diffusion in certain random lat- 6 tices,
P. Anderson, “Absence of diffusion in certain random lat- 6 tices,” Phys. Rev. 109, 1492–1505 (1958)
1958
-
[9]
Theory of the many-body localization transition in one-dimensional systems,
R. Vosk, D. A. Huse, and E. Altman, “Theory of the many-body localization transition in one-dimensional systems,” Phys. Rev. X 5, 031032 (2015)
2015
-
[10]
Universal dynam- ics and renormalization in many-body-localized sys- tems,
Ehud Altman and Ronen Vosk, “Universal dynam- ics and renormalization in many-body-localized sys- tems,” Annual Review of Condensed Matter Physics 6, 383–409 (2015), https://doi.org/10.1146/annurev- conmatphys-031214-014701
2015 doi
-
[11]
Anomalous diffusion and griffiths effects near the many-body localization transition,
K. Agarwal, S. Gopalakrishnan, M. Knap, M. M¨ uller, and E. Demler, “Anomalous diffusion and griffiths effects near the many-body localization transition,” Phys. Rev. Lett. 114, 160401 (2015)
2015
-
[12]
Critical properties of the many-body localization tran- sition,
V. Khemani, S. P. Lim, D. N. Sheng, and D. A. Huse, “Critical properties of the many-body localization tran- sition,” Phys. Rev. X 7, 021013 (2017)
2017
-
[13]
Entanglement critical length at the many-body localization transition,
F. Pietracaprina, G. Parisi, A. Mariano, S. Pascazio, and A. Scardicchio, “Entanglement critical length at the many-body localization transition,” J. Stat. Mech.: The- ory and Experiment 2017, 113102 (2017)
2017
-
[14]
Many-body localization and delocalization in large quantum chains,
E. V. H. Doggen, F. Schindler, K. S. Tikhonov, A. D. Mirlin, T. Neupert, D. G. Polyakov, and I. V. Gornyi, “Many-body localization and delocalization in large quantum chains,” Phys. Rev. B 98, 174202 (2018)
2018
-
[15]
Quan- tum chaos challenges many-body localization,
J. ˇSuntajs, J. Bonˇ ca, T. Prosen, and L. Vidmar, “Quan- tum chaos challenges many-body localization,” (2019), arXiv:1905.06345
2019 arXiv
-
[16]
Finite size scaling bounds on many-body localized phase tran- sitions,
A Chandran, CR Laumann, and V Oganesyan, “Finite size scaling bounds on many-body localized phase tran- sitions,” (2015), arXiv:1509.04285
2015 arXiv
-
[17]
Anomalous thermalization in ergodic systems,
D. J. Luitz and Y. Bar Lev, “Anomalous thermalization in ergodic systems,” Phys. Rev. Lett.117, 170404 (2016)
2016
-
[18]
Absence of diffusion in an interacting system of spinless fermions on a one-dimensional disordered lattice,
Y. Bar Lev, G. Cohen, and D. R. Reichman, “Absence of diffusion in an interacting system of spinless fermions on a one-dimensional disordered lattice,” Phys. Rev. Lett. 114, 100601 (2015)
2015
-
[19]
Self-consistent approach to many-body localization and subdiffusion,
P. Prelovˇ sek and J. Herbrych, “Self-consistent approach to many-body localization and subdiffusion,” Phys. Rev. B 96, 035130 (2017)
2017
-
[20]
Energy diffusion in the ergodic phase of a many body localizable spin chain,
V. Kerala Varma, A. Lerose, F. Pietracaprina, J. Goold, and A. Scardicchio, “Energy diffusion in the ergodic phase of a many body localizable spin chain,” (2015), arXiv:1511.09144
2015 arXiv
-
[21]
Diffusive and subdiffusive spin transport in the ergodic phase of a many-body localizable system,
M. ˇZnidariˇ c, A. Scardicchio, and V. K. Varma, “Diffusive and subdiffusive spin transport in the ergodic phase of a many-body localizable system,” Phys. Rev. Lett. 117, 040601 (2016)
2016
-
[22]
Griffiths effects and slow dynam- ics in nearly many-body localized systems,
S. Gopalakrishnan, K. Agarwal, E. A. Demler, D. A. Huse, and M. Knap, “Griffiths effects and slow dynam- ics in nearly many-body localized systems,” Phys. Rev. B 93, 134206 (2016)
2016
-
[23]
Spin transport of weakly disordered heisenberg chain at infinite temperature,
Ilia Khait, Snir Gazit, Norman Y. Yao, and Assa Auer- bach, “Spin transport of weakly disordered heisenberg chain at infinite temperature,” Phys. Rev. B 93, 224205 (2016)
2016
-
[24]
Phenomenology of anoma- lous transport in disordered one-dimensional systems,
Maximilian Schulz, Scott Taylor, Antonello Scardic- chio, and Marko Znidaric, “Phenomenology of anoma- lous transport in disordered one-dimensional systems,” (2019), arXiv:1909.09507
2019 arXiv
-
[25]
Den- sity propagator for many-body localization: Finite-size effects, transient subdiffusion, and exponential decay,
S. Bera, G. De Tomasi, F. Weiner, and F. Evers, “Den- sity propagator for many-body localization: Finite-size effects, transient subdiffusion, and exponential decay,” Phys. Rev. Lett. 118, 196801 (2017)
2017
-
[26]
Low-frequency conductivity in many-body local- ized systems,
Sarang Gopalakrishnan, Markus M¨ uller, Vedika Khe- mani, Michael Knap, Eugene Demler, and David A. Huse, “Low-frequency conductivity in many-body local- ized systems,” Phys. Rev. B 92, 104202 (2015)
2015
-
[27]
Slow dynamics and strong finite-size effects in many-body lo- calization with random and quasiperiodic potentials,
Felix Weiner, Ferdinand Evers, and Soumya Bera, “Slow dynamics and strong finite-size effects in many-body lo- calization with random and quasiperiodic potentials,” Phys. Rev. B 100, 104204 (2019)
2019
-
[28]
Dynamics of strongly interacting sys- tems: From fock-space fragmentation to many-body lo- calization,
Giuseppe De Tomasi, Daniel Hetterich, Pablo Sala, and Frank Pollmann, “Dynamics of strongly interacting sys- tems: From fock-space fragmentation to many-body lo- calization,” (2019), arXiv:1909.03073
2019 arXiv
-
[29]
Obtaining highly excited eigenstates of many-body lo- calized hamiltonians by the density matrix renormal- ization group approach,
Vedika Khemani, Frank Pollmann, and S. L. Sondhi, “Obtaining highly excited eigenstates of many-body lo- calized hamiltonians by the density matrix renormal- ization group approach,” Phys. Rev. Lett. 116, 247204 (2016)
2016
-
[30]
Spectral tensor networks for many-body localization,
A. Chandran, J. Carrasquilla, I. H. Kim, D. A. Abanin, and G. Vidal, “Spectral tensor networks for many-body localization,” Phys. Rev. B 92, 024201 (2015)
2015
-
[31]
Find- ing matrix product state representations of highly excited eigenstates of many-body localized hamiltonians,
Xiongjie Yu, David Pekker, and Bryan K. Clark, “Find- ing matrix product state representations of highly excited eigenstates of many-body localized hamiltonians,” Phys. Rev. Lett. 118, 017201 (2017)
2017
-
[32]
Many-body localization and transition by density matrix renormalization group and exact diagonalization studies,
S. P. Lim and D. N. Sheng, “Many-body localization and transition by density matrix renormalization group and exact diagonalization studies,” Phys. Rev. B 94, 045111 (2016)
2016
-
[33]
Delocalized glassy dynam- ics and many-body localization,
G. Biroli and M. Tarzia, “Delocalized glassy dynam- ics and many-body localization,” Phys. Rev. B 96, 201114(R) (2017)
2017
-
[34]
Critical behavior at the localization transition on random regular graphs,
K. S. Tikhonov and A. D. Mirlin, “Critical behavior at the localization transition on random regular graphs,” Phys. Rev. B 99, 214202 (2019)
2019
-
[35]
Many-body localization and new critical phenomena in regular random graphs and constrained Erd˝ os-R´ enyi net- works,
V. Avetisov, A. Gorsky, S. Nechaev, and O. Valba, “Many-body localization and new critical phenomena in regular random graphs and constrained Erd˝ os-R´ enyi net- works,” (2016), arXiv:1611.08531
2016 arXiv
-
[36]
Many-body localization in fock space: A local perspective,
D. E. Logan and S. Welsh, “Many-body localization in fock space: A local perspective,” Phys. Rev. B99, 045131 (2019)
2019
-
[37]
Exact solution of a percolation analog for the many-body localization transition,
S. Roy, D. E. Logan, and J. T. Chalker, “Exact solution of a percolation analog for the many-body localization transition,” Phys. Rev. B 99, 220201(R) (2019)
2019
-
[38]
Percolation in fock space as a proxy for many-body localization,
S. Roy, J. T. Chalker, and D. E. Logan, “Percolation in fock space as a proxy for many-body localization,” Phys. Rev. B 99, 104206 (2019)
2019
-
[39]
Quasiparticle lifetime in a finite system: A non- perturbative approach,
B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Lev- itov, “Quasiparticle lifetime in a finite system: A non- perturbative approach,” Phys. Rev. Lett. 78, 2803–2806 (1997)
1997
-
[40]
A selfconsistent theory of localization,
R Abou-Chacra, D J Thouless, and P W Anderson, “A selfconsistent theory of localization,” J. Phys. C: Solid State Physics 6, 1734 (1973)
1973
-
[41]
Anderson localization on the bethe lat- tice: Nonergodicity of extended states,
A. De Luca, B. L. Altshuler, V. E. Kravtsov, and A. Scardicchio, “Anderson localization on the bethe lat- tice: Nonergodicity of extended states,” Phys. Rev. Lett. 113, 046806 (2014)
2014
-
[42]
Nonergodic phases in strongly disordered random regular graphs,
B. L. Altshuler, E. Cuevas, L. B. Ioffe, and V. E. Kravtsov, “Nonergodic phases in strongly disordered random regular graphs,” Phys. Rev. Lett. 117, 156601 (2016)
2016
-
[43]
Mul- tifractal states in self-consistent theory of localization: analytical solution,
B. L. Altshuler, L. B. Ioffe, and V. E. Kravtsov, “Mul- tifractal states in self-consistent theory of localization: analytical solution,” (2016), arXiv:1610.00758. 7
2016 arXiv
-
[44]
Non- ergodic delocalized phase in anderson model on bethe lattice and regular graph,
V.E. Kravtsov, B.L. Altshuler, and L.B. Ioffe, “Non- ergodic delocalized phase in anderson model on bethe lattice and regular graph,” Annals of Physics 389, 148 – 191 (2018)
2018
-
[45]
Anderson model on Bethe lattices: density of states, localization proper- ties and isolated eigenvalue,
G. Biroli, G. Semerjian, and M. Tarzia, “Anderson model on Bethe lattices: density of states, localization proper- ties and isolated eigenvalue,” Prog. Theor. Phys. Suppl. 184, 187 (2010)
2010
-
[46]
Scaling theory of the anderson transition in random graphs: Ergodicity and universality,
I. Garc´ ıa-Mata, O. Giraud, B. Georgeot, J. Martin, R. Dubertrand, and G. Lemari´ e, “Scaling theory of the anderson transition in random graphs: Ergodicity and universality,” Phys. Rev. Lett. 118, 166801 (2017)
2017
-
[47]
Two localization lengths in the anderson transition on random graphs,
I Garc´ ıa-Mata, J Martin, R Dubertrand, O Giraud, B Georgeot, and G Lemari´ e, “Two localization lengths in the anderson transition on random graphs,” (2019), arXiv:1904.08869
2019 arXiv
-
[48]
On the anderson transition on the bethe lattice,
G. Parisi, S. Pascazio, F. Pietracaprina, V. Ros, and A. Scardicchio, “On the anderson transition on the bethe lattice,” (2018), arXiv:1812.03531
2018 arXiv
-
[49]
Ander- son localization on the bethe lattice using cages and the wegner flow,
Samuel Savitz, Changnan Peng, and Gil Refael, “Ander- son localization on the bethe lattice using cages and the wegner flow,” Phys. Rev. B 100, 094201 (2019)
2019
-
[50]
Dif- ference between level statistics, ergodicity and local- ization transitions on the Bethe lattice,
G Biroli, A C Ribeiro-Teixeira, and M Tarzia, “Dif- ference between level statistics, ergodicity and local- ization transitions on the Bethe lattice,” (2012), arXiv:1211.7334
2012 arXiv
-
[51]
Delocalization and ergodic- ity of the anderson model on bethe lattices,
G. Biroli and M. Tarzia, “Delocalization and ergodic- ity of the anderson model on bethe lattices,” (2018), arXiv:1810.07545
2018 arXiv
-
[52]
Large deviation func- tion for the number of eigenvalues of sparse random graphs inside an interval,
F. L. Metz and I. Perez Castillo, “Large deviation func- tion for the number of eigenvalues of sparse random graphs inside an interval,” Phys. Rev. Lett. 117, 104101 (2016)
2016
-
[53]
Level compressibility for the anderson model on regular random graphs and the eigenvalue statistics in the extended phase,
F. L. Metz and I. P. Castillo, “Level compressibility for the anderson model on regular random graphs and the eigenvalue statistics in the extended phase,” Phys. Rev. B 96, 064202 (2017)
2017
-
[54]
Extended states in a lifshitz tail regime for random schr¨ odinger operators on trees,
Michael Aizenman and Simone Warzel, “Extended states in a lifshitz tail regime for random schr¨ odinger operators on trees,” Phys. Rev. Lett. 106, 136804 (2011)
2011
-
[55]
Resonant delocalization for random Schr¨ odinger operators on tree graphs,
M. Warzel S. Aizenman, “Resonant delocalization for random Schr¨ odinger operators on tree graphs,” Journal of the European Mathematical Society 15, 1167–1222 (2013)
2013
-
[56]
The large connectivity limit of the Anderson model on tree graphs,
V. Bapst, “The large connectivity limit of the Anderson model on tree graphs,” Journal of Mathematical Physics 55, 092101 (2014), 10.1063/1.4894055
2014 doi
-
[57]
Localization transi- tion in the Anderson model on the Bethe lattice: spon- taneous symmetry breaking and correlation functions,
A. D Mirlin and Yan V Fyodorov, “Localization transi- tion in the Anderson model on the Bethe lattice: spon- taneous symmetry breaking and correlation functions,” Nucl. Phys. B 366, 507 (1991)
1991
-
[58]
Anderson localization and ergodicity on random regular graphs,
K. S. Tikhonov, A. D. Mirlin, and M. A. Skvortsov, “Anderson localization and ergodicity on random regular graphs,” Phys. Rev. B 94, 220203(R) (2016)
2016
-
[59]
Fractality of wave functions on a cayley tree: Difference between tree and locally treelike graph without boundary,
K. S. Tikhonov and A. D. Mirlin, “Fractality of wave functions on a cayley tree: Difference between tree and locally treelike graph without boundary,” Phys. Rev. B 94, 184203 (2016)
2016
-
[60]
Multi- fractality of wave functions on a cayley tree: From root to leaves,
M. Sonner, K. S. Tikhonov, and A. D. Mirlin, “Multi- fractality of wave functions on a cayley tree: From root to leaves,” Phys. Rev. B 96, 214204 (2017)
2017
-
[61]
Statistics of eigen- states near the localization transition on random regular graphs,
K. S. Tikhonov and A. D. Mirlin, “Statistics of eigen- states near the localization transition on random regular graphs,” Phys. Rev. B 99, 024202 (2019)
2019
-
[62]
Chaos and quantum thermalization,
M. Srednicki, “Chaos and quantum thermalization,” Phys. Rev. E 50, 888–901 (1994)
1994
-
[63]
Quantum statistical mechanics in a closed system,
J. M. Deutsch, “Quantum statistical mechanics in a closed system,” Phys. Rev. A 43, 2046–2049 (1991)
1991
-
[64]
Alternatives to eigenstate thermalization,
M. Rigol and M. Srednicki, “Alternatives to eigenstate thermalization,” Phys. Rev. Lett. 108, 110601 (2012)
2012
-
[65]
Many-body localization in a quantum simulator with programmable random disorder,
J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, “Many-body localization in a quantum simulator with programmable random disorder,” Nat. Phys.12, 907–911 (2016)
2016
-
[66]
Many-body localization and quantum ergodicity in disordered long-range ising mod- els,
P. Hauke and M. Heyl, “Many-body localization and quantum ergodicity in disordered long-range ising mod- els,” Phys. Rev. B 92, 134204 (2015)
2015
-
[67]
What Determines the Spreading of a Wave Packet?
R. Ketzmerick, K. Kruse, S. Kraut, and T. Geisel, “What Determines the Spreading of a Wave Packet?” Phys. Rev. Lett. 79, 1959–1963 (1997)
1997
-
[68]
Anomalous Diffusion at the Anderson Transitions,
Tomi Ohtsuki and Tohru Kawarabayashi, “Anomalous Diffusion at the Anderson Transitions,” J. Phys. Soc. Japan 66, 314–317 (1997)
1997
-
[69]
Notice the correct normalization ∑ x Π(x,t ) = 1, and Π(x,t )≥ 0
-
[70]
Integrals of mo- tion in the many-body localized phase,
V Ros, M M¨ uller, and A Scardicchio, “Integrals of mo- tion in the many-body localized phase,” Nucl. Phys. B 891, 420–465 (2015)
2015
-
[71]
The single-particle mobility edges have been proven to exist at W &K [54]
-
[72]
Mehta., Random Matrices, 3rd ed
M.L. Mehta., Random Matrices, 3rd ed. (Elsevier Inc., Amsterdam, 2004)
2004
-
[74]
Survival probability in General- ized Rosenzweig-Porter random matrix ensemble,
G. De Tomasi, M. Amini, S. Bera, I. M. Khaymovich, and V. E. Kravtsov, “Survival probability in General- ized Rosenzweig-Porter random matrix ensemble,” Sci- post Phys. 6, 14 (2019)
2019
-
[75]
We define a fully ergodic phase as the phase in which the system can be described by the random matrix theory
-
[76]
In Appendix B we explicitly show diffusive behavior in this disorder range W <0.16WAT
-
[77]
Exact closed form of the return prob- ability on the bethe lattice,
A. Giacometti, “Exact closed form of the return prob- ability on the bethe lattice,” J. Phys. A 28, L13–L17 (1995)
1995
-
[78]
ForW <0.16WAT the propagation is diffusive withβ = 1 and consistent with the fully-ergodic behavior given by Eq. (3)
-
[79]
Xfront(t) it is given by the value of x for which Π(x,t ) has a maximum
-
[80]
Numerical studies of localization in disordered systems,
J T Edwards and D J Thouless, “Numerical studies of localization in disordered systems,” Journal of Physics C: Solid State Physics 5, 807–820 (1972)
1972
-
[81]
(7) and (8), are consistent to each other provided the exponential behavior of f(z) in Eq
Note that both collapses, Eqs. (7) and (8), are consistent to each other provided the exponential behavior of f(z) in Eq. (7), with the maximal value Π( Xfront(t),t ) given byg(Xfront) [Π(0,t )− Π(0,∞)] due to the normalization condition of Π(x,t ) (see Appendix D for more details)
-
[82]
Thouless en- ergy and multifractality across the many-body localiza- tion transition,
M. Serbyn, Z. Papi´ c, and D. A. Abanin, “Thouless en- ergy and multifractality across the many-body localiza- tion transition,” Phys. Rev. B 96, 104201 (2017)
2017
-
[83]
Anomalous thouless energy and critical statistics on the metallic side of the many-body localization transition,
C. L. Bertrand and A. M. Garc´ ıa-Garc´ ıa, “Anomalous thouless energy and critical statistics on the metallic side of the many-body localization transition,” Phys. Rev. B 94, 144201 (2016). 8 2 4 6 8 10 12 14 W 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4X −1(0) f = 1/8 L = 218 220 FIG. 6...
2016
-
[84]
In the literature [87–90] this kind of phase is usually called weakly ergodic when the eigenfunctions occupy a finite, but small fraction fI ≪ 1 of the Hilbert space, meaning that the system splits into a finite number ∼ 1/fI of nearly independent sectors which are ergodic withi...
-
[85]
Absolutely continuous spectrum implies ballistic transport for quantum parti- cles in a random potential on tree graphs,
M. Aizenman and S. Warzel, “Absolutely continuous spectrum implies ballistic transport for quantum parti- cles in a random potential on tree graphs,” Journal of Mathematical Physics 53, 095205 (2012)
2012
-
[86]
Multifractality and its role in anomalous trans- port in the disordered xxz spin-chain,
David J Luitz, Ivan Khaymovich, and Yevgeny Bar Lev, “Multifractality and its role in anomalous trans- port in the disordered xxz spin-chain,” (2019), arXiv:1909.06380
2019 arXiv
-
[87]
Power-law random banded matrices and ultrametric matrices: Eigenvector distribution in the intermediate regime,
E. Bogomolny and M. Sieber, “Power-law random banded matrices and ultrametric matrices: Eigenvector distribution in the intermediate regime,” Phys. Rev. E 98, 042116 (2018)
2018
-
[88]
Correlation-induced localization,
P. A. Nosov, I. M. Khaymovich, and V. E. Kravtsov, “Correlation-induced localization,” Physical Review B 99, 104203 (2019)
2019
-
[89]
Robustness of de- localization to the inclusion of soft constraints in long- range random models,
P. A. Nosov and I. M. Khaymovich, “Robustness of de- localization to the inclusion of soft constraints in long- range random models,” Phys. Rev. B 99, 224208 (2019)
2019
-
[90]
Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems,
Arnd B¨ acker, Masudul Haque, and Ivan M Khaymovich, “Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems,” Phys. Rev. E 100, 032117 (2019). Appendix A: Initial wave-packet and dependence on f As we already discussed in the main text, the u...
2019
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