Disordered Quantum Spin Chains or Scratching at the Surface of Many Body Localization

Transcription

1 Disordered Quantum Spin Chains or Scratching at the Surface of Many Body Localization Günter Stolz University of Alabama at Birmingham Arizona School of Analysis and Mathematical Physics Tucson, Arizona, March 5 9, 2018

2 Contents I. Prelude: Towards Many-Body Localization II. MBL Properties of the Disordered XY Chain III. Interlude: The Quantum Ising Chain IV. Localization of the Droplet Spectrum in the XXZ Chain V. Some Proofs VI. Epilogue: Where to go from here? References

3 I. Prelude: Towards Many-Body Localization Localization Absence of Quantum Transport One Body Transport: Many Body Transport:

4 One-Body Transport: of 1 2/22/18, 11:55 AM

5 Many-Body Transport:

6 Want to get a paper into a really good journal? Take the Heisenberg model (for example): H = j Z(σ X j σ X j+1 + σ Y j σ Y j+1 + σ Z j σ Z j+1) Add a random field, e.g. H(ω) = H + j Z ω j σ Z j (i) Show that H(ω) is many-body localized in an extensive energy regime. (ii) Probably easier: Show that H(λω) is fully many-body localized at large disorder λ.

7 Some words of caution: In dimension d > 1 not even the one-particle case is settled: There is no rigorous proof of extended states for Anderson model in d = 3. There is no rigorous proof of non-existence of a mobility edge for Anderson model in d = 2. Hope: Starting with spin chains (d = 1) will help. Also, spins are much simpler than electrons. However: Dynamical systems methods (as for 1D ergodic Schrödinger operators) will not be equally useful for 1D many body models. In fact: For disordered Heisenberg model a many-body localization/delocalization transition is expected in 1D (for small disorder).

8 So what can we do? Will present some first results towards MBL in relatively simple models and regimes ( scratching the surface ). Focus on getting a sense of phenomena and relevant concepts. Have nothing to say about thermalization at high energy and the many-body Anderson transition. Have nothing to say about multidimensional disordered quantum spin systems. Conclusion will be: Field of localization/delocalization for many-body models is still wide open! 1 1 But be ready to be ridiculed by physicists...

9 A first (rough) glance at possible manifestations of MBL: Dynamical MBL: No many-body/information transport (e.g. Newton s cradle), quantum version can be formulated as zero velocity Lieb-Robinson bound. Spin systems are ideal models to study this (no one-particle transport).

10 A first (rough) glance at possible manifestations of MBL: Dynamical MBL: No many-body/information transport (e.g. Newton s cradle), quantum version can be formulated as zero velocity Lieb-Robinson bound. Spin systems are ideal models to study this (no one-particle transport). Localization of eigenstates (thermal states, etc.): All eigenstates of a many-body localized system should be close to the eigenstates of a non-interacting many-body system, i.e. product states. Can be detected through: Rapid decay of correlations Low entanglement (area laws)

11 II. MBL Properties of the Disordered XY Chain (In particular: Introduction of Relevant Concepts) Survey of some of the results of: Hamza/Sims/St Klein/Perez 1992 Sims/Warzel 2016 Pastur-Slavin 2014 Abdul-Rahman/St Abdul-Rahman/Nachtergaele/Sims/St. 2016, Survey 2017

12 XY (or XX) Chain in Random Field: L 1 H XY (ω) = (σj X σj+1 X + σj Y σj+1) Y j=1 L ω j σj Z j=1 in L j=1 C2 Assume: (ω j ) j=1 i.i.d. random variables, with distribution dµ(ω j ) = ρ(ω j ) dω j, where ρ is bounded and compactly supported. Jordan-Wigner transform: c 1 := a 1, c j := σ Z 1... σ Z j 1a j, j = 2,..., n, a := Canonical anti-commutation relations (CAR): {c j, c k } = δ jki, {c j, c k } = {c j, c k } = 0 ( )

13 Then H XY = 2c Mc + E 0 with E 0 = j ω j, c = (c 1,..., c L ) t, c = (c1,..., c L ) and effective Hamiltonian ω 1 1. M = D Anderson Model! 1 1 ω L Remarks: (i) In language of second quantization: H XY = 2dΓa (M) + E 0 on F a (l 2 (Z)) (ii) Jordan-Wigner transform is non-local. (iii) Spin chain has commuting degrees of freedom, while {c j } are anti-commuting degrees of freedom.

14 Nevertheless we will see: Anderson localization for M = MBL for H XY Known strong form of Anderson Localization: Eigencorrelator Localization: ( uniformly in L. E sup (g(m)) jk g 1 ) Ce µ j k In particular: Dynamical (One-Body) Localization: ( ) E sup (e itm ) jk t R Ce µ j k This implies various MBL properties for the disordered XY chain:

15 Zero-velocity Lieb-Robinson bound (Dynamical MBL) Local observables: For A, B C 2 2 let A j = I... A... I (in j-th position), B k =... Heisenberg dynamics: τ t (A j ) = e ith A j e ith Theorem 1 (Hamza/Sims/St. 2012) There exist C < and µ > 0 such that ( ) E sup [τ t (A j ), B k ] t R C A B e µ j k for all L, 1 j, k L, A, B C 2 2. Note: Requires averaging over disorder E( ).

16 Compare: Lieb-Robinson 1972 (recent explosion of improvements, extensions and applications: Nachtergaele/Sims 2006, Hastings/Koma 2006, Hastings 2007,...): For a quite general class of quantum spin systems (with bounded coefficients and bounded interaction range) it holds that [τ t (A j ), B k ] C A B e µ( j k v t ) v < Deterministic result! Lieb-Robinson (group) velocity Note: There are results on anomalous LR-bounds (quasi-periodic field: Damanik/Lemm/Lukic/Yessen 2014)

17 Key to proof of Theorem 1: Use basic properties of quasi-free and free Fermion systems to show τ t (c j ) = ( e 2iMt) c l jl l Thus: One-particle dynamics e 2iMt determines many-body dynamics τ t (c j ). Now undo Jordan-Wigner by two geometric summations to get dynamics of local operators such as a j, a j. Build other local observables from this.

18 Exponential Decay of Correlations ( Exponential Clustering ): Theorem 2 (Sims/Warzel 2016) There exist C < and µ > 0 such that ( E sup ψ, τ t (A j )B k ψ ψ, A j ψ ψ, B k ψ ψ,t ) C A B e µ j k for all L, 1 j, k k, and all A, B C 2 2. Here the supremum is taken over all normalized eigenfunctions ψ of H and all t R. Notes: (i) Same for thermal states ρ β = e βh /Tre βh, with expectations defined as A ρβ = Trρ β A. (ii) Earlier related work (ground state): Klein/Perez 1992

19 Area Law for the Entanglement Entropy: Bipartite decomposition: H L = H A H B, H A = l L C 2 j, H B = j=1 j=l+1 C 2 j ψ normalized eigenstate of H, ρ ψ = ψ ψ, reduced state: ρ A ψ = Tr B ρ ψ Bipartite entanglement entropy: E(ρ ψ ) := S(ρ A ψ ) := Tr ρa ψ log ρa ψ ( = S(ρ B ψ ) ) For generic pure states: E(ρ) l (volume law)

20 Uniform Area Law: Theorem 3: (Abdul-Rahman/St. 2015) There exists C < such that ( ) E sup E(ρ ψ ) ψ C for all L and all 1 l < L. Here the supremum is taken over all normalized eigenstates ψ of H. Notes: (i) Method due to Pastur/Slavin 2014, who proved the area law for the ground state of a disordered d-dimensional quasi-free Fermion system (bound Cl d 1 ). (ii) No logarithmic correction in l. (iii) Open problem: Analogue for thermal states? (Problem: E is not a good entanglement measure for mixed states.

21 Entanglement Dynamics under a Quantum Quench: Let H A, H B restrictions of H to A and B ψ A, ψ B normalized eigenstates of H A, H B, ρ A = ψ A ψ A, ρ B = ψ B ψ B ρ = ρ A ρ B (i.e. E(ρ) = 0) ρ t = e ith ρe ith (full) Schrödinger dynamics Theorem 4: (Abdul-Rahman/Nachtergaele/Sims/St. 2016) There exists C < such that ( ) for all l and L. E sup E(ρ t ) t,ψ A,ψ B C

22 Remarks on MBL Hierarchy: There are general results of the type: Zero-velocity LR bound = Exponential clustering (Hamza/Sims/St. 2012) = Area Law (Brandao/Horodecky 2013/2015) (Most likely no satisfying converses, I think. More math left to be understood here.)

23 Main tool in proofs of Theorems 2 to 4: Quasifree States and Correlation Matrices Fact: Eigenstates ρ = ρ α, α {0, 1} L, and thermal states ρ = ρ β, 0 < β <, of a quasifree Fermion system c Mc are quasifree (i.e. expectations of arbitrary products of the c j and cj can be calculated by Wick s Rule. Also: The reduced state ρ A of a quasifree state ρ is again quasifree. Thus ρ is uniquely determined by its correlation matrix Γ ρ = ( c j c k ρ) L j,k=1 and ρ A by the restricted correlation matrix Γ A ρ = ( c j c k ρ) l j,k=1

24 For Theorem 2 (Sims/Warzel): Calculate Pfaffians. For Theorems 3 and 4 use (Vidal, Latorre, Rico, Kitaev 2003): S(ρ A ) = Tr ρ A log ρ A = tr h(γ ρ A) where h(x) = x log x (1 x) log(1 x). If σ(m) = {λ j : j = 1,..., L} is simple, then Γ ρα = χ α (M), where α := {λ j : α j = 1} Rest of proof uses that, by Anderson localization of M, ( ) E sup (χ α (M)) jk α Ce µ j k

25 Most of the above can be extended to the anisotropic XY chain in random field (at least for large disorder λ): L 1 H γ = ((1 + γ)σj X σj+1 X + (1 γ)σj Y σj+1) Y λ j=1 = C MC + E 0 I L ω j σj Z Here C = (c 1,..., c L, c1,..., c L )t, C = (c1,..., c L, c 1,..., c L ), ( ) M K Block Anderson model: M = K M ω 1 M = 1 j=1 1 0 γ , K =. γ γ 1 γ 0 ω L

26 III. Interlude: The Quantum Ising Model Have seen: Heisenberg XYZ (XXX) model gets MUCH simpler if we drop the Z term! What happens for the other extreme, i.e., if we drop the XY terms? Model gets TRIVIAL! Quantum Ising model with disorder: H Ising (ω) = 1 σj 4 j Z(I Z σj+1) Z + ω j N j j Z ( ) 0 0 Here: N j = local number operator 0 1 j Have normalized so that Ω = is ground state ( vaccum ) with HΩ = 0. Assume here that ω j 0 for all j.

27 H Ising (ω) is diagonal in the product basis {ϕ X := j X a j Ω : X Z finite} ( ) 0 0 Creation operators : aj = (of down spins/particles) 1 0 { j 2 if {j, j + 1} X (surface of X ) Note: (I σj Z σj+1 Z )ϕ X = 0 else ( Thus: H Ising (ω)ϕ X = 1 2 X + ) j X ω j ϕ X In particular: σ(h Ising (0)) = {0, 1, 2,...}, eigenspace to E k = k: span{ϕ X : X consists of k clusters of down spins} k = 1: X interval, i..e., ϕ X forms a single down spin droplet

28 Philosophical Question: Pro: Is H Ising (0) many-body localized? H has an eigenbasis of product states. Dynamics is trivial, in particular [τ t (A), B] = 0 for all t if supp A supp B =. Contra: Eigenspaces are highly degenerate. There are eigenstates with long range correlations and high entanglement. A truly localized regime should be rather stable under (extensive) perturbations, which is arguably not the case here (as we will see in an example below).

29 Adding a random field overcomes these questionable issues: In finite volume [ L, L] and assuming that ω j are i.i.d. with absolutely continuous distributions: H [ L,L] Ising (ω) has almost surely non-degenerate spectrum, so that all eigenstates are product states. One way to think of main result of next chapter: MBL for H Ising (ω) is stable under a natural extensive perturbation, at least at low energies.

30 IV. Localization of the Droplet Spectrum in the XXZ Chain Idea: Perturb around the Ising model by bringing small XX-terms back! Droplets in the Ising phase of XXZ: Starr 2001, Nachtergaele/Starr 2001, Nachtergaele/Spitzer/Starr 2007 Fischbacher 2013, Fischbacher/St. 2014, 2017 Localization of the Droplet Spectrum: Beaud/Warzel 2017a, 2017b Elgart/Klein/St. 2017a, 2017b

31 The disordered infinite XXZ chain H XXZ (ω) = j Z h j,j+1 + λ j Z ω j N j on H = {ϕ X : X Z finite} (Hilbert space completion). h j,j+1 = 1 4 (I σz j σ Z j+1) 1 4 (σx j σ X j+1 + σ Y j σ Y j+1) Assume: (i) > 1 Ising phase (ii) ω j i.i.d., a.c. distribution, bounded density, support [0, ω max ] (iii) λ > 0 disorder parameter

32 Particle number conservation The subspaces H N = span{ϕ X : X = N}, N = 0, 1, 2,... are invariant under H XXZ (ω). Thus H XXZ (ω) = Identify H N = l 2 (V N ), where H N (ω) N=0 V N = {x = (x 1,..., x N ) Z N : x 1 < x 2 <... < x N }, i.e., ordered labelings of the down spin configurations X = {x 1,..., x N } Z.

33 The N-particle operators: H 0 (ω) = 0 on span Ω N 1: Here: H N (ω) = 1 2 A N D N + λv ω on l 2 (V N ) A N adjacency operator on V N (with l 1 -distance inherited from Z N ) D N multiplication operator ( potential ) on l 2 (V N ) by D N (x) = 2 {connected components in x} = X (D N is actually the degree function on the graph V N, see also talk by C. Fischbacher) V ω (x) = ω x ω xn N-particle Anderson potential

34 Happy Schrödinger :-)

35 Free XXZ (ω = 0): H XXZ (0) = N 0 H N(0) System of attractive hard core bosons (down spins) at sites x = {x 1 < x 2 <... < x N }: H N (0) = 1 2 A 1 N + }{{} 2 D N }{{} hopping (XX) interaction (Ising) on l 2 (V N ), N 1 Attractive interaction: D N (x) = N 1 k<l N Q( x k x l ), where Q(1) = 1, Q(r) = 0 for r 1. H XXZ (0) (meaning each H N (0)) can be exactly diagonalized by the Bethe Ansatz 2 (Babbitt/Thomas/Gutkin 70 s to 90 s, Borodin et al 2015): 2 But our proofs do actually NOT use this!

36

37 Nachtergaele/Starr 2001, Figure 2 ( = 2.125): "multiplicity of ground states" E/A( ) n n

38 Droplet bands: (with cosh(ρ) = ) [ δ N = tanh(ρ) cosh(nρ) 1, tanh(ρ) cosh(nρ) + 1 ] sinh(nρ) sinh(nρ) } δ = { 1 1/ 2 as N (exp. fast) Droplet spectrum: I = [1 1 (, 2 1 ) 1 ) =: [E 0, E 1 ) How do eigenstates to E I look like? They are exponentially close to droplets: See exact expressions in Nachtergaele/Starr 2001

39 The finite volume XXZ chain: On H [ L,L] = L j= L C2 : L 1 HXXZ L (ω) = j= L h j,j+1 + λ L ω j N j + β(n L + N L ) j= L Here: ω j 0 for all j, β 1 2 (1 1 ) ( droplet b.c. ) Particle number conservation: 2L 1 HXXZ L (ω) = N=1 2L 1 HN L (ω) on N=1 l 2 (V L N ), V L N := {x = (x 1,..., x N ) Z N : L x 1 <... < x N L}

40 Droplet configurations in V L N : V L N,1 := {x = (x 1, x 1 + 1, x 1 + N 1) V L N } Theorem 4: (Elgart/Klein/St. 2017a, Fischbacher/St. 2017) Let E < E 1, H L N (ω)ψ E = Eψ E, ψ E = 1, and let A V L N. Then χ A ψ E 4 E 1 E E 1 E e (log )d 2 N (A,VN,1 L ) χ V L ψ E. N,1 Here d N denotes the l 1 -distance of two subsets of Z N.

41 This is a consequence of a Combes-Thomas bound on the Green function shown in Elgart/Klein/St. 2017a. Constants are uniform in N. This becomes possible in the Combes-Thomas bound (where the decay rate usually depends on the dimension as 1/N) because the two terms in 1 2 A N D N balance one another. In particular, the bound holds for ω = 0.

42 Entanglement bounds: Bipartite decomposition: [ L, L] = Λ l Λ r, E(ρ ψ ) := S(ρ Λ l ψ ) Theorem 5: (essentially Beaud/Warzel 2017b) Let > 1 and δ > 0. Then there exist constants C 1 = C 1 (, δ) and C 2 = C 2 (, δ) such that (a) for every ω 0, every E E 1 δ and every normalized eigenvector ψ E of HXXZ L (ω) to E, E(ρ ψe ) C 1 log Λ l (b) If E( ) denotes disorder averaging, then ( ) E E(ρ ψe ) C 2 sup E E 1 δ ψ E = 1

43 Remarks: Part (a) holds, in particular, for ω = 0 (droplet states of free Ising phase XXZ)! Averaging over disorder kills the log-correction! Part (a) follows from Theorem 4 and summing up many geometric series. Part (b) follows quite easily from Part (a), because large down-spin clusters rarely have energy below E 1 (by large deviations for N j=1 ω j). Need to work harder for LR bounds and correlation bounds! (Remember MBL hiearchy.) Key result for everything else to come:

44 Droplet Localization: Theorem 6: (Elgart/Klein/St. 2017a) Let δ > 0, λ > 0 and > 1 be such that λ 1 is sufficiently large. Then there exist C and m > 0 such that E N j ψ E N k ψ E Ce m j k (1) E σ(h L XXZ ) I 1,δ uniformly in L > 0, j, k [ L, L]. Here ψ E is the (almost surely unique) normalized eigenstate to E σ(h L XXZ ) and I 1,δ := [E 0, E 1 δ]

45 Remarks (including on the proof): Interpretation: Eigenstates in the droplet spectrum are close to states with only one down-spin cluster! Special cases: (i) > 1 fixed, λ large, (ii) λ > 0 fixed, large. Particle number conservation reduces the proof to summing over eigencorrelators in the N-particle subspaces. In each N-particle space, the bound is proven by the fractional moments method. Uniformity of constants in N follows from uniformity of constants in the Combes-Thomas bound. Summability over N follows by a large deviations argument (IDS decays exponentially in N).

46 Consequences of Droplet Localization: Make all assumptions on H = HXXZ L (ω) from above, including the (, λ)-regime from Theorem 6. I := I 1,δ for any fixed δ > 0 (which constants below will depend on). P I = P I (H) = spectral projection for H onto I X local observable, supported on interval S X [ L, L] l-fattening of support: S X,l = S X + [ l, l]

47 Heisenberg evolution: τ t (X ) = e ith Xe ith Projection of local observable onto energy window: X I := P I XP I Energy restricted Heisenberg dynamics: τ t (X I ) = e ith P I XP I e ith = P I τ t (X )P I = (τ t (X )) I Compare with energy-restricted Schrödinger dynamics: e ith P I ψ

48 Zero-velocity Lieb-Robinson bound: Theorem 7: (Elgart/Klein/St. 2017b) There exist C < and m > 0, independent of L, such that, for all local observables X and Y, ( ) E sup [τ t (X I ), Y I ] t R C X Y e m dist(s X,S Y ).

49 Quasi-locality of the dynamics ( LR for gourmets ): Theorem 8: (Elgart/Klein/St. 2017b) There exist C < and m > 0 such that for all X, t and l there exists a (random) local observable X l (t) = X l (t, ω) with S Xl (t) = S X,l and ( ) E sup (X l (t) τ t (X )) I t R C X e ml. Note: In the energy restricted case zero-velocity LR and quasi-locality are NOT equaivalent!

50 Exponential clustering: Theorem 9: (Elgart/Klein/St. 2017a) There exist C < and m > 0 such that for all local observables X and Y, E sup R τt(xi ),Y I (ψ E ) C X Y e m dist(s X,S Y ). t R E σ(h) I Here R X,Y (ψ) = ψ, XY ψ ψ, X ψ ψ, Y ψ Note: According to our proposed hierarchy of MBL properties this should be the easiest result to prove.

51 V. Some Proofs The Combes-Thomas bound Exponential clustering How do dynamics come in?

52 A Combes-Thomas bound (in infinite volume) Fix N N (arbitrary) and drop all N s from notation. H = H N = 1 2 A D + V on l2 (V N ) = 1 2 L (1 1 )D + V, with any potential V 0, L = A + D graph Laplacian on V N. Recall: D(x) = 2 {connected components of x} = degree of x in V N P 1 := orthogonal projection onto l 2 (V N,1 ) Thus: H + (1 1 )P 1 2(1 1 ).

53 Theorem 10: (Elgart/Klein/St. 2017, Fischbacher/St. 2017) Let δ > 0, E (2 δ)(1 1 ) and A, B V N. Then χ A (H + (1 1 )P 1 E) 1 χ B ( δ( 1) ) δ( 1) dn (A,B) 8 Remarks: (1) No N-dependence in the constants!!! (2) Can be extended to complex energy, works in finite volume, etc.

54 Sketch of proof: As in the standard proof we use dilations: where η > 0, ρ A (x) := d N (A, x). (K η ψ)(x) = 1 2 K η := e ηρ A He ηρ A H y V N :y x (1 e η(ρ A(y) ρ A (x) )ψ(y) x y 1 e η(ρ A(y) ρ A (x) e η 1 Standard C-T: K η eη 1 2 2N (use 2N = max. degree of V N) Grows with N. Not good enough!

55 Instead: Borrow two factors D 1/2 and show D 1/2 K η D 1/2 C( )(e η 1). N-independence is a consequence of the balance of A and D. The two borrowed factors can be paid back via (due to L 0) D 1/2 (H (1 1 )P 1 E) 1 D 1/2 C(δ, ) Now continue as in the standard C-T proof (essentially a resolvent identity, e.g. Kirsch 2007)

56 Exponential Clustering Proof of Theorem 9 for t = 0 and one-site observables X A j, Y A k, j k: Can be built from ( ) ( ) ( ) ( ) X +,+ 1 0 =, X +, 0 1 =, X,+ 0 0 =, X, 0 0 = j j j j Y ±,± = same at site k 16 cases: R X a,b,y c,d (ψ E ), a, b, c, d {±} Case 1: X, = N j, Y, = N k R Nj,N k (ψ E ) = ψ E, N j (I ψ E ψ E )N k ψ E N j ψ E N k ψ E Thus Case 1 follows from Theorem 6.

57 Many other cases can be settled by particle number conservation: Assume ψ = ψ E H N for some N (holds a. s. for all eigenvectors by simplicity). Z {X, Y } = Z +, ψ has at most N 1 particles ( N 1 j=1 H j) Z,+ ψ has at least N + 1 particles ( 2L+1 j=n+1 H j) This settles five more cases: Z, preserves the particle number 0 = R X +,,Y +, (ψ) = R X,+,Y,+(ψ) = R X +,,Y, (ψ) = R X,,Y,+(ψ) = R X,,Y +, (ψ)

58 For example: ψ, X +, Y +, ψ ψ, X +, ψ ψ, Y +, ψ }{{}}{{}}{{} = X,+ ψ,y +, ψ =0 =0 =0 Another eight cases can be reduced to the previous six cases by using the properties (a) R X +,+,Z = R X,,Z (X +,+ = I X, ) (b) R Z,Y +,+ = R Z,Y, (c) R Z,W = R W,Z This leaves two cases: (i) R X,+,Y +, (ψ), (ii) R Y,+,X +, (ψ) Case (ii) reduces to Case (i) by commutation.

59 The remaining Case (i): R X,+,Y +, (ψ E ) = E E = E ψ E, X,+ (I P E )Y +, ψ E ψ E, N j X,+ (I P E )Y +, N k ψ E E N j ψ E N k ψ E The claim follows from Theorem 6. QED

60 VI. Epilogue: Where to go from here? (Gentle Ben s!) Reasonable (?) goals: Scratch deeper in the Ising phase of XXZ: States with k connected clusters of down-spins? Note: Theorem 6 (droplet localization) does NOT hold above [1 1/, 2(1 1/ )] (by a Theorem in Elgart/Klein/St. 2017b). But states with k downspin clusters have energy at least k(1 1/ ). Hunch: Need to think more in terms of scattering theory! Show that large disorder leads to a fully MBL regime for a general class of spin Hamiltonians with short range interaction (Imbrie 2016, uses assumption on level spacing).

61 Somewhat less reasonable (?) goals: Models without particle number conservation? Tempting thought: Can the methods for anisotropic XY and XXZ be combined to get a result for fully anisotropic XYZ? Remark on the side: Entanglement bounds for mixed (thermal) states in free Fermion systems?

62 Homework (unreasonable goals to impress physicists): We have shown zero-temperature localization for disordered XXZ in the Ising phase. MBL in an extensive energy regime would mean localization for energies in [ ρl, ρl] for some positive particle density ρ > 0 (L the system size). May currently be out of reach. Thermalization? Many-body mobility edge? Well, maybe better start with proving existence of extended states in 3D Anderson model... Higher-dimensional spin systems? (Even XY not understood.) Electron gases? (Basko/Aleiner/Altshuler 2006)

63 Cut the wires!

64 References: Abdul-Rahman/Nachtergaele/Sims/Stolz 2016, arxiv: Abdul-Rahman/Nachtergaele/Sims/Stolz 2017, arxiv: Abdul-Rahman/Stolz 2015, arxiv: Babbitt/Thomas/Gutkin 70s to 90s Basko/Aleiner/Altshuler 2006, arxiv:cond-mat/ Beaud/Warzel 2017a, arxiv: Beaud/Warzel 2017b, arxiv: Borodin et al 2015, arxiv: Brandao/Horodecky 2013, arxiv: Brandao/Horodecky 2015, arxiv: Damanik/Lemm/Lukic/Yessen 2014, arxiv: Elgart/Klein/Stolz 2017a, arxiv: Elgart/Klein/Stolz 2017b, arxiv: Fischbacher 2013, Master Thesis, LMU Munich Fischbacher/Stolz 2014, arxiv: Fischbacher/Stolz 2017, arxiv:

65 Hamza/Sims/Stolz 2012, arxiv: Hastings 2007, arxiv: Hastings/Koma, arxiv:math-ph/ Imbrie 2016, arxiv: Kirsch 2007, arxiv: Klein/Perez 1992, CMP 147, (1992) Lieb/Robinson 1972, CMP 28, (1972) Nachtergaele/Spitzer/Starr 2007, arxiv:math-ph/ Nachtergaele/Sims 2006, arxiv:math-ph/ Nachtergaele/Starr 2001, arxiv:math-ph/ Pastur/Slavin 2014, arxiv: Sims/Warzel 2016, arxiv: Starr 2001, arxiv:math-ph/ Vidal/Latorre/Rico/Kitaev 2003, arxiv:quant-ph/