Disordered Quantum Spin Chains or Scratching at the Surface of Many Body Localization
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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 Ising Chain IV. Localization of the Droplet Spectrum in the XXZ Chain V. Two Illustrative 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 ([1, L])) (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: Localization if Eigenvector Correlators: ( ) ( ) E sup (g(m)) jk g 1 = E ϕ l (j) ϕ l (k) Ce µ j k uniformly in L (ϕ l eigenvectors of M). 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: l
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 (e.g. 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 w.r.t. log. negativity (Vidal/Werner 2002). Note that 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, special case) 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, Friesdorf et al 2015) = Area Law (Brandao/Horodecky 2013/2015) For disordered XY chains the above direct results are stronger. Most likely no satisfying converses, I think. But 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 (in clever ways). 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). (Reduces dimension from 2 l to l.) 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 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! Not even 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 Note: { 2ϕX if {j, j + 1} X (surface of X ) (I σ Z j σ Z j+1 )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 sep. 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 =. Con: Eigenspaces are highly degenerate. There are eigenstates with long range correlations and high entanglement. A truly localized regime should be rather stable under (small but extensive) perturbations, which is arguably not the case here (as we will see in the example below). Corresponding 1-particle question: Is the identity operator I ϕ = ϕ Anderson localized in l 2 (Z)? Seems rather silly...
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. In infinite volume spectrum becomes dense pure point. 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 σj+1) Z 1 4 (σx j σj+1 X + σj Y σj+1) Y = 1 4 (I σ j σ j+1 ) (I 1 )(I σz j σj+1) Z Assume: (i) > 1 Ising phase (model frustration free) (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., x are ordered labelings of the down spin configurations X = {x 1,..., x N } Z, ϕ X = δ x.
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 of 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 disordered 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 Green s 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, uniformly in N. This bound is deterministic. In particular, it holds for ω = 0.
42 Entanglement bounds: (from now on H := H L XXZ (ω)) 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 H N of H to E, E(ρ ψe ) C 1 log min{ Λ l, N} 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)! C 1 is quite explicit. 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 (dep. on δ). 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) 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) 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. Droplet localization (??) is a form of many-body eigencorrelator localization in the droplet spectrum. In particular: E sup N j g(h)n k 1 E N j ψ E N k ψ E g χ I1,δ E σ(h) I 1,δ Note here that for a simple eigenvalue: N j ψ E ψ E N k 1 = N i ψ E N j ψ E
46 In particular, this is applicable to the (energy restricted) dynamics of H via g(h) = χ I1,δ (H)e ith. 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).
47 Consequences of Droplet Localization: We keep all assumptions on H = HXXZ L (ω) from above, including the (, λ)-regime from Theorem 6. Consequences of droplet localization are energy-restricted versions of several of our favorite MBL manifestations. More precisely: I := I 1,δ for any fixed δ > 0 (which constants below will depend on). P I = χ I (H) = spectral projection for H onto I
48 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 (as routinely used for one-particle models): e ith P I ψ, ψ H
49 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 ). Recall I = I 1,δ. X I = χ I (H) X χ I (H). S X and S Y are supports of X and Y (assumed to be intervals).
50 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 = S X + [ l, 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 equivalent!
51 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.
52 V. Two illustrative (sketches of) proofs The Combes-Thomas bound Exponential clustering
53 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 ).
54 Theorem 10: (Elgart/Klein/St. 2017a, 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.
55 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!
56 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)
57 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 P I (I ψ E ψ E )P I N k ψ E N j ψ E N k ψ E Thus Case 1 follows from Theorem 6.
58 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) Z, and P I preserve the particle number This settles five more cases: 0 = R X +,,Y +, (ψ) = R X,+,Y,+(ψ) = R X +,,Y, (ψ) = R X,,Y,+(ψ) = R X,,Y +, (ψ)
59 For example: ψ, X +, P I Y +, ψ ψ, X +, ψ ψ, Y +, ψ }{{}}{{}}{{} = X,+ ψ,p I 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.
60 The remaining Case (i): R X,+,Y +, (ψ E ) = E E = E ψ E, X,+ P I (I P E )Y +, ψ E ψ E, N j X,+ P I (I P E )Y +, N k ψ E E N j ψ E N k ψ E The claim follows from Theorem 6. QED
61 VI. Epilogue: Where to go from here? (Gentle Ben s!) Goals worthwhile trying: 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/ ). Will have to think much more in terms of scattering theory (of k quasi-particles)!
62 Also worthwhile: Find some other concrete models where MBL regimes can be identified! (Before attempting results for classes of quantum spin systems.) 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 (In Ising phase)? Unhappy Schrödinger! Models with spin > 1/2? (Higher spin XXZ, AKLT) Models and more models: beyond nearest neighbors, beyond quantum spin systems (harmonic oscillators, Abdul-Rahman 2017),...
63 A reasonable attempt at a first result for a class of spin systems: Show that large disorder leads to a fully MBL regime for a general class of spin chains with short range interaction (i.e., complete the program of Imbrie 2016, who uses an unproven assumption on level spacing). Even here it may initially help to focus on a concrete example such as the XXX Heisenberg chain. (Imbrie: i J iσ Z i σ Z i+1 + i h iσ Z i + i γ iσ X i )
64 Homework (unreasonable goals): 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. Will require much more groundwork for the non-random case. Electron gases? (Basko/Aleiner/Altshuler 2006)
65 Cut the wires!
66 References: Abdul-Rahman/Nachtergaele/Sims/Stolz 2016, arxiv: Abdul-Rahman/Nachtergaele/Sims/Stolz 2017, arxiv: Abdul-Rahman 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:
67 Friesdorf et al, arxiv: 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/ Vidal/Werner 2002, arxiv:quant-ph/
Disordered Quantum Spin Chains or Scratching at the Surface of Many Body Localization
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