Absence of thermalization after a local quench and strong violation of the eigenstate thermalization hypothesis

1. [1]

   The straight black lines in Fig

   obviously is (again as analytically predicted) not re- stricted to large system sizes L. The straight black lines in Fig. 3 depict the extrapo- lation of the numerically exact results (red crosses) for L ≤ 24 into the regime L >24. The main point is that these extrapolations are still quite close to the “true” numerical values at L = 10 4 (leftmost red cr...

2. [2]

   localization

   02 L − 10 L − 5 L 10− 4 10− 2 L − 10 L − 5 L ⟨sz α ⟩t − ⟨ sz α ⟩th α FIG. 4. Time-averaged expectation values ⟨sz α ⟩t − ⟨ sz α ⟩th vs α ∈ { L − 10, ..., L } for the same XX model as in Fig. 3(b). Red crosses: Extrapolations of the numerically exact resul ts for L = 8, 10, ..., 24 to L → ∞ by means of the same procedure as indicated by the black lines in ...

3. [3]

   and discussed in more detail in Sec. IV G. The inset as well as the green curve in the main plot corroborate an exponantial de- cay of the numerical results with L − α , as analytically predicted by Eq. ( 100). F. Temporal relaxation behavior While the focus in the previous sections was on time- averaged expectation values, the present section provides so...

4. [4]

   IV D [see also discussion below Eq

   Analytical approximations Similarly as in Sec. IV D [see also discussion below Eq. ( 16)], we take the approximation ( 15) for granted and focus on the behavior of the quantity ˜CVA(t) from Eq. (16). Moreover, the perturbations V and observables A are again assumed to be of the specific form ( 8), ( 11), and ( 42), hence we can employ the exact result ( 35...

5. [5]

   and (104)], we can exploit Eq. ( 79) to deduce the approxima- tion ˜CVA(t) ≃ b(t) +p2c(t) 4 for |p| ≪ 1, (105) b(t) := [ J0(t) +J2(t)]2 , (106) c(t) := [ J1(t) +J3(t)]2 − 2 [J0(t) +J2(t)] [J2(t) +J4(t)] , (107) where Jn(t) are Bessel functions of the first kind. Simi- larly, by expoiting Eq. ( 83) one obtains ˜CVA(t) ≃ J 2 0 (t) +J 2 1 (t) 4 for |p|= 1. (1...

6. [6]

   Finally, going beyond the approximations (

   is readily feasible in principle, but the analytical expressions would become even more involved than in the above findings. Finally, going beyond the approximations (

7. [7]

   ( 105), ( 108), and ( 109) be- comes very arduous by analytical means

   and ( 104) and beyond the p-values in Eqs. ( 105), ( 108), and ( 109) be- comes very arduous by analytical means

8. [8]

   fluctuations

   Numerical examples Here we illustrate the detailed temporal relaxation be- havior by employing the same numerical methods as in Sec. IV E. Moreover, we compare the analytical approxi- mations from the previous section with such numerically exact results. The red curves in Fig. 5 exemplify the numerical ex- pectation values ⟨A⟩t − ⟨A⟩th for A = sz L and th...

9. [9]

   already emulate very well the behavior in the thermodynamic limit L → ∞ up to times of about

10. [10]

    Indeed, our numerically exact results for L = 10 4 would be indistinguishable for t < 1

    75L. Indeed, our numerically exact results for L = 10 4 would be indistinguishable for t < 1. 75L from the blue curves in Fig. 5 and are therefore not shown. The intu- itive physical explanation is that the system’s behavior after a local quench at the right chain-end needs some 0 0.1 0.2 0 50 (c) 0 0.1 0.2 (b) 0 0.1 0.2 (a) 0 0.1 50 150 0 0.1 50 150 0 0....

11. [11]

    2 < pc in (a), p = 1 ≃ pc in (b), and p = 2 > pc in (c), where pc = (L + 1)/L = 1

    Note that p = 0. 2 < pc in (a), p = 1 ≃ pc in (b), and p = 2 > pc in (c), where pc = (L + 1)/L = 1. 0001, see also Eqs. ( 17) and ( 63). The black curves in Fig. 6 exemplify the approximations from Eqs. ( 105)-(110). In order to fulfill the requirement (103) in those approximations, we have chosen in Fig. 6 a smaller value of β than in Fig. 5. While the ag...

12. [12]

    critical point,

    is not yet very well fulfilled for the choice of p = 2 in Fig. 6(c). We numerically found that the agree- ment indeed becomes better upon further increasing p (not shown). On the other hand, for larger p-values, cor- respondingly smaller β -values must be chosen in order to still fulfill Eq. ( 103). Moreover, upon increasing p the overall temporal variation...

13. [13]

    n-spin excitations

    (apart from |0⟩) as some n-particle excitation of the general form |k1,...,k n⟩ :=f † k1 · · ·f † kn |0⟩ (112) with k1,...,k n ∈ { 1,...,L } and k1 < k2 < ... < kn. Sim- ilarly as above, one thus can conclude (see Appendix A in Ref. [13]) that |k1,...,k n⟩ = ∑′u(l1,...,l n)σ + l1 · · ·σ + ln |0⟩, (113) where ∑′ indicates a sum over all l1,...,l n ∈ { 1,.....

14. [14]

    in the middle of the spectrum of H

    and the com- plicated form of the coefficients in Eq. ( 114) make a more tangible physical interpretation quite difficult. There- fore, we now turn to the signatures of localization which do not manifest themselves directly in the system’s en- ergy eigenstates, but rather in certain characteristic fea- tures of some suitably chosen expectation values. A first ...

15. [15]

    indirect

    and ( 101). A quanti- tative numerical example is provided in Fig. 4. Note that the first two above signatures are of rather theoretical character in the sense that properties of sin- gle energy eigenstates are quite difficult to observe in an experimental many-body system. The third one is ex- perimentally unproblematic but is of a rather “indirect” charact...

16. [16]

    02 L − 10 L − 5 L 10− 3 10− 2 L − 10 L − 5 L ⟨sz α ⟩t − ⟨ sz α ⟩th α FIG. 8. Same as Fig. 4 but now for the corresponding XXZ model with Jz = 0. 3 in Eq. (115). 0 0.1 0.2 0 50 (c) 0 0.1 0.2 (b) 0 0.1 0.2 (a) 0 0.05 50 150 0 0.05 50 150 0 0.05 50 150 t ⟨sz L⟩t − ⟨ sz L⟩th FIG. 9. Same as Fig. 5 but now for the corresponding XXZ model with Jz = 0. 3 in Eq. ...

17. [17]

    ( 24)] with ν = L + 1 2 , (116) 18 0 0.02 0 0 .1 (b) 0 0.01 (a) ⟨ sz (L+1)/2 ⟩ t − ⟨ sz (L+1)/2 ⟩ th 1/L FIG

    and ( 8) [and thus in Eq. ( 24)] with ν = L + 1 2 , (116) 18 0 0.02 0 0 .1 (b) 0 0.01 (a) ⟨ sz (L+1)/2 ⟩ t − ⟨ sz (L+1)/2 ⟩ th 1/L FIG. 10. Same as in Fig. 3 but now for ν = α = ( L + 1)/ 2 (central impurity), g = 0 . 2 in (a), g = 0 . 4 in (b), and L = 10001, 25, 23, ..., 11, 9. The green bars indicate the analytical approximation for L → ∞ from Eqs. (11...

18. [18]

    and ( 42). General- izations to even L and arbitrary inner impurities are in principle straightforward, but require more lengthy cal- culations and case differentiations, and will therefore only be briefly summarized at the end of this section. For the rest, the detailed calculations are largely analogous to those in the previous Sec. IV, and are therefore ...

19. [19]

    (118) Likewise, a strong violation of the ETH immediately fol- lows by similar arguments as in Sec

    and ( 101) now assumes the form ⟨sz α ⟩t − ⟨sz α ⟩th =gβ p2e−|p||ν − α | 16 cosh 2(β/ 2) . (118) Likewise, a strong violation of the ETH immediately fol- lows by similar arguments as in Sec. IV D, and the con- comitant signatures of localization in the system’s Hamil- tonian (

20. [20]

    are analogous to those in Sec. IV G. The most important difference compared to the end- impurities considered in Sec. IV is that in the present case of a central impurity the critical value in Eq. ( 117) approaches zero for largeL. In particular, a quench in the form of an arbitrarily weak central impurity thus always gives rise to nonthermalization in a s...

21. [21]

    and ( 8) [and thus in Eq. ( 24)] with an arbitrary location ν ∈ { 1,...,L } one finds (after similar but more involved calculations) that the critical impurity strength is given by pc = 1 ν + 1 L + 1 − ν , (119) and that the same conclusions as in the previous para- graph apply whenever |p| exceeds this critical value. In particular, one readily recovers Eqs. (

22. [22]

    bulk” of the spin-chain in the first case and in one of the “end-regions

    and ( 117) in the special cases from Eqs. ( 42) and ( 116), respectively. Moreover,pc in Eq. ( 119) approaches zero whenever the impurity site ν scales with the system size L in such a way that both ν and L − ν diverge for large L. On the other hand, whenever either ν or L − ν remains finite as L → ∞ , the critical value in Eq. ( 119) approaches a non-vani...

23. [23]

    For- mally, this amounts to replacing the upper summation limit L − 1 in Eq

    Periodic boundary conditions A further, quite minor and natural extension is to con- sider periodic instead of open boundary conditions. For- mally, this amounts to replacing the upper summation limit L − 1 in Eq. (

24. [24]

    Obviously, in such a case the actual position ν of the im- purity in Eq

    by L and setting sa L+1 := sa 1. Obviously, in such a case the actual position ν of the im- purity in Eq. (

25. [25]

    quantum chaos

    becomes irrelevant. Moreover, it seems reasonable to expect that the system behaves similarly as an open system with a central impurity. A confirmation of this expectation follows upon comparing the numeri- cally exact findings in Fig. 11 with those in Fig. 10. We remark that also an analytical confirmation is in principle possible by a pertinent generalizat...

26. [26]

    ( 3) and ( 10) for sufficiently large L [see also the discussion above Eq

    and ( 10) In this Appendix we provide a detailed justification of why β can be chosen equal in Eqs. ( 3) and ( 10) for sufficiently large L [see also the discussion above Eq. ( 9)]. We adopt the same definition of the system’s initial stateρ(0) as in Eq. ( 3) with an arbitrary but fixed value of β . However, instead of ρcan in Eq. ( 10), we now consider more g...

27. [27]

    we will derive ( A4) even without this extra assumption. To begin with, let us define a set of auxiliary Hamilto- nians Hr and concomitant canonical ensembles ρr, Hr := H0 +r(H − H0) , (A6) ρr := e− βH r/Z r , (A7) Zr := tr {e− βH r } , (A8) where r ∈ [0, 1] is a parameter. Recalling the basic re- lations − ∂ ∂β lnZr = tr {ρrHr} and H = H0 +gV [see Eq. ( 5...

28. [28]

    partner state

    Evaluation of D and W (β m) Let us first remark that the derivation of ( A21) is still valid for entirely general Hamiltonians H and H0. Furthermore, our final result ( A5) follows immediately from ( A21) under the following three conditions: (i) v in (A15) remains bounded for L → ∞ , (ii) D in ( A16) re- mains bounded for L → ∞ , (iii) |W (βm)|in (A19) gro...

29. [29]

    Similarly as in Eqs

    can be recast by means of a Jordan- Wigner transformation into the form ( 26) and thus ( F1), and likewise for H0. Similarly as in Eqs. ( 27) and ( 28), there furthermore must exist unitary L × L matrices U and ˜U with entries Ukl and ˜Ukl, respectively, so that H = L∑ k=1 Ekf † kfk, (F3) H0 = L∑ k=1 ˜Ek ˜f † k ˜fk, (F4) where fk := L∑ l=1 Uklcl, (F5) ˜fk...

30. [30]

    (F15) To further evaluate the right-hand side of Eq

    that ⟨c† lcl⟩t = L∑ j,k =1 UjlU ∗ klei(Ej − Ek)tTjk, (F13) Tjk := L∑ m,n =1 S∗ jmSknpmn, (F14) pmn := tr {ρ(0) ˜f † m ˜fn} . (F15) To further evaluate the right-hand side of Eq. ( F15), we focus on our specific initial conditions from Eq. ( 3). By exploiting (F4) one thus finds after a straightforward cal- culation (see e.g. Eqs. (B21)-(B29) in Ref. [13] fo...

31. [31]

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    for our stan- dard observables A =sz α [see Eq. ( 11)] and Hamiltonians H [see Eq. ( 24)] can be analytically evaluated without any approximation, as shown in Ref. [13] [see Eq. (B41) therein], yielding ⟨sz α ⟩th = − 1 2 L∑ k=1 |Ukα |2 tanh(βEk/ 2) . (G1) We recall that Eq. ( G1) must be an odd of function p [see below Eq. ( 17)]. Furthermore, Eq. ( G1) i...

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