On Many-Body Localization for Quantum Spin Chains

Abstract

For a one-dimensional spin chain with random local interactions, we prove that many-body localization follows from a physically reasonable assumption that limits the amount of level attraction in the system. The construction uses a sequence of local unitary transformations to diagonalize the Hamiltonian and connect the exact many-body eigenfunctions to the original basis vectors.

Appendices

Appendix 1: Extending Graphs Across Gaps

Here we describe how to organize the graphical expansions of Sect. 4.3 to exhibit decay across gaps between graphs while limiting the duplication of denominators.

Consider the general expression for a term in $J^{(k)}$:

$$\begin{aligned} \left( \text {ad}\,A^{(k)}(g_{j'',n})\right) \cdots \left( \text {ad}\,A^{(k)}(g_{j'',1})\right) J^{(j')}(g_{j',0}). \end{aligned}$$

(6.1)

Divide the $g_{j'',p}$ into non-overlapping groups. The actions of a commutator of a graph in one group with a graph in another group is simple, as the operators in question involve disjoint sets of spin indices. As indicated in (4.48), the energy denominators are computed before and after a spin transition and the difference taken. We have the freedom to use either side of (4.48) when working with these denominator differences. We use the right-hand side of (4.48) and the graphical expansion (4.49) only for a minimal set of differences, sufficient to close the gaps between groups. For the remaining differences, we bound each term on the left-hand side of (4.48) separately: m differences result in $2^m$ terms; a factor of 2 per graph is easily controlled, as discussed in Sect. 4.2.1.

We use the following algorithm to generate differences and select ones that need to be expanded as in (4.49). We proceed through the sequence of graphs $g_{j'',1}, g_{j'',2},\ldots$ in (6.1) and the associated commutators. If a graph $g_{j'',p}$ appears as the first representative of one of the groups, the associated commutator is written as a sum of terms involving the commutators with each of the previous graphs $g_{j'',p-1}, g_{j'',p-2},\ldots , g_{j'',1},g_{j',0}$, by Leibniz’s rule for commutators. As $g_{j'',p}$ is the first in its group, there is no overlap between it and the previous graphs. Then, as was just explained, the commutator can be written as the sum of two differences. For example, one contains the effect of $A^{(k)}(g_{j'',p})$ on $J^{(j')}(g_{j',0})$ and the other contains the effect of $J^{(j')}(g_{j',0})$ on $A^{(k)}(g_{j'',p})$. This may be indicated graphically with a line from one graph to the other and an arrow indicating the direction of the effect. (The arrow can go in either direction, because each graph’s denominators are affected by changes in spin configuration induced by the other graph. However, the direction of the arrow is unimportant in what follows, as either direction will be sufficient to create a graphical connection across the gap between the graphs.) Note that the expanding set of lines ensures that each group is connected to its predecessors as soon as one of its graphs appears. Hence when all commutators have been performed, the graph of difference lines (the difference graph) connects all of the groups. Many of the lines are redundant, however.

We define a minimal subgraph of the difference graph using a lace construction similar to the one introduced in [51]. Starting from the left-most group, we take the first line of the lace to be the one reaching as far as possible to the right. (Note that there can be no more than one line between two groups, by construction, as our algorithm specifies that each line have one endpoint in a new group. Thus there is no ambiguity about which line reaches the farthest to the right.) The next line of the lace is taken as the one reaching the group farthest to the right from any of the groups spanned by the first line. (If two lines reach that group, we take the one originating from the group farthest to the right. This is unique because, as explained above, there cannot be two lines connecting the same two groups.) We continue, always choosing the line reaching the group farthest to the right from the groups spanned by the expanding lace graph, and breaking ties by choosing the line originating from the group farthest to the right. See Fig. 4.

Fig. 4

A lace connecting five groups. Each arch represents a difference operation that provides a graphical connection and exponential decay across gaps between groups

All lines not in the lace are left as is: a sum of two terms, as in the left-hand side of (4.48). They are not needed for generating decay across the gaps between graphs. The lines of the lace graph extend all the way from the left-most group to the right-most one. Therefore, by exhibiting decay between graphs connected by lace lines, we obtain decay across all gaps between groups.

The lace graph has the property that no more than two lines emanate from any group. There can be one line to the left and one to the right, but a second on either side is impossible, by the rules for constructing the lace graph. As a consequence, we see that no more than two differences will be applied to any graph. This is important because of the need to limit the extent of duplication of denominators.

The first difference creates a double denominator as in (4.48). Then (4.49) can be used to exhibit decay across the gap between the two graphs. The second difference creates a triple denominator as follows: When two differences are applied, we have four denominators, which may be written as $d_{11}$, $d_{12}$, $d_{21}$, $d_{22}$, with the first index indicating which spin configuration is involved on the left, and the second index indicating the one on the right. Then we may use Leibniz’s rule to write

$$\begin{aligned}&\left( \frac{1}{d_{22}} - \frac{1}{d_{21}}\right) - \left( \frac{1}{d_{12}} - \frac{1}{d_{11}}\right) = \frac{d_{21} - d_{22}}{d_{22}d_{21}} - \frac{d_{11} - d_{12}}{d_{12}d_{11}} \nonumber \\&\quad =\frac{d_{21} - d_{22}}{d_{22}d_{21}} - \frac{d_{21} - d_{22}}{d_{12}d_{21}} +\frac{(d_{21}-d_{22})-(d_{11}-d_{12})}{d_{12}d_{21}} +\frac{d_{11} - d_{12}}{d_{12}d_{21}} - \frac{d_{11} - d_{12}}{d_{12}d_{11}} \nonumber \\&\quad =\frac{(d_{21} - d_{22})(d_{21}-d_{22})}{d_{22}d_{21}d_{12} } +\frac{(d_{21}-d_{22})-(d_{11}-d_{12})}{d_{12}d_{21}} +\frac{(d_{11} - d_{21})(d_{11}-d_{12})}{d_{12}d_{21}d_{11} }. \end{aligned}$$

(6.2)

For each term in the final expression, we may obtain decay across the gaps to the left and to the right when denominator differences are expanded as in (4.49). (If a graph fails to cross both gaps, the result is zero because it cannot feel both changes in spin configuration.) Note that denominators can be tripled if differenced from both sides, but (as claimed in the main text) no power higher than three occurs.

We need to provide estimates on the number of terms produced from applications of Leibniz’s rule in the above algorithm. Recall that the range of dependence of denominators on the spin configuration is no greater than $\tfrac{15}{14}L_j$—see (4.50). This becomes a bound on the number of groups that can be reached with a difference line originating from a particular $\text {ad}\,A^{(k)}(g_{j'',p})$ operation. When a difference operator hits a group, there is a sum over the graphs in the group; this may be controlled most simply by assigning a combinatoric factor 2 to each graph in the group. The number of lines incident on a group cannot exceed $\tfrac{15}{14}L_j$ (the maximum number of groups in range). Hence each graph receives no more than a combinatoric factor $2^{(15/14)L_j}$. There are also factors of 2 per graph that were mentioned above, coming from the representation of a commutator as a sum of two terms or as a sum of two differences. The product of all these combinatoric factors is bounded by $c^{L_j} \le c^{|g|}$ for each $g \in \{g_{j',0},g_{j'',1}, \ldots , g_{j'',n} \}$, in view of the minimum graph size in this step. Thus the counting factors are in line with ones already considered in Sect. 4.2.1, and are controlled by the exponential bounds on graphs as in (4.2), (4.6). As one often finds in this type of argument, dangerous counting factors from applications of Leibniz’s rule are limited by geometrical considerations—in this case by the range limitation and the cap of one on the number of commutators per group that are expanded with Leibniz’s rule.

Appendix 2: Index of Definitions and Notations

Here we provide a list of important definitions and notations, along with locations where they are introduced in the text.

• $\rho _0$, bound on probability densities: after (1.2).

• $\rho _1$, constant governing probability of an energy difference lying in a small interval: above (3.5), start of Subsection 4.2.3.

• Level spacing assumptions. LLA($\nu , C$): (1.3); A1($c_b$): (4.16); A2($\nu$,$\varepsilon _0$): (5.1).

• $L_k =(15/8)^k$, length scales: start of Section 1.3.

• $\sigma ^{(i)}$, spin configuration flipped at i: (2.1).

• $E(\sigma )$, energy of spin configuration: (2.2).

• “i is resonant”; $\varepsilon = \gamma ^{1/20}$; ${\mathcal {S}}_1$, resonant set; $B^{(1)}$, its components: after (2.3).

• $J^{(0)}$, $J^{(0)\text {per}}$, $J^{(0)\text {res}}$, interaction and its perturbative and resonant parts: (2.4).

• A, A(i), generator of rotation: (2.5),(2.6).

• $\Omega$, the associated rotation; $H_0$, diagonal part of H: above (2.7).

• $J^{(1)}$, new interaction: (2.7); $J^{(1)}_{\sigma \tilde{\sigma }}(g_1)$, the term associated with a particular graph: (2.9).

• $|g_1|$, first step graph: above (2.10).

• $d_m=\exp (L_{m+m_0}^{1/2})$, extended separation distances: start of Section 2.3.

• $m_0$: chosen in Lemmas 4.1 and 4.2.

• $b^{(1)}$, small block; $\bar{b}^{(1)}$, collared version; $\overline{{\mathcal {S}}}_1$, collared version of ${\mathcal {S}}_1$; ${\mathcal {S}}_{1'}$, region of large blocks; $\overline{{\mathcal {S}}}_{1'}$, collared version; $\bar{B}^{(1')}$, $B^{(1')}$, large blocks; above (2.11). (Prime means large blocks only; bar means collar is included.)

• $J^{(1)\text {int}}$, interaction terms internal to blocks; $J^{(1)\text {sint}}$, $J^{(1)\text {lint}}$, internal to small, large blocks, respectively; $J^{(1)\text {ext}}$, non-internal terms: (2.11).

• O, small block rotation; $\bar{\bar{b}}^{(1)}$, second collar block: after (2.12).

• $H^{(1')}$, post-rotation effective Hamiltonian; $H_0^{(1')}$, its diagonal part; $J^{(1')}$, rotated interaction: (2.13)-(2.15).

• $g_{1'}$, graph extended by rotation matrix elements: above (2.16).

• $J^{(1')}(g_{1'})$, the associated interaction term: (2.17).

• $A^{(2)\text {prov}}$, provisional rotation generator; $E_\sigma ^{(1')}$, post-rotation energy: (3.1).

• $I(g_1)$, interval of $g_1$; $|I(g_1)|$, its size: after (3.2).

• $B^{(2)}$, new blocks: above (3.3).

• $b^{(2)}$, new step 2 small blocks; $|b^{(2)}|$, its size; separation conditions; ${\mathcal {S}}_{2'}$, new large block region; $\overline{{\mathcal {S}}}_{2'}$, collared version; $\bar{B}^{(2')}$, $B^{(2')}$, large blocks; ${\mathcal {S}}_{2}$ is ${\mathcal {S}}_{2'}$ plus small blocks $b^{(2)}$; $\overline{{\mathcal {S}}}_2$, $\bar{b}^{(2)}$, collared versions: (3.3) and after.

• $P_{ij}^{(2)}$, connectivity function for $B^{(2)}$ blocks; $|i-j|^{(1)}$, metric with $\bar{\bar{b}}^{(1)}$ blocks contracted to points; $s = \frac{2}{7}$: Proposition 3.1.

• $Q_{ij}^{(2)}$, connectivity function for small blocks; $\bar{b}^{(2)}$: above (3.9).

• $J^{(1')\text {per}}$, $J^{(1')\text {res}}$, perturbative and resonant interactions for step 2: (3.10).

• $A^{(2)}$, $A^{(2)}(g_{1'})$, generators of rotations: (3.11).

• Long graphs, short graphs, jump transitions; $g_{1''}$, graph with jump steps representing sums of long graphs; $|g_{1''}|$, its length: after (3.11).

• $A^{(2)}(g_{1''})$, generator of rotations with long graphs resummed: (3.12).

• $\Omega ^{(2)}$, the associated rotation: after (3.12).

• $H^{(2)}$, new Hamiltonian; $J^{(2)}$, new interaction: (3.13),(3.14).

• $g_2$, step 2 graph; $|g_2|$, its length; $J^{(2)}(g_2)$, the associated interaction term: (3.16) and above. $g_2!$: after (3.17).

• $O^{(2)}$, small block rotation matrix; $H^{(2')}$, new Hamiltonian; $H_0^{(2')}$, diagonal part; $E_\sigma ^{(2')}$, its diagonal entries (energies); $J^{(2')}$, off-diagonal part: (3.19)-(3.21) and after.

• $g_{2'}$, graph extended by rotation matrix elements: after (3.22).

• $J^{(2')}(g_{2'})$, the associated interaction term: (3.22).

• $R^{(1')}$, $R^{(2')}$, cumulative rotations: (3.23).

• Multigraphs; level i subgraphs; $g^{\text {s}}_{j'}$, spatial graph; $g^{\text {d}}_{j'}$, denominator graph: above (4.2).

• $g_{j''}!$, inductive definition of the factorial of $g_{j''}$; $|g_{i''}|$, length of $g_{j''}$; (4.3)-(4.4).

• $|x-y|^{(i)}$, metric with blocks $\bar{\bar{b}}^{(\tilde{i})}$, $\tilde{i}\le i$ contracted to points: after (4.4).

• $\bar{g}_{j'}$, reduced graph for probability sums; $|\bar{g}_{j'}|$, its length; $\bar{g}_{j'}!$, its factorial: above (4.7).

• $A^{(k)\text {prov}}$, provisional rotation generator; $\tilde{J}_{j'}$: (4.7) and after.

• “$g_{j'}$ is resonant”; $I(g_{j'})$, interval of $g_{j'}$; $|I(g_{j'})|$, its length: (4.8) and after.

• $B^{(k)}$, new block in step k; above (4.9).

• $b^{(k)}$, small block in step k; separation conditions; $|b^{(i)}|$, size of block: near (4.9).

• ${\mathcal {S}}_{k'}$, new large block region; $B^{(k')}$, large block; collared versions $\overline{{\mathcal {S}}}_{k'}$, $\bar{B}^{(k')}$, $\bar{b}^{(k)}$: after (4.9).

• Combinatoric factor: start of Sect. 4.2.1.

• $T_\alpha$, looping segment; $|T_\alpha |$, its weighted size: above (4.29).

• Erased subgraphs: after (4.29).

• $\bar{g}^{\text {e}}_{j'}!$, modified factorial eliminating those from erased sections: after (4.30).

• $P_{ij}^{(k)}$, connectivity function for $B^{(k)}$ blocks: Proposition 4.4.

• $Q_{ij}^{(k)}$, connectivity function for small blocks $\bar{b}^{(k)}$; $R_{ij}^{(k)}$, multiscale connectivity function for small blocks $\bar{b}^{(i)}, i \le k$: Proposition 4.5.

• $J^{(j')\text {per}}$, $J^{(j')\text {res}}$, perturbative and resonant interactions for step k: (4.37),(4.38).

• $A^{(k)}(g_{k''})$, generator of rotations: (4.39).

• Long graphs, short graphs, jump transitions; $g_{j''}$, graph with jump steps representing sums of long graphs; $|g_{1''}|$, its length: after (4.39).

• $A^{(k)}(g_{k''})$, generator of rotations with long graphs resummed: (4.40).

• $\Omega ^{(k)}$, the associated rotation: after (4.40).

• $H^{(k)}$, new Hamiltonian; $J^{(k)}$, new interaction: (4.41),(4.42).

• Gap graphs: after (4.50).

• $J^{(k)\text {ext}}$, $J^{(k)\text {sint}}$, $J^{(k)\text {lint}}$, interaction terms external to blocks, internal to small blocks, internal to large blocks: (4.51).

• $O^{(k)}$, small block rotation matrix; $H^{(k')}$, new Hamiltonian; $H_0^{(k')}$, diagonal part; $E^{(k')}_\sigma$, diagonal entry; $J^{(k')}$, rotated interaction: (4.52),(4.53).

• $g_{k'}$, graph extended by rotation matrix elements: after (4.53).

• $R^{(k')}$, cumulative rotation: (4.54).