Absence of Normal Heat Conduction in Strongly Disordered Interacting Quantum Chains

Wojciech De Roeck; Lydia Giacomin; Francois Huveneers; Oskar Prosniak

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Absence of Normal Heat Conduction in Strongly Disordered Interacting Quantum Chains

Wojciech De Roeck, Lydia Giacomin, Francois Huveneers, Oskar Prosniak

arXiv:2408.04338v2Aug 8, 2024math-phcond-mat.dis-nnmath.MP

TL;DR

The paper addresses whether conductivity robustly vanishes in strongly disordered, interacting quantum chains rather than only in fine-tuned localized models. It proves vanishing conductivity using time-averaged heat current and establishes that heat flow through length L vanishes faster than 1/L, with large probability.

Problem
The paper asks whether vanishing conductivity is a robust phenomenon in strongly disordered interacting chains, since fine-tuned free-spin and free-fermion examples are insufficient.
Source: §1.1 · p. 2
Method
The paper studies the time-averaged heat current in a non-equilibrium setup, taking a supremum over admissible finite-dimensional bath data and using locality properties of the interacting Hamiltonian.
Source: §2.2 · p. 4, §2.3 · p. 6
Results
Heat flow through a chain of length L vanishes faster than 1/L with large probability, proving that conductivity vanishes in these strongly disordered interacting systems.
Source: §1 · pp. 1–2
Takeaways & Limitations
Transport is anomalous rather than normally conducting in the strongly disordered interacting quantum chains covered by the result.
Source: §1 · pp. 1–2
The proof faces a technical obstruction because an inductive procedure introduces too many factors of 1/ε, while the needed resolvent bound appears implausible under the relevant maximization.
Source: §10.3 · p. 39

Abstract

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We prove that in strongly disordered, interacting, quantum chains, the conductance of a chain of length $L$ vanishes faster than $1/L$. This means that transport is anomalous in such chains. This phenomenon was first claimed in \cite{Basko2006,gornyi2005interacting} and a pioneering treatment appeared in \cite{Imbrie2016a}

1 Introduction

The paper proves anomalously weak heat transport in strongly disordered interacting one-dimensional quantum chains by establishing localization on rare logarithmic segments and relating this to vanishing conductivity.

1 Introduction

The time-averaged heat flow through a chain of length L vanishes faster than 1/L with large probability, so conductance vanishes and transport is anomalous.Source: §1 · pp. 1–2, §2.3 · p. 7

1.1 Robustness

The result targets robust interacting systems rather than fine-tuned examples, allowing fairly general interaction terms in the Hamiltonian.Source: §1.1 · p. 2

1.2 The Role of Spatial Dimension

The theorem is proved for one-dimensional systems, while extending the technique to higher dimensions is not expected by the authors.Source: §1.2 · p. 2

1.3 Many-Body Localization

Many-body localization is stronger than vanishing conductivity, and the paper uses localization in rare regions without determining whether the full chain is many-body localized.Source: §1.3 · p. 2

1.4 Earlier Mathematical Work

The proof is strongly inspired by a KAM-like method for extensive quantum systems and differs technically from work near spectral edges or in spin glasses.Source: §1.4 · p. 3

2.1 Model

The model is a finite spin chain with i.i.d. uniform random fields, bounded interaction coefficients, and norm-bounded Hermitian local interaction operators, with small coupling strength γ.Source: §2.1 · pp. 3–4

2.2 Result on Many-Body Localization

For γ below a threshold, the Hamiltonian admits a strongly local unitary with probability at least e^-γc1L; this yields quasi-local integrals of motion and exponentially decaying coefficients.Source: §2.2 · pp. 4–5

2.3 Result on Absence of Heat Conduction

The non-equilibrium setup couples the chain to finite-dimensional left and right heat baths, and the maximal long-time averaged current provides an upper bound on stationary current; Theorem 2 shows conduction is anomalous.Source: §2.3 · pp. 6–7

3 Outline of the Paper

The paper proves vanishing conductivity by finding logarithmic-sized, resonance-free segments where the Hamiltonian is many-body localized, then controls the transition to subdiffusive transport. Its proof combines a renormalization scheme, inductive expansion bounds, and probabilistic estimates of resonances.

3 Outline of the Paper

Theorem 2 follows from many-body localization on atypical resonance-free segments whose size is logarithmic relative to the total system size.Theorem 1 gives exponentially small-in-length probability for localization on such segments, making them likely to occur somewhere in the chain.Source: §3 · p. 7

3.1 Renormalization Scheme

The renormalization scheme applies infinitely many unitary basis changes to remove off-diagonal terms while preserving locality.The Hamiltonians evolve by H^(k+1) = e^(A^(k+1)) H^(k) e^(-A^(k+1)), and the limiting unitary is controlled for sufficiently small γ with exponentially small failure probability.Source: §3.1 · p. 7
Energy renormalization makes locality harder because later spin flips affect renormalized energies over increasingly long stretches.Gap diagrams and a second generator representation are introduced to maintain locality control at later scales.Source: §3.1 · p. 8, §3.2 · p. 8

3.2 Formal Set-Up and Inductive Control

The formal expansion uses diagrams, triads, factorial Taylor prefactors, and non-resonance conditions to establish convergence and inductive perturbation bounds.The algebraic setup precedes resonance control, while the 1/n! factor offsets diagram proliferation and NRII supplies a stringent bound needed for induction.Source: §3.2 · pp. 8–9

3.3 Probabilistic Estimates

NRII creates a summation problem because the Taylor prefactor cannot also control a union bound over all resonant triads.Equivalence classes group triads producing the same probability event, reducing the union bound and eliminating the need for an inverse factorial in that count.Source: §3.3 · p. 9
The resonance estimate splits denominators into inductively controlled and probabilistically controlled sets, using Jacobian changes of variables for the latter.A polymer-system partition function and cluster expansion then bound the probability of avoiding resonances at all scales.Source: §3.3 · p. 10

4 Technical Preliminaries

The preliminaries define the parameter hierarchy and operator language used throughout the proof. The parameters set resonance, decay, and scale conditions, while X-monomials and supports provide the working operator representation.

4.1 Parameters and Scales

ε is the resonance threshold, δ controls exponential decay with perturbative order, and β determines the spatial scales.A β close to 1 makes non-inductively estimated matrix elements rare while allowing them to cover large spatial domains.Source: §4.1 · pp. 10–11
The proof chooses β close to 1, then δ and ε small, and finally γ sufficiently small while keeping γ polynomial in ε.These choices make the arguments valid uniformly in the system size L.Source: §4.1 · pp. 10–11
The parameter constraints require δ and ε to be sufficiently large relative to γ and impose additional relations used later in the proof.The constants are independent of L and γ, and before β is fixed they are also independent of β.Source: §4.1 · p. 11

4.2 Operators

An X-monomial is a product of Pauli X operators multiplied by a function of the Pauli Z operators.The computational basis is labeled by configurations σ, and an X-monomial maps each configuration to at most one configuration with a nonzero matrix element.Source: §4.2 · p. 12
Hermiticity of an X-monomial is characterized by a conjugation condition on its coefficient function, and hermitian or skew-hermitian operators decompose into corresponding X-monomials.The Pauli commutation relations determine how coefficient functions move across X operators.Source: §4.2 · p. 12
The support of an operator is the smallest interval on which it acts nontrivially, and operator size is measured with the operator norm.The Hilbert-space norm underlies the operator-norm definition used in the preliminaries.Source: §4.2 · p. 12

5 Diagrams and Triads

The scheme defines diagrams and triads recursively across scales, assigning each objects orders, domains, active spins, and factorials for organizing the expansions. These constructions form rooted trees and preserve the required bounds under scale changes.

5.1 Diagrams

Diagrams carry an order, bare order, domain, active-spin set, diagonal/off-diagonal status, factorial, and crowdedness classification.Source: §5.1 · p. 13

5.2 Triads

Triads combine a central diagram with left and right components, defining their order, domain, active spins, factorial, and scale-dependent admissibility conditions.Source: §5.2 · p. 14

5.3 Diagrams at Scale k “ 0

At scale 0, diagrams are pairs g = (S, I) with S ⊂ I ⊂ ΛL and nonempty I, have factorial 1, and satisfy |g| = |I(g)|.Source: §5.3 · p. 15

5.4 From Diagrams at Scale k “ 0 to Diagrams at Scale k ` 1

Higher-scale diagrams are tuples of a diagram and triads whose orders and bare orders add, domains unite, active spins combine by odd parity, and factorials multiply with an n! factor.Source: §5.4 · pp. 15–16

5.5 Propagation of the Bounds (5.1), (5.2), and (5.3)

The bounds propagate from scale k to k + 1: the order reaches Lk+1, while additive bare order and domain relations preserve the remaining bounds.Source: §5.5 · p. 16

5.6 Tree Structure of Diagrams and Triads

Each diagram or triad is represented by a rooted tree whose vertices encode recursive components, with A-, V-, and gap-diagrams distinguished by their roles.Source: §5.6 · pp. 17–19

6 Hamiltonians

The Hamiltonian is decomposed scale by scale into diagonal energies and diagram-labelled perturbations, while generators remove selected off-diagonal terms through commutator equations. Locality and operator representations are propagated inductively across scales.

6.1 The Hamiltonian Hp0q

At scale 0, the Hamiltonian splits as H(0) = E(0) + V(0), with perturbations represented by scale-0 diagrams and bounded coefficients.Source: §6.1 · pp. 19–20

6.2 The Hamiltonian Hpkq

At every scale, H(k) = E(k) + V(k), with diagonal contributions collected into E(k) and diagram-labelled X-monomials forming V(k).Source: §6.2 · pp. 20–21

6.3 The Generator Apk`1q

The generator A(k+1) is chosen to satisfy [A(k+1), E(k)] = -V(k) for off-diagonal terms below the next scale.Source: §6.3 · p. 21

6.4 The Hamiltonian Hpk`1q

Expanding nested commutators constructs H(k+1), with factorial-dependent coefficients differing according to whether the retained component is below or above the scale threshold.Source: §6.4 · pp. 21–22
Locality is preserved because noncommuting terms must satisfy the adjacency constraint, allowing the diagram representation and support properties to propagate inductively.Source: §6.2 · p. 21, §6.4 · p. 22

7 Generators

The generators first solve the commutator equation using energy denominators, then expand those denominators into triad-indexed terms. This second representation restores locality and makes support and disorder dependence explicit.

7.1 A First Representation of Apk`1q

The first generator representation solves V(k)(g) + [A(k+1)(g), E(k)] = 0 by dividing the perturbation by the diagonal energy difference B_g E(k).Source: §7.1 · p. 23
Because the denominator can extend beyond the diagram domain, the first representation does not by itself preserve locality, motivating its expansion into triads.Source: §7.1 · p. 23, §7.3 · p. 24

7.2 Expanding Denominators

The text defines sets indexed by nonnegative integers u and v that constrain minimum and maximum interval endpoints via left and right offsets, then states an expansion lemma for any R ≥ 1.Source: §7.2 · pp. 23–24

7.3 A Second Representation of Apk`1q

The second representation assigns each triad offset indices r(t) and s(t), converting the generator into a sum of triad contributions with localized denominator factors.Source: §7.3 · pp. 24–25
The triad representation is supported in I(t) and depends only on disorder within I(t), thereby establishing the locality property inductively across scales.Source: §7.3 · p. 25

8 Counting Diagrams

This section establishes exponential-in-order bounds for diagram counts and transfers them to triad counts. An induction over scales closes the diagram-counting estimate using combinatorial bounds on locations, sizes, and partitions.

8 Counting Diagrams

N(x, k, w) ≤ C^w and N_T(x, k, w) ≤ w^8C^w, uniformly in x, k, and w.These are the main bounds of Proposition 1.Source: §8 · p. 26

8.1 From Diagram Counting to Triad Counting

Triad counting follows from diagram counting by decomposing each triad into a central diagram and two gap diagrams.Lemma 2 reduces the proposition to bounding N(x, k, w).Source: §8.1 · p. 26

8.2 Setup of a Running Constant

The induction uses an increasing bounded sequence C_k, whose boundedness follows from exponential growth of the scale lengths and convergence of the resulting infinite product.The base case has C_0 = 2, and the induction preserves the bound across scales.Source: §8.2 · p. 27

8.3 Preliminary Estimates for Diagram Counting

At each scale, the number of added triads is bounded by n ≤ (w − w_0)/L_k, while their locations and sizes are controlled through a bipartition into neighbors and non-neighbors of the base diagram.The distinction is needed because the base diagram lacks the scale-based interval bound available for triads.Source: §8.3 · p. 27

8.4 Induction Step for Diagram Counting

The induction step combines factorial, Stirling, location, and partition estimates to recover N(x, k+1, w) ≤ C^w.The summations over the number and bipartition of triads are bounded before the recursive estimate is applied.Source: §8.4 · pp. 28–29

9 Inductive Bounds

This section establishes inductive bounds for smooth tilded operators and their disorder derivatives under suitable non-resonance conditions. It develops scale-by-scale constructions and estimates, including bounds for crowded g_i and the prefactor needed in the argument.

9.1 Main Results

Proposition 2 constructs smooth tilded operators that coincide with the original operators on specified non-resonance events and satisfy bounds on the full sample space.The construction preserves active spins, support, and disorder dependence, while Corollary 1 extends the estimates to the associated observables.Source: §9.1 · pp. 30–31

9.2 Construction of the Tilded Operators

The tilded operators are defined inductively by smoothly extrapolating triad operators with cutoff functions for the non-resonance conditions, then recursively assembling them into diagram operators.The construction uses separate smooth indicators for the first and second non-resonance conditions and guarantees agreement with the original operators on the required events.Source: §9.2 · pp. 32–33

9.3 Bounds on the Tilded Operators

The operator bounds propagate across scales by combining the inductive estimates, smoothed-indicator properties, nested-commutator expansions, and the assumed parameter relations.The base scale is deterministic, triad bounds imply the next operator bounds, and the recursive diagram construction closes the induction.Source: §9.3 · pp. 33–34

9.4 Bounds on the Derivatives of Tilded Operators

The disorder-derivative estimates are proved by induction using Leibniz expansion, bounds on the smoothed indicators, and separate treatment of crowded and non-crowded diagrams.The argument controls six derivative contributions and uses scale-dependent size bounds to complete the estimate.Source: §9.4 · pp. 34–35
The subsection bounds the relevant derivative expressions for g_i and crowded g_i under the stated parameter conditions.The estimates require b ≥ 9 in the crowded case.Source: §9.4 · p. 36
The required prefactor is shown to be upper-bounded for any n ≥ 1, provided ε is sufficiently small.This is the concluding condition stated for the prefactor estimate.Source: §9.4 · p. 36

10 Probabilistic Estimates

This section constructs equivalence classes so resonance events depend on shared structural data, then develops probabilistic bounds that avoid an insufficient inductive estimate. The strategy partitions denominators and uses a carefully chosen subtree to expose enough independence while controlling inverse-ε factors.

10.1 An Equivalence Relation

The equivalence relation separates relevant from irrelevant colleagues using crowding, full overlap, and adjacency conditions, with overlap sets recording their shared locations.This classification determines which structural components must be retained in the class description.Source: §10.1 · p. 37
Equivalence classes preserve scale, interval, active-set, colleague, and recursive triad structure, making the associated non-resonance events class-invariant.For equivalent diagrams and triads, the events NR(g) and NR(t) depend only on their equivalence classes.Source: §10.1 · p. 37

10.2 Probability of Resonances for Equivalence Classes

The main probabilistic estimate bounds resonance probabilities uniformly over equivalence classes of diagrams and triads.The equivalence relation is designed so non-resonance events depend only on the class, enabling class-level estimates.Source: §10.1 · p. 37, §10.2 · p. 38

10.3 Outline of proof of (10.3)

A direct inductive estimate fails because the natural bound on the denominator norm grows as 2ε^-|g|, introducing too many factors of 1/ε.The desired small-resonance probability would require an unrealistically strong uniform denominator bound.Source: §10.3 · p. 39
The alternative strategy partitions denominators into probabilistic and inductive sets, choosing a subtree whose non-leaf factors admit independent integration while fully overlapping leaves contribute only a controlled inverse-ε power.The probabilistic set is arranged so non-hierarchically related diagrams yield independent variables after a change of variables.Source: §10.3 · pp. 39–41

11 Determining Integration Variables

This section constructs and assigns integration variables for probabilistic denominator bounds in non-crowded diagrams, using structural ordering and extremal active-spin properties. It organizes the relevant diagram families by class, left/right location, and parity to complete the proof of Proposition 4.

11.1 Probabilistic or Inductive Bounds for Denominators

Sind(g) contains fully overlapping, adjacent-to-fully-overlapping, crowded, and recursively inherited descendants, while Spro(g) is its complement within S(g).A-diagrams descending from crowded A-diagrams therefore belong to Sind(g).Source: §11.1 · pp. 41–42

11.2 Main Result

Proposition 4 partitions Spro(g) into six sets whose elements can be ordered with the structure needed for probabilistic estimates.The ordering property is stated to provide the independence required for the probabilistic bounds.Source: §11.2 · p. 42

11.3 Diagrams that Are Not Fully-Overlapping

For non-fully-overlapping, non-hierarchical diagrams, left-right order is equivalently determined by the ordering of interval and associated-triad endpoints.Among three non-gap diagrams with no hierarchical relations, pairwise weak adjacency forces at least one to be fully overlapping; related corollaries give further sufficient conditions.Source: §11.3 · p. 43

11.4 Decomposing Spro(g)

Spro(g) splits into S1(g), diagrams sharing an active spin with a colleague, and S2(g), all remaining diagrams.The sharing colleague cannot be fully overlapping, because that would place the diagram in Sind(g).Source: §11.4 · p. 44

11.5 Integration Variables for Diagrams in S1(g)

For S1(g), left diagrams receive nondecreasing shared active spins and right diagrams nonincreasing shared active spins, with each selected spin outside all preceding diagram intervals.Consequently, x1 < ... < xp and y1 > ... > yq.Source: §11.5 · p. 44

11.6 Integration Variables for Diagrams in S2(g)

For S2(g), diagrams are divided by adjacency direction, the right adjacent diagram is unique for a left-class member, and descendants of gap-diagrams belong to S1(g).The interval between an S2(g) diagram and its right neighbor consists of active spins for their common ancestor.Source: §11.6 · p. 45
Extending the left relation to descendants makes it a strict total order on Vnfo, allowing S2,l and S2,r to be ordered oppositely.The order is defined using either interval endpoint comparisons or the descendant relation.Source: §11.6 · pp. 45–46

11.6 Integration Variables for Diagrams in S2pgq.

For all 3 ≤ i ≤ p, an active spin x_i for h_i exceeds every point in I(h_1) ∪ ⋯ ∪ I(h_{i−2}); similarly, for all 3 ≤ i ≤ q, an active spin y_i for f_i lies below them.Source: §11.6 · p. 46
The families S2,l and S2,r are partitioned into even- and odd-indexed diagrams according to the established ordering.These partitions are denoted S2,l,e, S2,l,o, S2,r,e, and S2,r,o.Source: §11.6 · p. 47
The proof assembles S_prop = S1,l ∪ S1,r ∪ S2,l,o ∪ S2,l,e ∪ S2,r,o ∪ S2,r,e, with Proposition 5 covering S1,l/r and Proposition 6 covering S2,l/r,e/o.Together, these propositions fulfill the clauses of Proposition 4.Source: §11.6 · p. 47

12 Contribution of Fully Overlapping Descendants

This section controls contributions from fully overlapping descendants by pruning the descendant tree, isolating relevant diagram and triad subsets, and proving a bound for non-crowded diagrams.

12 Contribution of Fully Overlapping Descendants

The analysis targets descendants that are fully overlapping or adjacent to fully overlapping descendants because their denominators require an inductive estimate and they need not be crowded or scale zero.The tree vertices represent descendant diagrams or triads of the fixed diagram g.Source: §12 · p. 47

12.1 Pruning the Tree Υ

The tree Υ is pruned into ΥP, ΥP˚, and ΥP˚˚, with leaves respectively restricted by Sind(g), fully overlapping V-diagrams, and replacement of triad leaves by their children.ΥP˚˚ consists entirely of diagrams, while the other prunings retain triads under their stated rules.Source: §12.1 · pp. 47–48

12.2 Subsets Related to Fully Overlapping Diagrams

The subsets Pfo and P˚fo identify fully overlapping triads or diagrams and elements adjacent to fully overlapping diagrams, separating the cases needed for the contribution bound.The distinction between these sets motivates introducing the intermediate pruning ΥP˚.Source: §12.2 · pp. 48, 50

12.3 Bounding the Contribution from Pfo

For any non-crowded diagram g, the total contribution from the relevant pruned descendants is bounded by 156(1 − β)|g|.This proposition is obtained by combining bounds for fully overlapping diagrams and triads adjacent to them.Source: §12.3 · pp. 51–52

13 Expansion of Vertices into Their Children

This section expands diagram and triad operators into products over their children while controlling the number of resulting multiconfigurations through factorial and equivalence-class bounds.

13 Expansion of Vertices into Their Children

The expansion bounds an operator matrix element by a sum of products of matrix elements associated with its children, providing inputs for the later unpacking procedure.The number of terms is controlled by Proposition 8 and generalized to triads in Section 13.3.Source: §13 · p. 53

13.1 Expanding a Diagram into Its Children

Proposition 8 supplies a multiconfiguration set for diagram expansions and includes a 1/n! prefactor in the bound on the matrix element.The factorial controls the proliferation of terms generated by expanding a diagram into its children.Source: §13.1 · p. 53

13.2 Proof of Proposition 8

For diagrams in one equivalence class, the construction uses at most 2^|O(g)| distinct restricted-configuration values, yielding |W(g, σ)| ≤ 2^(n+|O(g)|).The set W is constant on equivalence classes.Source: §13.2 · pp. 54–55

13.3 Expanding a Triad into Its Children

Triad expansions enumerate multiconfigurations over the triad’s children, with four, two, or one configurations when the triad has three, two, or one children.The components are labeled by the children rather than by numerical positions.Source: §13.3 · p. 55

14 Bounding Matrix Elements

Section 14 constructs multiconfigurations on pruned descendant trees and proves bounds on matrix elements, their inductive factors, and class-dependent variability.

14.1 Sets of Multiconfigurations

C(g,σ) is constructed inductively from leaves to roots of the relevant tree, combining child multiconfigurations at diagram and triad vertices.Leaves start from σ, while triad steps combine the configurations associated with their children through K(t,σ).Source: §14.1 · p. 56

14 Bounding Matrix Elements

Proposition 9 bounds matrix elements using descendants in S_prop(g), leaf contributions collected in B(g), and multiconfigurations C(g,σ).The bound applies to g in G(k), z in Υ^P(g), and σ in Σ_0 on N_R(g).Source: §14 · p. 55, §14.2 · p. 57

14.2 Bound on the Matrix Elements

The proof of Proposition 9 proceeds by induction on scale, first for leaves, then triads, and finally diagrams decomposed into children handled by Proposition 8.The diagram step separates inductive and probabilistic children before recovering the claimed bound.Source: §14.2 · pp. 57–58

14.3 Bound on B(g)

Non-crowded triads occupy at most half the diagram order, yielding the stated bound on B(g) through the decomposition of diagram and bare orders.The key estimate is Σ_{z∈P_nc}|z| ≤ |g|/2, together with ε ≤ 1.Source: §14.3 · p. 58

14.4 Cardinality of C(g,σ) and Class-Dependence

The cardinality of C(g,σ) is controlled by a bounded increasing sequence in the scale, while Y(g,σ) takes at most C^|g| values across an equivalence class.The construction also introduces a class function Z(g,σ), from which the possible values of Y are obtained by selecting triad factors.Source: §14.4 · pp. 59–60

15 Conclusion of the Proof of Proposition 3

Section 15 completes the proof of Proposition 3 by combining matrix-element bounds with probabilistic estimates based on counting values and controlling a Jacobian determinant.

Conclusion of the Proof of Proposition 3

The relevant probability estimate reduces to at most C^|g| terms because Proposition 4 bounds |S_pro(g)| by 6|g| and the associated configurations have controlled value counts.For fixed σ, Y(g,σ) takes at most C^|g| values, while varying σ contributes at most 2^|I(t)| possibilities.Source: §15 · pp. 61–62
The integral estimate changes variables to denominator differences u_i = Δ_{h_i}(θ_x1,...,θ_xm), then lower-bounds the Jacobian determinant using diagonal dominance.The Jacobian entries have bounded off-diagonal terms and diagonal entries at least 2 − Cδ.Source: §15 · pp. 63–64
The determinant estimate gives |det J| bounded below by 3^m, and the resulting bound closes for α ≤ 1/19 when ε is sufficiently small.This bound is inserted into the preceding estimates to establish Lemma 19.Source: §15 · p. 64
Proposition 3 follows by combining Lemmas 18 and 19, whose estimates control the two components of the proof uniformly in the configurations and indices.Lemma 18 applies for 0 < α ≤ 1, while Lemma 19 applies uniformly for 0 < α ≤ 1/19.Source: §15 · pp. 61, 63, 64

16 Counting Equivalence Classes of Diagrams

Section 16 counts diagram and triad equivalence classes, showing that class counts grow exponentially in order with only polynomial corrections in the triad case.

16 Counting Equivalence Classes of Diagrams

Proposition 10 establishes N(x,k,w) ≤ C^w and N_T(x,k,w) ≤ w^8 C^w for fixed starting point, scale, and order.The proof uses induction on scale and bounded increasing constants C_k.Source: §16 · p. 64

16.1 From Diagrams to Triads

Triad-class counts follow from diagram-class counts by summing over the central and two side diagrams and their scale, position, and order parameters.For non-crowded central diagrams, the relevant orders satisfy w_c + w_l + w_r = w, producing the w^8 factor.Source: §16.1 · p. 65
Corollary 4 gives a cruder unrestricted triad count, which is acceptable because it is not used in an inductive argument.The proof sums over possible triad orders using βL^k ≤ |t| ≤ 3L^{k+1}.Source: §16.1 · p. 65

16.2 Preliminary Bounds for Counting Diagram Classes

Lemma 21 shows that, after the first relevant colleague, each relevant triad extends the accumulated support either to the left or to the right.This left-extender/right-extender dichotomy is the structural reason equivalence classes require fewer choices than individual diagrams.Source: §16.2 · p. 66

16.3 Proof of Proposition 10

The proof of Proposition 10 successively sums over colleague indices, active spins, support intervals, triad classes, extender positions, and scale-zero parameters.These constrained choices combine with the induction hypothesis to produce the exponential-in-order class bound.Source: §16.3 · pp. 67–68

17 Probability of Absence of Resonances

The section bounds the probability that all non-resonance conditions hold by organizing resonances into scale-dependent dressed events and controlling their interactions with a cluster expansion. The proof combines local resonance estimates, polymer counting, and weight bounds to establish the required global estimate.

17 Probability of Absence of Resonances

The probability that all non-resonance conditions in the chain are satisfied is bounded below by Proposition 11.The proof first truncates the scale range uniformly in kmax and then removes the truncation by monotone convergence.Source: §17 · p. 69

17.1 Dressed and Aggregated Resonance Events

Dressed aggregated resonance events are introduced for intervals containing triad supports, and each such event is shown to be locally unlikely.Intervals are assigned scales according to their cardinalities, with boundary-crossing intervals included to avoid separate boundary cases.Source: §17.1 · p. 70
Absence of all resonances coincides with absence of all dressed resonances, allowing the global event to be controlled through aggregated events across scales.The identification is applied successively from the largest scale down to scale 0.Source: §17.1 · pp. 70–71

17.2 Cluster Expansion

A cluster expansion exploits independence between sufficiently separated aggregated resonance events by representing connected collections as polymers.The adjacency relation is chosen so that nonadjacent events are independent, and polymer weights enter the expansion controlling the partition-function-like quantity Z.Source: §17.2 · p. 72

17.3 Proof of Proposition 12

The cluster-expansion condition is verified for small ε using bounds on polymer supports, decorated partitions, and nonzero polymer weights.The proof bounds the number of relevant polymers by C^|I| and separately controls their weights before completing Proposition 12 and Proposition 11.Source: §17.2 · p. 73, §17.3 · pp. 73–74

18 Proof of Main Theorems

This section proves the main theorems by constructing a localizing diagonalization on the full non-resonance event and applying it to establish the absence of heat conduction. It also derives the relevant current bound and expectation-value claim using norm control, an explicit time-integral treatment, and dominated convergence.

18.1 Constructing the Diagonalizing Unitary

On the full non-resonance event, the infinite-product unitary U is well-defined and diagonalizes the Hamiltonian.The limit exists because the product is represented by a time-dependent evolution with bounded generator; UHU† converges to a diagonal operator.Source: §18.1 · p. 77

18.2 Locality of the Diagonalizing Unitary

Lieb–Robinson bounds show that the diagonalizing unitary and its inverse preserve locality through the stated spatially decaying decompositions.The locality bounds extend from continuous generators to L1 generators and apply to both U and U†.Source: §18.2 · p. 79

18.3 Proof of Absence of Heat Conduction

Theorem 2 is reduced to a high-probability interval event where independent non-resonance conditions hold and Theorem 1 applies locally.For separated intervals, the corresponding non-resonance events are independent; with probability at least 1 − e^−L^(1−γc1), suitable sites ℓ1, ℓ2 exist.Source: §18.3 · p. 80
The proof decomposes the diagonal Hamiltonian and total Hamiltonian into left and right parts, then bounds their commutator contributions to control the current.The construction uses local expansions of D, its left/right components, and the corresponding Hamiltonian decomposition before estimating the terms entering the current bound.Source: §18.3 · pp. 80–81
The first right-hand-side term is bounded using (18.15), while the second is a time integral of a bath-independent total derivative.The observable in the total derivative does not involve the baths.Source: §18.3 · p. 81
The resulting estimate contains the terms 5 log(1/γ) and C_L T.Source: §18.3 · p. 81
The bound on |xJ_y^ness| follows by taking T → ∞, and the expectation-value claim follows from ||J|| ≤ C and dominated convergence.Source: §18.3 · p. 81

A Smooth Approximation of Some Indicator Functions

The appendix constructs smooth approximations to indicator sets and develops spectral bounds for perturbed triangular matrices. These tools provide controlled smoothness, resolvent convergence, and eigenvalue localization under the stated assumptions.

A Smooth Approximation of Some Indicator Functions

Smooth functions Q and S approximate joint threshold sets defined by |f_k| relative to η on the convex sample space Ω.Q approximates the region where all |f_k| are small, while S approximates the region where they are large.Source: A Smooth Approximation of Some… · p. 82
The smoothing kernel is supported within distance a, normalized, and chosen so boundary effects preserve the required properties of Q.The construction uses a symmetric compactly supported kernel and requires a ≤ η/(2Bn) for the Lipschitz estimates.Source: A Smooth Approximation of Some… · pp. 82–83
The resulting approximation satisfies 0 ≤ Q ≤ 1 and inherits the target-set implications through the derivative bound and convexity of Ω.If Q(θ)>0, nearby points satisfy the relaxed threshold bounds; points inside the η-threshold set yield Q(θ)=1.Source: A Smooth Approximation of Some… · p. 83

B Spectral Bounds for Perturbed Triangular Matrices

The matrix M has the decomposition M = D + N + E, with N upper triangular and off-diagonal, while E is lower triangular and off-diagonal with entries decaying by distance from the diagonal.Source: B Spectral Bounds for Perturbed… · p. 83