Tozer and Barford [Phys. Rev. B 89, 155434 (2014)] used density-matrix renormalization group (DMRG) calculations to map how a single particle in the one-dimensional quantum Holstein model with Gaussian off-diagonal (transfer-integral) disorder is localized, but only within the adiabatic regime (phonon-to-electronic energy ratio ω' = ℏω/4t₀ < 1) and the large-polaron regime (dimensionless coupling λ = g²ℏω/2t₀ ≲ 1). There they found a three-regime structure: Anderson-type localization by disorder (enhanced by polaron mass renormalization), self-localization by self-trapping (driven by divergence of the effective mass as ω' → 0), and a strong-disorder regime. The unresolved question is what happens outside this window: in the non-adiabatic regime ω' ≳ 1 and in the strong-coupling (small-polaron) regime λ ≳ 1, where different mass renormalization, band narrowing, and localization physics is expected, and where their own results and the Born-Oppenheimer picture no longer apply. Later work has covered adjacent territory — on-site disorder (both physics and rigorous mathematics), finite-density phases, and classical-phonon (adiabatic) dynamics — but no study has extended the localization-mechanism phase diagram of the off-diagonally disordered single-particle Holstein model to the complementary parameter regions. The problem remains open.

Background

The Holstein model is the minimal lattice model of electron-phonon coupling: a particle (an electron, hole, or Frenkel exciton) hops between sites while linearly coupled to a harmonic ("Einstein") oscillator of frequency ω on each site. Its two key dimensionless parameters are the adiabaticity ratio ω' = ℏω/4t₀, comparing the phonon quantum to the electronic bandwidth (t₀ is the bare transfer integral), and the coupling λ = g²ℏω/2t₀, where g measures the particle-phonon interaction. For λ ≪ 1 the polaron — the particle dressed by its phonon cloud — extends over many sites (a "large" polaron of classical radius ≈ 2/λ), while for λ ≳ 1 it shrinks toward a "small" polaron on one or two sites with an exponentially narrowed band. In the adiabatic limit ω' → 0 the phonons can be treated classically (the Born-Oppenheimer approximation, in which nuclear displacements are static classical variables), and the Holstein model famously admits self-localized, symmetry-breaking solutions at any λ > 0: the particle digs its own potential well and traps itself (Landau's self-trapping). In the opposite, non-adiabatic regime ω' ≳ 1, the phonons are intrinsically quantum and this classical self-localization argument fails. Disorder adds a second localization mechanism: Anderson localization, the exponential localization of all single-particle eigenstates by any amount of random potential in one dimension. Tozer and Barford (2014) studied the combined problem — the fully quantized Holstein model on a one-dimensional open chain with Gaussian random transfer integrals t_n (mean t₀, standard deviation σ), i.e., off-diagonal disorder acting on the hopping rather than on site energies — for a single particle at zero temperature, using the DMRG method (a controlled numerical truncation of the entangled particle-phonon Hilbert space). Within the adiabatic, large-polaron window (ω' < 1, λ ≲ 1, relevant to conjugated polymers) they established a qualitative three-regime phase diagram: regime I, where the polaron is localized by disorder like an Anderson insulator, only more so because the polaron effective mass m* > m₀ shrinks the localization length; regime II, where self-trapping self-localization dominates because m* → ∞ as ω' → 0 at fixed λ, so infinitesimal disorder suffices to pin the self-localized polaron; and regime III, strong disorder, where the disorder-induced localization length falls below the classical polaron size. Earlier and parallel work on disordered Holstein-type models treated diagonal (on-site) disorder or impurities: Bronold, Saxena, and Bishop (2001) studied the dynamics of a single electron with binary-alloy on-site disorder; Bronold and Fehske (2002) showed that extremely weak disorder Anderson-localizes small-polaron states; Hague, Kornilovitch, and Alexandrov (2008) and Ebrahimnejad and Berciu (2012) studied polaron trapping by impurities; and Berciu, Mishchenko, and Nagaosa (2010) developed an essentially exact method for a single polaron with on-site disorder and inhomogeneous coupling. None of these treated off-diagonal (transfer-integral) disorder, and none mapped the competition between disorder-induced localization and self-trapping across the (ω', λ) plane. The natural open question left by the Tozer-Barford study is therefore the completion of that map: what replaces their three-regime structure when the phonons are non-adiabatic (ω' ≳ 1) or the polaron is small (λ ≳ 1), where the effective-mass divergence that drives regime II may be cut off and the perturbative mass-renormalization formulas used in their analysis of regime I break down.

Problem Statement

For the one-dimensional, single-particle, zero-temperature quantum Holstein model with Gaussian off-diagonal disorder — the Hamiltonian of Tozer and Barford, in which a particle hops with random transfer integrals t_n (independent Gaussian variables of mean t₀ and standard deviation σ) and couples linearly with strength g to an Einstein oscillator of frequency ω on each site — determine the localization behavior of the polaron outside the previously computed adiabatic large-polaron window. Specifically: (i) in the non-adiabatic regime ω' = ℏω/4t₀ ≳ 1, and (ii) in the strong-coupling small-polaron regime λ = g²ℏω/2t₀ ≳ 1, compute (by DMRG or any other controlled method) the disorder-averaged localization length of the polaron — Tozer and Barford's ℓ = 2Δn_rms = 2√(⟨n²⟩ − ⟨n⟩²), twice the root-mean-square spread of the ground-state particle position, averaged over an ensemble of disorder realizations — and identify the dominant localization mechanism: Anderson localization by disorder, possibly enhanced by polaron mass renormalization and band narrowing, versus self-localization by self-trapping; and (iii) locate, as a function of (ω', λ, σ), the crossover boundaries between these mechanisms, thereby establishing whether the three-regime structure of Tozer and Barford's qualitative phase diagram (disorder-dominated, self-localization, strong disorder) extends to the complementary parameter regions, breaks down, or is qualitatively modified — in particular, whether the effective-mass divergence that produces self-localization for infinitesimal disorder in the adiabatic limit is cut off at large ω', and how the exponentially narrowed small-polaron band at λ ≳ 1 changes the disorder-induced localization length. All requested results jointly support one objective: a completed localization-mechanism phase diagram of this model over the (ω', λ) plane, consistent with the known limits (Born-Oppenheimer self-localization as ω' → 0, free-particle Anderson localization as λ → 0, and small-polaron band narrowing at large λ). All three parts (i)–(iii) are required, and together they constitute the single deliverable (the completed phase diagram), with (i) and (ii) supplying the localization lengths and mechanism identifications that (iii) assembles; a mechanism is counted as dominant where the computed disorder-averaged ℓ follows its signature — disorder-induced (Anderson-type) localization where ℓ decreases with the disorder strength σ, self-trapping where ℓ is (nearly) independent of σ and approaches the classical polaron size ≈ 2/λ — and the crossover boundaries are the loci in (ω', λ, σ) where this dominant-mechanism assignment changes.

References

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