&Bullet; physics 14, 57

An investigation of long-range interactions in disordered systems provides a surprising result: transport can increase with the disruption.

The physicist Philip Anderson introduced the concept of quantum localization in 1958 to explain why no diffusion is observed in impure semiconductors with spin excitations [1] . The localization is a quantum phase transition, in which coherent backscattering at disruption centers – depending on the situation – causes interference effects in electrons, spins, phonons or other quantum excitations. The resulting suppression of transport is particularly widespread in 1D systems [2] Therefore, researchers working with nanowires are interested in overcoming Anderson localization. A new theoretical study by Nahum Chávez of Meritorious Autonomous University in Puebla (BUAP), Mexico and colleagues examines the addition of long-range “hopping” interactions in a 1D perturbed system [3] . Their calculations of the current through the system show that if the disturbance is weak, Anderson localization occurs roughly as usual, with this disturbance hindering transport. The surprise occurs when the disturbance exceeds a certain value. Then the current shows a remarkable increase with the disturbance and finally reaches a plateau before decreasing again. Future experiments with trapped atoms in optical lattices and Bose-Einstein condensates (BECs) could investigate this predicted hop-induced delocalization.

The far-reaching hopping interaction that Chavez and colleagues are considering can occur in a number of different cases. An example is a chain of molecules in an optical cavity, where an excitation in a molecule can jump to a distant molecule in the chain through cavity coupling. The theorist Ugo Fano pointed out that far-reaching all-to-all interactions are the source of plasmonic vibrations and superconductivity [4] . In the latter case, it is the phonon-mediated all-to-all interactions between Cooper pairs on the Fermi surface that are responsible for the generation of the collective superconducting ground state [5] . In the case considered by Chávez and colleagues, a similar ground state occurs, which, however, is separated from the excited states by a large energy gap. One could assume that the delocalized ground state is “buried” under this gap and cannot weaken the localization of the excited states. In a loose metaphor, however, we can say that the ground state rises like a “phantom” in order to increase the transport probability of the excited states.

To understand this delocalization, we can look at a generic model of stimuli, or “excitons”. Using reasonable approximations, the dynamics of excitons are described by a linear chain of

$\mathrm{N.}$

“Orbitals” with Anderson perturbation. An exciton can occupy one of these orbitals or jump to an orbital of the nearest neighbor at a tunneling rate that is characterized by an energy term

$\Omega$

. Each orbital has a local energy that is drawn from a random distribution of latitude

$\mathrm{W.}$

this defines the disorder and discourages hopping. The excited states have localized wave functions, each of which peaks at a different orbital position and decays exponentially with distance from that peak position. The larger the perturbation, the steeper the exponential tails and the less likely it is that a stimulus can jump to a nearby orbital.

Chávez et al. Lead to this generic model. A superimposed effective all-to-all hopping with characteristic energy

$𝛾∕2$

. My student and I previously imagined this type of all-to-all hopping with a “star system”. [6] where the basic state

${\Psi }_{0}$

with energy

$\left(\left(1– –\mathrm{N.}\right)𝛾∕2$

is the completely symmetrical superposition of all orbitals, while the

$\mathrm{N.}– –1$

excited states form a band with energy

$𝛾∕2$

(Fig. 1). One could naively think that these (localized) excited states would be safely decoupled from the (delocalized) ground state due to the large energy gap

$\Delta =\mathrm{N.}𝛾∕2$

. Indeed, the localization is “almost” as prescribed by Anderson, but hopping over great distances causes the excited states to take on a hybrid character that mixes localized and delocalized states. To understand this hybridization, imagine one

$\mathrm{N.}$

-Star system with collective ground state

${\Psi }_{0}$

and then add an additional orbital with a localized wave function

${\Psi }_{\mathrm{N.}}$

. The “bonding” hybrid approximates the new basic state of the system

${\Psi }_{0}+{\Psi }_{\mathrm{N.}}∕\sqrt{\mathrm{N.}}$

while the “almost” is local excitement

${\Psi }_{\mathrm{N.}}– –{\Psi }_{0}∕\sqrt{\mathrm{N.}}$

. Thus, the strong correlations that form the ground state also place a very weak but inevitable delocalized bottom on each of the excited states. Back to our metaphor: residual ground states of smaller systems appear as a flat, noisy “phantom” background that can overcome the exponential tails at the extremes of the localized wave functions.

Chavez and colleagues investigate this effect by calculating the transport rate or current as the disturbance increases. With little disturbance, the wave functions are dominated by their relatively broad exponential tails, and the current decreases with increasing disturbance (Anderson localization). But when the disturbance reaches a value of

${\mathrm{W.}}_{1}$

The extremes of the exponential tails submerge under the “phantom” background. In this DET (Disorder Enhanced Transport) regime, the wave functions become spatially more extensive (more prone to hopping) with increasing disruption. With a higher disturbance value of

${\mathrm{W.}}_{2}$

If the localization length reaches a grid unit and an interference-independent transport (DIT) takes place exclusively via the “phantom” background. Finally, if there is a fault above

${\mathrm{W.}}_{\text{GAP}}$

the energy gap is closed and the transport decreases again with increasing disruption.

Since the observables that characterize the transport, such as B. currents, decrease as

$1∕{\mathrm{N.}}^{2}$

it might be difficult to get the delocalization phenomenon out of experiments with large ones

$\mathrm{N.}$

Systems. However, there are ways in which the “residual character” of the effect can be observed. On the one hand, experiments attempting to build long-range couplings into synthetic systems, such as trapped atoms in optical lattices and BECs, are relatively small

$\mathrm{N.}$

. On the other hand, if the overall coupling term is reduced to a finite length scale, that scale would break a large system down into a series of small ones.

$\mathrm{N.}$

Parts that behave like the model described. A similar situation can occur with weak interactions with many bodies [7] . Since these interactions could be viewed as a source of decoherent processes, they would impose a finite coherence length, which the system also breaks down into pieces with this length [8] . One might also wonder about the effect of remote coupling in more flexible perturbation models, such as incommensurate potentials, which were implemented experimentally to investigate the interplay between localization and many-body effects [9] . As weak as the “phantoms” that result from the ubiquitous collective basic state may be, they could have other unforeseen effects than those described in this paper.

## References

1. PW Anderson, “Local Moments and Localized States,” Rev. Mod. Phys.50191 (1978).
2. B. Kramer and A. MacKinnon, “Localization: Theory and Experiment”, Rep. Prog. Phys.561469 (1993).
3. NC Chavez et al., “Disturbance-Enhanced and Disturbance-Independent Transport with Long Distance Leap: Application to Molecular Chains in Optical Cavities” Phys. Rev. Lett.126153201 (2021).
4. U. Fano, “A Common Mechanism of Collective Phenomena” Rev. Mod. Phys.64313 (1992).
5. NH Chavez et al., “Real and Imaginary Energy Gaps: A Comparison Between Single Excitation Superradiance and Superconductivity and Resilience to Disturbances” EUR. Phys. JB92144 (2019).
6. FM Cucchietti and HM Pastawski, “Anomalous Diffusion in Quasi One-Dimensional Systems”, Physica A.283302 (2000).
7. M. Schreiber et al., “Observation of the many-body localization of interacting fermions in a quasi-random optical lattice” science349842 (2015).
8. JL D’Amato and HM Pastawski, “Conductivity of a disordered linear chain including inelastic scattering events”, Phys. Rev. B.417411 (1990).
9. PR Zangara et al., “Interaction disorder competition in a spin system assessed by the Loschmidt echo” Phys. Rev. B.88195106 (2013).

## About the author

Horacio M. Pastawski is Professor of Physics at the National University of Córdoba and Senior Researcher at the Enrique Gaviola Institute of Physics. He graduated from Instituto Balseiro in Argentina and worked at the Massachusetts Institute of Technology before joining Córdoba in 1993. He is also a member of the Argentine National Academy of Sciences. He has extensive experience in quantum aspects of molecular electronics and nuclear magnetic resonance. His interests include sonic lasers (SASERs) and adiabatic quantum motors. He develops time reversal experiments such as Loschmidt echoes and acoustic time reversal mirrors as sensors for quantum dynamic phase transitions, multi-body localization and heterogeneous catalysis.

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