**Some properties of the one-dimensional Schrödinger operator with random potential — Anderson localisation
**

* Introduction : *Real systems generally possess a certain amount of disorder (impurities or structural defects). Whether the disorder strongly affects the physical properties is sometimes a difficult question. It is well known that wave phenomenon are particularly sensitive to the presence of disorder. A striking manifestation of disorder is the localization of the electronic waves: whereas in a perfect cristal the eigenstates are extended Bloch waves, a sufficiently strong disorder can localize the electronic wave functions, as shown by Anderson in a pioneering article in 1958.

Dimensionality plays a crucial role. Strictly one-dimensional systems are somehow less rich than two- and three-dimensional cases: the rigorous proof of localization of all eigenstates in 1d, by Goldsheit, Molchanov & Pastur (1977) forbids the existence of a localization-delocalization transition in the spectrum, as in 3d (Note however that delocalization can occur in 1d if the random potential possesses some particular correlations: such an example is provided by the disordered SUSY quantum mechanics). The fact that the elastic mean free path is of the same order as the localization length in 1d leaves not room for the existence of a diffusive regime^{(1)}. However, despite they are less rich, strictly 1d disorder systems offer the possibility to use much more powerful (nonperturbative) techniques and study much finer properties. Several examples of such studies are developed in the following articles.

^{(1)} The disorder can be characterized by the elastic mean free path l_{e} which gives the length over which the momentum has relaxed. Another important length is the localization length l_{loc} that measures the exponential damping of the wave functions. If we call L the system size we can distinguish several regimes for a weak disorder:

- The ballistic regime, when L << l
_{e}, l_{loc} - The diffusive regime, when l
_{e}<< L << l_{loc}(weak localization). - The localized regime, when l
_{e}, l_{loc}<< L(strong localization).

### Scattering by a random potential: Distribution of the Wigner time delay

The Wigner time delay is a concept of scattering theory. Consider for example the scattering of a particle on some potential localised in space: the time delay is defined as the derivatives of scattering phase shifts with resspect with the energy. In a semiclassical picture, it measures the delay in time caused by the presence of the potential (note that the notion may be formulated in classical scattering theory, cf. S.-K. Ma, *Statistical mechanics*, World Scientific, chapter 14). The Wigner time delay can be related to the variation of the density of states due to the introduction of the scattering potential, through the Krein-Friedel sum rule. Therefore the time delay provides a measure of the density of states in a scattering situation.

- Alain Comtet and Christophe Texier,

**On the distribution of the Wigner time delay in one-dimensional disordered systems**

J. Phys. A: Math. Gen.**30**, 8017-8025 (1997).

cond-mat/9707046. - Christophe Texier and Alain Comtet,

**Universality of the Wigner time delay distribution for one-dimensional random potentials**

Phys. Rev. Lett.**82**(21), 4220-4223 (1999).

cond-mat/9812196. - Christophe Texier and Alain Comtet,

**Wigner time delay distribution for one-dimensional random potentials**

in « Quantum physics at mesoscopic scale », ed. by C.Glattli, M.Sanquer and J.Trân Thanh Vân, EDP Sciences, 2000, p. 475, Proceedings of the XXXIVth Moriond conference, 23-30 january 1999.

ps file.

A poster : **Wigner time delay distribution for one-dimensional random potentials**.

ps (1.3MB), pdf (0.5MB).

### Ordered (extreme value) statistics of energy levels for 1D disordered quantum mechanics

- Christophe Texier,

**Individual energy level distributions for one-dimensional diagonal and off-diagonal disorder**

J. Phys. A: Math. Gen.**33**, 6095-6128 (2000).

cond-mat/0004285.

Dirac equation with a random mass is known to support a critical extended state at zero energy (note that Dirac operator is directly related to the supersymmetric Schrödinger Hamiltonian). We analyse the average density of states for 1D Dirac operator in a bounded domain. Due to delocalisation, the DoS is sensitive to the boundaries for *E* → 0. This provides a possible definition of the localisation length, much larger than the inverse localisation length, that usually provides a definition of localisation in 1D problems. We argue that Lyapunov exponent is not able to capture the essential physics of low energy localisation near the delocalisation point (*E*=0).

- Christophe Texier and Christian Hagendorf,

**Effect of boundaries on the spectrum of a one-dimensional random mass Dirac Hamiltonian**

J. Phys. A: Math. Theor.**43**, 025002 (2010). (12pp)

cond-mat arXiv:0909.2205.

### How to break supersymmetry with a random scalar potential

Cf. paragraph on supersymmetric quantum mechanics

- Christian Hagendorf and Christophe Texier,

**Breaking of supersymmetry in one-dimensional a random Hamiltonian**

J. Phys. A: Math. Theor.**41**, 405302 (2008). (32pp)

cond-mat arXiv:0805.2883. - Christophe Texier and Christian Hagendorf,

**One-dimensional classical diffusion in a random force field with weakly concentrated absorbers**

Europhys. Lett.**86**, 37011 (2009). (6pp)

cond-mat arXiv:0902.2698. - Aurélien Grabsch, Christophe Texier and Yves Tourigny,

**One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers**

J. Stat. Phys.**155**, 237-276 (2014)

cond-mat arXiv:1310.6519.

**See also** the page on **Classical diffusion in a random force field (Sinai problem)**

### Localization properties with potential with large local fluctuations

Most of works on 1D Anderson localisation consider the case where the potential has relatively small local fluctuations, such that <*V*_{n} ^{2}> < ∞ (*V*_{n} is the on-site potenial for discrete models) or ∫_{-∞}^{+∞}d*x* <*V*(*x*)*V*(0)> < ∞ (for continuous models). Models where this condition is not fulfilled lead to non standard localization properties (super-localisation) with non exponential damping of wave function’s envelope, like exp(-|*x*|^{α}) for α>1.

- Tom Bienaimé and Christophe Texier,

**Localization for one-dimensional random potentials with large local fluctuations**

J. Phys. A: Math. Theor.**41**, 475001 (2008). (9pp)

cond-mat arXiv:0807.0772.

### One-dimensional quantum Hamiltonians for random potentials made of generalised point scatterers

Spectral properties of a one-dimensional Schrödinger Hamiltonian for a potential given by a random superposition of delta potentials have been studied for a long time : Schmidt (1957) for delta scatterers at regularly spaced positions with random weights, Lax & Philips (1958), Frisch & Lloyd (1960) and Bychkov & Dykhne (1966) for random positions with fixed weights, and Nieuwenhuizen (1983) for random positions and random weights (See also the book by Lifshits, Gredeskul & Pastur, 1988).

However the delta potential is a particular realisation of « point scatterer »: a general point scatterer may be described by a 2*2 S-matrix. The unitarity of the S matrix implies that it can be parametrised by four real parameters (the group U(2)=U(1)*SU(2) has 4 generators). The case of delta scatterer corresponds to a one-parameter subgroup. We have considered models of random generalised point scatterers at random positions. The close relation with the study of products of random matrices is emphasized.

- Alain Comtet, Christophe Texier and Yves Tourigny,

**Products of random matrices and generalised quantum point scatterers**

J. Stat. Phys.**140**, 427-466 (2010)

cond-mat arXiv:1004.2415.

### One-dimensional supersymmetric Hamiltonian with Lévy noises

In this work we consider Schrödinger supersymmetric Hamiltonians when the function φ(x) is a Lévy noise, i.e. when Φ(x)=∫_{0}^{x} dx’ φ(x’) is a Lévy process (a random process generalising the Brownian motion). The case of *subordinators* is considered (non decreasing processes). We have discovered a new exact model for a Lévy process with singular Lévy measure m(dy). Moreover, we have provided a general discussion of low-energy spectral properties for arbitrary subordinators. (i) For regular Lévy measure we show that the main exponential behaviour of the integrated density of states is N(E) ∼ exp[-πρ/√E] where ρ=∫_{0}^{∞}m(dy). (ii) For singular Lévy measures, ∫_{0}^{∞}m(dy)=∞, we obtain N(E) ∼ exp[-C E^{-η/2}] where the exponent is related to the singularity of the Lévy measure by η=1/(1-α), where m(dy) ∝ y^{-1-α}dy when y → 0+ (for 0<α<1).

- Alain Comtet, Christophe Texier and Yves Tourigny,

**Supersymmetric quantum mechanics with Lévy disorder in one-dimension**

J. Stat. Phys.**145**, 1291-1323 (2011)

math-ph arXiv:1105.5506

### General products of random 2*2 matrices in the continuum limit (random matrices close to the identity)

This general formulation allows us to exhaust all possible 1D disordered models mapped onto random matrix products of SL(2,R) ; we have provided a classification of solutions that can be understand as a **classification of 1D continuum disordered models**. Alain Comtet, Jean-Marc Luck, Christophe Texier and Yves Tourigny, J. Stat. Phys. **150**, 13-65 (2013) (and math-ph arXiv:1208.6430).

**See also page “ products of random matrices ”
**

### Fluctuations of random matrix products of SL(2,R) and localisation in the random mass Dirac equation

We study the *fluctuations* of the logarithm of the wave functions, a problem related to the analysis of the fluctuations of certain random matrix products, J. Stat. Phys. **157**, 497-514 (2014) (and cond-mat arXiv:1402.6943). This question is of importance in localisation problems : it is related to the discussion of the Single Parameter Scaling hypothesis ; this plays an important role when studying statistical properties of local density of states (Altshuler & Prigodin, 1989) or Wigner time delay (Texier & Comtet, Phys. Rev. Lett. **82**(21), 4220 (1999)).

**See also page “ products of random matrices ”
**

### The multichannel Dirac equation with a random (matricial) mass

We have considered the Dirac equation with a Gaussian white noise N*N matrical mass in Europhys. Lett. 116, 17004 (2016). The model presents interesting topological properties : as the ratio mass/disorder is tuned the systems exhibits a sequence of N topological phase transitions. The phase diagram (below) in the half plane (mass,disorder) presents *N*+1 sectors:

**See also page “ products of random matrices ”
**