Localization in random fractal lattices

TL;DR

This paper studies how eigenfunction localization in random fractal lattices depends on spectral and Hausdorff dimensions, revealing that spectral dimension critically influences localization while Hausdorff dimension has minimal effect.

Contribution

It demonstrates that spectral dimension governs localization properties in random fractal lattices with fixed Hausdorff dimension, highlighting the role of connectivity over fractal geometry.

Findings

Localization depends strongly on spectral dimension.

Hausdorff dimension has little effect on localization.

Superlocalization resonances and energy gaps occur at low spectral dimensions.

Abstract

We investigate the issue of eigenfunction localization in random fractal lattices embedded in two dimensional Euclidean space. In the system of our interest, there is no diagonal disorder -- the disorder arises from random connectivity of non-uniformly distributed lattice sites only. By adding or removing links between lattice sites, we change the spectral dimension of a lattice but keep the fractional Hausdorff dimension fixed. From the analysis of energy level statistics obtained via direct diagonalization of finite systems, we observe that eigenfunction localization strongly depends on the spectral dimension. Conversely, we show that localization properties of the system do not change significantly while we alter the Hausdorff dimension. In addition, for low spectral dimensions, we observe superlocalization resonances and a formation of an energy gap around the center of the spectrum.

Equations

$$P\Big{(}(i,j)\rightarrow(i^{\prime},j^{\prime})\Big{)}=\frac{(\phi_{i^{\prime},j^{\prime}})^{\eta}}{\sum(\phi_{i^{\prime},j^{\prime}})^{\eta}},$$

$${\varphi }_{i,j}=\frac{1}{4}\left({\varphi }_{i,j+1}+{\varphi }_{i,j-1}+{\varphi }_{i+1,j}+{\varphi }_{i-1,j}\right).$$

$$E{\psi }_{(i,j)}=-\sum _{i^{\prime },j^{\prime }}{\psi }_{(i^{\prime },j^{\prime })},$$

$$r_i=\frac{\min ({\delta }_i,{\delta }_{i-1})}{\max ({\delta }_i,{\delta }_{i-1})},$$

$$\Delta =0.114\pm \mathrm{0.017.}$$

$$PR(E)=\sum_{(i,j)}\Big{|}\langle(i,j)|\psi(E)\rangle\Big{|}^{4},$$

$$t(r,r^{\prime },E)=⟨{\left|⟨r|G^{+}(E)|r^{\prime }⟩\right|}^2⟩,$$

$$i{\partial }_t{\psi }_{(i,j)}=-\sum _{i^{\prime },j^{\prime }}{\psi }_{(i^{\prime },j^{\prime })},$$

Full text

I Introduction

Anderson localization (AL) is a single-particle disorder induced effect which leads to exponential localization of particles’ eigenfunctions Anderson (1958); Krame and MacKinnon (1993); Van Tiggelen (1999); Müller and Delande (2011); Lagendijk et al. (2009). In his groundbreaking work, Anderson considered non-interacting electronic gas in a tight-binding model in the presence of on-site disorder. Since then, AL was investigated in many different models, including off-diagonal disorder Eilmes et al. (1998); Sepehrinia (2010); Biddle et al. (2011), disorder correlations de Moura and Lyra (1998); Piraud and Sanchez-Palencia (2013); Kosior et al. (2015); Major (2016), random fluxes Kalmeyer et al. (1993); Sheng and Weng (2000), localization in the momentum space of classically chaotic systems Fishman et al. (1982); Garreau (2016) and, recently, localization in the time domain Sacha (2015); Sacha and Delande (2016).

The interest in AL renewed after the first experimental observation of the phenomenon in ultracold atomic gases Billy et al. (2008); Aspect and Inguscio (2009); Modugno (2010); Roati et al. (2008). Although the Anderson model was created to describe electronic gases, AL is difficult to observe in metals due to electron-phonon and electron-electron interactions. On the contrary, interatomic interactions can be switched off in a system of ultracold atomic gases trapped in optical lattice potentials Chin et al. (2010). An optical lattice serves as an artificial, phononless crystalline structure, whose geometry and properties can be easily changed Jaksch and Zoller (2005); Bloch et al. (2008). Therefore, in recent years, ultracold atoms in optical lattices have become a very important toolbox used to test diverse physical models and phenomena Feynman (1982); Buluta and Nori (2009); Hauke et al. (2012).

The dimensionality of a system plays an important role in the context of AL Krame and MacKinnon (1993); Van Tiggelen (1999); Lagendijk et al. (2009). In particular, in three dimensional (3D) space a phase transition occurs at a critical energy, called the mobility edge, separating localized and extended states Abrahams et al. (1979); Mott (1987). It is therefore natural to investigate AL phenomenon in systems with non-integer dimension, i.e. in fractals Nakayama et al. (1994); Garcia2010 . Whereas in Euclidean space it is sufficient to define one kind of space dimension, in the case of fractals one needs to distinguish: the dimension of the embedding Euclidean space D, the Hausdorff dimension $d_{H}$ and the spectral dimension $d_{s}$ Nakayama et al. (1994); Rammal and Toulouse (1983). Whereas the Hausdorff dimension describes how the number of sites scales with the system size, the spectral dimension is related to a random walk on the lattice: the number of distinct sites $S_{n}$ visited by a random walker in $n$ steps scales as $S_{n}\propto n^{d_{s}/2}$, provided that $d_{s}<2$. The studies proved that it is the spectral dimension $d_{s}$ that is relevant in AL and that $d_{s}=2$ is the lower critical dimension, below which all the states are localized in the presence of a disorder potential Schreiber and Grussbach (1996).

Another important theoretical model in study of localization properties is the quantum percolation model (QP) Kirkpatrick and Eggarter (1972); Nakanishi and Islam (2009); Schubert et al. (2005); Schubert and Fehske (2009). Anderson model describes particles in the presence of potential disorder (purely diagonal), whilst QP involves the binary kinetic disorder (purely off-diagonal). In QP models the disorder comes from random geometry: a QP lattice, which is a subset of the D-dimension lattice, arises by a removal of a number of sites or links with a probability $q$ (being the only parameter of the model). Despite its simplicity, QP models still arise controversies. The main concern is the question of existence of the localization-delocalization transition in 2D models for $q>0$ (see Islam and Nakanishi (2008); Gong and Tong (2009); Dillon and Nakanishi (2014) and references therein). In particular, this issue might be important in the context of application of QP models in the description of transport properties in e.g. manganite films Zhang et al. (2002), granular metals Feigel’man et al. (2004) and doped semiconductors Bergqvist et al. (2004).

In the present paper we investigate eigenfunction localization in random fractal lattices (RFL), i.e. in fractal objects with random site connectivity in the absence of any diagonal disorder, Fig.1. We would like to stress that QP systems do not belong to fractal objects. That is, in the QP case only an infinite cluster of a percolated lattice at the percolation threshold is a fractal object Christensen and Moloney (2005). Here, on the contrary, we consider a family of lattices with well defined Hausdorff dimension $d_{H}$ Niemeyer et al. (1984). Starting with the minimal, connected lattice (i.e. a lattice without loops) and adding links between nearest neighbors we can increase the spectral dimension $d_{s}$ of a lattice and keep the Hausdorff dimension $d_{H}$ fixed. Therefore, the main focus of this paper is to investigate the presence or absence of localization in RFLs while changing the spectral and Hausdorff dimensions independently. It is worth noting that the theoretical model investigated here can be realized in ultra-cold atoms laboratories where lattice geometry can be nearly arbitrarily shaped Bakr et al. (2009); Gaunt and Hadzibabic (2012); Bijnen et al. (2015).

This paper is organized as follows. In Section II we describe the growth algorithm of random fractal lattices and how the spectral dimension changes when new lattice links are created. In Section III we focus on localization properties of RFLs and on their dependence on the spectral and Hausdorff dimensions. Particularly, we analyze superlocalization resonances and formation of an energy gap which emerges in the system for small spectral dimension. In Section IV we investigate transmission probabilities through the system and quantum evolution of initially localized particles. Finally, in Section V we conclude.

II Random fractal lattices

We consider a family of lattices that we call random fractal lattices (RFLs), which first arose in a model of dielectric breakdown Niemeyer et al. (1984). The RFLs are lattices with random site connectivity and with well-defined Hausdorff $d_{H}$ and spectral $d_{s}$ dimensions. The Hausdorff dimension, or the capacity dimension, describes how the number of sites scales with the system size. In other words, if a lattice in Fig. 1 is a fractal object, then the number of sites inside a sphere o radius $r$ is proportional to $r^{d_{H}}$, where in general $d_{H}$ is a noninteger exponent Christensen and Moloney (2005). On the other hand, the spectral dimension is related to a random walk on the lattice. The number of distinct sites $S_{n}$ covered in $n$ steps of a random walk is proportional to $n^{d_{s}/2}$, if $d_{s}<2$. From a general analysis, the spectral dimension is never larger than the Hausdorff dimension Nakayama et al. (1994); Rammal and Toulouse (1983).

RFLs under consideration are embedded in the 2D Euclidean space and have their Hausdorff and spectral dimensions smaller than 2. A minimal RFL, i.e. a lattice with the smallest number of lattice links, is a single connected lattice generated by the growth algorithm defined in Ref. Niemeyer et al. (1984). In a nutshell, a new lattice site $(i^{\prime},j^{\prime})$ is chosen and linked to the existing lattice at site $(i,j)$ with the probability

[TABLE]

where the summation goes over all possible choices, $\eta$ is a free parameter and $\phi_{i,j}$ is a function fulfilling discrete Laplace equation:

[TABLE]

At start, we set $\phi_{i,j}=0.5$ everywhere. When a site $(i^{\prime},j^{\prime})$ is being connected to the lattice, then the value of $\phi_{i^{\prime},j^{\prime}}$ is changed to zero. Before linking another lattice site, the values of $\phi$ in the neighborhood of $(i,j)$ needs to be updated [typically in 5-20 iterations of Eq. (2)]. The algorithm is stopped after reaching $N$ lattice sites. The lattices grown in this manner have nonuniform geometry, both in the lattice sites and lattice links occurrence, as in Fig. 1. In particular, the two neighboring lattice sites do not necessarily need to be connected and the closed loops are forbidden, i.e. a minimal RFL is created. By adding links between nearest neighbors to a given minimal RFL, one opens up new possibilities for a random walker to explore and therefore increases the spectral dimension $d_{s}$ of the system while keeping the Hausdorff dimension $d_{H}$ intact.

The Hausdorff dimension of RFLs depends on the value of the parameter $\eta$ Niemeyer et al. (1984). For example, setting $\eta=1$ one can generate a minimal RFL with the Hausdorff dimension $d_{H}=1.75\pm 0.02$ and the spectral dimension $d_{s}=1.33\pm 0.03$. Adding links to a minimal RFL results in an increase of the spectral dimension, Fig. 2. At the same time the Hausdorff dimension $d_{H}$ remains unchanged.

III Localization properties

In the following we analyze solutions of the Schrödinger equation

[TABLE]

where $(i,j)$ denotes a position on a RFL embedded in the 2D Euclidean space and the sum runs over nearest neighbor sites if there is a link between $(i,j)$ and $(i^{\prime},j^{\prime})$. We assume that the tunneling amplitudes of a particle between neighboring sites and the Planck constant are equal to unity.

III.1 Analysis of energy level statistics

In our analysis we investigate two distinct scenarios:

• •

First, we fix the Hausdorff dimension by setting $\eta=1$ in (1), which corresponds to $d_{H}=1.75\pm 0.02$. Then, we change the spectral dimension in the range between 1.33 and 1.55 by adding links to the minimal RFLs, see Fig. 2.

• •

In the second scenario, we do the opposite, i.e. we fix the spectral dimension ($d_{s}\approx 1.35$ or $d_{s}\approx 1.5$) and change the Hausdorff dimension by varying the parameter $\eta$ in (1).

In order to explore the presence or absence of localization of eigenfunctions of a particle in RFLs, we use a convenient method engaging the energy level statistics obtained via direct diagonalization of finite systems Oganesyan and Huse (2007); Pal and Huse (2010); Tang et al. (2015). Since the localized states usually have very small overlaps (as they may be localized in different parts of the system), the energy levels can be nearly degenerate. This is why the localization of eigenstates can be observed directly from the spectrum. Therefore, we expect that in the localized phase the energy level statistics follow the Poisson distribution, and in the delocalized phase fulfill the Wigner Dyson distribution Oganesyan and Huse (2007); Pal and Huse (2010); Tang et al. (2015). Having the ordered spectrum of energy levels $\{E_{i}\}$, we can calculate a quantity

[TABLE]

where $\delta_{i}=E_{i}-E_{i-1}$. Next, we average the results over 2000 realizations of RFL and over neighboring energies $r(E)=\langle r_{i}\rangle$. The distinction between the two regimes is possible since for the Poisson distribution $r(E)\approx 0.3863$ localized phase) and for the Wigner Dyson distribution $r(E)\approx 0.5359$ (delocalized phase) Oganesyan and Huse (2007); Pal and Huse (2010); Tang et al. (2015).

To investigate the localization properties of wavefunctions under the change of the spectral dimension of RFLs, we plot $r(E)$ in Fig. 3 (top panel). The vertical axis expresses the spectral dimension via the dimensionless parameter $p$, see Fig. 2. The panel illustrates strong dependence of the localization properties on the spectral dimension $d_{s}$. While increasing the spectral dimension we observe a smooth transition from the localized to delocalized phase, what is evident in the middle panel of Fig. 3, where the averaged ratio $r(E)$ is plotted for two extreme values of $d_{s}$. Furthermore, we observe nonmonotonous dependence of $r(E)$ on energy (i.e. the localization is stronger near the edges and the center of the spectrum), which has been also observed in QP models Schubert et al. (2005); Gong and Tong (2009).

In the bottom panel of Fig. 3 we present $r(E)$ for a few different Hausdorff dimensions $d_{H}$ and very similar spectral dimension $d_{s}$. The data with different $d_{H}$ do not differ significantly apart from the edges of the spectra. The results show that, similarly to the AL models Schreiber and Grussbach (1996), it is the spectral dimension $d_{s}$ that is the relevant dimension in the context of the localization properties of the system.

III.2 Energy gap and superlocalization resonances

In Fig. 3 we can observe a peculiar narrow energy gap $\Delta$ around $E=0$ for $d_{s}\lesssim 1.4$, which eventually closes up when the spectral dimension is being increased. We find that the value of energy gap $\Delta$ for the minimal RFLs (i.e. $d_{s}=1.33\pm 0.03$) of length $N=5000$ is

[TABLE]

The energy gap (5) decreases slightly in larger systems, however, it seems that $\Delta$ survives in the thermodynamical limit, see Fig. 4. In order to illustrate the energy gap more clearly we show in Fig. 5 the participation ratio $PR(E)$

[TABLE]

where $|\psi(E)\rangle$ is an eigenstate corresponding to an energy $E$ and $|(i,j)\rangle$ is a state localized at a lattice site denoted by $(i,j)$ in the 2D Euclidean space. The participation ratio is yet another measure of the localization Krame and MacKinnon (1993); Van Tiggelen (1999). That is, the inverse of $PR$ estimates a number of fractal points on which an eigenstate is localized on.

What is the most striking in Fig. 5 is the emergence of peaks that are related to superlocalization resonances also observed in the QP model Schubert et al. (2005). The resonances appear for a discrete set of energies $E_{r}$ and they are dominant for $E_{r}=0,\pm\frac{\sqrt{5}\pm 1}{2},\pm 1,\pm\sqrt{2}$. The presence of the resonances is not visible in Fig. 3 because in Eq. (4) the degenerate levels are discarded to avoid divergence. The name superlocalization stems from the fact that the eigenstates localize on very small (a few lattice site) disjoint clusters. For example, a zero energy state can be localized on two sites only, as long as a certain building block appears on the lattice boundary. In Fig. 6 (a) we see a block with one vertex and four lattice sites. Note, that its zero energy eigenstate has non-zero values on two sites only and the other two sites form an ,,empty leg”. Now notice, that a structure in Fig. 6 (c) must have similar zero energy eigenstates because connecting empty lattice to an empty leg of a block like in Fig. 6(a) does not change a zero energy eigenstate localized on two sites. Therefore, if a lattice geometry allows for small blocks (like e.g. in panel (a) or (b) of Fig.6), then some eigenstates of the lattice coincide with those of small blocks and superlocalization resonances emerge.

If some blocks are frequently occurring in the lattice, the corresponding superlocalization resonances can be extremely degenerate. For example, for the minimal RFLs with $d_{H}=1.75\pm 0.02$ about 10% of eigenstates have zero eigenenergy. The zero energy manifold is thus extended over a substantial number of lattice points that is related to the appearance of the energy gap in the spectrum. That is, the zero energy manifold is large and other eigenstates with non-vanishing overlap on the manifold must necessary possess different energies.

The four-site structure shown in Fig. 6(a) has also non-zero energy eigensolutions, for instance corresponding to $E=-\sqrt{3}$. However, such eigenstates do not have an ,,empty leg” and therefore if the block is connected to an empty big lattice, these eigenstates are disturbed. Nevertheless, it is possible to build a 3-vertex block, see Fig. 6(b), where there are eigenstates corresponding to $E=-\sqrt{3}$ which possess ,,empty legs” and are not disturbed when the block is attached to a big empty lattice. Such a 3-vertex structure is far less common in RFLs, which explains low abundance of the $E=-\sqrt{3}$ superlocalization resonance (less then 1‰).

IV Transmission and quantum evolution

In this section we investigate transport properties of a quantum particle on RFLs: transmission probability through the lattice and evolution of a particle initially localized on a single lattice site. Here, we focus on small systems with 500 lattice sites only because smaller systems are closer to the experimental reality in ultra-cold atomic gases.

The transmission probability from site $r$ to site $r^{\prime}$ of a quantum particle with the energy $E$ is defined Krame and MacKinnon (1993) as:

[TABLE]

where $\langle..\rangle$ denotes average over different realizations of RFL, $G^{+}(E)=\lim_{\eta\rightarrow 0_{+}}\left(E+i\eta\,-H\right)^{-1}$ is the retarded one-particle Green’s function Economou (2006). We plot the transmission probability throughout considered lattices, i.e. between the most distant sites, in Fig. 7. The top panel presents the dependence of the transmission probability on energy for different spectral dimension of RFLs whereas in the bottom panel there are transmission probabilities averaged over entire energy spectra for different number of links in the system. In an agreement with the results presented in Fig. 3, we observe a drastic reduction (5 orders of magnitude) of the transport while decreasing the number of links in the system (bottom panel). Furthermore, we can see a number of strong dips in the plot of $t(E)$ (top panel), especially for $p=0.05$. These dips correspond to the energy gap around $E=0$ and the superlocalization resonances for discrete degenerate energies, see Fig. 5. The most pronounced dips are related to: $E=0$ (about 10% of all energy levels correspond to $E=0$), $E=\pm 1$ (4%) and $E=\pm\sqrt{2}$ (1%). Note, that an increase of $p$ (black dashed and red solid curves in the top panel) narrows the dip around $E=0$ significantly down because the gap is disappearing.

Furthermore, the transport properties of a quantum particle can be investigated more directly by solving the time dependent Schrödinger equation

[TABLE]

cf. Eq. (3).

In panels (a) and (b) of Fig. 8 we present the snapshots of the evolution of a quantum particle in lattices for the two extreme cases: the minimal and the maximal number of links for a given geometry. In panels (c) and (d) the time averaged results are shown. Starting from the same initial, fully localized (on a single lattice site) state, we obtain the opposing results: the probability density of finding a particles is either localized around the initial state or explores the whole lattice.

V Conclusions

We have investigated localization and transport of a quantum particle in lattices with a fractal structure. The lattices consist of points that form a connected cluster. Sites of the lattices are generated so that their fractal (Hausdorff) dimension $d_{H}$ is controlled. Independently one can control the spectral dimension $d_{s}$ of the systems by choosing how many nearest neighbor sites are linked to a given lattice point. It allows us to analyze how the localization properties vary with independent changes of the Hausdorff and spectral dimensions.

Analysis of energy level statistics and participation ratio of eigenstates shows that while the localization properties depend very weakly on $d_{H}$, they change strongly with $d_{s}$. For the smallest spectral dimension of the systems we observe strong localization of eigenstates. With an increase of $d_{s}$, eigenstates loose their localization properties and become extended over the entire finite lattices that we consider. Disorder in our systems stems from a non-uniform distribution of lattice points and from their random connections. When $d_{s}$ approaches $d_{H}$ all nearest neighbor sites become connected and the randomness is related to the non-uniform distribution of lattice points only. The latter introduces too weak dephasing and eigenstates do not localize.

We observe also eigenstates that are strongly localized on small parts of the random fractal lattices. The smaller part of the fractal, the higher chance for such eigenstates to occur. The zero energy eigenstates can occupy two sites only and consequently they form the largest degenerate manifold. At low spectral dimension they are so many that an energy gap around $E=0$ is created. The presence of strongly localized eigenstates is imprinted in the transport properties of the systems, i.e. the particle transmission probability drops at the corresponding energies.

Acknowledgments

We are grateful to Marcin Płodzień for encouraging discussions.

AK acknowledges a support of the National Science Centre, Poland via project DEC-2015/17/N/ST2/04006. KS acknowledges a support of the National Science Centre, Poland via project No.2015/19/B/ST2/01028.