Duality in power-law localization in disordered one-dimensional systems

TL;DR

This paper investigates localization properties in one-dimensional disordered systems with power-law hopping, revealing a duality between long-range and short-range hopping regimes and identifying a new universality class.

Contribution

It introduces a novel universality class of power-law hopping models exhibiting a duality between different decay regimes of wave functions.

Findings

Almost all eigenstates are power-law localized for any $a>0$

A duality exists between systems with $a<1$ and $a>1$

Wave function decay follows the same power law in dual regimes

Abstract

The transport of excitations between pinned particles in many physical systems may be mapped to single-particle models with power-law hopping, $1/r^a$. For randomly spaced particles, these models present an effective peculiar disorder that leads to surprising localization properties. We show that in one-dimensional systems almost all eigenstates (except for a few states close to the ground state) are power-law localized for any value of $a>0$. Moreover, we show that our model is an example of a new universality class of models with power-law hopping, characterized by a duality between systems with long-range hops ($a<1$) and short-range hops ($a>1$) in which the wave function amplitude falls off algebraically with the same power $\gamma$ from the localization center.

Equations

$$\gamma (a)=\gamma (2-a),$$

$$H_{XY}=-\frac{J}{2}\sum _{n,m}\frac{1}{|r_n-r_m{|}^a}\left(S_n^{+}{S}_m^{-}+S_n^{-}{S}_m^{+}\right),$$

$$H=\sum _n\sum _{m\ne n}{H}_{n,m}|n⟩⟨m|.$$

$$H_{n,m}=-J|r_n-r_m{|}^{-a}.$$

$$V_{n,m}$$

$$H_{n,m}=-(J+{\eta }_{n,m})/r_{nm}^a,$$

$$H_{n,m}=(-1{)}^{r_{nm}}/r_{nm}^a,$$

Full text

Long-range interactions are crucial in many disordered systems. In particular, dipole-induced transport of excitations, with a hopping amplitude decaying with interparticle distance $r$ as $1/r^{3}$, plays a key role in disparate scenarios, including magnetic atoms DePaz2013 , polar molecules Yan2013 , Rydberg atoms Saffman2010 , nitrogen-vacancy centers in diamonds Waldherr2014 , and nuclear spins in solid-state systems Alvarez2015 . Interestingly, systems with tunable power-law interactions, $1/r^{a}$, have been recently proposed. In the presence of laser-driven coupling, trapped ions may realize power-law-decaying spin interactions with tunable $0\!<\!a\!<\!3$ Richerme2014 ; Jurcevic2014 . Spin models with an arbitrary power law interaction can be also engineered between atoms trapped in a photonic crystal waveguide Hung2016 . These tunable systems can be realized in low dimensions, opening intriguing questions on transport with truly long-range interactions, i.e. for $a<d$, with $d$ the dimensionality.

Most of the previous works considered excitation transport amongst regularly spaced particles, e.g by pinning them in an uniformly filled lattice with on-site disorder. In addition, long-range hopping has been considered either anisotropic with zero angular average (e.g. dipole hopping Levitov1990 ; AltshAleiEfet2011 ) or random with zero ensemble average  MirlinFyod1996 . Under such conditions the interplay of on-site disorder and long-range hops may result in localization, critical behavior, or fully extended states. In his seminal work, Levitov suggested that these cases occur for $a\!>\!d$, $a\!=\!d$ and $a\!<\!d$, respectively Levitov1990 . Later studies for $d\!=\!1$ using a supersymmetric non-linear sigma-model approach confirmed this suggestion MirlinFyod1996 . Intense efforts have been devoted to the critical Power-Law Banded Random Matrix (PLBRM) ensemble ($d\!=\!a\!=\!1$) KrMut1997 ; KrYev2003 ; YevOss2007 ; Cuevas2007 ; Chalk2010 ; Evers2008 ; RMTbook . Dipolar excitations in a 2D lattice with on-site disorder ($d\!=\!a\!=\!2$) exhibit a purely critical behavior only for the time-reversal-invariant case and may have a ”metal-insulator” transition if the invariance is broken AltshAleiEfet2011 .

Notwithstanding these details, the common wisdom until recently was that for non-interacting particles with on-site disorder long-range hopping is a delocalizing factor that destroys localization for $a\leq d$. Only few recent works have stood out of this paradigm. Surprisingly, all states appeared to be localized for the exotic case of linear hopping ($a=-1$) between randomly placed points in a 1D system Bogomol2003 . A second example is provided by a simplex ($a=0$), where all excited states were found to be localized Ossipov2012 . This peculiar behavior was linked in Ref. Ossipov2012 to the macroscopic degeneracy of the perturbation matrix. The role of degeneracy in this model was recently further emphasized and extended to $a>0$ in Ref. Celardo2016 , where it was argued that the long-range hopping has no effect on the system dynamics as long as the width of the disorder-broadened $(N-1)$-fold degenerate level (with $N$ the number of sites) is smaller than the gap between the ground and excited states for $a=0$. Since this gap is proportional to $N^{1-a}$ and the bandwidth is $N$-independent, the long-range hopping is ”shielded” in the thermodynamic limit for all $0<a<1$ and the localization of excited states is preserved.

In this Letter we consider the physically relevant case of excitation transport due to power-law hops amongst randomly pinned particles. We call these models power-law Euclidean (PLE) models in connection with the Euclidean random matrices with no diagonal elements MezarParisiZee1999 . Disorder in PLE models is *purely off-diagonal * given by the Poisson distribution of the particle positions. However, it is crucially different from the off-diagonal disorder in nearest-neighbor hopping models Dyson1953 ; Lifshitz1964 , since long-range hops break down the chiral (sublattice) symmetry Brouwer2000 of the nearest-neighbor hopping case. It also differs from the off-diagonal disorder of Ref. MirlinFyod1996 because the angular or ensemble average of the hopping amplitude is not zero.

We focus on the 1D case, showing that PLE models possess features radically different from those with random-sign long-range hopping MirlinFyod1996 . Similar to the Anderson model on a lattice with diagonal disorder and non-random long-range hopping Celardo2016 , almost all states (except for a few states close to the lower edge of the spectrum, see Suppl. Material) are localized for any $a>0$. However, in contrast to the conventional Anderson model with exponential localization, in the PLE model localization is algebraic with a power-law decrease of wave functions, ${\rm exp}\left(\overline{\ln|\psi(x)|^{2}}\right)\sim|x-x^{\prime}|^{-\gamma}$, where $\gamma$ depends on $a$. This remains valid for the model of Ref. Celardo2016 , even for arbitrarily small but finite $a$. As discussed below, power-law localization emerges because long-range hops are not fully shielded, but rather become effectively short-range.

The main result of this paper is a surprising duality:

[TABLE]

for $0\!<\!a\!<\!1$. We obtain this duality numerically for $0<a<1$ (Fig. 1), and analytically for the PLE model at finite but small $a$ footnote-a0 . Strikingly, this duality is more general than our PLE models. It holds for the model of Ref. Celardo2016 , which presents on-site disorder in a regular lattice with long-range hops. It also holds for the PLE model with additional on-site disorder. Moreover, the exponent $\gamma$ depends only on $a$ but not on other details of the models, including the eigenstate energy, provided that it is in the bulk of the spectrum. Thus the considered models are not only experimentally relevant but represent a new class of long-range models where localization is algebraic and the duality Eq.(1) holds.

Model.–

We consider $N$ particles pinned at random positions $\{r_{n}\}$ by an external 1D potential. Each particle has two internal states, $\{\uparrow,\downarrow\}$, which can be treated as (pseudo-)spin states. The particles experience power-law spin exchange described by the XY Hamiltonian:

[TABLE]

with $S^{\pm}_{n}$ the spin operators associated to particle $n$. Assuming that all particles are in the state $\downarrow$, we denote $|n\rangle$ the state with a single spin-flipped particle $\uparrow$ at the site $n$ placed at position $r_{n}$. The propagation of a single excitation among the particles, determined by the XY exchange, can be modeled by the Hamiltonian:

[TABLE]

The hopping of the excitation between the $n$–th and $m$–th particles follows a power law

[TABLE]

Below we assume for simplicity $J=1$. Since the positions $r_{n}$ are randomly distributed, the excitation experiences a peculiar randomness that may significantly handicap its propagation in the system. The single-particle model is expected to describe well the case of more than one excitation, as long as the gas of excitations is dilute enough.

Localization and multifractality–

We are interested in the localization properties of the eigenstates $|\psi_{s}\rangle$ of $H$, $H|\psi_{s}\rangle=E_{s}|\psi_{s}\rangle$, close to the maximum of the density of states (see Suppl. Material), and, in particular, how these properties depend on $a$. The eigenfunctions and eigenenergies were obtained by exact diagonalization of Model (3) with $N$ up to $8.4\times 10^{4}$.

We characterize the eigenstates $|\psi_{s}\rangle=\sum_{n}\psi_{s}(n)|n\rangle$ by the moments $I_{q}(s)=\sum_{n}|\psi_{s}(n)|^{2q}\propto N^{-\tau(q)}$, where $D(q)=\tau(q)/(q-1)$ are the so-called fractal dimensions. Localized states are characterized by $D(q)=0$, ergodic extended states by $D(q)=1$, while extended, but non-ergodic, multifractal states have a set of non-trivial fractal dimensions $0<D(q)<1$ Wegner1981 ; Altshuler1986 ; Evers2008 ; DeLuca2014 ; DengAltshuler2016 . The Legendre transform $\tau(q)=q\alpha-f(\alpha)$ defines the multi-fractal spectrum (MFS) $f(\alpha)$ Wegner1981 , which characterizes the Hausdorff dimension of the manifold of sites where $|\psi_{s}(n)|^{2}=N^{-\alpha}$. Normalization, $\sum_{n}|\psi_{s}(n)|^{2}=1$, requires $\alpha\geq 0$.

An important difference between power-law- and short-range-hopping models is that in the former case localized states, if they exist at all, have power-law decaying tails $|\psi_{s}(n)|^{2}\propto 1/|n-n_{0}|^{\gamma}$, i.e. they are algebraically localized. One can show that the corresponding $f(\alpha)$ has a triangular form, where $f(\alpha)=k\alpha$ for $0<\alpha<1/k$ with $k=d/\gamma$. This is in contrast to exponentially localized states for a short-range hopping, where $k=0$. Note that $|\psi_{s}(n)|^{2}\propto|H_{n+n_{0},n_{0}}|^{2}\propto n^{-2a}$ in the perturbative regime at $a>d$, and hence $\gamma$ cannot be smaller than $\gamma_{a}=2a>2d$. Therefore, for algebraically localized states in 1D one has $k<1/2$ like for the exponentially localized states on the Bethe lattice DeLuca2014 .

We evaluate the spatial distribution of the eigenstates by setting the localization center placed at the maximum value of $|\psi|^{2}$, at index $n=n_{0}$, and averaging $\ln|\psi(n)|^{2}$ over disorder realizations and over an energy window. For the localized states discussed below the typical average ${\rm exp}(\overline{\ln|\psi(n)|^{2}})$ is well fitted by the expected algebraic dependence $|n-n_{0}|^{-\gamma}$ as seen in Fig.2, except for the special point $a=0$ (see Fig.2b) where the PLE model is disorder-free and the model [21] is exactly solvable [24].

Multi-fractal spectrum.–

We obtain the MFS in the thermodynamic limit by means of a linear extrapolation of our finite-size calculations in terms of $1/\ln N$ DeLuca2014 ; DengAltshuler2016 . For $a>1$ we expect localized states, and indeed we find linear $f(\alpha)$ with a slope $k<1/2$. At $a=1$, for all eigenstates the MFS displays a triangular shape with the slope $k_{c}\simeq 0.42$ (Fig. 3(b)). Since $k_{c}$ is very close to the critical value $1/2$, the eigenstates for $a=1$ are either weakly localized or critical. For $a<1$ one would naively expect extended eigenstates. However, this expectation appears to be wrong. The eigenstates inside a band emerging due to disorder-broadening of the macroscopically degenerate level remain localized for all $0<a<1$.

Note that the slope $k$ of the linear part of $f(\alpha)$ shown in Fig. 3 reflects the algebraic decay of $|\psi_{s}(n)|$ far from the localization center. For small $\alpha$ the MFS becomes concave, implying a reduced probability to find intermediate values $|\psi_{s}(n)|^{2}\propto N^{-\alpha}$. A similar concave MFS has been discussed for the insulating phase of the Rosenzweig-Porter random matrix ensemble KravtsovNJP2016 . The moments $I_{q}\propto N^{-\tau(q)}$ are however determined by finding the tangent $-\tau(q)+q\,\alpha$ to $f(\alpha)$ that does not cross the MFS at any point. Hence, $\tau(q)$ and fractal dimensions for $f(\alpha)$ with a concave segment are the same as those for the MFS in which the concave part is replaced by the red dashed line shown in Fig 3.

Duality.–

The localization properties of the eigenstates of Model (3) for $0<a<1$ and for $1<a<2$ exhibit a striking duality given by Eq.(1). The slope $k$ of the MFS is depicted in the upper panel of Fig. 1. The lower panel shows the exponent $\gamma$ extracted from the average $\overline{\ln(|\psi(n)|^{2})}\propto-\gamma\ln{|n-n_{0}|}$ (see Fig. 2). Both graphs are in good mutual agreement given by the relation $k=1/\gamma$. Remarkably, a similar duality is present for the model of Ref. Celardo2016 , where disorder is purely diagonal (see Fig.1).

Proof of duality for $a\rightarrow 0$ and $a\rightarrow 2$.–

As was already mentioned, for $a=0$ there is an $(N-1)$-fold degeneracy of the excited states, which may be chosen both extended and localized on equal footing. The localization properties of degenerate states are controlled by the perturbation $V(r)=r^{-a}-1$; namely, by the properties of the ”good zero-order wave functions” Landau-Lifshitz , which are eigenvectors of the perturbation operator in a certain convenient basis. Let us select a basis of $(N-1)$ states including $N/2$ non-overlapping dimers, $|\psi_{n,n+1}\rangle=(|n\rangle-|n+1\rangle)/\sqrt{2}$, $N/4$ non-overlapping tetramers, $|\psi_{n,n+1}^{n+2,n+3}\rangle=(|n\rangle+|n+1\rangle-|n+2\rangle-|n+3\rangle)/2$, $N/8$ octamers, and so on. The basis constructed in this way is orthonormal and complete for any $N=2^{k}$, where $k$ is a positive integer. Moreover, the $n$-mers are compact and thus represent localized states for any $n={\cal O}(1)$. The perturbation results in hopping between $n$-mers. The hopping amplitude $V_{n,m}$ from the dimer $|\psi_{n,n+1}\rangle$ to the dimer $|\psi_{m,m+1}\rangle$, located far away (and $m\gg n$), in the limit of small $a$ is given by:

[TABLE]

where $R_{nm}=(r_{m+1}+r_{m}-r_{n+1}-r_{n})/2$. A similar dependence holds for the hopping between tetramers and all $n$-mers (see Suppl. Mat.).

Therefore ”good zero-order wave functions” in the basis of $n$-mers obey the same equation as eigenvectors of the Hamiltonian with hopping $\sim 1/r^{2}$. Thus, we have reduced the problem of localization close to the point of macroscopic degeneracy $a=0$ to the problem of localization in a system of $n$-mers with hopping amplitude $\propto 1/R_{nm}^{2}$ for which localization is well understood. This proves the localization in our model for $a\rightarrow 0$ case footnote-a0 and its duality to localization at $a=2$.

Note finally that for $a<1$ at the edge of the spectrum at negative energy there are also delocalized states (see Suppl. Material). The measure of the number of such states is zero, but they will be responsible for the transport in the system.

Role of sign-alteration.–

The localization properties of our model and, in particular, the duality (2) are drastically different from those of PLBRM MirlinFyod1996 and Levitov’s scenario Levitov1990 . We show at this point that this difference is related to the sign-randomness in the long-range hopping. To this end we modify our model in such a way that $H_{n,m}$ acquires a randomness $\eta_{n,m}$:

[TABLE]

where $\eta_{n,m}$ are random bounded numbers, $|\eta_{n,m}|<W$, and $r_{nm}=|r_{n}-r_{m}|$. Model (3) corresponds to $W=0$, $J=1$, while that of Ref. MirlinFyod1996 corresponds to $W=1$, $J=0$. Model (6) can be physically realized for a spatially pinned two-component system, in which the spin-like excitations may be transferred either between particles of the same or of different species (see Suppl. Material).

Another possible modification is the staggering model:

[TABLE]

for which the sign of the product $H_{m,j_{1}}H_{j_{1},j_{2}}...H_{j_{\ell-1},n}$ over a path of $\ell$ hops from $m$ to $n$ is independent of the number of hops and equal to $(-1)^{r_{nm}}$.

The analysis presented in Suppl. Material shows that even at small $W=0.1$, Model (6) exhibits for $J=1$ the same properties as PLBRM and the eigenstates are extended for $a<1$. In contrast, Model (7) is in the same universality class as Model (3). This shows that localization in Model (3) is due to interference of long-range paths involving one or more hops, which is not affected by staggering, but destroyed by sign-randomness.

Conclusions.–

Systems with power-law, $\propto 1/r^{a}$, spin-exchange between randomly spaced particles represent a peculiar form of off-diagonal disorder that leads to surprising localization properties. We have shown that 1D spin excitations remain algebraically localized for any value of the hopping power $a>0$. Moreover, we show that our model is a representative of a new universality class of models with power-law hopping, characterized by a duality Eq.(1) between models with $0<a<1$ and $2>a>1$.

We acknowledge fruitful discussions with B.L. Altshuler, I.Khaymovich and E. Yuzbashyan. X. D. and L. S. thank the support of the DFG (SFB 1227 DQ-mat and FOR2247). G. V. S. acknowledges funding from the European Research Council under European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 341197).