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[EQUATION] Using a Taylor expansion in the small nonlinearity limit, the evolution operator contains a sum of a linear term, a second-order (in [MATH] ) term and a sequence of [MATH] -order terms [MATH] |
The full non-linear map reads [EQUATION] We will explicitly evaluate the first-order non-negligible term in nonlinearity only. Let us consider the evolution in the linear limit eigenmode basis, |
[EQUATION] With the wave function expanded in this basis [MATH] , the evolution equations read [EQUATION] with the overlap integral (or matrix element, or overalp integral) |
[EQUATION] I.2 Numerical Approach Let us discuss the details of simulations and data analysis. We directly propagate evolution equation ( 12 ). The initial conditions are uniformly spread over several neighboring sites to guarantee positivity and unitarity of the coins. |
The only source of the numerical error is the round-off errors of the finite dimensional computer algebra. We estimate the error by means of the total packet norm. It is equal to [MATH] for initial conditions and supposed to be constant due to unitary evolution. The relative value of the error never exceeded [MATH] |
In all the calculations we use [MATH] . The system size [MATH] is between [MATH] and [MATH] . The results which include ensemble averaging employ around [MATH] realizations of the random field [MATH] . The total evolution times reach up to [MATH] time steps, which exceeds the maximum previously gained limits for such a... |
The averaged curve of the second moment [MATH] is smoothed with the locally weighted regression smoothing (LOESS) cleavland1981lowess william1988Locally algorithm. The power-law exponent is then calculated as the two-point derivative of the smoothed data. To verify the smoothing procedure and exclude overfitting we als... |
# Source: arxiv 1806.06562 # Title: Deque languages, automata and planar graphs # Sections: all # Downloaded: 2026-03-03T01:49:02.744120+00:00 |
Deque languages, automata and planar graphs Abstract The memory of a deque automaton is more general than a queue or two stacks; to avoid overgeneralization, we consider quasi-real-time operation. Normal forms of such automata are given. Deque languages form an AFL but not a full one. We define the characteristic deque... |
Introduction This research pertains to the classical investigations on languages recognized by automata equipped with various types of auxiliary memory, such as pushdown stacks, queues, and combinations thereof. Introduced by D. Knuth DBLP:books/aw/Knuth68 , the deque data-type is common in computer science, where it i... |
First, the present study of deque automata is an attempt to fill such gap and to establish formal relations especially with the family of queue automata DBLP:journals/computing/Vollmar70 DBLP:journals/tcs/VauquelinF80 DBLP:journals/tcs/CherubiniCCM91 , which has recently attracted renewed interest (a survey is in Kutri... |
Paper content and contributions. Sect. defines deque automata (DA) consistently with existing definitions of pushdown and queue automata (PDA, QA). To prevent spontaneous moves to turn the deque without reading from input, thus simulating a Turing machine, we focus on quasi-real-time (QRT) operations as in QA DBLP:jour... |
To illustrate expressivity, we introduce a family of languages featuring any number and ordering of reversed and directed replications. Sect. 2.1 defines the characteristic deque language (CDL) which plays the role of the Dyck and the AntiDyck (a.k.a. FIFO) language, resp. for CF and queue languages and yields a Chomsk... |
Basic definitions and properties A double ended queue or deque is an unbounded tape containing a possibly empty string of symbols from an alphabet [MATH] . In a horizontal layout, the left end and the right end of the tape are resp. called front |
and tail , both equipped with a reading and writing head. The four deque operation types are: write/read a symbol at front, write/read a symbol at tail. Each reading operation cancels the symbol. After each writing (resp. reading) operation at front, the front head moves left (resp. right) by one case; the tail head re... |
deque automaton [MATH] is a nondeterministic acceptor with one-way read-only input tape, finite-state control, and deque memory tape, which initially is empty. |
[MATH] is restricted to operate in QRT. Definition 1 A QRT deque automaton DA ) is a 6-tuple [MATH] where [MATH] is the terminal alphabet, [MATH] the finite deque alphabet, [MATH] and [MATH] resp. the initial and final states; and the transition relation [MATH] is a finite subset: |
[MATH] For each 7-tuple, called a transition [MATH] [MATH] and [MATH] are resp. the present and next state, [MATH] are the words resp. read and written on deque front, and [MATH] are the words resp. read and written on deque tail. If [MATH] then the tuple is called an [MATH] -transition. |
A configuration is a 3-tuple [MATH] A move is a binary relation [MATH] on configurations defined as follows. We say that [MATH] if [MATH] [MATH] [MATH] |
[MATH] and the transition [MATH] . A move where [MATH] is called spontaneous We assume that every DA [MATH] operates in quasi-real-time (QRT) with a delay [MATH] . This means that every sequence of [MATH] moves shifts the input tape of at least one position. We say that [MATH] operates in real time (RT) if [MATH] |
[MATH] starts in the initial configuration [MATH] and accepts [MATH] if there is a sequence of moves ending in a configuration [MATH] with [MATH] i.e, [MATH] scans from left to right the input word starting with an empty tape and accepts upon reaching a final state provided the tape is empty. |
The language recognized by [MATH] , denoted by [MATH] , is the set of words that [MATH] accepts. The family of languages recognized by a DA is denoted by [MATH] |
Clearly, restricted types of DA correspond to QRT automata having one queue, i.e., a FIFO tape, or two stacks, i.e., two LIFO tapes with a common bottom. We illustrate Def. with some basic languages. |
Example 1 Any CF language can be implemented on a DA using either the front or the tail head, since the QRT condition does not restrict the recognition capability of nondeterministic PDA. We show in Fig. (left) a DA using the front stack for the language of palindromes |
[MATH] Then, since the front and tail stack operations are mutually independent, it is straightforward to define a DA accepting the intersection or the shuffle product ( |
[MATH] ) of two context-free languages. Moreover, any language accepted by a QRT QA (model named [MATH] in DBLP:journals/tcs/CherubiniCCM91 ) is obviously accepted by a DA that implements queue operations either in the direction front-to-tail ( ft ) or tail-to-front ( tf ). Incidentally, we observe that queue operation... |
Then, the concatenation [MATH] is easily recognized by a DA that starts as the recognizer of [MATH] then, upon reading [MATH] , switches to the transition relation of [MATH] Notice that if the deque is not empty when scanning the first [MATH] , the recognizer of |
[MATH] rejects the input. Clearly, [MATH] cannot be recognized by a nondeterministic QRT queue automaton DBLP:journals/tcs/CherubiniCCM91 (which cannot define the language of palindromes). |
At last, a bordered language DBLP:journals/acta/KariK17 such as [MATH] is easily recognized: the DA non-deterministically stores the left border in the tf queue, then recognizes a word of [MATH] using the front stack, then checks that the right border is identical to the stored border. Related examples inspired by the ... |
From the previous examples and from the properties of QRT queue automata DBLP:journals/tcs/CherubiniCCM91 we immediately have the following inclusions. |
Theorem 2.1 Family [MATH] strictly includes the families of languages recognized by QRT queue automata and by QRT double pushdown automata. |
Remarks. QRT double pushdown automata can easily define the shuffle product and the intersection of two CF languages. The shuffle product of two words contains all their interleavings, and, in the theory of parallel processes, represents all possible total orderings (i.e., serial scheduling) of two independent sequence... |
Several variants of Def. are possible, differing with respect to number of states, acceptance mode, QRT constant value, richness of operations per move, and in the structure of the tape alphabet. The following variants have the same power. |
Lemma 1 Let [MATH] be a DA with delay [MATH] as in Def. 1. realtime. There exists an RT DA equivalent to [MATH] 2. stateless. There exists a DA with one state recognizing [MATH] |
3. simple. There exists a so-called simple DA [MATH] equivalent to [MATH] such that all its moves have the form [MATH] (with [MATH] where for [MATH] [MATH] and at most one [MATH] is not [MATH] |
4. partitioned. There exists a DA [MATH] equivalent to [MATH] that uses four disjoint tape symbol alphabets, [MATH] [MATH] [MATH] and [MATH] |
resp. for operations on front stack (i.e., from front to front), tail stack (i.e., from tail to tail), front-to-tail queue, and tail-to-front queue,. i.e., for every move [MATH] of [MATH] , we have: |
[EQUATION] 5. [MATH] may have [MATH] moves testing the deque for emptiness without increasing DA’s expressive power. The proofs are standard and are in the Appendix. Space prevents detailed analysis of which normal forms can be combined together, and discussion of other accepting modes, such as acceptance by final stat... |
SDA . An example is in Def. Since at most one symbol is read or written in each move of an SDA, a shorthand for specifying moves (same in DBLP:journals/tcs/VauquelinF80 for queue automata) is convenient. We define two copies [MATH] [MATH] of [MATH] as follows: for every symbol [MATH] , a right (resp. left) pointing arr... |
[MATH] may be defined in short form as a subset of [MATH] For instance, the move [MATH] , which writes symbol [MATH] on the front, is now shortened to |
[MATH] Similarly, [MATH] , which reads [MATH] from the front, stands for [MATH] , and [MATH] is [MATH] , i.e., [MATH] is written to the front. Then, the transitions in short form of the DA for language |
[MATH] in Fig. are, without ambiguity: [MATH] Since the DA uses the front stack, the symbols are in [MATH] and should be renamed [MATH] and [MATH] |
Closure properties of [MATH] Proposition 1 The family [MATH] is an AFL, but it is not a full AFL. The proof is standard and is in the Appendix. Thus, family [MATH] enjoys a rich set of properties (e.g., closure under union, intersection with regular languages, nonerasing homomorphism, reverse homomorphism, Kleene’s sta... |
Definition 2 HoUl79 Given a language [MATH] , a homomorphism [MATH] with the property that [MATH] never maps more than [MATH] consecutive symbols of any sentence [MATH] to [MATH] is called a [MATH] -limited erasing on [MATH] A class of languages is said to be closed under [MATH] -limited erasing if, for any language [M... |
When a homomorphism [MATH] is a [MATH] -limited erasing on [MATH] , its effect is that, when applied to any word [MATH] of [MATH] , none of the factors of [MATH] of length [MATH] is entirely erased. |
2.1 Characteristic deque language and homomorphic characterization This section introduces the language that is to DA as Dyck and AntiDyck languages resp. are to PDA and to QA. The idea is that, as for simpler data structures, deque operations can be made into a terminal alphabet containing distinct copies of each oper... |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] The alphabet is [MATH] . Thus, the set of “open brackets” in [MATH] is denoted by [MATH] , the set of closing brackets of the form [MATH] as [MATH] , etc. |
The natural definition of CDL is by means of a deterministic RT DA with just one state, in analogy with the PDA recognizing the Dyck language. |
Definition 3 For each [MATH] , the characteristic deque language (CDL), denoted by [MATH] , is the language accepted by the SDA [EQUATION] |
where [MATH] and the transition relation is defined as follows (the only state of [MATH] is omitted): [MATH] Notice that this machine is deterministic, real time and “stateless”. To illustrate, word |
[MATH] is accepted with the computation: [MATH] In words, the sentences of CDL are the sequences that obey the natural schedule of deque operations. Other characterizations of CDL in terms of graphs, cancellation rules and shuffles will be given in Sect. LABEL:sect:AuerGraphs |
Using CDL we characterize deque languages [MATH] Chomsky-Schützenberger. Theorem 2.2 A language [MATH] is accepted by a deque automaton if, and only if, there exist [MATH] , a finite alphabet [MATH] , a homomorphism [MATH] a regular language [MATH] on [MATH] , and a [MATH] -limited erasing [MATH] on [MATH] such that |
[MATH] Proof By the closure properties of an AFL it is obvious that if [MATH] for some [MATH] verifying the statement of the theorem, then [MATH] can be recognized by a DA. Let now [MATH] be recognized by a DA [MATH] , which we assume to be in simple partitioned normal form. Therefore, [MATH] , where |
[MATH] and [MATH] are two disjoint copies of the tape alphabet [MATH] Since [MATH] is partitioned in [MATH] , etc., the alphabet [MATH] |
can be considered as a characteristic alphabet [MATH] , with [MATH] Define the finite alphabet [MATH] , i.e., [MATH] is the set of all quadruples in [MATH] Let [MATH] be the homomorphism defined by [MATH] for every [MATH] [MATH] Let [MATH] be the local language in [MATH] defined by the pairs of consecutive transitions ... |
(e.g., [MATH] , etc.) Let [MATH] be the projection of [MATH] on the 2nd component, i.e., [MATH] It is obvious that [MATH] is [MATH] Homomorphism [MATH] is [MATH] -erasing for [MATH] and thus for the language [MATH] , where [MATH] is the QRT constant of [MATH] , since it returns [MATH] only in correspondence to [MATH] -... |
it is possible to assume [MATH] . In fact, let [MATH] be the homomorphism defined by [MATH] , for every [MATH] , and [MATH] , for every [MATH] . Then [MATH] . ∎ |
2.2 Example of expressiveness of deque languages Deque automata have a noteworthy capability to define languages that replicate, any number of times, a factor or its reversal. We can introduce a schema for specifying replications, by means of a regular language [MATH] |
where [MATH] and [MATH] resp. stand for “direct” and “reverse”. Intuitively, a word such as [MATH] specifies that a given word [MATH] is followed by 4 replicas: [MATH] We can define languages parameterized by a replication schema, e.g., [MATH] A family [MATH] , of languages parameterized by a replication schema [MATH] ... |
# Source: arxiv 1806.06584 # Title: Spectral Functions of One-Dimensional Systems with Correlated Disorder # Sections: all # Downloaded: 2026-03-03T05:15:29.455030+00:00 |
Spectral Functions of One-dimensional Systems with Correlated Disorder Abstract We investigate the spectral function of Bloch states in an one-dimensional tight-binding non-interacting chain with two different models of static correlated disorder, at zero temperature. We report numerical calculations of the single-part... |
Introduction The one-electron spectral function is a key ingredient in the understanding of interacting and of disordered electronic systems. It can be thought of as the energy distribution of a state of momentum [MATH] |
[MATH] . In a non-interacting translationally invariant system it is simply a Dirac delta function of energy, peaked at the single particle energy [MATH] |
The spectral function has been the subject of intense study in correlated electronic systems, because it bears clear signatures of the low energy phases of interacting electron systems, whether it be a Fermi Liquid |
a marginal Fermi liquid as in high [MATH] cuprates or a one-dimensional Tomonaga-Luttinger liquid, with charge-spin separation . It is experimentally accessible by angle-resolved photo-emission spectroscopy (ARPES) |
Random disorder can introduce a finite width on the spectral function, averaged over disorder realizations, even in the absence of interactions. The more common approaches to its calculation rely on the Born approximation for the decay rate of a momentum state due to scattering by the disordered potential. They implici... |
The concept of spectral function, however, is not confined to weak disorder. Very efficient numerical methods are able to compute [MATH] |
for any strength of disorder . Trappe, Delande and Muller studied a continuum model with correlated disorder and argued that when the root mean square of the local random potential far exceeds the kinetic energy scale, [MATH] |
(where [MATH] is the spatial correlation length of the disorder), the non-commutativity of position and momentum can be ignored, and a classical limit is achieved, in which the spectral function portrays the probability distribution of the random potential. The coherent potential approximation, a well-known approximati... |
, which in its original formulation cannot account for spatially correlated disorder, has been generalized to treat spatially correlated disorder |
and can also go beyond perturbation theory and reproduce the classical limit results for strong disorder. The exquisite control that has become available in ultra cold atom experiments has renewed interest in the experimental study of disordered potentials, free of the complication of interactions, always present in el... |
included a measurement of the dependence of the mobility edge with the strength of the disordered potential The random potential implemented in ultra-cold atom experiments is correlated in space, in contrast with the standard Anderson model of site disorder. Disorder correlation studies of Anderson localization have be... |
Significant results were obtained in 1D, where it was found that extended states can exist at discrete energies in short-range correlated models |
and that a mobility edge appears in models with power-law decay of spatial correlation of the random potential Quite recently, a direct measurement of the one-particle spectral function in an ultra-cold atom experiment was reported |
By varying the intensity of the random potential, one observes a change from a perturbative lorentzian shape, to an asymmetric line shape, that reflects the probability distribution of the random potential. |
Our focus in this paper is also on the spectral function in 1D tight-binding models with correlated disorder. Unlike in the continuum case, band models have an intrinsic kinetic energy scale given by the bandwidth. It is relevant to consider whether the classical limit can be reached even when disorder is weak, in the ... |
This type of disorder has an infinite correlation length, and would appear to be always in the classical limit. Instead, we find that this limit for the averaged spectral function requires that [MATH] when the scattering really becomes local in momentum space. Our results are confirmed by numerical calculations. |
Localization properties have been studied for these power-law spectrum disorder models. While in the Anderson and other short-range correlated models, all states are localized in 1D, for these power-law spectrum models it has been claimed that a mobility edge appears for [MATH] |
This conclusion has been contested, on the grounds that in the thermodynamic limit this potential is not really disordered To investigate possible issues with the thermodynamic limit for these models, we investigated the statistical properties of the spectral function for different sized chains. We did find a transitio... |
The rest of this paper is organized as follows. In the next section, we start by defining our basic tight-binding model. Randomness is introduced in the site energies, and is characterized by its Fourier components, which have a prescribed magnitude, but randomly distributed independent phases. We then briefly review t... |
and its self-averaging properties are also discussed. Finally, in Section V we sum up our conclusions. The Disorder Model and The Kernel Polynomial Method |
2.1 The Disorder Model The Hamiltonian we use is an one-dimensional tight-binding model with nearest neighbor hopping and random site energies, |
[EQUATION] where [MATH] are the local Wannier states. In what follows, we impose periodic boundary conditions by setting [MATH] , the lattice parameter [MATH] is taken as [MATH] , and all energies are measured in units of the hopping [MATH] i.e., |
[MATH] ). If there were no disorder, the exact eigenstates of the previous Hamiltonian would be the Bloch states, defined as [EQUATION] |
The presence of static disorder causes scattering of [MATH] characterized by the matrix elements of the random potential [MATH] that connect two Bloch states, i.e. |
[EQUATION] seen here to depend only on the transferred momentum [MATH] . We easily invert Eq. to express the local energies as the Fourier sum |
[EQUATION] For the purposes of this paper, we choose to model the randomness by taking these matrix elements as [EQUATION] where [MATH] is a specified even function of [MATH] and [MATH] is a random phase with a uniform probability distribution in the circle [MATH] . The different phases are independent variables except... |
[MATH] , which ensure the hermiticity of the Hamiltonian. With these definitions, the mean of the site energies is [MATH] since the condition [MATH] fixes [MATH] and the individual phase averages are zero otherwise, [MATH] As [MATH] merely shifts the spectrum, we will always choose [MATH] , meaning that [MATH] |
In general, the values of the energies in different sites will be correlated in this model of disorder. The two-site covariance of the potential can be written as |
[EQUATION] where all the phase averages factorize (unless [MATH] ) and the average of a single phase is zero, [EQUATION] [EQUATION] |
Hence (using the property [MATH] [EQUATION] From Eq. , we see that [MATH] can be related to the Fourier transform of the spatial correlation function [MATH] |
of the disordered potential, as follows [EQUATION] In the case of an uncorrelated disorder, like in the usual Anderson’s model, we have |
[EQUATION] with [MATH] , or, equivalently [EQUATION] Thus, for these models, the magnitude of the scattering matrix element from [MATH] is independent of the transferred momentum, [MATH] |
2.1.1 Gaussian Correlated Disorder Our first model of correlated disorder is the gaussian case. For that, we choose [EQUATION] where [MATH] is a measure for the strength of disorder. The [MATH] |
factor in Eq. 12 is introduced in order to have a well-defined thermodynamic limit for the local variance and correlation functions of the disordered potential. |
In this model, the values of [MATH] are only significant inside an interval of linear size [MATH] , centered around [MATH] . This means that the disordered potential couples Bloch states with nearby momenta, more strongly . The statistical properties of the corresponding potential in the |
[MATH] limit, can be calculated through Eq. yielding [EQUATION] From these two equations, we notice that the normalized correlation function does not depend on the parameter [MATH] , i.e. |
[EQUATION] Finally, all the integrals above can be done analytically in the limit when [MATH] . In this case, the integration intervals may be extended to [MATH] and we get, |
[EQUATION] The correlation function of site energies is gaussian in real space with a decay length [MATH] . In this same limit, we can also relate the parameter [MATH] with the local disorder strength using Eq. 13a , i.e. |
[EQUATION] meaning that, [EQUATION] 2.1.2 Power-Law Correlated Disorder For our second model of disorder, we take the power-law potential defined by De Moura and Lyra |
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