Properties of 1D classical and quantum Ising models: rigorous results
In this paper we consider one-dimensional classical and quantum spin- quasi-periodic Ising chains, with two-valued nearest neighbor interaction modulated by a Fibonacci substitution sequence on two letters. In the quantum case, we investigate the energy spectrum of the Ising Hamiltonian, in presence of constant transverse magnetic field. In the classical case, we investigate and prove analyticity of the free energy function when the magnetic field, together with interaction strength couplings, is modulated by the same Fibonacci substitution (thus proving absence of phase transitions of any order at finite temperature). We also investigate the distribution of Lee-Yang zeros of the partition function in the complex magnetic field regime, and prove its Cantor set structure (together with some additional qualitative properties), thus providing a rigorous justification for the observations in some previous works. In both, quantum and classical models, we concentrate on the ferromagnetic class.
Key words and phrases:lattice systems, disordered systems, quasi-periodicity, quasicrystals, disordered materials, quantum Ising model, Fibonacci Hamiltonian, trace map.
2010 Mathematics Subject Classification:Primary: 82B20, 82B44, 82D30. Secondary: 82D40, 82B10, 82B26, 82B27.
Since the discovery of quasicrystals [53, 54, 71, 72], quasi-periodic models in mathematical physics have formed an active area of research. The method of trace maps, originally introduced in [45, 49, 59] (see also [38, 73, 80, 44, 48, 2, 66, 67] and references therein), has provided a means for rigorous investigation into the physical properties of one-dimensional quasi-periodic structures, leading, for example, to fundamental results in spectral theory of discrete Schrödinger operators and Ising models on one-dimensional quasi-periodic lattices (for Schrödinger operators: [49, 14, 74, 19, 20, 16, 18, 17, 10, 64, 22], for Ising models: [81, 11, 15, 33, 23, 12, 87, 77], and references therein).
Quasiperiodicity and the associated trace map formalism is still an area of active investigation, mostly in connection with their applications in physics. In this paper we use these techniques to investigate ergodic families of classical and quantum quasi-periodic one-dimensional Ising models.
1.1. One-dimensional quantum Ising chains
One-dimensional quasi-periodic quantum Ising spin- chains have been investigated (analytically and numerically) over the past two decades [11, 15, 33, 2, 23, 12, 87, 39, 82, 42, 40, 41]. Numerical and some analytic results suggest Cantor structure of the energy spectrum, with nonuniform local scaling (i.e. a multifractal). The multifractal structure and fractal dimensions of the energy spectrum have only recently been rigorously investigated in , and only in some special cases. In this paper we extend the results of  to cover the general case.
In the case of quantum Ising models, we investigate the energy spectrum of the Ising spin- chains with two-valued nearest neighbor couplings arranged in a quasi-periodic sequence, with uniform, transverse magnetic field. We shall concentrate on the quasi-periodic sequence generated by Fibonacci substitution on two symbols.
By mapping the single-fermion Ising Hamiltonian canonically to a Jacobi matrix and employing the techniques that we developed in , we rigorously demonstrate the multifractal structure of the energy spectrum (thus extending the results of –that is, answering [85, Conjecture 2.2]), and investigate its topological, measure-theoretic and fractal-dimensional properties. This provides analytic justification for the results that were observed through numeric experiments (and, in some cases, rather soft analysis) by some authors – we note, in particular, [11, 15, 12, 87, 23].
In fact, we consider a family of Ising models where interaction strength couplings are modulated by sequences from the hull of the Fibonacci substitution sequence, and by employing the results of , as mentioned above, we show that the spectrum is independent of the choice of a sequence in the hull.
After investigating the energy spectrum of a single fermion, we pass to topological description of the energy spectrum for many noninteracting fermions. In this case we show that for sufficiently many fermions, the spectrum is an interval whose length grows linearly as the number of particles is increased.
1.2. One-dimensional classical Ising chains
The classical Ising spin chain in one dimension with quasi-periodic interaction (nearest neighbor) and magnetic field have also been investigated both numerically and analytically (see, for example, [78, 79, 2, 3, 27, 28, 29, 33, 77, 81, 5], and references therein). A particularly curious problem is the analyticity of the free energy function in the case the interaction strength couplings as well as the external field are modulated by a quasi-periodic sequence (Fibonacci, say). Since the corresponding transfer matrices do not in general commute, this becomes a relatively nontrivial problem in comparison with the periodic case (we comment in more detail on this below, or see ). By employing the trace map as an analytic map on the three-dimensional complex manifold, we show that the free energy function is in fact analytic in the thermodynamic limit.
In fact, we consider a family of classical one-dimensional Ising models where interaction strength couplings and magnetic field are modulated by a quasi-periodic sequence from the hull of the Fibonacci substitution sequence. It is well known that the partition function can be computed as the trace of the associated transfer matrices. These transfer matrices form a cocycle over the shift map on the hull. We show that in the thermodynamic limit, the free energy is analytic, is independent of the choice of a sequence in the hull, and in fact coincides with the Lyapunov exponent of the associated cocycle.
Another problem is to describe topology of the Lee-Yang zeros of the partition function in the thermodynamic limit, when the former is viewed as a function of complex magnetic field. By the famous theorem of T. D. Lee and C. N. Yang , these zeros lie on when the couplings are ferromagnetic. Numerical experiments as well as some analysis in  suggest, among other things, that the distribution of zeros in the thermodynamic limit forms a Cantor set that exhibits gap structure similar to those that were investigated by J. Bellissard et. al. in  (see also [10, 7, 8] and ), when the sequence of interaction strength couplings is modulated by the Fibonacci substitution. This is also suggested by some results of J. C. A. Barata and P. S. Goldbaum in . Both aforementioned investigations, however, fall short of proving the postulated gap structure or fractal properties of the Lee-Yang zeros (albeit providing rather convincing heuristic arguments, in part based on numeric computation).
We prove that the Lee-Yang zeros in the thermodynamic limit do indeed form a Cantor set. In fact, we show that topological, measure-theoretic and fractal-dimensional properties of the set of Lee-Yang zeros on the circle are the same as those of the single-fermion energy spectra of quantum Ising quasicrystals, which are the same as those of the spectra of recently considered quasi-periodic Jacobi matrices.
A natural question remains: are the Lee-Yang zeros in the thermodynamic limit independent of the choice of a sequence in the hull? At present we are unable to answer this definitely, but we conjecture a positive answer.
2. The Fibonacci substitution sequence, symbolic dynamics on the hull and Lyapunov exponents of associated cocycles
Let us begin by constructing the Fibonacci substitution sequence, the hull of this sequence, and the associated symbolic dynamics (a left shift map on the hull). These constructions will play a crucial role in the rest of our investigation (to each sequence in the hull we shall associate an Ising model with interaction strength couplings and magnetic field modulated by the chosen sequence). We then investigate Lyapunov exponents of cocycles over the shift map on the hull (this will play a role in our investigation of the free energy function in the thermodynamic limit of classical quasi-periodic Ising models).
2.1. The Fibonacci substitution sequence and symbolic dynamics on the hull
Let ; denotes the set of finite words over . The Fibonacci substitution is defined by , . We formally extend the map to and by
There exists a unique substitution sequence with the following properties :
where is the sequence of Fibonacci numbers: . In fact, this sequence can be obtained constructively via the following recursive procedure. Starting with , we apply the substitution recursively:
From now on we reserve the notation for this specific sequence.
Let us remark that the sequence can also be obtained as a sampling of an irrational circle rotation on two intervals as follows. Let denote the golden mean. Define the sequence by
where denotes the set complement.
Observe that for any Fibonacci number , , the finite word contains symbols and symbols (where by we mean the th element of the sequence ).
So far is a sequence over . Let us extend arbitrarily to the left, and call the resulting sequence over , . Endow with discrete topology, and with the corresponding product topology, and let denote the left shift. Define the hull of , , as follows.
Note that there exists a sequence such that coincides with the sequence ; indeed, any limit point of the set , with the th Fibonacci number, would serve as such a sequence. From now on we fix such a sequence and reserve the notion for it.
It is easy to see that is compact and invariant under . Moreover, it is known that is strictly ergodic (that is, is topologically minimal and uniquely ergodic)—minimality follows from Gottschalk’s theorem  and unique ergodicity from Oxtoby’s ergodic theorem  and repetitive nature of ; for a comprehensive exposition see [63, Chapter 5]. Let us reserve the notation for the unique ergodic measure for on .
From now on whenever we make measure-theoretic statements about on , such as, for example, existence of Lyapunov exponents almost surely, we implicitly assume that the statement is made with respect to the measure . We shall refer to the dynamical system by , and when the measure needs to be emphasized.
2.2. Lyapunov exponents for cocycles over
Define a measurable map . For , define by
Observe that defines a subadditive process in the following sense. For any ,
According to the quite general subadditive ergodic theorem of J. F. C. Kingman (see ), the limit
exists -almost surely, is constant -almost surely, and belongs to . In the literature this limit is often called the Lyapunov exponent of the cocycle over . By historical convention, the Lyapunov exponent is often denoted by ; however, we reserve the notation for other things, and denote the Lyapunov exponent by :
In what follows, we shall be concerned with the situation where
For every , has strictly positive entries;
depends only on the first term of the sequence , hence is continuous.
Then the following theorem by P. Walters (see ) guarantees existence and finitude of the Lyapunov exponent not only almost surely, but everywhere.
Theorem 2.1 (P. Walters, 1984).
Let us mention quickly that according to A. Hof’s arguments [37, Proposition 5.2], for substitution dynamical systems, such as , generated by primitive and invertible substitutions, such as the Fibonacci substitution—see  for definitions and details— exists and is constant for all even when contains zero entries; as long as there exists such that depends only on the finitely many terms of the sequence .
3. Description of the models and main results
We begin with a discussion of the quantum Ising model on a one-dimensional integer lattice, with constant transverse magnetic field and interaction couplings modulated by a quasi-periodic sequence chosen freely from the hull of the Fibonacci substitution sequence , as constructed in the previous section.
3.1. One-dimensional quantum Fibonacci Ising model
Let , with and, unless stated otherwise, . For , set , and define a quantum Ising Hamiltonian on a lattice of size by
where is the external magnetic field in the direction transversal to the lattice. The matrices are spin- operators in the and direction, respectively, given by
where is the identity matrix. Here are the Pauli matrices given by
that we view as operators on . Thus is a linear operator on tensor-product of copies of . We shall use periodic boundary conditions.
Scaling by , by abuse of notation we can write , so that the Hamiltonian in (5) can be written as
The Hamiltonian in (7) can then be extended periodically to the infinite periodic lattice with unit cell . Let us denote the resulting Hamiltonian by .
The spin model in (7) can be transformed into a fermion model by applying the so-called Jordan-Wigner transformation (dating at least as far back as 1928 to the paper of P. Jordan and E. Wigner on second quantization ). This transformation transforms the Pauli spin operators into Fermi creation and annihilation operators, which in turn allows to diagonalize the Hamiltonian (7), and hence . The details of this method, due to E. Lieb, T. Schultz and D. Mattis, are rather technical and can be found in  (for the specialization of the results from  to our situation, see ). As a consequence, the energy spectrum of can be computed as the set of those values that satisfy the equations
where and are infinite tridiagonal periodic matrices given by
Observe that (8) is equivalent to
This equation can be written as
where for , and . Define an operator by
The operator is an infinite periodic Jacobi operator of period , and as , converges in strong sense to the operator defined by
(for more details see ). Consequently, the spectrum of can be investigated as the spectrum of , which, by Floquet theory, consists of compact intervals (see, for example, ). Let us denote the spectrum of by . We are interested in the structure of the spectrum in the thermodynamic limit (that is, as ).
Before stating one of our main results, let us quickly recall the definition of the Hausdorff metric, as a metric on and the definition of local Hausdorff dimension.
For any , define the Hausdorff metric by
Based on the connection mentioned above between and the spectrum of , one may postulate a connection between the spectrum of in (9) and the spectrum of in the thermodynamic limit. This indeed follows, since the Ising model is equivalent, via a unitary transformation, to the tight binding model in (9) (see, for example, [69, 51, 70] for an attempt to approach Ising models via these tight binding models—mostly numerically).
It was shown in  that the spectrum of is independent of . The same independence result follows for in the thermodynamic limit. It is therefore enough to study , where is as defined in Section 2.1.
Denote by the spectrum of , where is the th Fibonacci number. It was shown in , that as , the spectra of approach in Hausdorff metric the spectrum of . Again, via the aforementioned connection between the Ising Hamiltonian and Jacobi matrices, one would expect a similar result for the Ising Hamiltonian. This is indeed the case.
To avoid a rather opaque technical thicket, we shall proceed directly by explicitly constructing a compact set and showing that the sequence approaches this set in the Hausdorff metric. We then proceed to describe topological, measure-theoretic and fractal-dimensional properties of this limit set, which we call the energy spectrum of in the thermodynamic limit.
Before we continue, let us introduce the following notation. We shall denote the Hausdorff dimension of a set by . By we denote the local Hausdorff dimension of at a point :
The box-counting dimension of is denoted by .
In what follows, by a Cantor set we mean a (nonempty) compact set with no isolated points whose complement is dense.
Since a certain multifractal with specific properties will appear more than once, it is convenient to give this set a name:
We call a Cantor set a um-set (for uniform multifractal set) provided that satisfies the following properties.
The set has zero Lebesgue measure;
The function that maps to is continuous and for all and for , restricted to is non-constant (in particular, is ”uniformly”—i.e. localized at every —a multifractal);
The Hausdorff dimension of is strictly between zero and one.
If (c) fails, we call full um-set. If satisfies (a)–(c) and, in addition,
The box counting dimension of exists and coincides with its Hausdorff dimension,
then we call a moderate um-set. If satisfies (a)–(d) except (c), we call a full moderate um-set.
The terminology moderate set is not standard and, to the best of the author’s knowledge, has not appeared (at least in this context) before. This terminology is introduced here only for convenience.
Let us recall that, roughly speaking, a dynamically defined Cantor set is one which is a blowup of an arbitrarily small neighborhood of any under a map defined in a neighborhood of (here means continuously differentiable with Hölder continuous derivative whose Hölder exponent is ). For the technical definition and properties of dynamically defined Cantor sets, see [61, Chapter 4]. Let us only mention that dynamically defined Cantor sets are rather rigid. For example, the box-counting dimension exists and coincides with the Hausdorff dimension. Moreover, local box-counting and Hausdorff dimensions do not depend on the point of localization and coincide with the global dimensions.
It is convenient to give a name to a Cantor set which is, in a sense, arbitrarily close to being dynamically defined:
We call a Cantor set almost dynamically defined Cantor set provided that there exists a finite subset of (possibly empty) such that outside an arbitrarily small neighborhood of , the remainder of is a dynamically defined Cantor set.
We are now ready to state our main results for the quantum Ising quasicrystal.
There exists a compact (nonempty) set in such that the sequence converges to in the Hausdorff metric. The set implicitly depends on the choice of and , and has the following properties.
is a um-set, but not a full um-set;
With fixed, for all choices of sufficiently close to , but not equal to , is a moderate um-set, but not a full moderate um-set;
depends continuously on .
Next we discuss the topological structure of the energy spectrum for many noninteracting fermions. This can be thought of as identical Ising models on separate D lattices with no interaction between them. For noninteracting fermions, the corresponding energy spectrum is the –fold arithmetic sum of single-fermion spectra, excluding double-counting (as two fermions cannot occupy the same state). More precisely, if we denote by the –fermion spectrum, then we have
where denotes topological closure. On the other hand, since, as claimed in Theorem 3.5, is a Cantor set, it contains no isolated points. Therefore,
In this context, the term energy spectrum is synonymous with the more common excitation spectrum.
Regarding topological structure of , we have the following.
For every choice of positive and distinct , , there exists an such that for all , is the interval , where
Consequently, for ,
where is the upper bound of ; hence if , then . Moreover, for any and all sufficiently close to and not equal to , we can take .
The topological structure of for is an interesting question. At present we do not have any strong qualitative results in this direction. We believe that when and are sufficiently far, then for some , is a Cantorval—a compact set with dense interior, any connected component of the complement of which has its endpoints approximated by the endpoints of other connected components of the complement. In other words, it is in a sense a ”Cantor set with dense interior.” Sets of this nature, and their appearence in dynamical systems and number theory, have been under active investigation.
3.2. One-dimensional classical Fibonacci Ising model
We first consider the classical 1D Ising spin chain in a real magnetic field and real temperature regime. For the sake of brevity, we refer to it simply as the real regime, and in the complex magnetic field case as the complex regime.
3.2.1. Real regime
Let us consider a linear chain of spins , . The energy of a configuration is given by
with periodic boundary conditions (i.e. ). As mentioned in the introduction, we concentrate our attention on the ferromagnetic case (that is, the interaction strength couplings are positive). We take , the external magnetic field, non-negative.
The partition function of this system is given by
where the sum is taken over all possible configurations of spins. Here denotes the temperature and is the Boltzmann’s constant. The corresponding free energy is given by
expressed as a function of .
Let , and unless stated otherwise. Also, introduce with (we do not force, in general, that ). For any , , set
With these sequences of interaction strength couplings, , and magnetic field, , we obtain a classical (i.e. non-quantum) quasi-periodic Ising model in (11), which depends on the choice of (quasi-periodic in the sense that the interaction couplings and the magnetic field are modulated by a quasi-periodic sequence , and periodic in the sense that we use periodic boundary conditions). To emphasize this dependence, let us write and for the corresponding partition function and the free energy, respectively, on a lattice of size .
Let us remark that the partition function is obviously strictly positive, so that is well-defined. Regularity properties of in the thermodynamic limit (that is, as ), which we denote by (or by when dependence on needs to be emphasized), presents a curious problem in connection with critical phenomena (i.e. phase transitions). Since does not admit zeros for finite and , the free energy does not experience critical behavior in finite volume and finite temperature (this is of course expected in dimension one). In fact, for finite and , is easily seen to be analytic. Does this property survive the thermodynamic limit? It is expected that the answer is yes. In some cases can be computed analytically; in the present case, however, properties of (assuming is well-defined) were hitherto not known (see, for example, remarks in ). This brings us to
The sequence of functions converges, as , uniformly in to an analytic function of which is independent of and is strictly negative.
Our proof of Theorem 3.10 also provides a recursive algorithm for the numerical approximation of .
3.2.2. Complex regime
In this case we take constant—denote this constant by —and . For a choice of , the partition function can then be considered as a function of three variables:
(that is, and are functions of : , ). From now on we shall concentrate on fixed and variable (that is, are fixed, while the fugacity variable is varied over ).
While does not admit zeros in the real regime, it does (even in the finite volume and finite ) in the complex regime, as a function of (indeed, is a polynomial in ). These so-called Lee-Yang zeros  are guaranteed to lie on the unit circle if both couplings are ferromagnetic (that is, , which is the case considered here); similarly, in the purely antiferromagnetic case (that is, ), the zeros lie on the negative real axis; the mixed case is much more subtle and is not treated here (for further details and a more general exposition see, for example, ). We are interested here in distribution of zeros of in the thermodynamic limit, .
Let denote the set of zeros of in the variable . Some numerical experiments and soft analysis in  suggest that in the limit , accumulates in a Cantor set on , where is as defined in Section 2.1. Also, a gap structure similar to that studied by J. Bellissard et. al. in  was observed. We rigorously justify these observations, and add further detail to the qualitative description of the zeros in the thermodynamic limit in the following
There exists a compact, nonempty set such that in the Hausdorff metric. Moreover, the set has the following properties.
is a um-set together with (possibly) finitely many isolated points;
(as expected, of course);
With fixed, for all sufficiently close to , is a moderate um-set;
We believe that in statement i, there do not exist the mentioned isolated points, though at present we are unable to prove it; however, iii states that no isolated points are present provided that (assuming ).
Since in Theorem 3.12 statements are made only for the special choice , a natural question still remains: how is the conclusion of Theorem 3.12 affected as one considers general ? At present we do not have a general answer, but we conjecture that the conclusion of the theorem is not affected by any specific choice of , and, moreover, in Theorem 3.12 is independent of the choice of .
4. Proofs of main results
For proofs of main results we rely heavily on dynamics of the so-called Fibonacci trace map (to be defined momentarily), and the geometry of some dynamical invariants. We recall here only those properties of the aforementioned objects that are necessary to prove our results, and only briefly. For a survey on trace maps (including the Fibonacci trace map) associated to substitution sequences, the reader may consult, for example, [67, 66, 4]. For the necessary notions from hyperbolic and partially hyperbolic dynamics (including notation and terminology), see a brief outline in Appendix B, and references therein for more details. For model-independent results (some of which appear below), see [21, Section 2].
4.1. Dynamics of the Fibonacci trace map: a brief outline
Define the so-called Fibonacci trace map on by
It turns out that is invariant under the action of , that is,
are also invariant under , and in fact . The sets are complex-analytic surfaces.
In what follows, we sometimes consider the real part of separately (that is, the subset of ). Moreover, as far as the real part, we only need to consider for and . In this case, if , then is a smooth, connected, unbounded two-dimensional submanifold of without boundary. It is homeomorphic to the two-dimensional sphere with four points removed. If , then has four conic singularities. Minus these singularities, it is a smooth manifold with five connected components: one bounded, diffeomorphic to the two-dimensional sphere with four points removed, and four unbounded, diffeomorphic to the two-dimensional disc with one point removed.
To distinguish between the ”full complex surface” and , we denote the former by and the latter by . The surface is called the Cayley cubic.
It isn’t difficult to see that is also invariant under . In what follows, we are concerned mostly with those points in , that have bounded forward semi-orbit under the action of ; that is, such that
is bounded (we define the backward semi-orbit of , , similarly by considering iterates of in place of iterates of ). Following convention from the theory of holomorphic dynamical systems, let us define the following sets:
Regarding the topological structure of and dynamics of restricted to (clearly is -invariant) we have the following theorem. For definition of hyperbolicity, see Section B.1.
For all , is a compact Cantor set of zero Lebesgue measure. It is precisely the nonwandering set for restricted to and on is hyperbolic (the nonwandering set is the set of those points such that for every open neighborhood of and there exists such that ). Moreover, is locally maximal and is topologically transitive.
Proposition 4.2 ([85, Combination of Corollary 4.8 and Proposition 4.9]).
For any , with , is partially hyperbolic for . Thus, is a family of pairwise disjoint smooth open two-dimensional injectively immersed submanifolds of without boundary, called the center-stable manifolds. The center-stable manifolds intersect the surfaces , , transversally.
We shall also need regularity of escape rate of those points whose forward semi-orbit is unbounded (the set of such points is open in ).
Let be a domain in such that no point of has bounded forward semi-orbit under . Then the map defined by
where is the golden mean, is well-defined and analytic. Moreover, , where
where denotes projection onto the first coordinate. We assume that , which is guaranteed for sufficiently large by hypothesis.
Proof of Proposition 4.3.
Existence of the limit in (17) follows from [13, Corollary 3.3]. That the sequence is uniformly Cauchy in , and hence converges uniformly in , follows from arguments similar to those given in the proof of [6, Lemma 3.2]. Since for every , the map is pluriharmonic on , it follows that its uniform limit, , is also pluriharmonic, and hence analytic, on .
Next we state one of the main results of  in a slightly modified, more general geometric form. This result will play a central role in the proof of Theorem 3.12 below. Although proofs in  can be easily extended to cover this more general case, we invite the reader to see a discussion on model-independent results in [21, Section 2] for the technical details.
Theorem 4.4 ([86, Theorem 2.3]).
Suppose that is a compact interval and is a regular analytic curve that intersects the center-stable manifolds. Then either lies entirely on a center-stable manifold, or it intersects the center-stable manifolds transversally in all but, possibly, finitely many points. Denote the set of tangential intersections by . Let denote the set . If contains points outside of other than just the endpoints of , then the following properties hold.
is a um-set together with (possibly) finitely many isolated points if and only if does not lie entirely in some , for some ; otherwise is an almost dynamically defined Cantor set;
If , then is a moderate um-set if does not lie entirely on some , , and a dynamically defined Cantor set otherwise;
is a full um-set if and only if there exists a point in lying on .
Suppose depends analytically on a parameter . Write for the corresponding set of intersections with the center-stable manifolds. If for some , is a um-set together with (possibly) finitely many isolated points, then for all sufficiently close to , is also a um-set together with (possibly) finitely many isolated points. Moreover, if does not contain isolated points, then for all sufficiently close to , does not contain isolated points and is continuous. However, the properties of being a full um-set and almost dynamically defined Cantor set do not survive arbitrarily small -perturbations.
We are now equipped with all the tools needed to prove the main results.
4.2. Proof of Theorem 3.5
For any choice (as usual, ), define the line in parameterized by by
It was shown in  that is compact (and, of course, nonempty) and that in Hausdorff metric. Thus the first claim of Theorem 4.2 follows. We now investigate properties of . We start by noting that contains infinitely many points none of which are isolated (this can be seen from relation of with the spectrum of the tight-binding Hamiltonian in (9), which was investigated rigorously in ).
Let us remark that properties of were investigated in  in the special case that for fixed , is sufficiently close to one.
It is of course enough to consider
in place of , with . Evaluating the Fricke-Vogt invariant from (16) along , we get
where . Clearly and . Moreover, we have
Since the image of coincides with that of , it follows that is transversal to every , , and from (19) it follows that intersects each , in only one point.
The points of intersection of with the center-stable manifolds lie on a compact line segment along , since for all such that is sufficiently large, satisfies the hypothesis of [85, Proposition 4.1-(2)], and hence escapes. We are therefore working in the setting of Theorem 4.4.
Observe that intersects at , in the point
This completes the proof of Theorem 3.5.
4.3. Proof of Theorem 3.8
First we need to introduce the notion of the Newhouse thickness (see ).
Let be a Cantor set and denote by its convex hull. Any connected component of is called a gap of . A presentation of is given by an ordering of the gaps of . If is a boundary point of a gap of of , we denote by the connected component of (with chosen so that ) that contains , and write
With this notation, the thickness of is given by
Next we define what S. Astels in  calls normalized thickness. For a Cantor set , define the normalized thickness of by
A specialized version of the more general [1, Theorem 2.4] gives the following remarkable result.
Given a Cantor set , if , , then the -fold arithmetic sum of with itself is equal to the -fold sum of with itself, where is the convex hull of .
In order to apply Theorem 4.6 in our case, it is enough to show that for any positive and distinct , , we have .
For a fixed , define the local thickness at by
(that the limit exists follows easily from the fact that the thickness is decreasing with respect to set inclusion). From [86, Lemma 3.6] it follows that for any , the local thickness at coincides with the thickness of some dynamically defined Cantor set. On the other hand, the thickness of any dynamically defined Cantor set is strictly positive (see [61, Chapter 4]). Hence . This proves the first statement of Theorem 3.8.
To prove the second statement, it is of course enough to show that for fixed and all sufficiently close to , is bigger than one. By [85, Theorem 2.1-iv], we know that can be forced arbitrarily close to one by taking sufficiently close to . This, together with the following relation between Hausdorff dimension and thickness (see  for details) gives :
4.4. Proof of Theorem 3.10
Let and . Then the partition function can be written as
(recall that we are using periodic boundary conditions; that is, ). For a detailed derivation, see Theorem C.1 in Appendix C. Here we adopt the convention . Note that, unlike in the periodic case, the matrices do not commute and so cannot be diagonalized simultaneously. This is precisely what complicates the analysis, as the partition function, and hence the free energy function, cannot be computed explicitly.