ELSEVIER Physica D 107 (1997) 30--42

PHYSICA

Frenkel-Kontorova model with pinning cusps

Hsien-Chung Kao a, Shih-Chang Lee b, Wen-Jet Tzeng b,, a Department of Physics, Tamkang University, Tamsui, Taiwan 251, ROC

b Institute of Physics, Academia Sinica, Taipei, Taiwan 115, ROC

Received 13 January 1997; received in revised form 12 March 1997; accepted 26 March 1997 Communicated by J.D. Meiss

Abstract

A Frenkel-Kontorova model with a piecewise concave parabolic potential with downward cusps such that atoms could be pinned at the potential minima is exactly solved. An infinite series of first-order transitions, which may be understood as the dissociation of a large "molecule" into two smaller ones, is found as the strength of the potential increases from zero. The minimum energy configurations need not have a well-defined winding number. Given the winding number, the ground state configurations in general are highly degenerate and it is shown that one of them can be depicted by an increasing hull function. A novel type of non-recurrent minimum energy configuration, which may be viewed as defects carrying "fractional charge", exists.

PACS: 03.20.+i; 05.45.+b; 64.60.Ak Keywords: Frenkel-Kontorova model; Farey fraction; Fractional charge; Area-preserving maps; Harmless staircase

1. Introduct ion

Periodic modulated structures are quite common in condensed matter physics. In general there is a tendency for

the periodicity to lock into values which are commensurable with the lattice constant [l] . As external parameters

are changed, the system may pass through several commensurate phases with or without incommensurate phases

between them. If the commensurate phases fill up the whole phase diagram, the periodicity as a function of the

external parameter is said to form a complete staircase [2].

The Frenkel-Kontorova (FK) model provides such an example of commensurate- incommensurate transition.

This model describes a linear chain of coupled atoms in an external periodic potential. The enthalpy [3] of the

system has the form

1 H({un]) = ~n [-~(Un+l -- Un)2 "~- ~.g(un) --tY(Un+l -- Un) ] , (1)

* Corresponding author. Tel.: +886 2 7899688; fax: +886 2 7834187; e-mail: tzeng@phys.sinica.edu.tw.

0167-2789/971517.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PH S0167-2789(97)00055-9

H.-C. Kao et a l . /Physica D 107 (1997) 30-42 31

where Un is the position of the nth atom. The periodic potential )~V(u) has amplitude ~. and period l, which can

be set to 1 with suitable choice of the length scale, tr is a tensile force which allows the atomic distance between neighboring atoms to be changed continuously in the absence of the periodic potential. For a stationary configuration

one has OH/~u, , = 0 and thus

Un+l - 2un + Un-l = ~.V'(un), (2)

which can be formulated as a two-dimensional area-preserving twist map [4]. In this language, ~. can be looked

upon as a measure of the nonlinearity. One notes that in Eq. (2) the tensile force term drops out and is irrelevant in

determining the stationary configuration. Hence in deriving the system energy, this term will be discarded; however, it is indespensable in studying the phase diagram. For smooth potential V(u) , Aubry [5,6] defines a minimum-

energy configuration {Un } as the one in which H cannot be decreased by altering a finite number of un. Under this circumstance, there is a well-defined winding number (but not in our case)

U N -- U_NI ~o = lim , (3)

N,N'-+~ N + N t

which is also the mean atomic distance. A ground state is by definition a recurrent minimum-energy configuration, whose atomic positions can be characterized by [5,6]

u . = f~o(nw + or).

Here the hull function f ~ ( x ) is increasing and satisfies

f ,o(x + 1) = f w ( x ) + 1,

(4)

(5)

i.e., f ,o(x) - x is a periodic function with the period of the potential and u is a phase variable which determines the relative position of the atomic chain with respect to the periodic potential.

The FK model with the potential in a sinusoidal form and its accompanying twist map - the standard map - have been extensively studied due to their relevance to miscellaneous branches in physics, e.g., nonlinear dynamics (especially, as an example of the KAM theorem) (see, for example, reprints in [7]), condensed matter physics [1], and even accelerator theory [8]. Most of the studies have been via numerical simulation and ananlytical results are scarcely obtained [6].

Aubry first studied an interesting model with piecewise parabolic potential and obtained the ground state configu- rations as well as the phase diagram rigorously [4,9,10]. This model was also employed to study the one-dimensional ionic conductor [11-13]. The potential of this model is convex almost everywhere with cusps at the potential max-

ima. In a given ground state configuration, there can be no atom in the close vicinity of the cusps [14,15]. This property greatly facilitates the analytical analysis. If the potential is inverted, the cusps will be at the bottom of the potential wells. The effect resulting from the possibility of atoms being pinned at those cusps, as we shall see, is

drastic. Specifically, the ground state configurations are highly degenerate and the lowest enthalpy configurations need not have a definite winding number. In this paper, we would like to focus on the case with cusps at the bottom and derive the exact solution of the ground state configuration. Some results obtained here should also be applicable to any potential concave almost everywhere or with downward cusps.

The paper is organized as follows. In Section 2, the exactly solvable model is defined and some useful theorems accompanying this model are derived. The Farey fractions is also introduced there to facilitate further discussions. The ground state configuration is determined in Section 3. In Section 4, the evolution of the ground state configuration with fixed winding number is described and some low energy excitations, as well as defects, are also discussed. A specific ground state configuration which can be described using an increasing hull function is built up in Section 5.

32 H.-C. Kao et al./Physica D 107 (1997) 30--42

In Section 6 the lowest enthalpy configuration is derived and the phase diagram is presented. Part of the physical implications of these results have been reported previously [ 16] while many mathematical details were in [ 17].

2. The model

In the exactly solvable model the potential V is a scalloped (piecewise parabola) function,

V (u) = - l (u - Int[u] - ½)a, (6)

where Int[u] is the largest integer not greater than u. The average energy per atom in the ground state configuration

is given by

life(O) ~--- ( l ( U n + l -- Un) 2) n t- )~(V(un)}, (7)

where (- • .) denotes the average over the whole system. For a fixed tensile force ~r, the lowest enthalpy state is the

one with minimal ~e (~o) - aoJ. The derivative of the potential V is discontinuous at each integer. These discontinuities, unlike the case where

they are on the maxima of the potential [14], have conspicuous effect on the minimum energy configuration. First,

they locate at the bottom of the potential wells and the atoms have a tendency to approach them to lower the energy of the system. For example, as ~. ---> oo, one expects that all the atoms should sit at the cusps. Second, the atoms

located at the cusps can sustain force with magnitude smaller than 1),; namely, these atoms, instead of satisfying

Eq. (2), obey

--½)~ ~ Un+l n t -un-I --2Un < ½)v (8)

for the nth atom that is pinned. As shown below, this flexibility in assigning forces for the pinned atoms results in

a large number of degenerate ground state configurations. From Eq. (8) we know that the stationary configurations need not be symmetrical with respect to an atom pinned

at the cusps. A pinned atom thus allows us to join two different stationary configurations, each of which has at least

one atom located at the cusps. Therefore, for a configuration with u0 = 0, Uq = p and none of the in-between

atoms pinned at the cusps, we shall say that the q atoms, {u0, ul . . . . . Uq-1}, form a "p/q-section".

Consider a stationary configuration with a segment of q - 1 consecutive atoms, {u l, ua . . . . . Uq_ l }, none pinned

at the cusps. One can study the phonon spectrum of this segment with all the other atoms fixed. The energy of this

system is given by

q-1 q-I 2n({un})e~ui~uj ' on({un}) . 0 H({u'n}) = g({un}) + Z "7"-- oui + Z (9)

i= l OUi UiDUj i,j=l

where 8ui ~ u I - ui is the displacement of the ith atom. For a stationary configuration, the second term on the right- hand side vanishes. Denoting the column vector 8u = col(Sul, 8u2 . . . . . 8Uq_l), the third term can be expressed as a quadratic form, (Su)WM(Su), where M is a (q - l) x (q - l) matrix, given by

1 2 7 ~ . i f / = j , M(i , j ) = - i f / = j 4- 1, (10)

0 otherwise,

for 1 < i, j _< q - 1. One can derive that

d e t M = s i n q x (11) sin X

H.-C. Kao et a l . /Physica D 107 (1997) 30-42 33

with X = arccos(1 - ½L), 0 < ~ < 4. )~ is a monotonically increasing function of ~. and can be used, instead of

L, as the parameter characterizing the nonlinearity. As ~. increases from 0, M is positive definite and these q - 1

atoms are in stable positions until X = ~(q ~ 7E/q. At this point, there appears a zero frequency phonon mode with

3ui (x sin i X. When X > Xq, there is a phonon mode with imaginary frequency; i.e., the segment becomes unstable. Therefore, we have the following theorem.

Theorem 1. For a stationary configuration of the FK model determined by Eqs. (2), (6) and (8), there cannot be

consecutive q - 1 atoms, none pinned at the cusps, in a minimum-energy configuration for ;( > Xq.

For the convenience of the following discussions, we would like to introduce the notion of the "Farey frac-

tions" [ 18], which is defined as the collection of all irreducible fractions in [0, 1], arranged in an ascending order.

The "qth row" of Farey fractions is defined as the Farey fractions where irreducible fractions with denominator

larger than q are truncated. In a qcth row of Farey fractions, an irreducible fraction co = p / q has two nearest

neighbors, o91 = p l / q l and are = P2/q2, such that

xp = pl + p2, xq = ql + q2, o91 < w < o92 (12)

for some positive integer x and

pql = p lq + 1, P2q = Pq2 + 1, P2ql = Plq2 + x. (13)

It should be noted that for given p / q , the values of p l / q l , pz/q2, and tc are dependent on qc. For instance, assume

p / q = 3/5, then Pl /q l = 1/2, Pz/q2 = 2/3, and x = 1 forqc = 6; however, p l / q l = 4/7, pz/q2 = 5/8, and tc -- 3 for qc = 9. When qc = q, x is I and the corresponding Pi, qi, wi are denoted by p/0, q/0, coo, respectively.

From Theorem 1, we know that for Xqc+l < X _< Xqc, we only need to consider those p/q-sections with p / q in the qcth row of Farey fractions in searching for the minimum-energy configuration.

For a section begining at u0 = 0, a formal solution consistent with Eqs. (2), (6) and (8) is given by

u0 = 0, (14)

ul = so sin X, (15)

u2 = so sin 2X + ll sin X, (16)

u3 = so sin 3X + lj sin 2X + 12 sin X, (17)

q - I

Uq = so s inqx + Z l i sin(q -- i )x , (18) i=1

where so is a parameter that can be determined by ul and, for 1 < i < q - 1, I i ~ ~.(Int[ui] + ½)/sin X if the ith

atom is not pinned, and li sin X c [)~(ui - ½), )~(ui + ½)] if the ith atom is pinned. Let us consider Eq. (16) first.

If the coefficients, sin 2X and sin X, are positive, so (or u l) and l l can be regarded as increasing functions of u2 as shown in Fig. 1. Namely, if ul is pinned, then so is a constant and Ii increases with u2; on the other hand, if ul is

not pinned, then Ii is a constant and so increases with u2. Any combination of so and Ii with positive coefficients is a monotonically increasing function of u2.

Now consider Eq. (17). I f sin 3X is still positive, then the sum of the first two terms in the right-hand side is a

monotonically increasing function of u2. With the same argument as above, u2 and 12 can be regarded as increasing functions of u3. If u2 is pinned, then u2 is a constant, so are so and Ii, and 12 increases with u3; on the other hand, if u2 is not pinned, then 12 is a constant and uz increases with u3, accompanied with the increase in either so or 11.

34 H.-C. Kao et al./Physica D 107 (1997) 30-42

3.5

2.5

1.5

0.5

S O

11

j~

. , /

#l I 18d//ll

-0.5 0.0 1.0 2.0 3.0 4.0 5.0

U 2

Fig. 1. s o and II as functions of u2 with k = 1.

Therefore, so and ll are also increasing functions of u3 and any combination of s0, II, and 12 with positive coefficients

is a monotonical ly increasing function of u3.

Similar arguments can be carried on up to Uq if all the coefficients of li 's are positive. Therefore, we have the

following theorem.

Theorem 2. In a stationary configuration of the FK model determined by Eqs. (2), (6) and (8) with u0 = 0 and

X < Xq, Ur is an increasing function of Us for 0 < r < s < q.

An obvious corollary of Theorem 2 is that the p/q-sec t ions are uniquely determined for X < Xq. Furthermore,

for reducible p / q , we can find some positive integer x so that p = xpo and q = ~cqo with po/qo irreducible. K

po/qo-sections are able to coalesce into a segment with u0 = 0 and Uq ---- p. For X < Xq, this is the only way to

satisfy these two conditions, but this segment cannot be regarded as a "section" since some of the in-between atoms

are pinned. Therefore, in the remaining part of this paper, all the fractions mentioned are assumed to be irreducible.

3. G r o u n d s ta te configurations

From Eqs. (14)-(18), it is easy to see that a p/q-sec t ion with u0 = 0 and gq ---~ p is expressible as [17]

q , ( , , ) Un = I )qZ Mnq-k--Mk q -~k - -~n+k-q c ° s [ ( k - 2) X ]

k=0 " (19)

H.-C. Kao et al./Physica D 107 (1997) 30-42 35

(a)

(b)

(c)

(d)

Fig. 2. The ground state configurations for w = 3 at: (a) X -- I j r , (b) and (c) X = lzr (degenerate), and (d) X = ¼zr. Note that the

configurations in (c) and (d) is composed of 1/2- and 2/3-sections. At X = ½rr, the ground state configuration in (b) can be continuously deformed to (c). The curves describe V (u) and the black circles denote the atomic positions.

for 0 < n < q, where pq ~ tan(½g) csc(½qg), ~k is the Kronecker delta function (80 = 1 and 8k = 0 for k ~ 0) and Mk -- Int[kw] + ½ denoting which potential branch the kth atom locates at. For n = 0, the term, ~n+k-q,

should be replaced with 6n+k. The average energy per atom of a p/q-section, qJs (w), can be obtained by expressing the configuration in terms of the hull function and carrying out the integration over the phase variable as was done explicitly in [17]. The result is

q-I {~ - ( n o ) Int[n0)]) 2 } 0) 2 )~Pq ff-~cos [(k _ q ) g ] 1 ~.l)q q(s(W)-- 2 4 n=0 - 2 - - 8q c ° s ( q x ) " (20)

For any three consecutive fractions 0)1 < 0) < 0)2 in the qcth row of Farey fractions, a straightforward calculation shows that [17]

k tan ~ cot ~ _ cot qlg q2g ql~s(0)l) + q2kOs(0)2) -- xq~s(0)) = ~ _ _ _ ~ - - cot --~-, (21)

where x is defined in Eq. (12). Eq. (21) gives the energy difference between two segments of x q atoms, one composed of a 0)l-section joined with a 0)2-section and the other composed of x 0)-sections. The right-hand side is always positive for X < re/max(q, q l, q2). Therefore, qJs (0)) is a convex function of co for those 0) in the qcth row of Farey fractions. This indicates that for 0 < g < Xq, the ground state configuration with winding number 0) = p/q is given by the one composed ofp/q-sections (see Fig. 2(a) for an example). Namely, ~e(0)) = q(~ (0)) for those 0) in the qcth row of Farey fractions for gqc+l < g < Xqc.

From Theorems 1 and 2, those p/q-sections given by Eq. (19) with p/q in the qcth row of Farey fractions are all the constitutional elements of the minimum-energy configurations for gq,+l < g < Xqc. For a general winding number not in the qcth row of Farey fractions, the ground state configurations must be composed of different kinds of sections. Consider Pa/qa < PO/qb to be non-successive fractions in the qcth row of Farey fractions. Assume there are both Pa/qa- and pb/qb-sections in a stationary configuration. By interchanging them with other sections

36 H.-C. Kao et aL /Physica D 107 (1997) 30-42

and moving them rigidly, they can be placed next to each other without changing the energy of this configuration.

One can find an irreducible p/q such that x'p = Pa + Pb and x~q = qa + qb with some positive integer x and

Pa/qa < P/q < Pb/qb. If p/q is in the qcth row of Farey fractions, then transforming the consecutive Pa/qa- and pb/qb-sections into K p/q-sections will lower the energy. If not, we can find a unique pair of successive fractions

in the qcth row of Farey fractions, Pl/ql < P2/q2, such that p = tClPl + tc2P2 and q = tClql + tc2q2 with some positive integers tel and ~c2 and there must be either Pa/qa < Pl/ql or Pz/q2 < Pb/qo. From Eq. (21), transforming the consecutive Pa/qa- and PO/qb-sections into xP~cl Pl/ql-sections and x'x2 pz/qz-sections will lower the energy. Therefore, we have the following theorem.

Theorem 3. In the FK model determined by Eqs. (2), (6) and (8) with Xqc+l < X < Xqc, the minimum-energy configuration can be composed of at most two kinds of sections with their corresponding fractions being successive in the qcth row of Farey fractions.

In summary, for a given X, there exists a positive integer qc such that ~r/(qc + 1) < X < zr/qc. The minimum- energy configurations can only be composed of the w-sections with 09 in the qcth row of Farey fractions in the

following way. For an arbitrary irrational winding number 09 or rational but not in the qcth row Farey fractions, we

can find a unique pair of consecutive fractions o91 and 092 in the qcth row of Farey fractions such that o91 < 09 < 092.

The ground state configuration with this given 09 can be constructed with a fraction fl of atoms associated with the

Wl-sections and a fraction f2 associated with 092-sections such that

fl -3-f2 = 1, (22)

and

f lO) l "{'- t"2°92 = 09. (23)

More specifically, the ground state configuration is composed of wl- and w2-sections with their abundance in the

ratio flq2 : f2ql. The average energy per atom in this configuration is given by

0)2 - - 0 9 0 9 - - 0 9 1 t/)'e (09) - - - - 1/-/9 (O91) q - - - I/-/s (092). ( 2 4 )

O92 - - 091 O92 - - 091

Eq. (21) then indicates that qJe (w) is a convex piecewise linear function ofw with kinks at those w's in the qcth row

of Farey fractions. Here we would like to elaborate on the ground state configurations and the minimum-energy configurations. For

a given ,k and an accompanying qc, if the winding number is fixed at a value not in the qcth row of Farey fractions,

then the ground state configurations are highly degenerate and there are no other distinguishable minimum-energy configurations. If the winding number is fixed at a value, o9, in the qcth row of Farey fractions, the ground state configuration is composed solely of w-sections. However, according to the definition in [6], a minimum-energy configuration allows of a finite number of either 601-sectiOns (solitons) or w2-sections (anti-solitons), but not a mix of them, being created, in consistent with Theorem 3.

It is interesting to note that a configuration composed of two kinds of sections with fractions consecutive in the qcth row of Farey fractions for Xqc+l < X <_ Xqc must be a minimum-energy configuration. One is then able to construct minimum-energy configurations without well-defined winding numbers. For example, one can, starting from u = 0 to both sides, put 2 0 091-section, then adding 2 1 092-sections, then 2 2 Wl-sections, 2 3 092-sections . . . . . and so forth. The winding number defined in Eq. (3) does not have a well-defined value for this configuation. Therefore, some of Aubry's results [6] for a minimum-energy configuration do not apply in this case.

H.-C. Kao et al./Physica D 107 (1997) 30-42 37

4. Discussions

It is instructive to learn how the ground state configuration with winding number p/q evolves as X increases just

over Xq. For 0 < X <- Xq, the ground state configuration with winding number p/q is composed of p/q-sections (see Fig. 2(a) for an example). From Theorem 1, we know that these p/q-sections will lose its stability for X > Xq. At X = Xq, there is a zero frequency phonon mode assosiated with a p/q-section with displacements of each atoms,

{&un}, given by

?/2"/" &Un = a sin - - , (25)

q

where the amplitude a is restricted so that the in-between atoms, {Ul, u 2 , . . . , Uq-1}, can only barely touch the cusps and Eq. (2) still holds for them. The extreme case happens when either UqO = pO or UqO = pO. In either case,

say UqO = pO, the qO atoms from u0 to Uq20_, is exactly a p°/q°-section, i.e., a consecutive qO atoms chain with

the first atom sitting at the cusp, while the q0 atoms from UqO to Uq-1 is a p°/q°-section (see Figs. 2(b) and (c)

for an example). For a general o9 = p/q, if we regard a p/q-section as a "molecule" of size (q, p), i.e., composed of q atoms and p wells, then gq can be considered as a critical point of a first-order transition when a molecule of size (q, p) is just about to break up into two molecules of sizes (q0, p0) and (qO, p0), respectively. There the

three types of molecules, whose corresponding o90, to, and o9 o are related as consecutive Farey fractions in the qth row of Farey fractions, coexist. Moreover, the zero frequency phonon mode allows us to continuously deform the

co-sections into degenerate configurations.

It should be noted that at this critical point, the cusp can barely provide the force to pin the atom at the conjunction between coo_ and og°-sections. For ~( just below Xq, the cusps are not able to provide the force needed to pin the atom at this conjunction. If we put an o90_ and an og°-section next to each other forcibly, then this non-stationary

configuration will automatically evolve to a og-section once the external force on the conjunction atom is released.

On the other hand, when X increases just above Xq, the co-section becomes unstable and any small perturbation will cause one more atom to roll down to the cusp in each period, resulting in a o9 o- and 0 a o92-sections. Therefore, the molecules of size (q, p) completely dissociate and the ground state now is a mixture of o90_ and og°-sections (see

Fig. 2(d) for an example) with the right proportion, given by Eqs. (22) and (23). One should note that there is a great deal of degeneracy due to the arbitrary ordering of these two kinds of sections.

It is interesting to observe that for Xq < ~( < )(q = n~ max(q °, q~), even though p/q-sections no longer

describes the ground state configuration, it continues to evolve as an unstable configuration until X reaches ~q. Hence molecules of size (q, p) may be regarded as an unstable resonance state of two smaller molecules of sizes (q0, p0) and (qO p0) when g lies in the interval (Xq, f(q).

In summary, as g increases from O, the ground state configuration with winding number p/q is composed of p/q-sections. As X increases above Xq, the configuration evloves to the one composed of o90_ and og2°-sections. As X increases further, co O- and og°-sections will evolve into sections with winding numbers equal to o90 and og°'s

neighboring fractions in the qlth and q2th row of Farey fractions, respectively, when Xql and Xq2 are reached. It should be noted that, assuming q0 > qO, as g increases over XqO, the molecule of size (ql 0' pl 0) will dissociate

into two molecules of respective sizes, (qO, pO) and (q0 _ q02, p0 _ p0). Therefore, Theorem 3 still holds after the dissociation process. This process continues until all the atoms are pinned at the cusps; namely, the ground state configuration consists of 0/1- and 1/1-sections. This occurs at g > ½re or ~ > 2.

From the view point of an area-preserving twist map, as X increases above gq, minimizing cycles of period q split into two cycles of periods ql and q2, respectively, in a manner dictated by the Farey fractions.

38 H.-C. Kao et aL/Physica D 107 (1997) 30-42

In this system, discommensurations will be relaxed via replacing some w I -sections with og2-sections or vice versa. Therefore, soliton-like distortion can be confined in a finite region. This is contrary to the case when the potential

is convex almost everywhere [10]. More specifically, adding or removing one atom from the system will turn pl

"w2-section" into p2 "ogj-sections" or vice versa. This can be seen from p lpz (1 /Wl - 1/oge) = 1. Starting from

pure o92-sections, i.e., the ground state configuration with winding number we, we can approach the ground state configuration with winding number wl by adding atoms one by one. Solitons and anti-solitons in the usual sense of local "defects" do not exist. Efforts to create them merely induce transitions between wl - and o92-sections. If we

insist on calling a col -section in a background of o92-sections "defect", this "defect" will carry a "fractional charge"

1/p2. Consider the low energy excited configurations for this system. A molecule of size (ql + q2, Pl + P2) is an

unstable resonace state. Stable excitations can be created by the introduction ofwo- or o93-sections with o90 < ogl <

o92 < w3, being successive in the qcth row of Farey fractions. These o9o- and w3-sections are locally stable if they

are, respectively, surrounded by col- and wz-sections. These are the "fractionally charged" molecules mentioned above. They will be annihilated when a coo-section (w3-section) encounters a we-section (Wl-section).

To keep the winding number fixed, creation of a o90-section must be accompanied by the creation of a we-section

and the annihilation of K1 COl-sections with Kr =- (qo + q2)/ql . The excitation energy accompanying the creation

of a wo-section is given by )~ tan(½x) cot(½qox) cot(½ql X)~ cot(½qzX). Similarly, that of a o93-section is given by

~. tan (½ X) cot(½ q l X) cot(½ q2 X ) ½ cot(½ q3 g). Therefore, the lowest energy excitation is determined by which of qo

and q3 is larger.

5. The hull function

In [6] Aubry derived a theorem, stating that in a ground state configuration with winding number 09, there exists

a phase ot such that

Int[nw + a] < Un < Int[no9 + or] + 1 (26)

holds for every atom. This theorem holds for any smooth potential. In the model under consideration, for some co there

can be degeneracies obtained by permutations of the ogl- and o92-sections, which does not cost energy. Therefore, Eq. (26) is easy to violate. Nevertheless, there is always some way to modify the cusps in an arbitrary small region

so that the potential becomes smooth. In this case, Eq. (26) should hold for the ground state configuration. Indeed, it is possible to arrange these two sections in an appropriate way so that Eq. (26) still holds and one can use a hull function, denoted by fo~(x), to describe the atomic positions through Eq. (4). It is natural to conjecture that this ground state configuration can be carried over, with minor modifications, to the system with a modified smooth

potential. A configuration composed solely of p / q (---- og)-sections trivially satisfies Eq. (26). There are only q allowable

positions in each period of the potential, hence the hull function is made up of plateaux with widths 1/q. One can thus fix the phase variables to discrete values; i.e., x = i /q . The hull function, f~o(x), can be constructed as

f W ( q ) = U n - I n t [ u n ] , (27)

with i = n p - q Int[nog] for i, n = 0, 1, 2 . . . . . q - 1 and Un is defined in Eq. (19). For the ground state configuration with winding number o9 and composed of o91- and w2-sections, define vi =

foJl (iogl) and wi = f~o2 (io92). The hull function fo~(X) should have ql + q2 - 1 plateaux between f~o(0) = 0 and

H.-C. Kao et a l . /Physica D 107 (1997) 30-42 3 9

foo (1) = 1. Note that, if for every n the atom at Un is shifted by an integer value - In t [un ] to the interval [0, 1 ), the

width of each plateau of f(o(x) taking value in [0, 1) equals the fraction of atoms shifted to that position.

Assuming ql > q2, one has A _~ Wq2 -- Vq2 = P2 -- fwj (q2Pl/ql) > 0. From Theorem 2, one has wi > vi and

P2 - Wq2--i > Pl -- Vq~_i for 0 < i < q2. More specifically, one can derive from Eq. (2) that

sin i X tO i - - V i = A - - ( 2 8 )

s i n q 2 X

and

s in ix 1)ql-i -- ll)q2--i = Pl -- P2 + A - -

sin q2x

for 0 < i < q2. Similar relations hold if qi

one can prove that

(29)

< q2. With fo,, ( i /qt) and f~o2 (i/q2) being increasing functions of i,

( ~ 1 ) ( j ) i j f~ol > fo~2 i f - > - - , " qi q2

( ~ 1 ) ( j ) i j f~ol < fw2 i f - < - - , qt q2

( ~ 1 ) ( j ) i j f,,l = fa'2 if - - = - - , ql q2

(30)

for X < min(Xq~, Xq2) and the last case occurs only at integer value of i/ql and j /q2. To construct the hull function with this winding number w, one should connect these plateaux so that the hull

function is increasing. After assigning widths f l /ql to each step f,o = fo)t ( i /qD for i = 1, 2 . . . . . q] - 1, f2/q2 to

each step fo, = f~oz(J/q2) for j = 1, 2 . . . . . q2 - 1, and (f l /ql + f2/q2)/2 tO fw ----- fo~l (0) = f ~ ( 0 ) = 0 as well as f~o = f,o~ (1) = f~o2 (1) = 1 with ft and f2 defined in Eqs. (22) and (23), one can construct fo)(x) by coalescing

these q I + q2 steps in an order of ascending magnitude into 0 < x < 1. This hull function fo,(x) describes, through

Eq. (4), a specific ground state configuration with winding number w, which satisfies Eq. (26). Fig. 3 shows how

this hull function evolves as X increases for w --- J r /2 . For X = Jr/100, zr/20, ~r/6, 2zr/5, and 2zr/3, one has

o)t = 0 / 1 , 0 / l , 1/3, 6/17, 35/99 and a)2 = 1/1, 1/2, 2/5, 5/14, 29/82, respectively.

6. Phase diagram

From the enthalpy of the system, given in Eq. (1), one can derive the w(~r), which specifies the winding number

a) of the lowest enthalpy configuration for each or, to reveal how the periodicity of the atomic chain locks into that

of the external potential. For 0 < X < Jr, one can always find a positive interger qc such that gqc+l < X <_ Xq,.. From Eqs. (21) and (24), one finds that the average energy per atom, qJ~ (o9), for the ground state configuration with

winding number o) is a convex piecewise linear function of a) with kinks locating at winding numbers belonging

to the q,:th row of Farey fractions. Except for cr belonging to a set of zero measure, composed of the slopes of the

linear segments of the function tPe(W), only those winding numbers in the qcth row of Farey fractions need to be

considered for the lowest enthalpy configuration. This suggests that o)(cr) should be a complete harmless staircase. The phase diagram is given in Fig. 4.

For X < Xq, the lowest enthalpy configuration is locked to w = p/q for

t/./s ( 0 ) ) - - t f f s ( O ) l ) I/is ( 0 ) 2 ) - - t/.ts ( 0 ) ) < (7 < , (31)

o) - - (Ol o92 - - (.o

4 0

1.5

1.0

0.5

0.0 -0.5

H.-C. Kao et al./Physica D 107 (1997) 30-42

° ~ i

X=2~3

. . . . . . . . . . . 2n/5

. . . . . . ~ 6 !

I , j n / 2 0 . . . . .

/ S O 0 i .--,---:

: - . . . . ;

I - - t , 'd ,.....,: _ . / / / / 1 , I J ." ..... / / ! i : . . . . / i 1 1 0 I I , - - ' ;" ..... / / " , I , , ...... . . / / /

r . . . . . . . t ,I- 1 ..... " / - / " t .; j / / / /

- - ~ I / / 1 r - . . . . . . . . i - - - ,....~ .t "a- /

I : 1 s f ' ~ / 1

" - - - - . . . . . . ) ..... / 1 " 0.-: f 1 /

/ / / .

/ i i i i I ~ i i i , ,i | m i 1 i

0.0 0.5 1.0

X

Fig. 3. The hull func t ions for the recurrent g round state configurat ion with wind ing n u m b e r (o = ~ ~/2. For the sake of clarity, the origin is shif ted to a cor responding closed circle for each X.

where O)] < o9 < O)2 are consecutive fractions in the qcth row of Farey fractions with Xqc+] < X <- Xqc. The

equation for the coexistent curve of the two phases corresponding to molecules of sizes (ql, p l ) and (q2, P2) is

given by ~r = [q6 (o)2) - qJs ((Ol)]/(o)2 - O)1 ) for o91 and O)2 being two consecutive Farey fractions.

At the coexistent curve of the two phases corresponding to molecules of sizes (ql, Pl) and (q2, P2), different

proportions of these two kinds of molecules will lead to varieties of configurations with the same average en-

thalpy per atom but different winding numbers. The winding number of this lowest enthalpy configuration is thus

not uniquely specified. Moreover, as discussed in Section 3, the minimum energy configuration without a well-

defined winding number also has lowest enthalpy in this case. Therefore, a lowest enthalpy configuration need

not have a well-defined winding number. Nevertheless, the set of all ~r with this kind of property is of measure zero.

When X = Xq, the step given by Eq. (31) shrinks to a point and we have

IPs(O) ) - - I/-/s (O)1) tPs(O)2 ) - - Ills(O) ) ~r = = , (32)

O9 - - O)I O)2 - - O)

which gives the equations for the triple point, where three types of molecules, (ql, Pl), (q, P), and (q2, P2), co-exist. Actually, at this triple point, there is a non-denumerable degeneracy due to the fact that the larger molecule (q, p)

can be deformed into (ql, Pl) and (q2, P2) molecules continuously, as discussed in Section 4.

H.-C. Kao et al./Physica D 107 (/997) 30-42 4l

1.0

2Z/ ;

0.5

0/1 1/1

1/5 4/5

/ \ i

0°0 t I

0.0 0.5 (Y .0

Fig. 4. The domains of stability in the X-tr plane. The number in each domain denotes its winding number.

7. Conclusions

Piecewise concave potential with pinning cusps could be a good first approximation to model some potentials occurring in nature. We showed that FK models (or, equivalently, area preserving twist maps) with pinning cusps

exhibit many interesting physical phenomena which did not show up in the previously studied model. In particular,

as the potential strength increases from zero, an infinite series of first-order phase transitions occurs. These phase transitions are not nucleated by solitions but rather resemble the dissociation of a molecule into two smaller ones. Minimum-energy configurations in such models need not have a definite winding number. For a given winding number, the ground state configurations are often highly degenerate and increasing hull functions are found only

for a particular ground state configuration. The low lying excitations consist of"fractionally charged" defects rather

than the usual solitons. The generality of FK models suggests that the interesting physical phenomena that we

discussed could very well be observed in nature. The sharp contrast between the FK models with an almost everywhere concave potential with pinning cusps and

those with an almost everywhere convex potential with repelling cusps [10] suggests that we investigate FK models with potential interpolating between the two extremes. This is currently under study.

Acknowledgements

The authors would like to express their gratitute to Prof. R.B. Griffiths for stimulating discussions and valuable comments. This work is supported in part by grants from the National Science Council of Taiwan-Republic of China under Contract No. NSC-86-2112-M001-003.

42 H.-C. Kao et al./Physica D 107 (1997) 30-42

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