Ground states of one-dimensional long-range ferromagnetic ising model with external field

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Vol. 26, No. 3 (2012) 1150014 (6pages) c

World Scientific Publishing Company DOI:10.1142/S021798491150014X

GROUND STATES OF ONE-DIMENSIONAL LONG-RANGE FERROMAGNETIC ISING MODEL WITH EXTERNAL FIELD

AZER KERIMOV

Bilkent University, Department of Mathematics, 06800 Bilkent, Ankara, Turkey

kerimov@fen.bilkent.edu.tr

Received 16 August 2011 Revised 6 October 2011

A zero-temperature phase-diagram of the one-dimensional ferromagnetic Ising model is investigated. It is shown that at zero temperature spins of any compact collection of lattice points with identically oriented external field are identically oriented.

Keywords: Hamiltonian; ground state; Gibbs state; phase transition. Mathematics Subject Classification (2000): 82B20, 82B23

1. Introduction

Consider the one-dimensional ferromagnetic Ising model with long range interac-tion: H(φ) = − X x,y∈Z1_;x>y U(x − y)φ(x)φ(y) − X x∈Z1 hxφ(x) , (1)

where spin variables φ(x) associated with the one-dimensional lattice sites x take

values −1 and +1, the pair potential U (x − y) = (x − y)−γ, γ > 1 and {hx, x∈ Z1}

is an external field. The condition γ > 1 is necessary for the existence of the

thermodynamical limit. If γ > 2 then P_x∈Z1_,x>0xU(x) < ∞ and the model (1)

has a unique Gibbs state1–3 _{at any non-zero value of the temperature.}

In the absence of the external field (hx ≡ 0) the model exhibits a phase

tran-sition. Suppose that a positive decreasing potential U (r) = U (|x − y|) satisfies the

conditions: P∞

r=1U(r) < ∞ and

P∞

r=1(ln ln(r + 4)) (r

3_U_(r))−1_<_{∞ (the model}

(1) with 1 < γ < 2 readily satisfies these conditions). Then there exists a value

of the inverse temperature β1 such that if β > β1 then there are at least two

ex-treme Gibbs states P+ _{and P}− _{corresponding to the ground states φ(x) = +1 and}

φ(x) = −1.4,5 _{This result is related to the phenomenon of surface tension in}

one-dimensional models. The phase transition also takes place in the borderline case

Mod. Phys. Lett. B 2012.26. Downloaded from www.worldscientific.com

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γ= 2.6–8 _{In the case of unbiased random external field the model (1) exhibits low}

temperature phase transition with probability one.9

In this paper we investigate the phase diagram of the model (1) under nonzero external fields.

The following two results show that if the absolute value of the external field is

sufficiently strong or the aligned vectors hxare “well organized” and constitute long

blocks, then the external field exterminates the pair interaction and the dependence on the boundary conditions disappears in the limit:

Theorem 1.10 _{At any fixed value of the inverse temperature β there exists a}

con-stant h0 such that for all realizations of the external field {hx, x ∈ Z1} satisfying

|hx| > h0, x∈ Z1 the model (2) has at most one limiting Gibbs state.

Theorem 2.11 Let hx be a periodic function of period 2r: hrx = hx+2rk for all

integer values of k and for some sufficiently small fixed positive ǫ.

hr

x= (

+ǫ if x= 1, . . . , r ,

−ǫ if x= r + 1, . . . , 2r .

There exist natural numbers R1 = R1(ǫ) and R2 = R2(ǫ) such that at all

sufficiently small temperatures the model (1) has at least two limiting Gibbs states

for all r ≤ R1 and at most one limiting Gibbs state for all r > R2.

Thus, if the absolute value of the external field is not sufficiently strong and

the aligned vectors hxdo not constitute long blocks then the set of limiting Gibbs

states of the model (1) may have one or more than one element.11

2. The Structure of Ground States

In this section, we investigate the set of ground states of the model (1), where h > 0

and hx= ±h.

A configuration φgr_{is said to be a ground state of the model (2), if for any finite}

set A ⊂ Z1

H(φ′_{) − H(φ}gr_{) ≥ 0, where φ}′ _{is a perturbation of φ}gr _{on the set A.}

Let us define a configuration ϕh by ϕh(x) = sign(hx). The ground state

config-uration φgr _{results ferromagnetic “struggle” between spins of ϕ}

h.

For integer a and b, [a, b] denotes the set of all integers lying between a and b, including a and b. Let φ be any fixed configuration. We say that an interval [a, b] with integer endpoints is a block if ϕ([a, b]) is a constant and for any other interval [c, d] with integer endpoints satisfying [a, b] ⊂ [c, d], ϕ([c, d]) is not a constant (the case a = b is not excluded). The collection of all blocks of φ we order in the following

way: 0 ∈ [a0, b0] and for each integer i bi+ 1 = ai+1. Thus, Z1=S∞<i<∞[ai, bi].

The block [ai, bi] of the configuration φ we denote by ∆i(φ).

The blocks ∆(φh) of the configuration ϕh we call h-blocks.

It is known that that the restriction of a ground state to any “long” block

coincides with ϕh: There is a constant L such that for all h-blocks ∆(φh) with

lengths exceeding L, φgr_(∆(φ

h)) = ϕh(∆(φh)).11

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It turns out that the restriction of a ground state to any block is a constant configuration.

Theorem 3. Let φgr _{be a ground state of the model} _{(1) with arbitrary decreasing}

pair potential U(·) and ∆(φh) be a h-block. Then φgr(∆(φh)) = ϕh(∆(φh)) or

φgr_(∆(φ

h)) = −ϕh(∆(φh)).

Proof. Suppose that there is a h-block ∆(φh) = [ak, bk] such that φgr(∆(φh)) 6=

ϕh(∆(φh)) and φgr(∆(φh)) 6= −ϕh(∆(φh)). Then without loss of generality, we

can suppose that ϕh([ak, bk]) ≡ 1 and there are two points z0and z0+ 1 both from

[ak, bk] such that φgr(z0) = 1 and φgr(z0+1) = −1. Let {∆l(φgr); l ∈ Z1} be the set

of all blocks of φgr_{. By definitions, z}

0and z0+ 1 belong to two neighboring blocks of

φgr_{: for some l}

0there are blocks ∆l0(φ

gr_{) = [a}

l0, bl0] and ∆l0+1(φ

gr_{) = [a}

l0+1, bl0+1]

such that bl0 = z0and al0+1= z0+ 1.

Let us define two configurations: e φgrz0 = ( −φgr _{if x = z} 0, φgr _{if x 6= z} 0, and e φgrz0+1= ( −φgr _{if x = z} 0+ 1 , φgr _{if x 6= z} 0+ 1 .

Since φgr _{is a ground state, H( e}_φgr

z0) − H(φ gr_{) ≥ 0 and H( e}_φgr z0+1) − H(φ gr_{) ≥ 0.} Therefore, H( eφgrz0) − H(φ gr ) + H( eφgr_z₀+1) − H(φ gr ) ≥ 0 . (2)

It can be readily seen that

H( eφgr z0)−H(φ gr_{) = 2h+2} ∞ X i=0 (−1)i X x∈∆_l0−i U(bl0−x)+2 ∞ X j=1 (−1)j X x∈∆_l0+j U(x−bl0) , (3) and H( eφgr_z₀+1) − H(φ gr ) = −2h + 2 ∞ X i=0 (−1)i+1 X x∈∆_l0−i U(al0+1− x) + 2 ∞ X j=1 (−1)j+1 X x∈∆_l0+j U(x − al0+1) . (4)

The sum of (3) and (4) after canceling of opposite terms yields

H( eφgrz0) − H(φ gr ) + H( eφgr_z₀₊₁) − H(φgr) = 2 ∞ X i=0 (−1)i+1U(al0+1− al0−i) + 2 ∞ X j=1 (−1)jU(bl0+j− bl0) .

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Now by pairing of neighboring terms in both sums we get H( eφgr z0) − H(φ gr_{) + H( e}_φgr z0+1) − H(φ gr_{) = 2} ∞ X i=0 Ai+ 2 ∞ X j=1 Bi,

where Ai = U (al0+1− al0−2i−1) − U (al0+1− al0−2i) and Bj = U (bl0+2j− bl0) −

U(bl0+2j−1− bl0).

Finally we note that since the potential function U (·) is decreasing, all terms Ai

and Bj of both sums are negative and consequently H( eφ

gr

z+1) − H(φgr) + H( eφgrz ) −

H(φgr_{) < 0, which contradicts (2). The proof of Theorem 3 is completed.}

By Theorem 3, any ground state φgr _{can be obtained by reversing of all spins}

of some not long h-blocks of φh. In other words, spins of φgrbelonging to the same

h-block are strongly correlated and change their values synchronously. This fact considerable simplifies the structure of the zero-temperature phase diagram of (1).

The following theorem shows that if in a ground state spins of some h-block of φh

are reversed, then the spins of its neighbor blocks are not reversed:

Theorem 4. Let ∆1(φh), ∆2(φh) and ∆3(φh) be three consecutive h-blocks.

Sup-pose that φgr_(∆

2(φh)) = −ϕh(∆2(φh)). Then φgr(∆1(φh)) = ϕh(∆1(φh)) and

φgr_(∆

3(φh)) = ϕh(∆3(φh)).

Proof. Without loss of generality we suppose that hx = −h for all x ∈ ∆1(φh),

x∈ ∆3(φh) and hx= h for all x ∈ ∆2(φh). Let ∆1(φh) = [ak−1, bk−1], ∆2(φh) =

[ak, bk], ∆3(φh) = [ak+1, bk+1] and at least one of the relations φgr(∆1(φh)) =

ϕh(∆1(φh)); φgr(∆3(φh)) = ϕh(∆3(φh)) is not held. Then without loss of generality

we suppose that φgr_(a

k+1) = 1. Let z0= bk and z0+ 1 = ak+1. Thus, φgr(z0) = −1

and φgr_(z

0+ 1) = 1. Let {∆l(φgr); l ∈ Z1} be the set of all blocks of φgr. By

definitions, z0and z0+ 1 belong to two neighboring blocks of φgr: for some l0there

are blocks ∆l0(φ gr_{) = [a} l0, bl0] and ∆l0+1(φ gr_{) = [a} l0+1, bl0+1] such that bl0 = z0 and al0+1= z0+ 1.

As in the proof of Theorem 3, we define configurations eφgr

z0 and eφ

gr

z0+1. Since φ

gr is a ground state, (2) holds. It can be readily seen that

H( eφgrz0) − H(φ gr_{) = −2h + 2} ∞ X i=0 (−1)i X x∈∆_l0−i U(bl0− x) + 2 ∞ X j=1 (−1)j X x∈∆_l0+j U(x − bl0) , (5)

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and H( eφgrz0+1) − H(φ gr_{) = −2h + 2} ∞ X i=0 (−1)i+1 X x∈∆_l0−i U(al0+1− x) + 2 ∞ X j=1 (−1)j+1 X x∈∆_l0+j U(x − al0+1) . (6)

Again as in the proof of Theorem 3, we get that

H( eφgrz0) − H(φ gr_{) + H( e} φgrz0+1) − H(φ gr_{) = −4h + 2} ∞ X i=0 Ai+ 2 ∞ X j=1 Bi

where Ai = U (al0+1− al0−2i−1) − U (al0+1− al0−2i) and Bj = U (bl0+2j − bl0) −

U(bl0+2j−1− bl0).

Now since the potential function U (·) is decreasing, all terms Ai and Bj of both

sums are negative and consequently H( eφgrz+1)− H(φgr)+ H( eφgrz )− H(φgr) < −4h <

0, which contradicts (2). The proof of Theorem 4 is completed.

In the case when the external field takes three values ±h and 0 one can define

φhby

φh(x) =

(

sign(hx) if hx6= 0 ,

0 if hx= 0 ,

and set h-blocks ∆(φ) as above. In this case the structure of a ground state is also analogous to the above-mentioned ones:

Theorem 5. Let φgr _{be a ground state of the model}_{(1) and ∆(φ}

h) be a h-block.

Then φgr_(∆(φ

h)) is a constant.

The proof is analogous to the proof of Theorem 3 and is omitted.

If the ground state is unique and stable then at sufficiently low values of the

temperature the limiting Gibbs state is also unique.12 _{In general, the investigation}

of the non-zero temperature phase diagram of the model (1) is a rather complicated problem.

Acknowledgments

The author thanks the referees for their corrections and useful suggestions. References

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