Ground state criteria in one-dimensional
antiferromagnetic Ising model with long
range interaction
Cite as: J. Math. Phys. 61, 043301 (2020);doi: 10.1063/5.0001423 Submitted: 18 January 2020 • Accepted: 26 March 2020 • Published Online: 15 April 2020
Azer Kerimova) AFFILIATIONS
Department of Mathematics, Bilkent University, Ankara, Turkey
a)Author to whom correspondence should be addressed:kerimov@fen.bilkent.edu.tr
ABSTRACT
One-dimensional long-range Ising models with antiferromagnetic, convex pair interactions are investigated. A new criterion characteriz-ing ground states is given. The criterion and a new transformation yield short proofs identifycharacteriz-ing and characterizcharacteriz-ing the ground states. The uniqueness of periodic ground states up to shifts is shown.
Published under license by AIP Publishing.https://doi.org/10.1063/5.0001423
I. INTRODUCTION
We consider the one-dimensional antiferromagnetic Ising model with the Hamiltonian
H(φ) = ∑
x,y∈Z1,x>y
U(x − y)φ(x)φ(y) − μ∑ x∈Z1
φ(x), (1)
where the spin variables φ(x) = ±1 and μ ∈ R is a chemical potential. The antiferromagnetic potential U(⋅) > 0 satisfies the following conditions:
(1) U(x + y) + U(x − y) > 2U(x) for all x, y ∈ Z1,x > y. (2) ∑∞
x=1U(x) < ∞.
The first condition is a convexity condition, which, to a large extent, determines the structure of the set of ground states. The second condition is necessary for the existence of the thermodynamic limit.
The initial studies of model (1) have been made in Refs.1–3. It was established that if the configuration satisfies Hubbard’s crite-rion,1then it is a periodic ground state for some chemical potential. The configurations satisfying Hubbard’s criterion were constructed in Ref.4. The existence of configurations satisfying Hubbard’s criterion ensures that the criterion is also a necessary condition for periodic ground states. It turns out that these periodic ground states are unique up to translations.5Ground states satisfying Hubbard’s condition in the context of non-periodic long-range order, instead of periodic long-range order, are recently investigated in Ref.6. It is well known that the models with potentials satisfying∑x∈Z+xU(x) < ∞ have a unique Gibbs state.7–9The uniqueness of Gibbs states of (1) for a wider class of
potentials was established in Refs.10–12.
In this paper, we formulate a criterion equivalent to Hubbard’s criterion. Particularly, the criterion allows us to present a very short construction of periodic ground states and prove the uniqueness of those periodic ground states of model (1). Surprisingly, the presented new proof of uniqueness does not even use the structure of ground states.
A configuration φ is a function Z1→ {0, 1}. Let a rational number q/p be fixed. The set of all periodic configurations with period p and ∑x+p
y=x+1φ(y) = q will be denoted byΦ per
η(φ) is an average distance between neighboring points with φ(x) = 1. For each φ ∈Φperp,q, h(φ) =1 p x+p ∑ y=x+1 ϕ(y)∑ z>0 U(z)φ(y + z), not depending onx, which is the mean energy of φ.
II. GROUND STATE CRITERION
The following definition is useful for the zero temperature phase diagram of model (1). A configuration φ0∈ Φperp,qis a periodic ground state of (1) if
h(φ0) = min
φ∈Φperp,q
h(φ).
For each φ, if φ(x) = 1, we say that there is a particle at x ∈ Z1. Let φ ∈ Φperp,qand φ(x) = 1. The distance between x and the i-th particle on the right ofx will be denoted by ri(x, φ).
Hubbard’s criterion: Let φ ∈ Φperp,q. If for eachx with φ(x) = 1 and for each positive integer i,
[iη] ≤ ri(x, φ) ≤ [iη] + 1, (2)
then φ is a ground state.
Theorem 1 (Ref.4). For each rational 0 <q/p < 1, there exists φ ∈ Φperp,qsatisfying Hubbard’s criterion.
The construction of configurations satisfying Hubbard’s criterion uses the decomposition of the inverse density η into continued frac-tions.4 Hubbard’s criterion provides a sufficient condition for the ground state. As soon as a configuration satisfying Hubbard’s criterion is constructed, condition (2) also becomes a necessary condition for characterizing ground states. Hubbard’s criterion allows us to get an explicit expression of the mean energyh(ϕ) in terms of the reciprocal density η(φ). Since h(ϕ) continuously depends on rational values of η(φ), it can be extended to the whole interval [0, 1]. It turns out that at each rational η, the mean energy h(η) has a left-hand derivative μ−
η and
a right hand derivative μ+η. and the complement of the set⋃η(μ−η, μ+η) has Lebesgue measure zero. The dependence of the reciprocal density
η on the chemical potential is characterized by a devil’s staircase type function: for μ ∈ (μ−
η, μ+η), the reciprocal density of the periodic ground
state is η.3,4
We say that a configuration φ′is a shift of a configuration φ, if φ′(x) = φ(x + d) for all x ∈ Z1and some integerd. Theorem 2 (Ref.5). For each rational 0 <q/p < 1, the ground state φ ∈ Φperp,qis unique up to shifts.
For given φ ∈ Φperp,qand intervalU = [s, t] of length ∣U∣ = t − s, let σU(φ) be the total number of particles in U, σU(φ) =∑
x∈U φ(x).
Definition. The configuration φ is said to be nearly uniform, if for two arbitrary intervals, U and V with equal lengths (∣U∣ = ∣V∣), ∣σU(φ) − σV(φ)∣ ≤ 1.
Theorem 3. A configuration φ ∈ Φperp,qis a ground state if and only if it is nearly uniform.
Proof. In order to prove this theorem, we will show that a configuration satisfies Hubbard’s criterion if and only if it is nearly uniform. Suppose that φ ∈ Φperp,q satisfies Hubbard’s criterion. On the contrary, assume that φ is not nearly uniform: there are two intervals U and V such that ∣U∣ = ∣V∣, but σU(φ) = i, σV(φ) = j, and i − j ≥ 2. Let x1be the rightmost particle among all particles lying strongly on the left ofV andx2be the leftmost particle among all particles lying strongly on the right ofV. Thus, x2is thej + 1-th particle on the right of x1and x2− x1≥ ∣V∣ + 2. Now, if we apply Hubbard’s criterion to the particle x1andj + 1, then we get
[(j + 1)η] ≤ rj+1(x1, φ) ≤ [(j + 1)η] + 1. (3)
Letx3andx4be the leftmost and rightmost particles, respectively, inside the intervalU. Now, if we apply Hubbard’s criterion to the particlex3andi = a − 1, then we get
[(i − 1)η] ≤ ri−1(x3, φ) ≤ [(i − 1)η] + 1. (4) Sincei − j ≥ 2, we have i − 1 ≥ j + 1, and by (3) and (4),
ri−1(x3, φ) − rj+1(x1, φ) ≥ −1. (5)
On the other hand, since ∣U∣ = ∣V∣, ri−1(x3, φ) ≤ ∣U∣, and rj+1(x1, φ) ≥ ∣V∣ + 2, we get
ri−1(x3, φ) − rj+1(x1, φ) ≤ −2. (6)
The contradiction between inequalities (5) and (6) shows that φ is nearly uniform.
Now, suppose that the configuration φ ∈ Φperp,q is nearly uniform. Considerri(x1, φ) and ri(x2, φ) for some i and particles x1andx2, and assume thatri(x1, φ) − ri(x2, φ) ≥ 2. Let V = [x2,x2+ri(x2, φ)] and U be any subinterval of [x1,x1+ri(x1, φ)] not including the endpoints x1 andx1+ri(x1, φ) of length ri(x2, φ). Then, ∣U∣ = ∣V∣, but ∣σU(φ) − σV(φ)∣ ≥ 2, a contradiction. Thus, for any x1andx2,
∣ri(x1, φ) − ri(x2, φ)∣ ≤ 1. (7)
Now, suppose that for somei, x1ri(x1, φ) ≥ [iη] + 2. Then, by (7),ri(x, φ) ≥ [iη] + 1 for all x. Consider a sequence x1,x2, . . . ,xq, where xn+1= ri(xn, φ) for n = 1, . . . , q − 1. Then,∑l=ql=1ri(xl, φ) > ([iη] + 1)q = ([ip/q] + 1)q ≥ ip. On the other hand,∑l=ql=1ri(xl, φ) = ip, a contradiction. Similarly, we get a contradiction in the case when for somex1ri(x, φ) ≤ [iη] − 1. Indeed, in this case, again by (7),ri(x, φ) ≤ [iη] for all x. Consider a sequencex1,x2, . . . ,xq, wherexn+1= ri(xn, φ) for n = 1, . . . , q − 1. Then,∑l=ql=1ri(xl, φ) < ([iη])q = ([ip/q])q ≤ ip. On the other hand, ∑l=q
l=1ri(xl, φ) = ip, a contradiction. Thus, (2) and, hence, Theorem 3 is proved. □
Now, we investigate the set of nearly uniform configurations. The restriction of a configuration φ to interval [a, b] is called a configuration with support [a, b].
Let φ be nearly uniform configuration and x′∈ Z1 such that φ(x′− 1) = 1 and φ(x′) = 0. We say that B = {φ(x′), φ(x′+ 1), . . . , φ(x′
+t − 1)} is a block containing t points if φ(x′ ) = φ(x′
+ 1) = ⋅ ⋅ ⋅ = φ(x′
+t − 1) = 0 and φ(x′
+t) = 1. By Theorem 1, each block con-tains either [η] or [η] + 1 points. We say that a block is normal or long if it concon-tains [η] or [η] + 1 points. Clearly, the restrictions of the configuration φ to both [x′, ∞) and (−∞,x′− 1] are constituted by arrayed normal and long blocks. Thus, φ is an arrayed collection of blocks. Leta = a(ϕ) and b = b(φ) be the total number of normal and long blocks in a period of φ. Since
a + b = q, aη + b(η + 1) = p, (8)
the values ofa and b are uniquely determined by p and q.
Let us define a transformationTηdefined on Φperp,q, where η = p/q. This transformation plays a crucial role in the proof of Theorems 1 and 2. In order to numerate blocks constituted φ by numbers . . . − 2, −1, 0, 1, 2, . . ., we assign a number 0 to the block B0containing the origin ofZ1and after that numerate all blocks from the right ofB0one by one by numbers 1, 2, . . . and all blocks from the left ofB0one by one by numbers −1, −2, . . .,
φ = ⋅ ⋅ ⋅ B−2,B−1,B0,B1,B2, . . . . Let us define a configuration φ′= T
η(φ) by
φ′
(k) = {1 ifBk is long 0 ifBkis normal.
Thus, in order to obtain φ′, we put φ′(0) = 0(1), ifB0is normal (long) and after that replace each long block by 1 and each normal block by 0. For example, the piece {. . . 000 100 010 001 000 100 010 001 000 010 001 . . .} of φ will be transformed to the piece {. . . 000 000 10 . . .}. Clearly, φ′∈ Φper
a+b,b. LetU be any interval constituted by normal and long blocks of φ. By definition, Tη(φ(U) is the restriction of TηtoU. The support of the configurationTη(φ(U), we denote by Tη(U).
Clearly, the transformationTηis not one to one. Let φ″∈ Φperp,q such thatTη(φ″) = φ′, and for somen, the B0block of φ″is [0,n]. We
define the functionT−1η byTη−1(φ′) = φ″. LetU be any interval constituted by normal and long blocks of φ′. By definition,Tη−1(φ(U) is the
restriction ofTη−1toU. The support of the configuration Tη−1(φ(U) is denoted by Tη−1(U). Let ∣Tη(φ(U)∣ and ∣Tη−1(φ(U)∣ be the total number
of points in configurationsTη(φ(U) and Tη−1(φ(U), respectively.
Lemma 1. Suppose that φ ∈ Φperp,qis nearly uniform. Then,Tη(φ) ∈ Φpera+b,bis also nearly uniform.
Proof. On the contrary, suppose that φ is nearly uniform, but φ′= T
η(φ) is not nearly uniform: there are two intervals U′andV with
∣U′∣ = ∣V∣ such that σ
σU′(Tη(φ)) − σV(Tη(φ)) = 2. Let us consider the pre-images of φ′(U) and φ′(V). Clearly, since ∣U∣ = ∣V∣, we get σT−1
η U(φ) − σTη−1V(φ) = 0. On
the other hand, since φ′
(U) contains two more particles than φ′
(V), we get ∣T−1η (U)∣ − ∣Tη−1(V)∣ = 2. Let x1be the rightmost point ofTη−1(U))
[by definitions φ″(x1) = 1] andx2be the rightmost point lying on the left (outside) ofT−1
η (V) [by definitions φ″(x2) = 1]. Now, we have a
contradiction: the intervalsW1= Tη−1(U) − x1andW2= Tη−1(V) ∪ x2have equal lengths, but σW1(φ
″) − σW
2(φ
″) = 2 stating that φ″, which is
just the shift of φ, is not nearly uniform. □
Lemma 2. Suppose that φ ∈ Φpera+b,bis nearly uniform. Then,Tη−1(φ) ∈ Φ
per
p,qis also nearly uniform. Proof. On the contrary, suppose that φ is nearly uniform, but φ′= T−1
η (φ) is not nearly uniform: there are two intervals U and V with
∣U∣ = ∣V∣ such that σU(φ′) − σV(φ′) ≥ 2. Let σV(φ′) =c and σU(φ′) ≥c + 2. Then, φ′(U) contains at least c + 1 blocks and one possibly not complete block 0 . . . 01 of length at least 1. Suppose thatd of these c + 1 blocks are long blocks. Then, readily
∣U∣ ≥ (c + 1)[η] + d + 1. (9)
On the other hand, φ′(V) contains at most c blocks and possibly one not complete block 0 . . . 0 of length at most [η] − 1. Under trans-formation,T c + 1 blocks (all except possibly not complete first block 0 . . . 01) of φ′(U) transfer to the configuration φ(W1) consisting ofc + 1 points. This configuration containsd particles. Under transformation, T at most c blocks (c blocks if each out of c particles is included into some block) of φ′
(V) transfer to the configuration φ(W2) consisting ofc points. Since φ is nearly uniform and ∣W1∣ > ∣W2∣, we get that φ(W2) contains at mostd + 1 particles. Therefore, we readily get
∣V∣ ≤ c[η] + d + 1 + ([η] − 1) = (c + 1)[η] + d, (10)
the inequalities (9) and (10) contradict with ∣U∣ = ∣V∣. □
The following corollary readily follows from the definition ofTη.
Corollary 1. φ is a shift of φ′, if and only ifT
η(φ) is a shift of Tη(φ′).
The transformationT allows us to present alternative and very short proofs of Theorems 1 and 2.
Proof of Theorem 1. We prove the theorem by induction on p. The case p = 2 is obvious. Suppose that Theorem 2 is proved for all p < n. Let us prove the existence of ground state φ ∈ Φperp,q. For givenp, q, let a and b be unique solutions of (8). Sincea + b < p by inductive hypothesis, there exists a ground state φ1∈ Φpera+b,b. Then, φ1is nearly uniform, and by Lemma 2, the configuration φ = Tη−1(φ1) is also nearly
uniform. Thus, φ is a ground state. Done.
Proof of Theorem 2. We prove the theorem by induction on p. The case p = 2 is obvious. Suppose that Theorem 2 is proved for all periods less thanp. Let φ1, φ2∈ Φperp,q be two ground states. By Theorem 3, these configurations are nearly uniform. Consider configurations Tη(φ1),Tη(φ2) ∈ Φpera+b,b, wherea and b are unique solutions of (8). By Lemma (1),Tη(φ1) andTη(φ2) are nearly uniform and therefore are
also ground states. Sincea + b < p by the inductive hypothesis, Tη(φ2) is a shift ofTη(φ2). Therefore, by Corollary 1, φ2is a shift of φ1.
Done.
III. CONCLUDING REMARKS
In contrast to Hubbard’s criterion, the presented criteria do not involve any explicit condition depending on particle density. The employ-ment of similar ground state criteria may be very efficient in the investigation of antiferromagnetic one-dimensional models with a fixed number of clusters or in higher-dimensional models.
ACKNOWLEDGMENTS
The author would like to thank the referee for valuable suggestions. REFERENCES
1
J. Hubbard, “Generalized Wigner lattices in one dimension and some applications to tetracyanoquinodimethane (TCNQ)-Salts,”Phys. Rev. B17, 494–505 (1978).
2
V. L. Pokrovsky and G. V. Uimin, “On the properties of monolayers of absorbed atoms,”J. Phys. C: Solid State Phys.11, 3535–3549 (1978).
3
P. Bak and R. Bruinsma, “One-dimensional Ising model and the complete devil’s staircase,”Phys. Rev. Lett.49, 249–251 (1982).
4
S. E. Burkov and Ya. G. Sinai, “Phase diagrams of one-dimensional lattice models with long-range antiferromagnetic interaction,” Usp. Mat. Nauk 38, 205–225 (1983).
5
A. Kerimov, “On the ground states of one dimensional antiferromagnetic models with long range interaction,” Teor. Mat. Fiz. 58, 473–480 (1984).
6
7R. L. Dobrushin, “Description of a random field by means of conditional probabilities and conditions for its regularity,” Teor. Veroyatn. Primen. 18, 201–229 (1968). 8R. L. Dobrushin, “The problem of uniqueness of a Gibbsian random field and the problem of phase transitions,” Funk. Anal. Pril. 2, 44–57 (1968).
9D. Ruelle, “Statistical mechanics of a one-dimensional lattice gas,”Commun. Math. Phys.
9, 267–278 (1968).
10A. Kerimov, “Absence of phase transitions in one-dimensional antiferromagnetic models with long-range interactions,”J. Stat. Phys.
72, 571–620 (1993).
11A. Kerimov, “Uniqueness of Gibbs states in one-dimensional antiferromagnetic model with long-range interaction,”J. Math. Phys.
40, 4956–4974 (1999).
12M. Fannes, P. Vanheuverzwijn, and A. Verbeure, “Energy-entropy inequalities for classical lattice systems,”J. Stat. Phys.
