On the ground state of Ising chains with finite range interactions

Bundaru, M; Angelescu, N; Nenciu, G.

doi:10.1016/0375-9601(73)90518-5

Cited by 12 publications

(18 citation statements)

References 1 publication

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“…A type (2) can add an even number of JAs by having only one color at I, or an odd number by having X serve as source or sink. The same holds true for type (3).…”

Section: Preliminariessupporting

confidence: 57%

“…However only four of these are distinct, since the structure of Sai 3 ) constrains noz = na:O by conservation of flow. Hence S P l (3) is a polyhedron in 3D with four faces: it is a tetrahedron. The (single) D-pair for this case is the lowest-energy SC in the subvolume bounded by the planes Jd2 + J 2 + J a < 0, Jd2 -J 2 + J a < 0, and J 1 > 0.…”

Section: Discussion Example and Summarymentioning

confidence: 99%

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Disordered ground states for classical discrete-state problems in one dimension

Canright

Watson

1996

J Stat Phys

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It is known that one-dimensional lattice problems with a discrete, finite set of states per site 'generically' have periodic ground states (GSs). We consider slightly less generic cases, in which the Hamiltonian is constrained by either spin (S) or spatial (1) inversion symmetry (or both). We show that such constraints give rise to the possibility of disordered GSs over a finite fraction of the coupling-parameter space-that is, without invoking any nongeneric 'fine tuning' of coupling constants, beyond that arising from symmetry. We find that such disordered GSs can arise for many values of the number of states (k) at each site, and the range r of the interaction. The Ising (k = 2) case is the least prone to disorder: I symmetry allows for disordered GSs (without fine tuning) only for r 2 5, while S symmetry 'never' gives rise to disordered GSs.

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“…A type (2) can add an even number of JAs by having only one color at I, or an odd number by having X serve as source or sink. The same holds true for type (3).…”

Section: Preliminariessupporting

confidence: 57%

Section: Discussion Example and Summarymentioning

confidence: 99%

Disordered ground states for classical discrete-state problems in one dimension

Canright

Watson

1996

J Stat Phys

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“…It is very useful to represent the Hamiltonian pictorially as a directed graph G (k) r with energy weights assigned to the arcs. 3,[6][7][8] The graph has k r nodes, each representing a possible sequence of r spins in the system. The arcs in the graph correspond to the operation of spatial translation in the chain by one unit: a directed arc connects two nodes if the rightmost r − 1 spins of one agree with the leftmost r − 1 spins of the other.…”

Section: Graphs Correlation Polytopes and D-pairsmentioning

confidence: 99%

“…By construction, the given D-pair phase is the ground state of H * : since the energy is the average arc weight, any of the D-pair spin configurations has energy −1, while other configurations use some weight 0 arcs and have higher energy. In (8), H * is written in terms of arc densities. The latter are related to the correlations as follows.…”

Section: Explicit D-pair Hamiltonianmentioning

confidence: 99%

Reasonable and robust Hamiltonians violating the third law of thermodynamics

Watson¹,

Canright²,

Somer³

1997

Phys. Rev. E

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It has recently been shown that the third law of thermodynamics is violated by an entire class of classical Hamiltonians in one dimension, over a finite volume of coupling-constant space, assuming only that certain elementary symmetries are exact, and that the interactions are finite-ranged. However, until now, only the existence of such Hamiltonians was known, while almost nothing was known of the nature of the couplings. Here we show how to define the subvolume of these Hamiltonians-a 'wedge' W in a d-dimensional space-in terms of simple properties of a directed graph. We then give a simple expression for a specific Hamiltonian H * in this wedge, and show that H * is a physically reasonable Hamiltonian, in the sense that its coupling constants lie within an envelope which decreases smoothly, as a function of the range l, to zero at l = r + 1, where r is the range of the interaction.

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“…They are realized in type I. The energy of type I is expressed as Next, 2r1!J1I + 2r2IJ2I + 2raiJsl =rl(21Jli-IJ21-21Jsl) + h+2r2) (IJ21-21Jsl) +2(2rl+2r2+rs)IJal· (22) Taking ,r1 = 0 and using (3), ( 4) and (21), we rewrite (8) HFB= -4Jl+J2+2Ja, [1][2][3][4] 2<J1/Js, J2/Ja+2>2J1/Ja, A~F~I~C, The third term of (26) can be transformed into q1J1 +2r2IJ2I +2rsiJsl =t(ql+ 2ra) (2J1-IJ2I) +t(ql + 4r2) IJ2I +rs(IJ21 +21Jsi-2Jl). (27) Taking r 8 = 0 and using (6) and (7), we rewrite (26) as ,…”

Section: ) Ja=jamentioning

confidence: 99%

Ground State and the Magnetization Process of the Body-Centered Tetragonal Lattice with Ising Spins

Narita

Katsura²

1974

Progress of Theoretical Physics

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The ground state and the magnetization process of the body-centered tetragonal (bet) lattice with Ising spins up to the third neighbor interactions (J,, J,, J,) are investigated by the method of inequalities. Eight types of spin ordering are found of which three are new compared to the case J 3 =0. The magnetization process at T=O is determined and classified in several classes according to combinations of signs and magnitudes of J,, J, and J,. The system with J 8 =0 is topologically reduced to the linear chain and the system with J,=O and that with J,=J, to the square lattice with the 1irst. and the second neighbor interactions. The cause of degeneracies appearing in both cases is clarified by the reduction to lower dimensions. The three-dimensional character is realized only by. taking J, into account. Comparison of the ground states between the case of the ~sing model and the case of the classical Heisenberg model is made in the case where the magnetic field is zero. § I. IntroductionThe studies of spin orderings in the ground state of a spin system with several neighbor interactions are important in connection with phase transitions and the magnetization process. Review of the ground state (other than ferroand antiferromagnetic states) of classical spin systems (Ising and classical Heisenberg model) for several crystal lattices is given shortly. We denote the k-th neighbor interaction by Jk.The ground state and the magnetization process in: the linear chain were discussed by Oguchi,I> Morita and Horiguchi, 2 > Katsura and Narita, 8 > Bandaru,Angelescu anq Nenciu/> and Morita. 5 > For the square lattice with J 1 and J 2 they were discussed by Domb and Potts, 6 > Fan and Wu/> Kanamori, 8 > Horiguchi and Morita, 9 > and Kaburagi and Kanamori. 10 > Ter Haar and Lines 11 > discussed the ground state of sc, bee, fcc, bet lattices with several neighbor interactions and anisotropy in the case where the magnetic field is zero by assuming the sublattice structure. The ground state in the magnetic field of sc, bee, fcc, bet with J 1 and J 2 were obtained by Kanamori 8 > by the method of inequalities. The magnetization process in the J 3 results is expressed in phase diagrams 11y Fig. 1. Body-centered tetragonal lattice.

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On the ground state of Ising chains with finite range interactions

Cited by 12 publications

References 1 publication

Disordered ground states for classical discrete-state problems in one dimension

Disordered ground states for classical discrete-state problems in one dimension

Reasonable and robust Hamiltonians violating the third law of thermodynamics

Ground State and the Magnetization Process of the Body-Centered Tetragonal Lattice with Ising Spins

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