Partition function zeros of the one-dimensional Blume-Capel model in transfer matrix formalism

Ghulghazaryan, Ruben; Sargsyan, Karen; Ananikian, N. S.

doi:10.1103/physreve.76.021104

Cited by 20 publications

(45 citation statements)

References 49 publications

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“…In this case the linear density was already known exactly [2] furnishing σ = −1/2. Numerical works have confirmed σ = −1/2 in several one-dimensional spin models [13,14,15,16] including, see [17], the same model discussed here. An early exception is a special type of three-state Potts model [18].…”

Section: Introductionsupporting

confidence: 67%

Unusual Yang-Lee edge singularity in the one-dimensional axial-next-to-nearest-neighbor Ising model

Dalmazi

2010

Phys. Rev. E

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We show here for the one-dimensional spin-1/2 axial-next-to-nearest-neighbor Ising model in an external magnetic field that the linear density of Yang-Lee zeros may diverge with critical exponent =−2/ 3 at the Yang-Lee edge singularity. The necessary condition for this unusual behavior is the triple degeneracy of the transfer-matrix eigenvalues. If this condition is absent we have the usual value =−1/ 2. Analogous results have been found in the literature in the spin-1 Blume-Emery-Griffths model and in the three-state Potts model in a magnetic field with two complex components. Our results support the universality of =−2/ 3 which might be a one-dimensional footprint of a tricritical version of the Yang-Lee edge singularity possibly present also in higher-dimensional spin models.

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Section: Introductionsupporting

confidence: 67%

Unusual Yang-Lee edge singularity in the one-dimensional axial-next-to-nearest-neighbor Ising model

Dalmazi

2010

Phys. Rev. E

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“…To do so we closely follow the method developed in Refs. [34,45]. With increase of system size, Lee-Yang zeros z 1 = |z 1 |e iθ terminate in the complex plane at the Yang-Lee edge z e 1 = |z e 1 |e iθe .…”

Section: Appendix B Duality Relationsmentioning

confidence: 99%

“…The exponent σ, like the other critical exponents, is characteristic of a given universality class. For the 1D Ising and q−state Potts models its exact value is σ = − 1 2 [14,45,48]. Another known exact value for the σ−exponent has been obtained for the spherical model, where σ = 1 2 independently of the type of interaction (short-or long-range) and space dimensionality [61].…”

Section: Appendix B Duality Relationsmentioning

confidence: 99%

Classical phase transitions in a one-dimensional short-range spin model

Sarkanych¹,

Holovatch²,

Kenna³

2018

J. Phys. A: Math. Theor.

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Ising's solution of a classical spin model famously demonstrated the absence of a positive-temperature phase transition in one-dimensional equilibrium systems with short-range interactions. No-go arguments established that the energy cost to insert domain walls in such systems is outweighed by entropy excess so that symmetry cannot be spontaneously broken. An archetypal way around the no-go theorems is to augment interaction energy by increasing the range of interaction. Here we introduce new ways around the no-go theorems by investigating entropy depletion instead. We implement this for the Potts model with invisible states. Because spins in such a state do not interact with their surroundings, they contribute to the entropy but not the interaction energy of the system. Reducing the number of invisible states to a negative value decreases the entropy by an amount sufficient to induce a positive-temperature classical phase transition. This approach is complementary to the long-range interaction mechanism. Alternatively, subjecting positive numbers of invisible states to imaginary or complex fields can trigger such a phase transition. We also discuss potential physical realisability of such systems.

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“…Knowing the behaviour of a system in one dimension (1D) can help one to understand and predict its behaviour in higher dimensions too. Also some chemical compounds are effectively described by quasi-1D models [2,3,4,5,6]. For these reasons, 1D models are interesting from both theoretical and experimental points of view.…”

Section: Introductionmentioning

confidence: 99%

“…In the thermodynamic limit they approach the real axis at the transition point. For the 1D system, Fisher zeros can be obtained by equating (at least) the two eigenvalues which are largest by modulus [39,5,6]:…”

Section: Partition Function Zerosmentioning

confidence: 99%

Exact solution of a classical short-range spin model with a phase transition in one dimension: The Potts model with invisible states

2017

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We present the exact solution of the 1D classical short-range Potts model with invisible states. Besides the q states of the ordinary Potts model, this possesses r additional states which contribute to the entropy, but not to the interaction energy. We determine the partition function, using the transfer-matrix method, in the general case of two ordering fields: h 1 acting on a visible state and h 2 on an invisible state. We analyse its zeros in the complex-temperature plane in the case that h 1 = 0. When Im h 2 = 0 and r ≥ 0, these zeros accumulate along a line that intersects the real temperature axis at the origin. This corresponds to the usual "phase transition" in a 1D system. However, for Im h 2 = 0 or r < 0, the line of zeros intersects the positive part of the real temperature axis, which signals the existence of a phase transition at non-zero temperature.

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Partition function zeros of the one-dimensional Blume-Capel model in transfer matrix formalism

Cited by 20 publications

References 49 publications

Unusual Yang-Lee edge singularity in the one-dimensional axial-next-to-nearest-neighbor Ising model

Unusual Yang-Lee edge singularity in the one-dimensional axial-next-to-nearest-neighbor Ising model

Classical phase transitions in a one-dimensional short-range spin model

Exact solution of a classical short-range spin model with a phase transition in one dimension: The Potts model with invisible states

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