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Phase transition in spin systems with various types of fluctuations.

Miyashita S - Proc. Jpn. Acad., Ser. B, Phys. Biol. Sci. (2010)

Bottom Line: Generally, the so-called order-disorder phase transition takes place in competition between the interaction causing the system be ordered and the entropy causing a random disturbance.As to the critical property of phase transitions, the concept "universality of the critical phenomena" is well established.However, we still find variety of features of ordering processes.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, The University of Tokyo, Japan. miya@spin.phys.s.u-tokyo.ac.jp

ABSTRACT
Various types ordering processes in systems with large fluctuation are overviewed. Generally, the so-called order-disorder phase transition takes place in competition between the interaction causing the system be ordered and the entropy causing a random disturbance. Nature of the phase transition strongly depends on the type of fluctuation which is determined by the structure of the order parameter of the system. As to the critical property of phase transitions, the concept "universality of the critical phenomena" is well established. However, we still find variety of features of ordering processes. In this article, we study effects of various mechanisms which bring large fluctuation in the system, e.g., continuous symmetry of the spin in low dimensions, contradictions among interactions (frustration), randomness of the lattice, quantum fluctuations, and a long range interaction in off-lattice systems.

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Configurations of the quantum Monte Carlo simulation for the transverse Ising model. (a) Thermal fluctuation, and (b) Quantum fluctuation.
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fig15: Configurations of the quantum Monte Carlo simulation for the transverse Ising model. (a) Thermal fluctuation, and (b) Quantum fluctuation.

Mentions: Usually, quantum fluctuation tends to destroy the classical ordered state (DLRO). The ground state order–disorder transition in the transverse Ising model97)[45]is the most typical example of the quantum phase transition. When Γ exceeds a critical value ΓC, the ground state becomes disordered, and it is called “quantum disordered state”. The effect of quantum fluctuation is taken into account by the Suzuki–Trotter expression of the partition function which is a path-integral representation of the canonical weight.98) This formula was proposed for the quantum Monte Carlo method.99) The ground state property of a d-dimensional quantum system is expressed by the partition function[46]This can be expressed by a path-integral or Suzuki–Trotter formula in a (d +1)-dimensional space,[47]where Trd+1 denotes the sum all over the spin states including those inserted as a complete set of states {∑i/{σim}〉〈{σim}/\quad(m=1,…n)}, and we call n “Suzuki–Trotter number”. If we regard the direction of n as a new spatial direction, the model [45] is transformed into a (d + 1)-dimensional Ising model with the coupling constants:[48]where[49]Thus, the critical property of a d-dimensional quantum system is related to that of a (d + 1)-dimensional classical system at finite temperature.98) In Fig. 15, we depict configurations in the (d + 1)-dimensional system for an one dimensional transverse Ising model. The left and right panels show typical configurations for the thermal fluctuation and the quantum fluctuation, respectively. The thermal fluctuation gives disorder in the real space while the quantum fluctuation gives disorder in the imaginary time space.


Phase transition in spin systems with various types of fluctuations.

Miyashita S - Proc. Jpn. Acad., Ser. B, Phys. Biol. Sci. (2010)

Configurations of the quantum Monte Carlo simulation for the transverse Ising model. (a) Thermal fluctuation, and (b) Quantum fluctuation.
© Copyright Policy - open-access
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3066537&req=5

fig15: Configurations of the quantum Monte Carlo simulation for the transverse Ising model. (a) Thermal fluctuation, and (b) Quantum fluctuation.
Mentions: Usually, quantum fluctuation tends to destroy the classical ordered state (DLRO). The ground state order–disorder transition in the transverse Ising model97)[45]is the most typical example of the quantum phase transition. When Γ exceeds a critical value ΓC, the ground state becomes disordered, and it is called “quantum disordered state”. The effect of quantum fluctuation is taken into account by the Suzuki–Trotter expression of the partition function which is a path-integral representation of the canonical weight.98) This formula was proposed for the quantum Monte Carlo method.99) The ground state property of a d-dimensional quantum system is expressed by the partition function[46]This can be expressed by a path-integral or Suzuki–Trotter formula in a (d +1)-dimensional space,[47]where Trd+1 denotes the sum all over the spin states including those inserted as a complete set of states {∑i/{σim}〉〈{σim}/\quad(m=1,…n)}, and we call n “Suzuki–Trotter number”. If we regard the direction of n as a new spatial direction, the model [45] is transformed into a (d + 1)-dimensional Ising model with the coupling constants:[48]where[49]Thus, the critical property of a d-dimensional quantum system is related to that of a (d + 1)-dimensional classical system at finite temperature.98) In Fig. 15, we depict configurations in the (d + 1)-dimensional system for an one dimensional transverse Ising model. The left and right panels show typical configurations for the thermal fluctuation and the quantum fluctuation, respectively. The thermal fluctuation gives disorder in the real space while the quantum fluctuation gives disorder in the imaginary time space.

Bottom Line: Generally, the so-called order-disorder phase transition takes place in competition between the interaction causing the system be ordered and the entropy causing a random disturbance.As to the critical property of phase transitions, the concept "universality of the critical phenomena" is well established.However, we still find variety of features of ordering processes.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, The University of Tokyo, Japan. miya@spin.phys.s.u-tokyo.ac.jp

ABSTRACT
Various types ordering processes in systems with large fluctuation are overviewed. Generally, the so-called order-disorder phase transition takes place in competition between the interaction causing the system be ordered and the entropy causing a random disturbance. Nature of the phase transition strongly depends on the type of fluctuation which is determined by the structure of the order parameter of the system. As to the critical property of phase transitions, the concept "universality of the critical phenomena" is well established. However, we still find variety of features of ordering processes. In this article, we study effects of various mechanisms which bring large fluctuation in the system, e.g., continuous symmetry of the spin in low dimensions, contradictions among interactions (frustration), randomness of the lattice, quantum fluctuations, and a long range interaction in off-lattice systems.

Show MeSH
Related in: MedlinePlus