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Physica A Statistical Mechanics and its Applications

Soft mode in the class of exactly soluble models of phase transitions

Andrzej Radosz

Published Oct 1, 1990• 5 citations
Publisher
DOI
10.1016/0378-4371(90)90034-p

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Quantum phase transitions in the discrete φ4 model: The crossover between two types of transition

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Quantum phase transitions in the discrete ${\ensuremath{\varphi}}^{4}$ model are studied in two dimensions (2D) and 3D at $T=0.$ Scenario of the transition can be changed continuously from the soft mode to transverse-field Ising case by the varying of a model parameter. This change is studied by the quantum Monte Carlo (QMC) simulation. To describe the numerical result, we use an extension of the mean-field scheme that takes nonlinearity of the on-site response into account. It reproduces QMC data with a 5% accuracy and becomes exact in the soft-mode limit. For the transverse-field Ising model, it gives the value of critical coupling with a 3% accuracy.

Physical review. B, Condensed matter•Apr 9, 2001•12 citations
Zero-temperature or quantum phase transitions in itinerant electronic systems both with and without quenched disordered are discussed. Phase transitions considered include, the ferromagnetic transition, the antiferromagnetic transition, the superconductor-metal transition, and various metal-insulator transitions. Emphasis is placed on how to determine the universal properties that characterize these quantum phase transitions. For the first three of the phase transitions listed above, one of the main physical ideas established is that in zero-temperature systems there are soft or slow modes that exist in addition to the soft order parameter fluctuations, and that these modes can couple to the critical modes. These extra soft modes are shown to have a profound effect on the quantum critical properties. For quantum phase transitions involving zero wavenumber order parameters, i.e., the ferromagnetic and superconductor-metal transitions, these extra modes effectively lead to long-ranged effective interactions between order parameter fluctuations, which in turn lead to exactly soluble critical behaviors. For the antiferromagnetic case, we argue that while in low enough dimensions disorder fluctuation effects tend to destroy long-range order, quantum fluctuations counteract this effect and in some parameter regions manage to re-establish antiferromagnetic long-range order. For the metal-insulator transition, some recent new ideas are reviewed. In particular, it is pointed out that for interacting disordered electrons, one expects that in high dimensions the metal-insulator transition is related to the phase transition that occurs in random-field magnets in high dimensions. If the analogy also holds in three dimensions this suggests that the metal-insulator transition might have glassy characteristics.
Quantum phase transitions in the discrete ${\ensuremath{\varphi}}^{4}$ model are studied in two dimensions (2D) and 3D at T=0.T=0.T=0. Scenario of the transition can be changed continuously from the soft mode to transverse-field Ising case by the varying of a model parameter. This change is studied by the quantum Monte Carlo (QMC) simulation. To describe the numerical result, we use an extension of the mean-field scheme that takes nonlinearity of the on-site response into account. It reproduces QMC data with a 5% accuracy and becomes exact in the soft-mode limit. For the transverse-field Ising model, it gives the value of critical coupling with a 3% accuracy.