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arXiv:cond-mat/9811235 · Physical Review Letters

Strongly Asymmetric Tricriticality of Quenched Random-Field Systems

Alkan Kabakçıoğlu, A. Nihat Berker

Published Mar 22, 1999• 7 citations
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Abstract

In view of the recently seen dramatic effect of quenched random bonds on tricritical systems, we have conducted a renormalization-group study on the effect of quenched random magnetic fields on the tricritical phase diagram of the spin-1 Ising model in $d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3$. We find that random fields convert first-order phase transitions into second-order phase transitions more effectively than random bonds. The coexistence region is extremely flat, attesting to an unusually small tricritical exponent ${\ensuremath{\beta}}_{u}$. The random-field phase diagram is strikingly asymmetric. To accommodate this asymmetry, the second-order boundary exhibits reentrance.

DOI
10.1103/physrevlett.82.2572

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In view of the recently seen dramatic effect of quenched random bonds on tricritical systems, we have conducted a renormalization-group study on the effect of quenched random magnetic fields on the tricritical phase diagram of the spin-1 Ising model in d=3d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3d=3. We find that random fields convert first-order phase transitions into second-order phase transitions more effectively than random bonds. The coexistence region is extremely flat, attesting to an unusually small tricritical exponent \ensuremathβu{\ensuremath{\beta}}_{u}\ensuremathβu​. The random-field phase diagram is strikingly asymmetric. To accommodate this asymmetry, the second-order boundary exhibits reentrance.
Temperature-driven first-order phase transitions that involve symmetry breaking are converted to second order by the introduction of infinitesimal quenched bond randomness in spatial dimensions d≤2 or d≤4, respectively, for systems of n=1 or n≥ (R18)2 component microscopic degrees of freedom. Even strongly first-order transitions undergo this conversion to second order! Above these dimensions, this phenomenon still occurs, but requires a threshold amount of bond randomness. For example, under bond randomness, the phase transitions of q-state Potts models are second order for all q in d≤2. If no symmetry breaking is involved, temperature-driven first-order phase transitions are eliminated under the above conditions. Another consequence is that bond randomness drastically alters multicritical phase diagrams. Tricritical points and critical endpoints are entirely eliminated (d≤2) or depressed in temperature (d≳2). Similarly, bicritical phase diagrams are converted (d≤2) to reentrant-disorder-line or decoupled-tetracritical phase diagrams. These quenched-fluctuation-induced second-order transitions (a diametric opposite to the previously known annealed-fluctuation-induced first-order transitions) should lead to a multitude of new universality classes of criticality, including many experimentally accessible cases.
The effect of annealed antiferromagnetic bond randomness on the phase transitions of the Quantum Ising Model (QIM) is studied by using mean-field renormalization group method. It is argued that bond randomness drastically alters multicritical phase diagram via transverse field. Multicritical points and coexistence region of ferromagnetic and antiferromagnetic case exist only at weak transverse field, and are entirely eliminated at strong transverse field. The coexistence region diminishes in reducing the fluctuation interaction. This physical picture demonstrates that the competition between transverse field and exchange interaction and fluctuation interaction via bond randomness play an important role in generating multiphase structure. Another consequence of competition is that tricritical points of first-second order phase transitions are not entirely eliminated by bond randomness in two-dimensional QIM.
The effect of quenched impurities on systems undergoing first-order phase transitions is studied within the framework of the qqq-state Potts model. For large qqq a mapping to the random-field Ising model explains the absence of any latent heat in 2D, and suggests that for d>2d>2d>2 such systems exhibit a tricritical point with an exponent $\ensuremath{\nu}$ related to those of the random-field model by $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\nu}}_{\mathrm{RF}}/(2\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{RF}}\ensuremath{-}{\ensuremath{\beta}}_{\mathrm{RF}})$. In 2D we analyze the model using finite-size scaling and conformal invariance, and find a continuous transition with a ratio $\ensuremath{\beta}/\ensuremath{\nu}$ which varies continuously with qqq, and a weakly varying exponent $\ensuremath{\nu}\ensuremath{\approx}1$. We find strong evidence for the multiscaling of the correlation functions.
Mean-field theory and renormalization-group arguments are used to show that the phase transition in a system with a random ordering field becomes first order at sufficiently low transition temperature, provided the (symmetric) random-field distribution function has a minimum at zero field. The first-order region is separated from the second-order region by a tricritical point. Both the critical and the tricritical exponents at d>4d>4d>4 dimensions are shown to be the same as for the pure system at $d\ensuremath{-}2$ dimensions. The relevance to spin glasses and other systems is discussed. The new tricritical point is very different from all previously studied tricritical points, as it deviates from mean-field theory at d=5d=5d=5, and not at d=3d=3d=3. Although quantitative results are calculated only at $d=5\ensuremath{-}\ensuremath{\epsilon}$ dimensions, the qualitative results are expected to apply at d=3d=3d=3.
It is argued on general grounds that bond randomness drastically alters multicritical phase diagrams via a random-field mechanism. For example, tricritical points and critical end points are entirely eliminated (d\ensuremath{\le}2) or depressed in temperature (d>2). These predictions are confirmed by a renormalization-group calculation. Another consequence of this phenomenon is that, under bond randomness, the phase transitions of q-state Potts models are second order for all q at dimensionality d\ensuremath{\le}2.
It is argued on general grounds that bond randomness drastically alters multicritical phase diagrams via a random-field mechanism. For example, tricritical points and critical endpoints are entirely eliminated (in spatial dimension d≤2) or depressed in temperature (d>2). These predictions are confirmed by a renormalization-group calculation.1 Another consequence of this phenomenon is that, under bond randomness, the phase transitions of q-state Potts models are second order for all q at dimensionality d≤2.