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arXiv:2307.05165 · Physical review. B./Physical review. B

ϕ4 lattice model with cubic symmetry in three dimensions: Renormalization group flow and first-order phase transitions

Martin Hasenbusch B

Published Feb 15, 2024• 4 citations
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Abstract

We study the three-component ${\ensuremath{\phi}}^{4}$ model on the simple cubic lattice in the presence of a cubic perturbation. To this end, we perform Monte Carlo simulations in conjunction with a finite-size scaling analysis of the data. The analysis of the renormalization group (RG) flow of a dimensionless quantity provides us with the accurate estimate ${Y}_{4}\ensuremath{-}{\ensuremath{\omega}}_{2}=0.000\phantom{\rule{0.16em}{0ex}}81(7)$ for the difference of the RG eigenvalue ${Y}_{4}$ at the $\text{O}(3)$-symmetric fixed point and the correction exponent ${\ensuremath{\omega}}_{2}$ at the cubic fixed point. We determine an effective exponent ${\ensuremath{\nu}}_{\text{eff}}$ of the correlation length that depends on the strength of the breaking of the $\text{O}(3)$ symmetry. Field theory predicts that depending on the sign of the cubic perturbation, the RG flow is attracted by the cubic fixed point, or runs to an ever increasing amplitude, indicating a fluctuation-induced first-order phase transition. We demonstrate directly the first-order nature of the phase transition for a sufficiently strong breaking of the O(3) symmetry. We obtain accurate results for the latent heat, the correlation length in the disordered phase at the transition temperature, and the interface tension for interfaces between one of the ordered phases and the disordered phase. We study how these quantities scale with the RG flow, allowing quantitative predictions for weaker breaking of the $\text{O}(3)$ symmetry.

DOI
10.1103/physrevb.109.054420

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We study the three-component \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 model on the simple cubic lattice in the presence of a cubic perturbation. To this end, we perform Monte Carlo simulations in conjunction with a finite-size scaling analysis of the data. The analysis of the renormalization group (RG) flow of a dimensionless quantity provides us with the accurate estimate Y4\ensuremath−\ensuremathω2=0.00081(7){Y}_{4}\ensuremath{-}{\ensuremath{\omega}}_{2}=0.000\phantom{\rule{0.16em}{0ex}}81(7)Y4​\ensuremath−\ensuremathω2​=0.00081(7) for the difference of the RG eigenvalue Y4{Y}_{4}Y4​ at the O(3)\text{O}(3)O(3)-symmetric fixed point and the correction exponent \ensuremathω2{\ensuremath{\omega}}_{2}\ensuremathω2​ at the cubic fixed point. We determine an effective exponent \ensuremathνeff{\ensuremath{\nu}}_{\text{eff}}\ensuremathνeff​ of the correlation length that depends on the strength of the breaking of the O(3)\text{O}(3)O(3) symmetry. Field theory predicts that depending on the sign of the cubic perturbation, the RG flow is attracted by the cubic fixed point, or runs to an ever increasing amplitude, indicating a fluctuation-induced first-order phase transition. We demonstrate directly the first-order nature of the phase transition for a sufficiently strong breaking of the O(3) symmetry. We obtain accurate results for the latent heat, the correlation length in the disordered phase at the transition temperature, and the interface tension for interfaces between one of the ordered phases and the disordered phase. We study how these quantities scale with the RG flow, allowing quantitative predictions for weaker breaking of the O(3)\text{O}(3)O(3) symmetry.
We study the cubic fixed point for N=3N=3N=3 and 4 by using finite-size scaling applied to data obtained from Monte Carlo simulations of the NNN-component ${\ensuremath{\phi}}^{4}$ model on the simple cubic lattice. We generalize the idea of improved models to a two-parameter family of models. The two-parameter space is scanned for the point, where the amplitudes of the two leading corrections to scaling vanish. To this end, a dimensionless quantity is introduced that monitors the breaking of the O(N)O(N)O(N) invariance. For N=4N=4N=4, we determine the correction exponents ${\ensuremath{\omega}}_{1}=0.763(24)$ and ${\ensuremath{\omega}}_{2}=0.082(5)$. In the case of N=3N=3N=3, we obtain Y4=0.0142(6){Y}_{4}=0.0142(6)Y4​=0.0142(6) for the renormalization group exponent of the cubic perturbation at the O(3)O(3)O(3)-invariant fixed point, while the correction exponent ${\ensuremath{\omega}}_{2}=0.0133(8)$ at the cubic fixed point. Simulations close to the improved point result in the estimates $\ensuremath{\nu}=0.7202(7)$ and $\ensuremath{\eta}=0.0371(2)$ of the critical exponents of the cubic fixed point for N=4N=4N=4. For N=3N=3N=3, at the cubic fixed point, the O(3)O(3)O(3) symmetry is only mildly broken and the critical exponents differ only by little from those of the O(3)O(3)O(3)-invariant fixed point. We find $\ensuremath{-}0.000\phantom{\rule{0.16em}{0ex}}01⪅{\ensuremath{\eta}}_{\text{cubic}}\ensuremath{-}{\ensuremath{\eta}}_{O(3)}⪅0.000\phantom{\rule{0.16em}{0ex}}07$ and ${\ensuremath{\nu}}_{\text{cubic}}\ensuremath{-}{\ensuremath{\nu}}_{O(3)}=\ensuremath{-}0.000\phantom{\rule{0.16em}{0ex}}61(10)$.
The detailed analysis of the global structure of the renormalization-group (RG) flow diagram for a model with isotropic and cubic interactions is carried out in the framework of the massive field theory directly in three dimensions (3D) within an assumption of isotropic exchange. Perturbative expansions for RG functions are calculated for arbitrary N up to four-loop order and resummed by means of the generalized Pad\'e-Borel-Leroy technique. Coordinates and stability matrix eigenvalues for the cubic fixed point are found under the optimal value of the transformation parameter. Critical dimensionality of the model is proved to be equal to ${N}_{c}=2.89\ifmmode\pm\else\textpm\fi{}0.02$ that agrees well with the estimate obtained on the basis of the five-loop $\ensuremath{\varepsilon}$ expansion [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)] resummed by the above method. As a consequence, the cubic fixed point should be stable in 3D for N> 3,N>~3,N> 3, and the critical exponents controlling phase transitions in three-dimensional magnets should belong to the cubic universality class. The critical behavior of the random Ising model being the nontrivial particular case of the cubic model when N=0N=0N=0 is also investigated. For all physical quantities of interest the most accurate numerical estimates with their error bounds are obtained. The results achieved in the work are discussed along with the predictions given by other theoretical approaches and experimental data.
We study the critical behavior of a generalized icosahedral model on the simple cubic lattice. The field variable of the icosahedral model might take 1 of 12 vectors of unit length which are given by the normalized vertices of the icosahedron as value. Similar to the Blume-Capel model, where in addition to $\ensuremath{-}1$ and 1, as in the Ising model, the spin might take the value 0, we add in the generalized model (0,0,0) as allowed value. There is a parameter DDD that controls the density of these voids. For a certain range of DDD, the model undergoes a second-order phase transition. On the critical line, O(3)\text{O}(3)O(3) symmetry emerges. Furthermore, we demonstrate that within this range, similar to the Blume-Capel model on the simple cubic lattice, there is a value of DDD, where leading corrections to scaling vanish. We perform Monte Carlo simulations for lattices of a linear size up to L=400L=400L=400 by using a hybrid of local Metropolis and cluster updates. The motivation to study this particular model is mainly of technical nature. Less memory and CPU time are needed than for a model with O(3)\text{O}(3)O(3) symmetry at the microscopic level. As the result of a finite-size scaling analysis we obtain $\ensuremath{\nu}=0.711\phantom{\rule{0.16em}{0ex}}64(10)$, $\ensuremath{\eta}=0.037\phantom{\rule{0.16em}{0ex}}84(5)$, and $\ensuremath{\omega}=0.759(2)$ for the critical exponents of the three-dimensional Heisenberg universality class. The estimate of the irrelevant renormalization group eigenvalue that is related with the breaking the O(3)\text{O}(3)O(3) symmetry is ${y}_{\mathrm{ico}}=\ensuremath{-}2.19(2)$.
The critical behavior of a model describing phase transitions in cubic and tetragonal anti-ferromagnets with 2N-component (N>1) real order parameters as well as the structural transition in NbO 2 crystal is studied within the field-theoretical renormalization-group (RG) approach in three and (4-∊)-dimensions. Perturbative expansions for RG functions are calculated up to three-loop order and resummed, in 3D, by means of the generalized Padé–Borel procedure which is shown to preserve the specific symmetry properties of the model. It is found that a stable fixed point does exist in the three-dimensional RG flow diagram for N>1, in accordance with predictions obtained earlier within the ∊-expansion. Fixed-point coordinates and critical-exponent values are presented for physically interesting cases N=2 and N=3. In both cases critical exponents are found to be numerically close to those of the 3DXY model. The analysis of the results given by the ∊-expansion and by the RG approach in three dimensions is performed resulting in a conclusion that the latter provides much more accurate numerical estimates.
We study a generalized clock model on the simple cubic lattice. The parameter of the model can be tuned such that the amplitude of the leading correction to scaling vanishes. In the main part of the study, we simulate the model with Z8{Z}_{8}Z8​ symmetry. At the transition, with increasing length scale, O(2)O(2)O(2) symmetry emerges. We perform Monte Carlo simulations using a hybrid of local Metropolis and cluster algorithms of lattices with a linear size up to L=512L=512L=512. The field variable requires less memory and the updates are faster than for a model with O(2)O(2)O(2) symmetry at the microscopic level. Our finite-size scaling analysis yields accurate estimates for the critical exponents of the three-dimensional XYXYXY-universality class. In particular, we get $\ensuremath{\eta}=0.03810(8),\ensuremath{\nu}=0.67169(7)$, and $\ensuremath{\omega}=0.789(4)$. Furthermore, we obtain estimates for fixed point values of phenomenological couplings and critical temperatures.
The Monte Carlo renormalization group is applied to the three-dimensional Ising model on simple cubic lattices with 83{8}^{3}83, 163{16}^{3}163, 323{32}^{3}323, and 643{64}^{3}643 sites. The comparison of block-spin correlation functions from the largest lattices yields the nearest-neighbor critical coupling K1c=0.221654(6){K}_{1}^{c}=0.221654(6)K1c​=0.221654(6). After allowing for (i) interpolation to this best estimate for K1c{K}_{1}^{c}K1c​, (ii) an apparent finite-size effect in the renormalization-group transformation due to the measurement of correlation functions of too few (seven) operators, and (iii) the extrapolation for the effect of a slow transient towards the fixed point, the values $\ensuremath{\nu}=0.629(4)$ and $\ensuremath{\eta}=0.031(5)$ are obtained for the thermal and magnetic exponents. The correction-to-scaling exponent $\ensuremath{\omega}$ is estimated to be around 1; to obtain an accuracy competitive with other methods requires measurements with more than seven operators. We briefly review the problem of redundant operators and indicate the future prospects for this kind of calculation.