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Physical review. B, Condensed matter

Asymptotic symmetry: Enhancement and stability

Charles M. Newman, Lawrence S. Schulman

Published Oct 1, 1982• 6 citations
Publisher

Abstract

The scaling-limit symmetry of ${Z}_{q}$-invariant spin systems is studied by renormalization-group methods based on qualitative bifurcation theory rather than a series expansion in $\ensuremath{\epsilon}$. Under the plausible assumption of the absence of secondary bifurcations, it is shown that an ordinary critical phase with $q\ensuremath{\ne}2 or 4$ must have full $S{O}_{2}$ invariance in the scaling limit; moreover, any such asymptotically isotropic critical phase is stable under ${Z}_{q}$-invariant perturbations for $q>4$. Previously obtained results about two-dimensional Kosterlitz-Thouless phases for clock models have a natural interpretation within this group-theoretic bifurcation analysis.

DOI
10.1103/physrevb.26.3910

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The scaling-limit symmetry of Zq{Z}_{q}Zq​-invariant spin systems is studied by renormalization-group methods based on qualitative bifurcation theory rather than a series expansion in \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ. Under the plausible assumption of the absence of secondary bifurcations, it is shown that an ordinary critical phase with q\ensuremath≠2or4q\ensuremath{\ne}2 or 4q\ensuremath=2or4 must have full SO2S{O}_{2}SO2​ invariance in the scaling limit; moreover, any such asymptotically isotropic critical phase is stable under Zq{Z}_{q}Zq​-invariant perturbations for q>4q>4q>4. Previously obtained results about two-dimensional Kosterlitz-Thouless phases for clock models have a natural interpretation within this group-theoretic bifurcation analysis.
Motivated by recent attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we define an extended-O(2) model by adding a $\ensuremath{\gamma}\mathrm{cos}(q\ensuremath{\varphi})$ term to the ordinary O(2) model with angular values restricted to a $2\ensuremath{\pi}$ interval. In the $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, the model becomes an extended qqq-state clock model that reduces to the ordinary qqq-state clock model when qqq is an integer and otherwise is a continuation of the clock model for noninteger qqq. By shifting the $2\ensuremath{\pi}$ integration interval, the number of angles selected can change discontinuously and two cases need to be considered. What we call case 1 has one more angle than what we call case 2. We investigate this class of clock models in two space-time dimensions using Monte Carlo and tensor renormalization group methods. Both the specific heat and the magnetic susceptibility show a double-peak structure for fractional qqq. In case 1, the small-$\ensuremath{\beta}$ peak is associated with a crossover, and the large-$\ensuremath{\beta}$ peak is associated with an Ising critical point, while both peaks are crossovers in case 2. When qqq is close to an integer by an amount $\mathrm{\ensuremath{\Delta}}q$ and the system is close to the small-$\ensuremath{\beta}$ Berezinskii-Kosterlitz-Thouless transition, the system has a magnetic susceptibility that scales as $\ensuremath{\sim}1/(\mathrm{\ensuremath{\Delta}}q{)}^{1\ensuremath{-}1/{\ensuremath{\delta}}^{\ensuremath{'}}}$ with ${\ensuremath{\delta}}^{\ensuremath{'}}$ estimates consistent with the magnetic critical exponent $\ensuremath{\delta}=15$. The crossover peak and the Ising critical point move to Berezinskii-Kosterlitz-Thouless transition points with the same power-law scaling. A phase diagram for this model in the $(\ensuremath{\beta},q)$ plane is sketched. These results are possibly relevant for configurable Rydberg-atom arrays where the interpolations among phases with discrete symmetries can be achieved by varying continuously the distances among atoms and the detuning frequency.
The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute mmm-vector model, the n=0n=0n=0 limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic mmm-component fixed point, which characterizes the critical behavior of the pure system, is stable. For m<mcm<{m}_{c}m<mc​, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for m=1m=1m=1. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.
The critical behavior of the general isotropic, ferromagnetic two-dimensional spin system with openZ(5) symmetry is studied with use of high-temperature expansions of its mass gap. On the basis of these expansions we propose a simple analytic representation of the mass gap which naturally reproduces all the different phase transitions exhibited by this model (first order and second order of the Ising and of the Kosterlitz-Thouless types). In addition, the bifurcation point where the soft phases originate is clearly identified with the Fateev-Zamolodchikov value.
The critical behavior of three-state statistical models invariant under the full symmetry group S 3 and its dependence on space dimension have been a matter of interest and debate. In particular, the phase transition of the 3-state Potts model in three dimensions is believed to be of the first order, without a definitive proof of absence of scale invariance in three-dimensional field theory with S 3 symmetry. This scale invariance should appear as a non-trivial fixed point of the renormalization group, which has not been found. Our new search, with the non-perturbative renormalization group, finds such a fixed point, as a bifurcation from the trivial fixed point at the critical space dimension d = 10 / 3 , which extends continuously to d = 3 . It does not correspond to a second-order phase transition of the 3-state Potts model , but is interesting in its own right. In particular, it shows how the ε -expansion can fail.
The critical behavior of three-state statistical models invariant under the full symmetry group S3 and its dependence on space dimension have been a matter of interest and debate. In particular, the phase transition of the 3-state Potts model in three dimensions is believed to be of the first order, without a definitive proof of absence of scale invariance in three-dimensional field theory with S3 symmetry. This scale invariance should appear as a non-trivial fixed point of the renormalization group, which has not been found. Our new search, with the non-perturbative renormalization group, finds such a fixed point, as a bifurcation from the trivial fixed point at the critical space dimension d = 10/3, which extends continuously to d = 3 . It does not correspond to a second- order phase transition of the 3-state Potts model, but is interesting in its own right. In particular, it shows how the E-expansion can fail.
We investigate the general features of the renormalization-group flow at the Berezinskii-Kosterlitz-Thouless (BKT) transition, providing a thorough quantitative description of the asymptotc critical behavior, including the multiplicative and subleading logarithmic corrections. For this purpose, we consider the RG flow of the sine-Gordon model around the renormalizable point which describes the BKT transition. We reduce the corresponding $\ensuremath{\beta}$ functions to a universal canonical form, valid to all perturbative orders. Then we determine the asymptotic solutions of the RG equations in various critical regimes: the infinite-volume critical behavior in the disordered phase, the finite-size scaling limit for homogeneous systems of finite size, and the trap-size scaling limit occurring in two-dimensional bosonic particle systems trapped by an external space-dependent potential.