Study of two dimensional anisotropic Ising models via a renormalization group approach

Taherkhani, Farid; Akbarzadeh, Hamed; Abroshan, Hadi; Ranjbar, Shahram; Fortunelli, Alessandro; Parsafar, Gholamabbas

doi:10.1016/j.physa.2013.07.026

Cited by 4 publications

(1 citation statement)

References 33 publications

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“…In order to systematically investigate the effects of anisotropy we study a wide range of values of λ. Studies such as these of the critical behavior have been carried out numerically for the Ising model [65], where it was confirmed that the low-energy universal physics at the phase transition is not affected by the strength of the anisotropy. There is no reason to doubt the irrelevance of the anisotropy in the conventional critical behavior of the XY model, and we indeed find consistency with 3D XY exponents (which we will not discuss here), including the Z q scaling dimensions y q .…”

Section: B Monte Carlo Resultsmentioning

confidence: 91%

Unconventional U(1) to $\mathbf{Z_q}$ cross-over in quantum and classical ${\bf q}$-state clock models

Patil,

Shao,

Sandvik

2020

Preprint

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We consider two-dimensional q-state quantum clock models with quantum fluctuations connecting states with all-to-all clock transitions with different choices for the matrix elements. We study the quantum phase transitions in these models using quantum Monte Carlo simulations and finite-size scaling, with the aim of characterizing the cross-over from emergent U(1) symmetry at the transition (for q ≥ 4) to Zq symmetry of the ordered state. We also study classical three-dimensional clock models with spatial anisotropy corresponding to the space-time anisotropy of the quantum systems. The U(1) to Zq symmetry cross-over in all these systems is governed by a so-called dangerously irrelevant operator. We specifically study q = 5 and q = 6 models with different forms of the quantum fluctuations and different anisotropies in the classical models. In all cases, we find the expected classical XY critical exponents and scaling dimensions yq of the clock fields. However, the initial weak violation of the U(1) symmetry in the ordered phase, characterized by a Zq symmetric order parameter φq, scales in an unexpected way. As a function of the system size (length) L, close to the critical temperature φq ∝ L p , where the known value of the exponent is p = 2 in the classical isotropic clock model. In contrast, for strongly anisotropic classical models and the quantum models we find p = 3. For weakly anisotropic classical models we observe a cross-over from p = 2 to p = 3 scaling. The exponent p directly impacts the exponent ν governing the divergence of the U(1) to Zq cross-over length scale ξ in the thermodynamic limit, according to the relationship ν = ν(1 + |yq|/p), where ν is the conventional correlation length exponent. We present a phenomenological argument for p = 3 based on an anomalous renormalization of the clock field in the presence of anisotropy, possibly as a consequence of topological (vortex) line defects. Thus, our study points to an intriguing interplay between conventional and dangerously irrelevant perturbations, which may affect also other quantum systems with emergent symmetries.

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Section: B Monte Carlo Resultsmentioning

confidence: 91%