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arXiv:2112.14485 · Journal of Mathematical Physics

Quantum symmetry on Potts model

Debashish Goswami, Sk Asfaq Hossain

Published Apr 1, 2022• 2 citations
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Abstract

We formulate the notion of quantum group symmetry of the Hamiltonian corresponding to the Potts model and compute it for few simple models. Our examples illustrate how a slight change of the model parameter may result in a drastic change of the quantum symmetry group (in some cases, the classical symmetry group remains unaffected), signifying a case of phase transition.

DOI
10.1063/5.0083709

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We formulate the notion of quantum group symmetry of the Hamiltonian corresponding to the Potts model and compute it for few simple models. Our examples illustrate how a slight change of the model parameter may result in a drastic change of the quantum symmetry group (in some cases, the classical symmetry group remains unaffected), signifying a case of phase transition.
The phase transition of the classical two-dimensional Potts model, in particular the order of the transition as the number of components qqq increases, is studied by constructing renormalization-group transformations on the equivalent one-dimensional quantum problem. It is shown that the block transformation with two sites per cell indicates the existence of a critical qc{q}_{c}qc​ separating the small-qqq and large-qqq regions with different critical behaviors. The physically accessible fixed point for q>qcq>{q}_{c}q>qc​ is a discontinuity fixed point where the specific-heat exponent $\ensuremath{\alpha}=1$, and therefore the transition is of first order.
Relations between two-dimensional physical systems, continuous-spin Hamiltonians, and discrete spin or Potts models are established by Landau symmetry and renormalization-group arguments. Experimental realizations of XY and Heisenberg models with cubic anisotropy as well as of different Potts models are reviewed. Then a unified model encompassing these classes of models is introduced. For the special case of a six-state discrete model a phase diagram is proposed based on duality and Migdal renormalization-group transformations.
We use our finite-lattice approach to study the phase transitions of the Hamiltonian formulation for the (1 + 1)-dimensional Z(q)Z(q)Z(q) models. We confirm the known result that these models possess two phases for q<qcq<{q}_{c}q<qc​ and three phases for q> qcq>~{q}_{c}q> qc​. Our calculation, however, gives qc=6{q}_{c}=6qc​=6, while the perturbative calculation predicts qc=5{q}_{c}=5qc​=5. Similar calculations for the (1 + 1)-dimensional qqq-state Potts models for q=4,5,and8q=4, 5, \mathrm{and} 8q=4,5,and8 failed to differentiate the second-order transition expected for q< 4q<~4q< 4 from the first-order transition expected for q> 5q>~5q> 5.
Symmetries in a Hamiltonian play an important role in quantum physics because they correspond directly with conserved quantities of the related system. In this Letter, we propose quantum algorithms capable of testing whether a Hamiltonian exhibits symmetry with respect to a group. We demonstrate that familiar expressions of Hamiltonian symmetry in quantum mechanics correspond directly with the acceptance probabilities of our algorithms. We execute one of our symmetry-testing algorithms on existing quantum computers for simple examples of both symmetric and asymmetric cases.
Spontaneous symmetry breaking is well understood through the classical "Mexican Hat" picture, which describes many quantum phases of matter. Recently, several new classes of quantum phases of matter, such as topological orders and symmetry protected topological (SPT) orders, were discovered. In an attempt to address the transitions between all those phases of quantum matter under the same framework, we introduced an analogous yet very simple picture for phase transitions in the context of tensor-networks. Using a very simple iteration process, we found that both symmetry breaking and some topological phase transitions (for topological orders described by gauge theory and 1D SPT orders) could be marked by a sudden change in the symmetry structure of the so-called "environment matrix". In this process, the environment matrix serves as an "order parameter" that captures patterns of entanglement in topological phases. The symmetry change in the environment matrix is very much like the symmetry breaking of conventional order parameters. We applied this method to both the transverse Ising model (1D and 2D honeycomb), spin-1 model (1D), and the Toric Code model in a magnetic field (2D honeycomb), and explored the corresponding symmetry structure changes in their environment matrices in details. With just a few variational parameters and a few minutes' run time on a laptop, we could get the corresponding phase transition points within a few percent error compared with the Quantum Monte Carlo results.