Groupe de Physique Statistique

Equipe 106, Institut Jean Lamour

                     
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Séminaire de groupe

$1D$ Potts model with invisible states
Petro Sarkanych
ICMP
jeudi 14 janvier 2016 , 10h25
Salle de séminaire du groupe de Physique Statistique

We present an exact solution of the $1D$ Potts model with invisible states. The model was introduced a few years ago [1] to explain some untypical phase transitions with spontaneous symmetry breaking. In addition to $q$ ordinary Potts states this model possesses $r$ states which do not interact, and thus contribute to the entropy, but not to the internal energy. The number of invisible states plays a role of a parameter, which changes the order of a phase transition. At presence of two ordering fields, $h_1$ and $h_2$, the generalised Hamiltonian of the model reads: \begin{equation*} H=-\sum_{i} \delta_{s_i,s_{i+1}}\sum_{\alpha=1}^q\delta_{s_i,\alpha}-h_1\sum_i \delta_{s_i,1}-h_2\sum_i\delta_{s_i,q+1}, \end{equation*} where the Potts variable $s_i=1, \ldots ,q,q+1, \ldots ,q+r$, $q$ and $r$ are the numbers of visible and invisible states respectively and the sums span $N$ sites of $1D$ lattice. Using transfer matrix method we obtain the partition function of the model and further analyse partition function zeros in the complex $T$ and $h$ planes (Fisher and Lee-Yang zeros). We find the locus of Yang-Lee edge and generalize the duality transformation which maps Lee-Yang zeros to Fisher zeros [2] for $r>0$. At $h_1=0$ and ${\rm Im}\, h_2=0$ Fisher zeros accumulate along the line that intersects real $T$-axis at $T=0$. This corresponds to the usual phase transition in a $1D$ system. However, for the $r+e^{h_2}<0$ the line of zeros intersects the positive part of the real $T$-axis, which means an existence of a phase transition at non-zero temperature. This surprising result can be achieved if ${\rm Im}\, h_2=i\pi/2$. As it has been shown recently, a complex magnetic field maps to the quantum coherence time [3]. Therefore, our approach shows a connection between criticality in quantum and classical models. R. Tamura, S. Tanaka, and N. Kawashima, Prog. Theor. Phys. {\bf 124} (2010) 381. Z. Glumac and K. Uzelac J. Phys. A {\bf 27} (1994) 7709. B.-B Wei and R.-B. Liu, Phys. Rev. Lett. {\bf 114} (2015) 010601.



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