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arXiv:1608.04887 · Physical review. E

Entanglement entropy of the Q≥4 quantum Potts chain

Péter Lajkó, Ferenc Iglói

Published Jan 4, 2017• 6 citations
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Abstract

The entanglement entropy $\mathcal{S}$ is an indicator of quantum correlations in the ground state of a many-body quantum system. At a second-order quantum phase-transition point in one dimension $\mathcal{S}$ generally has a logarithmic singularity. Here we consider quantum spin chains with a first-order quantum phase transition, the prototype being the $Q$-state quantum Potts chain for $Q>4$ and calculate $\mathcal{S}$ across the transition point. According to numerical, density matrix renormalization group results at the first-order quantum phase transition point $\mathcal{S}$ shows a jump, which is expected to vanish for $Q\ensuremath{\rightarrow}{4}^{+}$. This jump is calculated in leading order as $\mathrm{\ensuremath{\Delta}}\mathcal{S}=lnQ[1\ensuremath{-}4/Q\ensuremath{-}2/(QlnQ)+O(1/{Q}^{2})]$.

DOI
10.1103/physreve.95.012105

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The entanglement entropy S\mathcal{S}S is an indicator of quantum correlations in the ground state of a many-body quantum system. At a second-order quantum phase-transition point in one dimension S\mathcal{S}S generally has a logarithmic singularity. Here we consider quantum spin chains with a first-order quantum phase transition, the prototype being the QQQ-state quantum Potts chain for Q>4Q>4Q>4 and calculate S\mathcal{S}S across the transition point. According to numerical, density matrix renormalization group results at the first-order quantum phase transition point S\mathcal{S}S shows a jump, which is expected to vanish for Q\ensuremath→4+Q\ensuremath{\rightarrow}{4}^{+}Q\ensuremath→4+. This jump is calculated in leading order as \ensuremathΔS=lnQ[1\ensuremath−4/Q\ensuremath−2/(QlnQ)+O(1/Q2)]\mathrm{\ensuremath{\Delta}}\mathcal{S}=lnQ[1\ensuremath{-}4/Q\ensuremath{-}2/(QlnQ)+O(1/{Q}^{2})]\ensuremathΔS=lnQ[1\ensuremath−4/Q\ensuremath−2/(QlnQ)+O(1/Q2)].
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We study the scaling properties of entanglement entropy (EE) near the quantum critical points in interacting random antiferromagnetic (AFM) spin chains. Using density-matrix renormalization group, we compute the half-chain EE near the topological phase transition between the Haldane and random singlet phases in a disordered spin-1 chain. It is found to diverge logarithmically in system size with an effective central charge ceff=1.17(4){c}_{\mathrm{eff}}=1.17(4)ceff​=1.17(4) at the quantum critical point (QCP). Moreover, a scaling analysis of EE yields the correlation length exponent $\ensuremath{\nu}=2.28(5)$. Our unbiased calculation establishes that the QCP is in the universality class of the infinite-randomness fixed point predicted by previous studies based on the strong disorder renormalization group technique. However, in the disordered spin-1/2 Majumdar-Ghosh chain, where a valence bond solid phase is unstable to disorder, the crossover length exponent obtained from a scaling analysis of EE disagrees with the expectation based on the Imry-Ma argument. We provide a possible explanation.
The qqq-state Potts chain with ferromagnetic couplings, J=1J=1J=1, in the presence of a transverse field $\mathrm{\ensuremath{\Gamma}}$ has a quantum phase transition at $\mathrm{\ensuremath{\Gamma}}/q=1$, which is continuous for $q\ensuremath{\le}4$ and of first order for q>4q>4q>4. Here we introduce a qqq-periodic alternating longitudinal field of strength hhh and study the phase diagram and the critical properties of the model. For $h<q/(q\ensuremath{-}1)$, there is a ferromagnetic ordered phase, for $\mathrm{\ensuremath{\Gamma}}<{\mathrm{\ensuremath{\Gamma}}}_{c}(h)$, and at $h=q/(q\ensuremath{-}1)$, there is a classical endpoint at $\mathrm{\ensuremath{\Gamma}}=0$, with finite entropy at T=0T=0T=0. We considered the q=3q=3q=3 model and using density matrix renormalization group techniques, we calculated the low-lying spectrum of the Hamiltonian, the transverse magnetization, and the spin-spin correlation function, all of which signal a diverging correlation length at the transition point with the exponent of the three-state Potts model. In the vicinity of the classical endpoint, the model is mapped to a quantum hard rod model, which also belongs to the universality class of the ferromagnetic three-state Potts model. Also, the spectrum of the critical Hamiltonian is found in agreement with conformal invariance. At the same time, the correlation function shows a jump at the transition point, and thus the transition is of mixed order for $h<q/(q\ensuremath{-}1)$.
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