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arXiv:1901.00278 · Physical review. B./Physical review. B

Quantum phase diagram of the two-dimensional transverse-field Ising model: Unconstrained tree tensor network and mapping analysis

M. Sadrzadeh, Reza Haghshenas, A. Langari

Published Apr 16, 2019• 11 citations
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Abstract

We investigate the ground-state phase diagram of the frustrated transverse-field Ising (TFI) model on the checkerboard lattice (CL), which consists of N\'eel, collinear, quantum paramagnet, and plaquette-valence bond solid (-VBS) phases. We implement a numerical simulation that is based on the recently developed unconstrained tree tensor network ansatz, which systematically improves the accuracy over the conventional methods as it exploits the internal gauge selections. At the highly frustrated region $({J}_{2}={J}_{1})$, we observe a second-order phase transition from the plaquette-VBS state to the paramagnet phase at the critical magnetic-field ${\mathrm{\ensuremath{\Gamma}}}_{c}=0.28$ with the associated critical exponents $\ensuremath{\nu}=1$ and $\ensuremath{\gamma}\ensuremath{\simeq}0.4$, which are obtained within the finite-size scaling analysis on different lattice sizes $N=4\ifmmode\times\else\texttimes\fi{}4,6\ifmmode\times\else\texttimes\fi{}6,8\ifmmode\times\else\texttimes\fi{}8$. The stability of the plaquette-VBS phase at low magnetic fields is examined by a spin-spin correlation function, which verifies the presence of plaquette-VBS at ${J}_{2}={J}_{1}$ and rules out the existence of a N\'eel phase. In addition, our numerical results suggest that the transition from the N\'eel (for ${J}_{2}<{J}_{1})$ to the plaquette-VBS phase is a deconfined phase transition. Moreover, we introduce a mapping, which renders the low-energy effective theory of the TFI on the CL to be the same model on the ${J}_{1}\ensuremath{-}{J}_{2}$ square lattice (SL). We show that the plaquette-VBS phase of the highly frustrated point ${J}_{2}={J}_{1}$ on the CL is mapped to the emergent string-VBS phase on the SL at ${J}_{2}=0.5{J}_{1}$.

DOI
10.1103/physrevb.99.144414

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We use the recently developed tensor network algorithm based on infinite projected entangled pair states (iPEPS) to study the phase diagram of frustrated antiferromagnetic J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model on a checkerboard lattice. The simulation indicates a N\'eel ordered phase when J2<0.88J1{J}_{2}<0.88{J}_{1}J2​<0.88J1​, a plaquette valence bond solid state when 0.88<J2/J1<1.110.88<{J}_{2}/{J}_{1}<1.110.88<J2​/J1​<1.11, and a stripe phase when J2>1.11J1{J}_{2}>1.11{J}_{1}J2​>1.11J1​, with two first-order transitions across the phase boundaries. The calculation shows that the cross dimer state proposed before is unlikely to be the ground state of the model, although such a state indeed arises as a metastable state in some parameter region.
We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg J1{J}_{1}J1​-J2{J}_{2}J2​ model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with LLL === 8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different J2/J1{J}_{2}/{J}_{1}J2​/J1​ values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations, and plaquette order parameters. In addition to the well-known N\'eel order and collinear order at low and high J2/J1{J}_{2}/{J}_{1}J2​/J1​, we observe a plaquettelike order at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.5$. A continuous transition between the N\'eel order and the plaquettelike order near ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.40{J}_{1}$ is observed. The collinear order emerges at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.62{J}_{1}$ through a first-order phase transition.
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We study the ground-state (gs) phase diagram of the frustrated spin-12\frac{1}{2}21​ J1{J}_{1}J1​--J2{J}_{2}J2​--J3{J}_{3}J3​ antiferromagnet with ${J}_{2}={J}_{3}=\ensuremath{\kappa}{J}_{1}$ on the honeycomb lattice using coupled-cluster theory and exact diagonalization methods. We present results for the gs energy, magnetic order parameter, spin-spin correlation function, and plaquette valence-bond crystal (PVBC) susceptibility. We find a N\'eel antiferromagnetic (AFM) phase for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}\ensuremath{\approx}0.47$, a collinear striped AFM phase for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}\ensuremath{\approx}0.60$, and a paramagnetic PVBC phase for ${\ensuremath{\kappa}}_{{c}_{1}}\ensuremath{\lesssim}\ensuremath{\kappa}\ensuremath{\lesssim}{\ensuremath{\kappa}}_{{c}_{2}}$. The transition at ${\ensuremath{\kappa}}_{{c}_{2}}$ appears to be of first-order type, while that at ${\ensuremath{\kappa}}_{{c}_{1}}$ is continuous. Since the N\'eel and PVBC phases break different symmetries our results favor the deconfinement scenario for the transition at ${\ensuremath{\kappa}}_{{c}_{1}}$.
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