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arXiv:1407.0201 · Physical Review B

Edge modes in a frustrated quantum Ising chain

Somenath Jalal, Brijesh Kumar

Published Nov 17, 2014• 7 citations
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Abstract

The ground state properties of an Ising chain with nearest (${J}_{1}$) and next-nearest neighbor (${J}_{2}$) interactions in a transverse field are investigated using the density matrix renormalization group and cluster mean-field theory methods. Its quantum phase diagram has two ordered regions (separated by a quantum disordered region), one of which has simple ferromagnetic or N\'eel order depending upon the sign of ${J}_{1}$, and the other is the double-staggered antiferromagnetic phase. The presence of the edge modes in these ordered phases is inferred by calculating the end-to-end spin-spin correlations on open chains. It is found that there occur four edge modes in the double-staggered phase, while the ferromagnetic and N\'eel phases support two edge modes, except very near ${J}_{1}=0$, where it seems they have four edge modes like the exact case at ${J}_{1}=0$ itself.

DOI
10.1103/physrevb.90.184416

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The ground state properties of an Ising chain with nearest (J1{J}_{1}J1​) and next-nearest neighbor (J2{J}_{2}J2​) interactions in a transverse field are investigated using the density matrix renormalization group and cluster mean-field theory methods. Its quantum phase diagram has two ordered regions (separated by a quantum disordered region), one of which has simple ferromagnetic or N'eel order depending upon the sign of J1{J}_{1}J1​, and the other is the double-staggered antiferromagnetic phase. The presence of the edge modes in these ordered phases is inferred by calculating the end-to-end spin-spin correlations on open chains. It is found that there occur four edge modes in the double-staggered phase, while the ferromagnetic and N'eel phases support two edge modes, except very near J1=0{J}_{1}=0J1​=0, where it seems they have four edge modes like the exact case at J1=0{J}_{1}=0J1​=0 itself.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1FJ_1^FJ1F​) and antiferromagnetic (J1AJ_1^AJ1A​) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2FJ_2^FJ2F​). In this model frustration is induced by the non-zero J2FJ_2^FJ2F​ and the effect of dimerization is realized by changing the strength of J1AJ_1^AJ1A​ for a fixed value of J1FJ_1^FJ1F​. The model with site spin sss behaves like a Haldane spin chain with site spin 2sss in the limit of vanishing J2FJ_2^FJ2F​ and large J1F/J1AJ_1^F/J_1^AJ1F​/J1A​. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2FJ_1^A-J_2^FJ1A​−J2F​ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.
Using the coupled-cluster method for high orders of approximation and Lanczos exact diagonalization we study the ground-state phase diagram of a quantum spin-12\frac{1}{2}21​ J1{J}_{1}J1​-J2{J}_{2}J2​ model on the square lattice with plaquette structure. We consider antiferromagnetic (J1>0{J}_{1}>0J1​>0) as well as ferromagnetic (J1<0{J}_{1}<0J1​<0) nearest-neighbor interactions together with frustrating antiferromagnetic next-nearest-neighbor interaction J2>0{J}_{2}>0J2​>0. The strength of interplaquette interaction $\ensuremath{\lambda}$ varies between $\ensuremath{\lambda}=1$ (which corresponds to the uniform J1{J}_{1}J1​-J2{J}_{2}J2​ model) and $\ensuremath{\lambda}=0$ (which corresponds to isolated frustrated 4-spin plaquettes). While on the classical level ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) both versions of models (i.e., with ferro- and antiferromagnetic J1{J}_{1}J1​) exhibit the same ground-state behavior, the ground-state phase diagram differs basically for the quantum case s=1/2s=1/2s=1/2. For the antiferromagnetic case (J1>0{J}_{1}>0J1​>0) N\'eel antiferromagnetic long-range order at small J2/J1{J}_{2}/{J}_{1}J2​/J1​ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.47$ as well as collinear striped antiferromagnetic long-range order at large J2/J1{J}_{2}/{J}_{1}J2​/J1​ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.30$ appear which correspond to their classical counterparts. Both semiclassical magnetic phases are separated by a nonmagnetic quantum paramagnetic phase. The parameter region, where this nonmagnetic phase exists, increases with decreasing $\ensuremath{\lambda}$. For the ferromagnetic case (J1<0{J}_{1}<0J1​<0) we have the trivial ferromagnetic ground state at small J2/∣J1∣{J}_{2}/|{J}_{1}|J2​/∣J1​∣. By increasing J2{J}_{2}J2​ this classical phase gives way for a semiclassical plaquette phase, where the plaquette block spins of length s=2s=2s=2 are antiferromagnetically long-range ordered. Further increasing of J2{J}_{2}J2​ then yields collinear striped antiferromagnetic long-range order for $\ensuremath{\lambda}\ensuremath{\gtrsim}0.38$, but a nonmagnetic quantum paramagnetic phase $\ensuremath{\lambda}\ensuremath{\lesssim}0.38$.
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The ground-state phase diagram of a spin-12\frac{1}{2}21​ XXZ chain with competing ferromagnetic nearest-neighbor (J1<0{J}_{1}<0J1​<0) and antiferromagnetic second-neighbor (J2>0{J}_{2}>0J2​>0) exchange couplings is studied by means of the infinite time evolving block decimation algorithm and effective field theories. For the SU(2)-symmetric (Heisenberg) case, we show that the nonmagnetic phase in the range $\ensuremath{-}4<{J}_{1}/{J}_{2}<0$ has a small but finite ferromagnetic dimer order. We argue that this spontaneous dimer order is associated with effective spin-1 degrees of freedom on dimerized bonds, which collectively form a valence bond solid state as in the spin-1 antiferromagnetic Heisenberg chain (the Haldane spin chain). We thus call this phase the Haldane dimer phase. With easy-plane anisotropy, the model exhibits a variety of phases including the vector chiral phase with gapless excitations and the even-parity dimer and N\'eel phases with gapped excitations, in addition to the Haldane dimer phase. Furthermore, we show the existence of gapped phases with coexisting orders in narrow regions that intervene between the gapless chiral phase and any one of Haldane dimer, even-parity dimer, and N\'eel phases. Possible implications for quasi-one-dimensional edge-sharing cuprates are discussed.
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