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Physical review. B./Physical review. B

Quantum order by disorder in frustrated spin nanotubes

João C. Getelina, Zekun Zhuang, Premala Chandra, Piers Coleman, Peter P. Orth, Shivaji L. Sondhi

Published Feb 16, 2024• 3 citations
Publisher

Abstract

We investigate quantum order by disorder in a frustrated spin nanotube formed by wrapping a ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model at ${45}^{\ensuremath{\circ}}$ around a cylinder. Using Schwinger boson theory and density matrix renormalization group (DMRG), we have computed the ground-state phase diagram to reveal a ${\mathbb{Z}}_{2}$ phase in which collinear spin stripes form a right- or left-handed helix around the nanotube. We have derived an analytic estimate for the critical ${\ensuremath{\eta}}_{c}={J}_{1}/2{J}_{2}$ of the ${\mathbb{Z}}_{2}$-helical phase transition, which is in agreement with the DMRG results. By evaluating the entanglement spectrum and nonlocal string order parameters we discuss the topology of the ${\mathbb{Z}}_{2}$-helical phase.

DOI
10.1103/physrevb.109.075137

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We investigate quantum order by disorder in a frustrated spin nanotube formed by wrapping a J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ Heisenberg model at 45\ensuremath∘{45}^{\ensuremath{\circ}}45\ensuremath∘ around a cylinder. Using Schwinger boson theory and density matrix renormalization group (DMRG), we have computed the ground-state phase diagram to reveal a Z2{\mathbb{Z}}_{2}Z2​ phase in which collinear spin stripes form a right- or left-handed helix around the nanotube. We have derived an analytic estimate for the critical \ensuremathηc=J1/2J2{\ensuremath{\eta}}_{c}={J}_{1}/2{J}_{2}\ensuremathηc​=J1​/2J2​ of the Z2{\mathbb{Z}}_{2}Z2​-helical phase transition, which is in agreement with the DMRG results. By evaluating the entanglement spectrum and nonlocal string order parameters we discuss the topology of the Z2{\mathbb{Z}}_{2}Z2​-helical phase.
We investigate quantum order by disorder in a frustrated spin nanotube formed by wrapping a J1J_1J1​-J2J_2J2​ Heisenberg model at 45∘^\circ∘ around a cylinder. Using Schwinger boson theory and Density Matrix Renormalization Group (DMRG), we have computed the ground-state phase diagram to reveal a Z2\mathbb{Z}_2Z2​ phase in which collinear spin stripes form a right or left handed helix around the nanotube. We have derived an analytic estimate for the critical ηc=J1/2J2η_c=J_1/2J_2ηc​=J1​/2J2​ of the Z2\mathbb{Z}_2Z2​ helical phase transition, which is in agreement with the DMRG results. By evaluating the entanglement spectrum and nonlocal string order parameters we discuss the topology of the Z2\mathbb{Z}_2Z2​-helical phase.
Strongly correlated systems with geometric frustrations can host the emergent phases of matter with unconventional properties. Here, we study the spin S=1S=1S=1 Heisenberg model on the honeycomb lattice with the antiferromagnetic first- (J1{J}_{1}J1​) and second-neighbor (J2{J}_{2}J2​) interactions ($0.0\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$) by means of density-matrix renormalization group. In the parameter regime ${J}_{2}/{J}_{1}\ensuremath{\lesssim}0.27$, the system sustains a N\'eel antiferromagnetic phase. At the large J2{J}_{2}J2​ side ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.32$, a stripe antiferromagnetic phase is found. Between the two magnetic ordered phases $0.27\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.32$, we find a nonmagnetic intermediate region with a plaquette valence-bond order. Although our calculations are limited within a six unit-cell width on cylinder, we present evidence that this plaquette state could be a strong candidate for this nonmagnetic region in the thermodynamic limit. We also briefly discuss the nature of the quantum phase transitions in the system. We gain further insight into the nonmagnetic phases in the spin-1 system by comparing its phase diagram with the spin-1/21/21/2 system.
We study the phase diagram of the frustrated Heisenberg model on the triangular lattice with nearest- and next-nearest-neighbor spin-exchange coupling, on three-leg ladders. Using the density-matrix renormalization-group method, we obtain the complete phase diagram of the model, which includes quasi-long-range ${120}^{\ensuremath{\circ}}$ and columnar order, and a Majumdar-Ghosh phase with short-ranged correlations. All these phases are nonchiral and planar. We also identify the nature of phase transitions.
We study the spin-1 Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor J1{J}_{1}J1​ and the next-nearest-neighbor J2{J}_{2}J2​ couplings by using the infinite projected entangled-pair state (iPEPS) ansatz and the density matrix renormalization group (DMRG) calculation. The iPEPS simulation, which studies the model directly in the thermodynamic limit, finds a crossing of the ground state from the N\'eel magnetic state to the stripe magnetic state at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$, showing a direct phase transition. In the finite-size DMRG calculation on the cylinder geometry up to the cylinder width Ly=10{L}_{y}=10Ly​=10, we find that around the same critical point the N\'eel and the stripe orders are strongly suppressed, which implies the absence of an intermediate phase. Both calculations identify that the stripe order comes with a first-order transition at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$. Our results indicate that unlike the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square model, quantum fluctuations in the spin-1 model may not be strong enough to stabilize an intermediate nonmagnetic phase.
We study a spin-half frustrated coupled ladder system, in which ladders with leg, rung, and diagonal interactions are linked via nearest-neighbor coupling. By introducing a leg-symmetric inter-ladder interaction that connects the left-to-left and right-to-right legs of adjacent ladders, the model is found to possess an exact dimer ground state, characterized by a product of two-spin singlets forming a columnar dimer phase. We analyze this model using bond-operator mean-field theory (BOMFT) and the density matrix renormalization group (DMRG) to probe the phase transitions that occur as one traverses the coupling space. The BOMFT reveals three distinct phases: a double-stripe ordered phase, a Néel ordered phase, and a quantum disordered dimerized phase. The critical points for the transitions are at $ J_1 = -0.81 $ (double-stripe to dimerized) and at $ J_1 = 2.81 $ (dimerized to Néel phase). Further, the DMRG results corroborate the exact ground state and refine the critical points to $ J_1 = -0.79 $ and $ J_1 = 2.29 $ for the respective transitions. Additionally, another transition is identified as the Néel order vanishes for $ J_1 \ge 4.5 $. The model can alternatively be represented as a network of orthogonal zigzag and fully frustrated spin ladders, offering a structural framework conducive to quantum materials engineering.
We study an asymmetric ${J}_{1}\ensuremath{-}{J}_{2}$ zigzag ladder consisting of two different spin-12\frac{1}{2}21​ antiferromagnetic (AFM; ${J}_{2}, \ensuremath{\gamma}{J}_{2}>0$) Heisenberg legs coupled by zigzag-shaped ferromagnetic (FM; J1<0{J}_{1}<0J1​<0) interleg interaction. On the basis of density-matrix renormalization group based calculations, the ground-state phase diagram is obtained as functions of $\ensuremath{\gamma}$ and J2/∣J1∣{J}_{2}/|{J}_{1}|J2​/∣J1​∣. It contains four kinds of frustration-induced ordered phases except a trivial FM phase. Two of the ordered phases are valence bond solid (VBS) with spin-singlet dimerization, which is a rather conventional order by disorder. Still, it is interesting to note that the VBS states possesses an Affleck-Kennedy-Lieb-Tasaki--type topological hidden order. The remaining two phases are ferrimagnetic orders, each of which is distinguished by commensurate or incommensurate spin-spin correlation. It is striking that the ferrimagnetic orders are not associated with geometrical symmetry breaking; instead, the global spin-rotation symmetry is broken. In other words, the system lowers its energy via the FM interleg interaction by polarizing both of the AFM Heisenberg legs. This is a rare type of order by disorder. Besides, the incommensurate ferrimagnetic state appears as a consequence of the competition between a polarization and a critical Tomonaga-Luttinger--liquid behavior in the AFM Heisenberg legs.