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Numerical Diagonalization Study on Frustrated Quantum Spin Systems

Tôru Sakai

Published Jun 8, 2019• 0 citations
OpenAlex

Abstract

Frustrated quantum spin systems are investigated by the large-scale numerical exact diagonalization based on the Lanczos algorithm. Our recent results based on the calculation for finite-size clusters up to 45 spins are introduced about the following topics; (1) spin gap issue of the kagome-lattice antiferromagnet[1.2], (2) magnetization plateaux of the triangular- and kagome-lattice antiferromagnets[3,4]. (3) quatnum phase transitions of the Shastry-Sutherland model[5]. References [1]H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 80 (2011)053704;arXiv:1103.5829 [2]T. Sakai and H. Nakano, Physica B 536 (2018)85; arXiv:1801.04458 [3]H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 87 (2018)063706. [4]H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 86 (2017)114705;arXiv:1708.07248. [5]H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 87 (2018) 123702; arXiv: 1810.11533.

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Frustrated quantum spin systems are investigated by the large-scale numerical exact diagonalization based on the Lanczos algorithm. Our recent results based on the calculation for finite-size clusters up to 45 spins are introduced about the following topics; (1) spin gap issue of the kagome-lattice antiferromagnet[1.2], (2) magnetization plateaux of the triangular- and kagome-lattice antiferromagnets[3,4]. (3) quatnum phase transitions of the Shastry-Sutherland model[5]. References [1]H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 80 (2011)053704;arXiv:1103.5829 [2]T. Sakai and H. Nakano, Physica B 536 (2018)85; arXiv:1801.04458 [3]H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 87 (2018)063706. [4]H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 86 (2017)114705;arXiv:1708.07248. [5]H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 87 (2018) 123702; arXiv: 1810.11533.
The S = 1/2 Heisenberg antiferromagnet on the orthogonal-dimer lattice (Shastry-Sutherland model) is studied by the Lanczos diagonalization method.The properties of this model are determined by the ratio of two interactions, namely, r = J 2 /J 1 , where J 1 denotes the amplitude of spin interactions at orthogonal dimers and the interactions represented by J 2 form the square lattice.We focus our attention on the edge of the phase in which the dimer state is realized as the exact ground state.Our large-scale calculations of diagonalizations treating finite-size clusters including 44 and 48 spin sites successfully detect the target edge.Our conclusion is that the ratio for the edge is r = 0.6754(2).This estimate is compared with an experimental result from electron spin resonance measurements.
We study the system size dependence of the singlet-triplet excitation gap in the S=1/2S=1/2S=1/2 kagome-lattice Heisenberg antiferromagnet by numerical diagonalization. We successfully obtain a new result of a cluster of 42 sites. The two sequences of gaps of systems with even-number sites and that with odd-number sites are separately analyzed. Careful examination clarifies that there is no contradiction when we consider the system to be gapless.
Frustrated quantum magnets may exhibit fascinating collective phenomena. The main goal of this dissertation is to provide conclusive evidence for the emergence of novel phases of matter like quantum spin liquids in local quantum spin models. We develop novel algorithms for large-scale Exact Diagonalization computations. Sublattice coding methods for efficient use of lattice symmetries in the procedure of diagonalizing the Hamiltonian matrix are proposed and suggest a randomized distributed memory parallelization strategy. Benchmarks of computations on various supercomputers with system size up to 50 spin-1/2 particles have been performed. Results concerning the emergence of a chiral spin liquid in a frustrated kagome Heisenberg antiferromagnet are presented. The stability and extent of this phase are discussed. In an extended Heisenberg model on the triangular lattice, we establish another chiral spin liquid phase. We discuss the special case of the Heisenberg J1J_1J1​-J2J_2J2​ model and present a scenario where the critical point of phase transition from the 120-degree Néel to a putative Z2\mathbf{Z}_2Z2​ spin liquid is described by a Dirac spin liquid. A generalization of the SU(2) Heisenberg model with SU(N) degrees of freedom on the triangular lattice with an additional ring-exchange term is discussed. We present our contribution to the project and the final results that suggest a series of chiral spin liquid phases in an extended parameter range. Finally, we present preliminary data from a Quantum Monte Carlo study of an SU(N) version of the J-Q model on a square lattice for N=2,...,10, and multi-column representations. We establish the phase boundary between the Néel ordered phase and the disordered phases. The disordered phase in the four-column representation is expected to be a two-dimensional analog of the Haldane phase for the spin-1 Heisenberg chain.
Plateaus can be observed in the zero-temperature magnetization curve of quantum spin systems at rational values of the magnetization. In one dimension, the appearance of a plateau is controlled by a quantization condition for the magnetization which involves the length of the local spin and the volume of a translational unit cell of the ground state. We discuss examples of geometrically frustrated quantum spin systems with large (in general unbounded) periodicities of spontaneous breaking of translational symmetry in the ground state. In two dimensions, we discuss the square, triangular and Kagomé lattices using exact diagonalization (ED) for up to N = 40 sites. For the spin-1/2 XXZ model on the triangular lattice we study the nature and stability region of a plateau at one third of the saturation magnetization. The Kagomé lattice gives rise to particularly rich behaviour with several plateaus in the magnetization curve and a jump due to local magnon excitations just below saturation.
The study of quantum frustrated systems remains one of the most challenging subjects of quantum magnetism, as they can hold quantum spin liquids, whose characterization is quite elusive. The presence of gapped quantum spin liquids possessing long-range entanglement while being locally indistinguishable often demands highly sophisticated numerical approaches for their description. Here we propose an easy computational method based on exact diagonalization with engineered boundary conditions in very small plaquettes. We apply the method to study the quantum phase diagram of diverse antiferromagnetic frustrated Heisenberg models in the triangular lattice. Our results are in qualitative agreement with previous results obtained by means of sophisticated methods like density matrix renormalization group (2D-DMRG) or variational quantum Monte Carlo.
We discuss recently introduced numerical linked-cluster (NLC) algorithms that allow one to obtain temperature-dependent properties of quantum lattice models, in the thermodynamic limit, from exact diagonalization of finite clusters. We present studies of thermodynamic observables for spin models on square, triangular, and kagomé lattices. Results for several choices of clusters and extrapolations methods, that accelerate the convergence of NLCs, are presented. We also include a comparison of NLC results with those obtained from exact analytical expressions (where available), high-temperature expansions (HTE), exact diagonalization (ED) of finite periodic systems, and quantum Monte Carlo simulations. For many models and properties NLC results are substantially more accurate than HTE and ED.