竹篩晶格上海森堡反鐵磁模型及反對稱交互作用之無能隙自旋液

Abstract

在竹篩晶格上的1/2自旋海森堡反鐵磁模型的基態是一個根本且具爭議性的問題。這個問題同時也是礦物Herbersmithite的理論研究起點。Herbersmithite是一個極有可能實現量子自旋液的物質，實驗上低至50mK的溫度都沒有觀察到磁有序。在Herbersmithite中已知存在有最近鄰反鐵磁交互作用之外的微擾，包括反對稱交互作用，XXZ非均相性，次近鄰交互作用以及雜質。在本研究中我們探討反對稱交互作用所造成的影響。 過去關於海森堡反鐵磁模型考慮反對稱交互作用影響的結果，是自旋液的基態會存在於一個小的反對稱交互作用強度。其中包括精確對角化法，密度矩陣重整化群，施溫格玻色子之平均場理論，泛函重整化群，皆得到相變是出現在反對稱交互作用約等於0.1J時。然而此自旋液的本質是有爭議性的，主要的兩派說法是其為有能隙的Z2自旋液以及無能隙的U(1)自旋液。 考量到過去的許多研究大部分都是基於平均場理論或是有限大小系統的計算，我們使用張量網態的算法，可以計算無限大小的系統。首先，在不考慮反對稱交互作用的情況下，我們在竹篩晶格上看到無能隙的自旋液，與之前張量網態的結果吻合。其次，因為在竹篩晶格上數值計算的限制，我們引入Husimi晶格作為輔助，因為其具有易於計算，並且局部相似於竹篩晶格的特性。最終，結合兩者的結果，我們推測相變化出現在反對稱交互作用約等於0.012(2)J的時候，遠小於之前研究的結果，也小於實驗上推測Herbersmithite內反對稱交互作用的大小。此結果意味著在Herbersmithite上觀察到的自旋液的特性可能有其他的原因。

The ground state of S=1/2 kagome Heisenberg antiferromagnet (KHAF) is one of the most fundamental and controversial problems in frustrated quantum magnetism. The KHAF problem is also the starting point of the mineral Herbertsmithite, a promising candidate material to realize quantum spin liquid. Experimental studies observe no magnetic order down to temperature as low as 50 mK. It is known that Herbertsmithite might be perturbed by interactions beyond nearest-neighbor, such as Dzyaloshinskii-Moriya (DM) interactions, XXZ anisotropy, next-nearest-neighbor interactions and impurities. In this work we investigate the effect of DM interactions. Previous studies of KHAF with DM interactions suggest the spin-liquid ground state is robust against small Dz. Calculations including exact diagonalization (ED), density-matrix renormalization-groups (DMRG), Schwinger-boson method and functional renormalization-groups (FRG), all suggest the transition to magnetic ordered phase take place at Dc~0.1J. However the interpretation of spin-liquid ground state for Dz<Dc is controversial. Possible candidate are gapped Z2 spin-liquid and gapless U(1) spin-liquid. In consideration of that most of previous studies are either based on mean-field approach or finite-size calculation, we were motivated to investigate the problem by tensor network state methods, where infinite-size calculation is performed. Firstly we find gapless spin liquid for KHAF problem, consistent with previous tensor network state study. Secondly, because of the limitation of numerical calculation on kagome lattice, we introduce the Husimi lattice where the extremely accurate calculation is possible. The Husimi lattice is locally similar to the kagome lattice and can serve as a benchmark of the kagome data. Finally, combining the result of both lattice, we deduced that a weak but finite DM interaction is required to destabilize the gapless spin-liquid state. The critical magnitude, Dc~0.012(2)J, lies well below the critical value proposed in previous studies, and also smaller than DM strength deduced in herbertsmithite. Therefore, our work implies that there might be other origins of the spin-liquid nature in herbertsmithite.Previous studies of KHAF with DM interactions suggest the spin-liquid ground state is robust against small Dz. Calculations including exact diagonalization (ED), density-matrix renormalization-groups (DMRG), Schwinger-boson method and functional renormalization-groups (FRG), all suggest the transition to magnetic ordered phase take place at Dc ∼ 0.1J. However the interpretation of spin-liquid ground state for Dz < Dc is controversial. Possible candidate are gapped Z2 spin-liquid and gapless U(1) spin-liquid. In consideration of that most of previous studies are either based on mean-field approach or finite-size calculation, we were motivated to investigate the problem by tensor network state methods, where infinite-size calculation is performed. Firstly we find gapless spin liquid for KHAF problem, consistent with previous tensor network state study. Secondly, because of the limitation of numerical calculation on kagome lattice, we introduce the Husimi lattice where the extremely accurate calculation is possible. The Husimi lattice is locally similar to the kagome lattice and can serve as a benchmark of the kagome data. Finally, combining the result of both lattice, we deduced that a weak but finite DM interaction is required to destabilize the gapless spin-liquid state. The critical magnitude, Dc ~ 0.012(2)J, lies well below the critical value proposed in previous studies, and also smaller than DM strength deduced in herbertsmithite. Therefore, our work implies that there might be other origins of the spin-liquid nature in herbertsmithite.