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arXiv:1506.01379 · Physical Review Letters

Spin-Peierls Instability of Three-Dimensional Spin Liquids with Majorana Fermi Surfaces

Maria Hermanns, Simon Trebst, Achim Rosch

Published Oct 22, 2015• 47 citations
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Abstract

Three-dimensional (3D) variants of the Kitaev model can harbor gapless spin liquids with a Majorana Fermi surface on certain tricoordinated lattice structures such as the recently introduced hyperoctagon lattice. Here, we investigate Fermi surface instabilities arising from additional spin exchange terms (such as a Heisenberg coupling) which introduce interactions between the emergent Majorana fermion degrees of freedom. We show that independent of the sign and structure of the interactions, the Majorana surface is always unstable. Generically, the system spontaneously doubles its unit cell at exponentially small temperatures and forms a spin liquid with line nodes. Depending on the microscopics, further symmetries of the system can be broken at this transition. These spin-Peierls instabilities of a 3D spin liquid are closely related to BCS instabilities of fermions.

DOI
10.1103/physrevlett.115.177205

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Motivated by the recent synthesis of $\ensuremath{\beta}$-Li2{}_{2}2​IrO3{}_{3}3​---a spin-orbit entangled j=1/2j=1/2j=1/2 Mott insulator with a three-dimensional lattice structure of the Ir4+{}^{4+}4+ ions---we consider generalizations of the Kitaev model believed to capture some of the microscopic interactions between the iridium moments on various trivalent lattice structures in three spatial dimensions. Of particular interest is the so-called hyperoctagon lattice---the premedial lattice of the hyperkagome lattice, for which the ground state is a gapless quantum spin liquid where the gapless Majorana modes form an extended two-dimensional Majorana Fermi surface. We demonstrate that this Majorana Fermi surface is inherently protected by lattice symmetries and discuss possible instabilities. We thus provide the first example of an analytically tractable microscopic model of interacting SU(2) spin-12\frac{1}{2}21​ degrees of freedom in three spatial dimensions that harbors a spin liquid with a two-dimensional spinon Fermi surface.
In quantum magnetism, Kitaev's honeycomb model is well known for its gapped and gapless spin liquid ground states in which the elementary spin degrees of freedom fractionalize into Majorana fermions and a Z2{\mathbb{Z}}_{2}Z2​ gauge field. In this paper, the authors discuss the gapless spin liquids arising in three-dimensional Kitaev models and show that the gapless Majorana fermions form metallic states which, depending on the underlying lattice structure, exhibit Fermi surfaces, nodal lines, or Weyl points.
The formation of quantum spin liquids in frustrated magnets represents an exciting possibility due to the rather exotic features they harbor, including fractionalized excitations and emergent gauge fields. \nUnfortunately, they are notoriously difficult to study as there are often no good analytical methods available and quantum Monte Carlo simulations are hindered by the negative sign problem. \nThe Kitaev honeycomb model is a notable exception of a frustrated quantum model which is exactly solvable and which hosts a number of distinct quantum spin liquid ground states. \nAs such, it allows for a rare opportunity to study the physics of spin liquids with full analytical control. \n \nIn this thesis, we study the fractionalization of spin-1/2 moments into Majorana fermions and an emergent Z2 gauge field in a generalization of the Kitaev honeycomb model to a number of three-dimensional lattices. \nWhile the excitations of the gauge field are always gapped, the fermionic quasiparticles may exhibit a gapless dispersion, forming fully two-dimensional Fermi surfaces, symmetry protected nodal lines, or topological Weyl nodes. \nWe show that one can deduce rather general constraints on the possible gapless excitations by making use of an object called the projective symmetry group. \nIn doing so we provide a scheme for classifying the various gapless Kitaev spin liquids. \nA thorough analysis is carried out for a number of these spin liquids, primarily investigating the stability of the gapless modes and the novel features resulting from their sometimes non-trivial topology, as well as their effects on certain equal-time correlation functions.
A three-dimensional Kitaev model on a hyperhoneycomb lattice is investigated numerically at finite temperature. The Kitaev model is one of the solvable quantum spin models, where the ground state is given by gapped and gapless spin liquids, depending on the anisotropy of the interactions. This model can be rewritten as a free Majorana fermion system coupled with Z 2 variables. The density of states of Majorana fermions shows an excitation gap in the gapped region, while it is semimetallic in the gapless region reflecting the Dirac node. Performing the Monte Carlo simulation, we calculate the temperature dependence of the Majorana spectra. We find that the semimetallic dip is filled as temperature increases in the gapless region, but surprisingly, the spectrum develops an excitation gap in the region near the gapless-gapped boundary. Such changes of the low-energy spectrum appear sharply at the transition temperature from the spin liquid to the paramagnetic state. The results indicate that thermal fluctuations of the Z 2 fields significantly influence the low-energy state of Majorana fermions, especially in the spin liquid formation.
The excitation spectrum of Kitaev quantum spin liquids consists of itinerant Majorana fermions, which can be gapless or gapped, and vortices of a Z2\mathbb{Z}_2Z2​ gauge field, known as visons, which are \new{gapped within a stable Z2\mathbb Z_2Z2​ spin liquid phase}. In this work, we investigate visons in Kitaev-type models where the Majorana fermions form a Fermi surface. In this case, the creation of a vison pair is analogous to introducing a local impurity potential in a metal. Since the gapless modes lead to strong finite-size effects, we compare the numerical calculation of the \new{two-vison gap} on finite lattices with the result from an analytical approach based on Green's function techniques. We find that the \new{two-vison gap} decreases as the size of the Fermi surface increases, signalling an instability of the quantum spin liquid ground state. We also show that larger Fermi surfaces tend to suppress the change in local spin correlations due to the Majorana-vison scattering potential.
Quantum spin liquids are highly fascinating quantum liquids in which the spin degrees of freedom fractionalize. An interesting class of spin liquids axe the exactly solvable, three-dimensional Kitaev spin liquids. Their fractionalized excitations are Majonara fermions, which may exhibit a variety of topological band structures-ranging from topologically protected Weyl semimetals over nodal semi-metals to systems with Majorana Fermi surfaces. We study the entanglement spectrum of such Kitaev spin liquids and verify that it is closely related to the topologically protected edge spectrum. Moreover, we find that in some cases the entanglement spectrum contains even more information about the topological features than the surface spectrum, and thus provides a simple and reliable tool to probe the topology of a system.