Detecting crystal symmetry fractionalization from the ground state: Application toZ2spin liquids on the kagome lattice

Abstract

In quantum spin liquid states, the fractionalized spinon excitations can carry fractional crystal symmetry quantum numbers, and this symmetry fractionalization distinguishes different symmetry-enriched spin liquid states with identical intrinsic topological order. In this work we propose a simple way to detect signatures of such crystal symmetry fractionalizations from the crystal symmetry representations of the ground state wave function. We demonstrate our method on projected Z 2 spin liquid wave functions on the kagome lattice, and show that it can be used to classify generic wave functions. Particularly our method can be used to distinguish several proposed candidates of Z 2 spin liquid states on the kagome lattice. It is well known that anyons in topologically ordered phases can carry symmetry quantum numbers that are quantized to fractional values. In the celebrated example of fractional quantum Hall states, Laughlin quasiparticles carry fractional charge-the quantum number of the U (1) symmetry [1]. In recent years, great progress has been made in understanding the interplay between symmetry and fractionalization in other topologically ordered states. In particular, topological spin liquids exhibit a more subtle kind of symmetry fractionalization, associated with the crystal symmetry of the underlying lattice instead of internal symmetries [2][3][4]. While some aspects of it have been studied for quite a while, crystal symmetry fractionalization has now received renewed attention, due to an increased interest in the role of crystal symmetry in topological phases of matter. This topic is also becoming timely in view of strong numerical evidence for spin liquids on kagome lattice found in the last few years [5][6][7][8][9]. In order to fully pin down the topological nature of the numerically found spin liquid liquid, the complete pattern of crystal symmetry fractionalization needs to be determined.In this work, we offer our perspective on crystal symmetry fractionalization in Z 2 spin liquids. We find that the nontrivial way that crystal symmetry acts on an individual anyon is directly related to the symmetry representation of the topologically ordered ground states, as labeled by the crystal momentum and parity of many-body wave functions. Given that states with different symmetry labels cannot be adiabatically connected, our finding immediately makes it clear that the classification of spin liquids is refined and enriched by taking into account crystal symmetries [10]. Our theoretical result also provides a straightforward method to classify and detect different spin liquids in numerical studies. As a concrete example, we demonstrate that our method can be used to easily distinguish various Z 2 spin liquids on the kagome lattice [11][12][13].We begin by briefly reviewing what is known about crystal symmetry fractionalization in Z 2 spin liquids, and setting up the terminology for our work. A Z 2 spin liquid [14,15] supports three types of anyon excitations: bosonic spinons, fermionic spinons, and visons. ...

Symmetry fractionalization in the topological phase of the spin-12J1−J2triangular Heisenberg model

“…However, the properties of these states are unclear, since, depending on the sector chosen, the state may develop chiral order [21], or breaking of C 6 rotational symmetry [20,21,28], leading to a nematic SL. Recent studies [11,29,30] focused on the kagome lattice show that the time-reversal symmetric Z 2 SL can be fully characterized by the symmetry properties of lattices on tori or infinite cylinders via the projective symmetry group (PSG) classifications [1,11,29,31,32].…”

Anomalies and symmetry fractionalization in reflection-symmetric topological order

“…Additionally, while we focus on the generating fields a and b, we should mention that composite quasiparticles which possess both nonzero Z N gauge charge and nonzero Z N gauge flux may transform under P 2 with a possible additional phase factor which depends on the self-statistics of the quasiparticle in question 30,38 . However, this detail will not play an important role in our discussion.…”

Frustrated spin systems, an MPS approach

“…see Essin and Hermele [2013], Qi and Fu [2015], Reuther et al [2014], Wen [2002]) have been trying to present a new material classification. This came after the realization of robust paramagnetic groundstates (with zero magnetization at temperatures much lower than bond energies) in some quasi-2D crystals including κ-(BEDT-TTF) 2 Cu 2 (CN) 3 [Kurosaki et al, 2005, Shimizu et al, 2003, Yamashita et al, 2009 and EtMe 3 Sb[Pd(dmit) 2 ] 2 [Itou et al, 2007[Itou et al, , 2008 on triangular, and ZnCu 3 (OH) 6 Cl 2 (herbertsmithite) [Helton et al, 2007] and Cu 3 BaV 2 O 8 (OH) 2 (vesignieite) on kagomé lattices (see also Chapter 5).…”

“…This is equivalent to creating . Given a particular MES on the cylinder, the action of a global symmetry group (g) member, T g , on the state can be considered as two independent actions on each anyon: , 2013, Qi and Fu, 2015] the symmetry, g, by factorizing an identity member of the group (i.e. square rooting of g).…”

“…Since, T y is just a product of swap operations, the eigenvalue per site can be easily calculated for the variational wavefunctions (even keeping a relatively small m). However, θ y results of Qi and Fu [2015] for topological invariants with explicit preservation of the SU(2)-symmetry. Furthermore, the commutators give information about the self-statistics; in particular, the obtained topological invariants are incompatible with the double-semion topological order (with anyonic content of {î,ŝ,ŝ ,b}, whereŝ,ŝ , andb are the semion, the anti-semion, and a composite bosonic quasi-particle; fusion rules can be constructed using ∀â ∈ {î,ŝ,ŝ ,b},â ×â = 1,î ×â = a,ŝ ×ŝ =b) [Freedman et al, 2004].…”