Quadrupole Order in the Frustrated PyrochloreTb2+xTi2−xO

Abstract

A hidden order that emerges in the frustrated pyrochlore Tb 2+x Ti 2−x O 7+y with T c = 0.53 K is studied using specific heat, magnetization, and neutron scattering experiments on a high-quality single crystal. Semi-quantitative analyses based on a pseudospin-1/2 Hamiltonian for ionic non-Kramers magnetic doublets demonstrate that it is an ordered state of electric quadrupole moments. The elusive spin liquid state of the nominal Tb 2 Ti 2 O 7 is most likely a U(1) quantum spin-liquid state. PACS numbers: 75.40.Cx, 78.70.Nx, 75.10.Kt, 75.30.Ds Geometrically frustrated magnets have been actively investigated in condensed matter physics [1]. In particular, spin ice (SI), e.g. R 2 Ti 2 O 7 (R = Dy or Ho) [2,3], provides prototypical frustrated Ising magnets with the pyrochlore lattice structure [4], consisting of a three-dimensional network of cornersharing tetrahedra [ Fig. 1(b)]. It displays fascinating features such as a finite zero-point entropy [5] and thermally excited emergent magnetic or SI monopoles [6,7]. An intriguing theoretical proposal for a U(1) quantum spin liquid (QSL) state [8] has been made for variants of SI endowed with quantum spin fluctuations [9][10][11][12][13][14]. The U(1) QSL state [8-10] is characterized by an emergent U(1) gauge field producing gapless fictitious photons and by gapped bosonic spinon excitations carrying the SI magnetic monopole charge [8,9,13,15]. By increasing the transverse interaction, the system can undergo a phase transition from the U(1) QSL to a long range ordered (LRO) state of transverse spins or pseudospins representing electric-quadrupole moments for non-Kramers ions [9][10][11]. This state can be described as a Higgs phase [16][17][18][19][20].In a quest to QSL states in frustrated magnetic systems from both theoretical [21][22][23] and experimental [24,25] viewpoints, an Ising-like pyrochlore Tb 2 Ti 2 O 7 (TTO) is a potential candidate for a U(1) QSL: it has been reported to remain in a fluctuating spin state down to 50 mK without magnetic LRO [26,27]. However, the origin of this spin liquid state of TTO has been elusive for more than a decade despite many investigations (see Refs. [4,13,28] and references therein, and recent Refs. [29-31]), and is still under hot debate [13,28]. To solve this challenging problem of TTO, we start this investigation by postulating that the theoretically-proposed interaction between electric quadrupole moments of non-Kramers ions including Tb 3+ [the fourth term of Eq. interactions between magnetic dipole moments [the first three terms of Eq. (1)] and the perturbation through first excited crystal-field (CF) states [14,32], and by taking another assumption of Jahn-Teller (JT) distortion [28,33]. Under the present postulation, two ground states of off-stoichiometric Tb 2+x Ti 2−x O 7+y samples [34] will possibly be accounted for by the U(1) QSL (x < x c ) and electric quadrupolar (x > x c ) states of Ref. [9].In this Letter, we investigate the hidden order of Tb 2+x Ti 2−x O 7+y (x = 0.005 > x c ), because the electric qua...

Composite Spin and Quadrupole Wave in the Ordered Phase ofTb2+xTi2-xO7+y

“…The studies 7,23,25 of the effective Hamiltonian of Eq. (16) showed that there are two electric quadrupolar states: the PAF state (planar antiferropseudospin) and the PF state (planar ferropseudospin) depending on the two parameters (δ, q) (see Fig.…”

“…(17)) along several symmetry directions in the FCC Brillouin zone using the interaction parameters J nn = 1 K, q = 0.85, and δ = 0 adopted in Ref. 23. One can see two flat excitation branches in Fig.…”

Double vibronic process in the quantum spin ice candidate Tb2Ti2O7 revealed by terahertz spectroscopy

“…As a result, phonons (and lattice zero point motions) are now the source of spin fluctuations and by their virtual exchange may give rise to new quantum terms in the effective Hamiltonian of Tb 2 Ti 2 O 7 . Depending on their strength, a long-range quadrupolar order or Coulombphase-like state might be stabilized, provided the Ising term dominates [7, 10,21]. Furthermore, an interesting feature of the vibronic process is that the acoustic phonon crosses the two doublets, which themselves being close in energy, will also partially mix.…”

“…This signal likely corresponds to fluctuations between the two |± electronic states of the crystal electric field (CEF) ground state doublet. It was interpreted as a signature of interactions between quadrupoles, highlighting the importance of those degrees of freedom [18,[20][21][22]. This is because the non-Kramers nature of Tb 3+ ions prevents magnetic exchange alone from inducing such fluctuations within the Ising-like ground state [22][23][24].…”

Neutron Larmor diffraction investigation of the rare-earth pyrochlores R2Ti2O7 ( R=Tb , Dy, Ho

“…Our study covered the temperature range 0.5 K < T < 300 K, as it was convenient to use the 3 He refrigerator to study all three compounds. Recently, it has been suggested that there is a quadrupole ordering transition in Tb 2 Ti 2 O 7 at a temperature of T ≈ 0.5 K [46], only observable in crystals of very high quality or powders. Larmor diffraction could be used to study the transition in a small crystal with directly verified heat capacity-since the signal is proportional to Bragg intensity, even a relatively small sample can be used.…”