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...
