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Read [18] Quantized Hall (QH) State Selected for a Viewpoint in Physics week ending PRL 107, 077201 (2011) PHYSICAL REVIEW LETTERS 12 AUGUST 2011 Kaleidoscope of Exotic Quantum Phases in a Frustrated XY Model Christopher N. Varney,1,2 Kai Sun,1,3 Victor Galitski,1,3 and Marcos Rigol2 1Joint Quantum Institute and Department of Physics, University of Maryland, College Park, Maryland 20742, USA 2Department of Physics, Georgetown University, Washington, D.C. 20057, USA 3Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA (Received 19 May 2011; revised manuscript received 17 June 2011; published 8 August 2011; publisher error corrected 10 August 2011) The existence of quantum spin liquids was first conjectured by Pomeranchuk some 70 years ago, who argued that frustration in simple antiferromagnetic theories could result in a Fermi-liquid-like state for spinon excitations. Here we show that a simple quantum spin model on a honeycomb lattice hosts the long sought for Bose metal with a clearly identifiable Bose surface. The complete phase diagram of the model is determined via exact diagonalization and is shown to include four distinct phases separated by three quantum phase transitions. DOI: 10.1103/PhysRevLett.107.077201 PACS numbers: 75.10.Kt, 05.30.Rt, 21.60.Fw, 75.10.Jm We learn early in our education that as matter is cooled low-lying fermionic spinon excitations that resemble a down to low temperatures it normally experiences transi- Fermi-liquid state. Also, the work of Motrunich, Fisher, tions into ordered states of various kinds—crystalline solid and Sheng [14,15] provides strong arguments in favor of structures, ordered magnetic phases, superfluid and super- the existence of such putative two-dimensional Bose met- conducting states, etc. It is also common knowledge that als and suggests that the strong singularity in the spin upon heating the matter, the ordered phases melt into the structure factor at a Bose surface is one of the hallmark familiar gaseous or liquid classical states that we encounter phenomena of this exotic state. routinely in our everyday lives. A more specialized but The main idea is that, despite the fact that the underlying equally well-established result is that no order that breaks a particles are bosons, the collective behaviors in these continuous symmetry can survive in one-dimensional sys- strongly correlated Bose-metal states show a strong anal- tems [1], because quantum zero-point motion acts there ogy to a Fermi liquid formed by fermionic particles. In a similarly to thermal effects and ‘‘quantum-melts’’ ordered Fermi liquid, the fermion statistics dictate the formation of phases even at zero temperature. Fermi surfaces, which possess singular behavior. In a Bose It has been a long-standing and important question in metal, despite the absence of Pauli’s principle, similar physics whether quantum fluctuations in higher- singularities also arise and define a surface in momentum dimensional quantum spin or boson systems can have the space, known as a Bose surface [14,15]. The existence of a same debilitating effect, giving way to quantum liquids [2]. Bose surface is the key property and most striking experi- The interest in such a hypothetical spin liquid, also known mental feature of a Bose metal. However, unlike a Fermi as a Bose or spin metal [3], has experienced multiple liquid, where the Luttinger theorem requires that the Fermi revivals with the most prominent one associated with the wave vector depends on the density of fermions, the Bose discovery of high-temperature superconductivity [4,5]. wave vector in a Bose metal depends on the control pa- However, despite the decades of intensive search, no con- rameters and can vary continuously even at fixed particle vincing examples of a gapless spin liquid have been found density. in any realistic two-dimensional quantum model. Here we provide strong evidence that a model as simple In models that are fermionizable via the Jordan-Wigner as the XY-spin model on a honeycomb lattice with nearest- transformation [6,7], the existence of spin liquids has now neighbor (NN) and next-to-nearest-neighbor (NNN) inter- been firmly established [7,8], but the physics there mimics actions hosts, among other phases, a Bose metal with a somewhat the one-dimensional result [9–11]. What re- clearly identifiable Bose surface. Although we came across mains of crucial importance is whether a truly higher- this finding serendipitously, we would like to provide dimensional spin system may host a quantum liquid. qualitative arguments that could potentially guide searches Among the influential recent results here are the stability for other such interesting spin models. Note that the de- argument by Hermele et al. [12], who showed that there is scription of a spin Fermi-liquid-like state is necessarily a no fundamental obstacle to the existence of quantum spin gauge theory [5,16,17], which is very similar to that of the liquids, and a complete classification of quantum orders by Halperin-Lee-Read [18] quantized Hall (QH) state. In the Wen [13], who demonstrated that an amazing variety of latter gapless phase, the interacting electron system in a hypothetical gapless spin liquids can all be divided into large classical external field gives rise to composite fermi- several distinct classes, which include stable phases with ons in zero classical field but coupled to a fluctuating 0031-9007=11=107(7)=077201(4) 077201-1 Ó 2011 American Physical Society week ending PRL 107, 077201 (2011) PHYSICAL REVIEW LETTERS 12 AUGUST 2011 quantum field—the Chern-Simons field that implements of this Hamiltonian is an antiferromagnet [Fig. 1(b)]. flux attachment. The natural question here, considered When J2=J1 !1, however, the ground state is a spin before, e.g., in Refs. [19,20], is whether a fractional QH wave with 120 order [Fig. 1(d)]. Because the system is state of this or any other type is possible in a sensible lattice highly frustrated, there is a strong possibility of intermedi- model. ate phases. In Fig. 1(a), we show the phase diagram for The above remarks are relevant to our work because our 24-site clusters as a function of J2=J1, finding two inter- simple Hamiltonian, see Eq. (1) below, can be viewed as a mediate phases: (1) a quantum spin liquid and (2) an exotic natural ‘‘trial model’’ for such a possible fractional lattice spin wave state [Fig. 1(c)]. QH state per the following construction. Take the Haldane To pin down the phase boundaries, we consider the model [21] of noninteracting electrons on a honeycomb ground-state fidelity metric g, which has been shown to lattice with simple NN hoppings and complex NNN hop- be an unbiased and sensitive indicator of quantum phase i pings, jJ2je .If is nonzero it realizes a topological transitions [23,24]. In Fig. 2(a), we show the fidelity metric insulator or lattice ‘‘integer’’ QH state. Replace the fermi- for three different 24-site clusters [22,25]. There are three ons with hard-core bosons at half-filling [22] and it be- peaks in g, indicating three quantum phase transitions. As comes a promising strongly interacting model. Notably, the we discuss in greater detail below, the three clusters have most frustrated limit corresponds to ¼ , which maps at slightly different momentum space representations, result- half-filling to the following Hamiltonian ing in the second and third transitions to occur at slightly X X different values of J2=J1 for each cluster. ¼ ð y þ H c Þþ ð y þ H c Þ H J1 bi bj : : J2 bi bj : : ; (1) Another indicator of a phase transition can be seen in the h i hh ii ij ij NN energy E1 (the NNN energy is denoted by E2). In y Fig. 2(b), we show the ratio E1=E (here E is the ground- where bi (bi) is an operator that creates (annihilates) a state energy) for the 24D cluster. The transition points are hard-core boson on site i. Here, we require the sign of J2 to directly connected with the inflection points in E1=E.To ¼ be positive (that is, ), while the sign of J1 is in fact demonstrate this more clearly, we also show the derivative irrelevant because of the particle-hole symmetry of the of E1=E, whose minima coincide with the transitions de- honeycomb lattice (bi !bi for one of the two sublatti- ¼ 1 termined by the fidelity metric. ces). In what follows, J1 sets our unit of energy. From the mapping between spins and hard-core bosons, Note that Hamiltonian (1) maps to a frustrated it follows that the antiferromagnetic and the other two y ! þ ! À antiferromagnetic-XY model (bi Si and bi Si ), ordered states correspond to Bose-Einstein condensates X X þ À þ À (BECs) in which bosons condense into quantum states H ¼ J1 ðS S þ H:c:ÞþJ2 ðS S þ H:c:Þ: (2) i j i j with different momenta. To characterize these phases, we hiji hhijii measure the condensate fraction fc ¼ Ã1=Nb (Nb is the The properties of this Hamiltonian are governed by the total number of bosons) by computing the largest eigen- Ã ¼h y i dimensionless control parameter J2=J1. The limits of this value 1 of the one-particle density matrix ij bi bj .If model are well understood. For J2=J1 ¼ 0, the ground state fc scales to a nonzero value in the thermodynamic limit, then the system exhibits Bose-Einstein condensation [26]. This is the case in three of our phases, as depicted in (a) I II III IV Spin Anti-ferromagnet Spin Wave 120˚ Order 102 Liquid (a) 1 24C 10 24D 10-1 100 J2 / J1 0 24E g 10 (b) (c) (d) 10-1 10-2 1.2 (b) 0 0.8 / E 1 E -3 0.4 Derivative E1 / E Derivative FIG.
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