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Kaleidoscope of Exotic Quantum Phases in a Frustrated XY Model

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Kaleidoscope of Exotic Quantum Phases in a Frustrated XY Model 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, DC 20057, USA and 3Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA The existence of quantum spin liquids was first conjectured by Pomeranchuk some seventy 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. PACS numbers: 75.10.Kt, 67.85.Jk, 21.60.Fw, 75.10.Jm We learn early in our education that as matter is cooled (a) I II III IV down to low temperatures it normally experiences tran- Spin Anti-ferromagnet Spin Wave 120Ê Order sitions into ordered states of various kinds - crystalline Liquid solid structures, ordered magnetic phases, superfluid and 10-1 100 superconducting states, etc. It is also common knowledge J2 / J1 that upon heating the matter, the ordered phases melt (b) (c) (d) into the familiar gaseous or liquid classical states that we encounter routinely in our everyday lives. A more specialized but equally well-established result is that no order can survive in one-dimensional systems [1], because quantum zero-point motion acts there similarly to ther- mal effects and “quantum-melts” ordered phases even at zero temperature. It has been a long-standing and important ques- FIG. 1: (a) Phase diagram of the model in Eq. (1) as a function of J2/J1, (b) anti-ferromagnetic ordering (Phase I), tion in physics whether quantum fluctuations in higher- (c) spin wave ordering with wavevector k = M (Phase III), (d) dimensional quantum spin or boson systems can have collinear spin wave ordering with wavevector k = K (Phase the same debilitating effect, giving way to quantum IV). The phase boundaries are (J2/J1)I→II = 0.210 ± 0.008, liquids [2]. The interest to such a hypothetical spin liq- (J2/J1)II→III = 0.356 ± 0.009, and (J2/J1)III→IV = 1.320 ± uid, also known as a Bose or spin metal, has experienced 0.020. multiple revivals with the most prominent one associ- ated with the discovery of high-temperature supercon- ductivity [3, 4]. However, despite the decades of intensive results here are the stability argument by Hermele search, no convincing examples of a gapless spin-liquid et al. [11] who showed that there is no fundamental has been found in any truly two-dimensional quantum obstacle to the existence of quantum spin liquids, model. We note that only boson and spin systems are and a complete classification of quantum orders by of interest here, as the non-superconducting metals are Wen [12], who demonstrated that an amazing variety of abundant in nature and provide explicit examples of elec- hypothetical gapless spin liquids can all be divided into tronic Fermi liquids. We also remark that since spin-1/2 several distinct classes, which include stable phases with models can be mapped onto hardcore boson models and low-lying fermionic spinon excitations that resemble a vice versa, one can use the terms “spin liquid” and “Bose- Fermi-liquid state. Also, the work of Motrunich, Fisher, liquid” (also “Bose-metal”) interchangeably when refer- and Sheng [13, 14] provides strong arguments in favor of ring to such states, as we do below. existence of such putative two-dimensional Bose metals In models that are fermionizable via the and suggests that the strong singularity in the spin Jordan-Wigner transformation [5, 6], the existence structure factor at a Bose surface is one of the hallmark of spin liquids has now been firmly established [6, 7], but phenomena of this exotic state. the physics there mimics somewhat the one-dimensional The main idea is that despite the fact that the underly- result [8–10]. What remains of crucial importance is ing particles are bosons, the collective behaviors in these whether a truly higher-dimensional spin system may strongly correlated Bose-metal states show a strong anal- host a quantum liquid. Among the influential recent ogy to a Fermi liquid formed by fermionic particles. In 2 102 a Fermi liquid, the fermion statistics dictate the forma- (a) tion of Fermi surfaces, which possess singular behavior. 1 24C 10 24D In a Bose metal, despite the absence of Pauli’s princi- 0 24E ple, similar singularities also arise and define a surface g 10 in momentum space, known as a Bose surface [13, 14]. 10-1 The existence of a Bose surface is the key property and 10-2 most striking experimental feature of a Bose metal. How- 1.2 (b) ever, unlike a Fermi liquid, where the Luttinger theorem 0 requires that the Fermi wave vector depends on the den- 0.8 / E 1 sity of fermions, the Bose wave vector in a Bose metal E -3 0.4 E1 / E Derivative depends on the control parameters and can vary contin- Derivative uously even at fixed particle density. 0 -6 Here we provide strong evidence that a model as 0.1 1 simple as the XY -spin model on a honeycomb lat- J2 / J1 tice with nearest-neighbor (NN) and next-to-nearest- FIG. 2: (a) Fidelity metric vs. J2/J1 for clusters 24C, 24D, neighbor (NNN) interactions hosts, among other phases, and 24E. (b) Ratio of the NN energy to the total energy E1/E a Bose metal with a clearly-identifiable Bose surface. Al- and its derivative (right axis) for the 24D cluster. though, we came across this finding serendipitously, we would like to provide qualitative arguments that could † + − potentially guide searches for other such interesting spin anti-ferromagnetic-XY model (bi → Si and bi → Si ), models. Note that the description of a spin Fermi-liquid- like state is necessarily a gauge theory [4, 15, 16], which is + − + − H = J1 X(Si Sj + H.c.) + J2 X(Si Sj + H.c.). (2) very similar to that of the Halperin-Lee-Read [17] quan- ij ij tized Hall (QH) state. In the latter gapless phase, the interacting electron system in a large classical external The properties of this Hamiltonian are governed by field gives rise to composite fermions in zero classical field the dimensionless control parameter J2/J1. The limits but coupled to a fluctuating quantum field - the Chern-Si- of this model are well understood. For J2/J1 = 0, the mons field that implements flux attachment. The natural ground state of this Hamiltonian is an anti-ferromagnet question here, considered before e.g. in Refs. 18 and 19, [Fig. 1(b)]. When J2/J1 → ∞, however, the ground state is whether a fractional QH state of this or any other type is a spin wave with 120◦ order [Fig. 1(d)]. Because the is possible in a sensible lattice model. system is highly frustrated, there is a strong possibility The above remarks are relevant to our work because of intermediate phases. In Fig. 1(a), we show the phase our simple Hamiltonian, see Eq. (1) below, can be viewed diagram for 24-site clusters as a function of J2/J1, finding as a natural “trial model” for such a possible fractional two intermediate phases: (1) a quantum spin liquid and lattice QH state per the following construction. Take (2) an exotic spin wave state [Fig. 1(c)]. the Haldane model [20] of non-interacting electrons on a To pin down the phase boundaries, we consider the honeycomb lattice with simple NN hoppings and com- ground-state fidelity metric g, which has been shown to iφ plex NNN hoppings, |J2|e . If φ is non-zero it re- be an unbiased and sensitive indicator of quantum phase alizes a topological insulator or lattice “integer” QH transitions [22, 23]. In Fig. 2(a), we show the fidelity met- state. Replace the fermions with hardcore bosons at ric for three different 24-site clusters [21, 24]. There are half-filling [21] and it becomes a promising strongly in- three peaks in g, indicating three quantum phase tran- teracting model. Notably, the most frustrated limit cor- sitions. As we discuss in greater detail below, the three responds to φ = π, which maps at half-filling to the fol- clusters have slightly different momentum space repre- lowing Hamiltonian sentations, resulting in the second and third transitions to occur at slightly different values of J2/J1 for each clus- † † ter. H = J1 X(bi bj + H.c.) + J2 X(bi bj + H.c.), (1) Another indicator of a phase transition can be seen ij ij in the NN energy E1 (the NNN energy is denoted by E2). In Fig. 2(b), we show the ratio E1/E (here E is the † where bi (bi ) is an operator that creates (annihilates) a ground state energy) for the 24D cluster. The transition hard-core boson on site i. Here, we require the sign of points are directly connected with the inflection points J2 to be positive (that is, φ = π), while the sign of J1 is in E1/E. To demonstrate this more clearly, we also show in fact irrelevant because of the particle-hole symmetry the derivative of E1/E, whose minima coincide with the of the honeycomb lattice (bi → −bi for one of the two transitions determined by the fidelity metric.
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