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. sublattices). In what follows, J1 = 1 sets our unit of From the mapping between spins and hard-core energy. Note that Hamiltonian (1) maps to a frustrated bosons, it follows that the anti-ferromagnetic and the 3

0.5

0.4 J1 J2 1.00 0.00 0.3 1.00 0.30 c

f 1.00 0.80 0.00 1.00 0.2

0.1

0 0 0.05 0.1 0.15 0.2 0.25 1 / L

FIG. 3: Finite-size scaling of the condensate fraction fc for parameters that are representative of each phase depicted in Fig. 1 (J2/J1 = 0.00, 0.30, 0.80, ∞). The color of each curve is consistent with the color coding in Fig. 1. In the limit L →∞, the condensate fraction is non-zero in the anti-ferromagnetic, ◦ k k spin wave, and 120 ordered phases. FIG. 4: n( ) vs for (a) the Néel state (J2/J1 = 0), (b- d) Bose metal (J2/J1 = 0.27, 0.30, and 0.32), (e) collinear ◦ spin wave (J2/J1 = 0.80), and (f) the 120 ordered state other two ordered states correspond to Bose-Einstein (J2/J1 = ∞). In panels (b-d), the Bose surface is indicated condensates (BECs) in which bosons condense into quan- by the dashed red line and has a radius of magnitude kB = 0.9, 1.0, and 1.4, respectively. Each plot contains 19, 600 k-points. tum states with different momenta. To characterize these phases, we measure the condensate fraction fc =Λ1/Nb (Nb is the total number of bosons), by computing the largest eigenvalue Λ1 of the one-particle density matrix kB at which the maxima of n(k) occurs changes (in- † ρij = bi bj . If fc scales to a non-zero value in the ther- creases) with increasing J2/J1. The important distinc- modynamic limit, then the system exhibits Bose-Einstein tion to be made here is that those maxima do not reflect condensation [25]. This is the case in three of our phases, Bose-Einstein condensation, i.e. they do not scale with as depicted in Fig. 3. For the Bose metal phase, on the system size as the ones in the other three phases do. other hand, the condensate fraction vanishes in the ther- We should add that, in order to exclude other or- modynamic limit, indicating the absence of BEC. dering tendencies in Phase II, we also examined the To further examine the properties of these phases, we z z z z z S correlation function Ci,j = (Sia − Sib)(Sja − Sjb) calculate the single particle occupation at different mo- and the dimer-dimer correlation function Dij,kℓ = mentum points (Si Sj )(Sk Sℓ) and their corresponding structure fac- † † tors. Finite-size scaling of these structure factors (not n(k)= αkαk + βkβk . (3) shown) made evident that neither charge density wave Here, αk and βk are boson annihilation operators at mo- formation nor dimer formation occurs. We also com- mentum k for the A and B sublattices. Because we are puted the excitation gap in the Bose metal phase and studying finite-sized clusters, we utilize twisted bound- found it to be much smaller than the (finite-size) exci- ary conditions [26] and average over 40 × 40 boundary tation gap in the anti-ferromagnetic state. In the anti- conditions to fully probe the Brillouin zone. ferromagnetic phase, the system is gapless in the ther- In Fig. 4, we show the momentum distribution function modynamic limit, due to the spontaneous breaking of as a function of k for select values of J2/J1 that are the spin-rotation symmetry and the resulting Goldstone representative of each phase. In the first phase [Fig. 4(a)], modes. Since the gap in the Bose metal phase is signifi- the momentum distribution function is sharply peaked cantly smaller, we believe this gap is also due to finite-size at k = Γ, indicating an anti-ferromagnetically ordered effects and will close in thermodynamic limit. Our small state. The third and fourth phases [Figs. 4(e) and 4(f), system-sizes prevent us from reaching conclusive results respectively] also exhibit shark peaks in n(k), but this for this quantity after a finite size extrapolation. Never- time at the edges of the Brillouin zone. For Phase III theless, all the phenomena we observed in this phase are [Fig. 4(e)], n(k) is maximal at k = M, corresponding to consistent with and indicate a Bose metal phase. the collinear spin wave state illustrated in Fig. 1(c). For In Fig. 5, we illustrate how both the momentum Phase IV [Fig. 4(f)], n(k) is maximal at k = K, as one distribution function and the largest eigenvalue of ◦ † † would expect for a 120 ordered phase [Fig. 1(d)]. αi αj + βi βj , λ1, evolve over the entire parameter space The momentum distribution function in the Bose for two different 24 site clusters with periodic boundary metal is depicted in Figs. 4(b-d) for three different val- conditions. The maximum of n(k) perfectly matches λ1, ues of J2/J1. One can see there that n(k) in this and it is clear that the momenta of the condensates in phase exhibits a remarkable J2/J1-dependent Bose sur- Phases I, III, and IV are k = Γ, M, and K, respectively. face. Namely, the magnitude of the Bose wave vector In addition, it can be seen that the momentum distribu- 4

Γ λ (a) k = k = K k = k2 1 to be studied in condensed matter systems. As a matter k = M k = k k = k 6 1 3 of fact, a gapped spin liquid phase has been recently ob-

4 served in a numerical study of the Hubbard model in the n(k) honeycomb lattice [29]. Further studies of the eﬀective 2 Heisenberg model that emerges in such systems support the early results, see e.g. Ref. 30 in which a phase dia- 0 Γ λ (b) k = k = M k = K k = k1 1 gram with similarities to ours has been reported. Exper- 6 imental set-ups dealing with strongly correlated bosons

4 are available in optical lattice systems [1].

n(k) This research was supported by NSF through JQI-PFC 2 (C.N.V. and K.S.), ONR (C.N.V. and M.R.), US-ARO (V.G.), and NSF under Grant No. PHY05-51164. The 0 0.1 1 authors thank M.P.A. Fisher and L. Fu for discussions J2 / J1 and A. Läuchli and M.M. Ma´ska for useful comments on the manuscript. FIG. 5: Momentum distribution function (symbols) for the different values of k and clusters (a) 24C and (b) 24D. Also shown are the largest eigenvalue λ1 of the single-particle density matrix (line). The inset of each panel illustrates the k- points for each cluster. [1] M. A. Cazalilla, et al., arXiv:1101.5337 (2011). [2] I. Y. Pomeranchuk, Zh. Eksp. Teor. Fiz., 11, 226 (1941). [3] P. W. Anderson, Science, 235, 1196 (1987). tion in Phase II exhibits a peak inside the Brillouin zone [4] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys., that shifts to larger momenta as J2/J1 is increased. 78, 17 (2006). Phases III and IV exhibit an interesting phenomenon [5] P. Jordan and E. Wigner, Z. Phys. A, 47, 631 (1928). that can be unveiled by examining the degeneracy of [6] V. Galitski, Phys. Rev. B, 82, 060411(R) (2010). the largest eigenvalues of the one-particle density ma- [7] A. Paramekanti, L. Balents, and M. P. A. Fisher, Phys. 66 trix. In an ordinary BEC state, condensation occurs Rev. B, , 054526 (2002). [8] V. J. Emery, et al., Phys. Rev. Lett., 85, 2160 (2000). to a unique effective single particle state, and thus the [9] A. Vishwanath and D. Carpentier, Phys. Rev. Lett., 86, largest eigenvalue of the density matrix is nondegenerate 676 (2001). and O(Nb), while the second largest eigenvalue is already [10] R. Mukhopadhyay, C. L. Kane, and T. C. Lubensky, O(1). This is what we find in the anti-ferromagnetic state Phys. Rev. B, 64, 045120 (2001). (Phase I). However, in general, condensation can occur [11] M. Hermele, et al., Phys. Rev. B, 70, 214437 (2004). [12] X.-G. Wen, Phys. Rev. B, 65, 165113 (2002). to more than one effective one-particle state [27, 28], and 75 various largest eigenvalues of the one-particle density ma- [13] O. I. Motrunich and M. P. A. Fisher, Phys. Rev. B, , 235116 (2007). trix may become O(Nb) and degenerate. This fragmen- [14] D. N. Sheng, O. I. Motrunich, and M. P. A. Fisher, Phys. tation occurs in phases III and IV. For phase IV, it is Rev. B, 79, 205112 (2009). trivial to realize that the condensate must be degener- [15] L. B. Ioffe and A. I. Larkin, Phys. Rev. B, 39, 8988 ate in the limit J2/J1 → ∞, where the system consists (1989). of two disconnected triangular lattices. Interestingly, in [16] O. I. Motrunich, Phys. Rev. B, 72, 045105 (2005). [17] B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B, this model, fragmentation occurs for all values of J2/J1 47 in phases III and IV, and is related to the number of M , 7312 (1993). [18] A. Seidel, et al., Phys. Rev. Lett., 95, 266405 (2005). or K points present in the clusters under consideration. [19] A. A. Burkov, Phys. Rev. B, 81, 125111 (2010). In summary, we have studied a frustrated XY model [20] F. D. M. Haldane, Phys. Rev. Lett., 61, 2015 (1988). on a honeycomb lattice. We find that this model exhibits [21] C. N. Varney, et al., Phys. Rev. B, 82, 115125 (2010). four phases [see phase diagram in Fig. 1(a)]: (I) a BEC [22] P. Zanardi and N. Paunković, Phys. Rev. E, 74, 031123 at k = Γ (anti-ferromagnetism), (II) a Bose-metal (spin (2006). 80 liquid), (III) a BEC at k = M (a collinear spin wave), [23] M. Rigol, B. S. Shastry, and S. Haas, Phys. Rev. B, , k ◦ 094529 (2009). and (IV) a BEC at = K (120 order). The Bose metal [24] See supplemental material for a description of the ground phase is characterized by a parameter dependent peak in state fidelity and the clusters used in this study. n(k), and a lack of condensation, solid order, and dimer [25] O. Penrose and L. Onsager, Phys. Rev., 104, 576 (1956). order. [26] D. Poilblanc, Phys. Rev. B, 44, 9562 (1991). Due to the simplicity of the model considered here, we [27] A. J. Leggett, Rev. Mod. Phys., 73, 307 (2001). believe that there is no fundamental challenge to realize [28] T. D. Stanescu, B. Anderson, and V. Galitski, Phys. 78 the Bose metal phase in experimental systems dealing Rev. A, , 023616 (2008). [29] Z. Y. Meng, et al., Nature, 464, 847 (2010). with spins or with bosons in the regime of large on-site [30] A. F. Albuquerque, et al., arXiv:1102.5325 (2011). Hubbard repulsion. The first case may be more suitable