A Quantum Dipolar Spin Liquid
Norman Y. Yao, Michael P. Zaletel, Dan M. Stamper-Kurn, Ashvin Vishwanath
AA Quantum Dipolar Spin Liquid
N. Y. Yao, M. P. Zaletel, D. M. Stamper-Kurn, and A. Vishwanath Physics Department, University of California Berkeley, Berkeley, CA 94720, U.S.A. Station Q, Microsoft Research, Santa Barbara, CA 93106, U.S.A.
Quantum spin liquids are a new class of magnetic ground state in which spins are quantummechanically entangled over macroscopic scales. Motivated by recent advances in the control ofpolar molecules, we show that dipolar interactions between S = 1 / PACS numbers: 37.10.Jk, 75.10.Jm, 75.10.Kt
In strongly frustrated systems, competing interactionscan conspire with quantum fluctuations to prevent classi-cal order down to zero temperature. In an antiferromag-net, frustration allows magnetic moments to evade theformation of conventional long-range order, leading to themagnetic analog of liquid phases. Such quantum spin liq-uids are characterized by entanglement over macroscopicscales and can exhibit a panoply of exotic properties,ranging from emergent gauge fields and fractionalized ex-citations to robust chiral edge modes [1–3]. Definitivelyfinding and characterizing such an exotic paramagnet re-mains one of the outstanding challenges in strongly in-teracting physics.When antiferromagnetic interactions are short-ranged,frustration relies on geometry: for example, lattices con-taining plaquettes with an odd number of sites may frus-trate N´eel order. This route is most pertinent in solid-state magnets, where exchange interactions are short-ranged, and has led to the discovery of a number of ex-citing spin liquid candidates in layered two-dimensionalMott insulators [3–7]. An alternate route to frustra-tion is provided by longer range interactions. An ar-ray of numerical studies have demonstrated that addingfurther-neighbor couplings can destabilize classical orderand lead to spin liquid phases. Unfortunately, liquidphases are often found only for a narrow range of fur-ther neighbor couplings comparable to the nearest neigh-bor exchange, making it challenging to identify relevantphysical systems.The recent emergence of polar-molecular gases opens anew route toward long-range interactions [8–11]: in con-trast to both their atomic cousins and conventional quan-tum materials, polar molecules exhibit strong, dipolarinteractions [12–14]. However, these interactions are nei-ther isotropic nor obviously frustrated, leading to manyproposals which ‘engineer’ frustrated phases via the useof multiple molecular states, microwave dressing fields,and spatially varying optical potentials [15–20].Furthermore, although long-ranged, the dipolar cou-plings are not easily fine-tuned; rather, scale invariancedictates that the simplest effective Hamiltonian one could ⇥ ⇡/ ⇡/ ⇡/ ⇡/ XY Magnet Spin Liquid ⇡/ ⇡/ ⇡/ ⇡/ XY Magnet Ferrimagnet Striped Neel Neel + XY Magnet Chiral Spin Liquid ⇥ XY Magnet Striped Neel Striped Neel (a) (b) Kagome Triangular . .
20 0 . . . . . . . . FIG. 1. a) Phase diagram of the dipolar Heisenberg modelon the a) kagome lattice (YC8 geometry truncated at J ) andthe b) triangular lattice (YC6 geometry truncated at J ) asa function of the XXZ anisotropy ∆ (which is controlled bythe magnitude of the applied electric field [see Fig. 2]) and thepolar tilt, Θ , of the applied electric field (the azimuthal angleis given by the green arrow). Near Θ = 0, where the modelis fully frustrated, we observe quantum spin liquid groundstates on both geometries. Ordered phases for Θ > hope for is a ‘dipolar Heisenberg antiferromagnet:’ H = X i,j ~S i · ~S j | R ij | . (1)Two fundamental questions arise: is H naturally realized,and what is its ground state?Here, we answer both of these questions. First, weconsider synthetic quantum magnets constructed from a r X i v : . [ c ond - m a t . s t r- e l ] O c t an array of lattice-trapped, polar molecules interactingvia dipole-dipole interactions. We demonstrate that thissystem easily realizes the dipolar Heisenberg antiferro-magnet, requiring only a judicious choice of two molec-ular rotational states (to represent a pseudo-spin) anda constant electric field [18]. The simplicity of our pro-posal stems from using rotational states with no angularmomentum about the electric field axis. This contrastswith previous works where non-zero matrix elements ap-pear for the transverse electric dipole operator, unavoid-ably generating ferromagnetic spin-spin interactions be-cause of the inherent anisotropy of the dipolar interaction[19, 20].Second, motivated by this physical construction, weperform a large-scale density matrix renormalizationgroup [21, 22] and exact diagonalization study of thedipolar Heisenberg model and find evidence for quantumspin liquid ground states on both triangular and kagomelattices (Fig. 1). Because of the long-range interactionsand the need for time-reversal breaking complex wave-functions, our model is one to two orders of magnitudemore challenging to simulate numerically than earliernearest-neighbor models. The further-neighbor dipolarcouplings play a crucial role, leading to a different phaseof matter for both lattice geometries when compared totheir nearest-neighbor counterparts realized in Mott in-sulating materials. We compute the phase diagram of thedipolar Heisenberg model as a function of experimentalparameters (the electric field strength and tilt) for anyultracold polar molecule. Realization —We consider a two-dimensional array ofpolar molecules trapped in an optical lattice. Thelattice freezes the translational motion, leaving eachmolecule to behave as a simple dipolar rigid rotor [16–20].The Hamiltonian governing these molecular rotations is H m = B J + ~E · d , where B is the rotational constant, J is the angular momentum operator, ~E is the externalelectric field, and d is the dipole operator. For | E | = 0,each molecule has eigenstates indexed by | J, M i , where M is the z -component of angular momentum. An ap-plied electric field, ~E = E ˆ z , weakly aligns the moleculesalong the field direction, mixing states with identical M .Each | J, M i evolves adiabatically with E , picking up adipole moment and splitting the degeneracy within each J manifold at order ( dE ) /B (inset Fig. 2).The molecules interact with one another via the elec-tric dipole-dipole interaction, H dd = g X i = j R ij h d i · d j − d i · ˆR ij )( d j · ˆR ij ) i , (2)where g = 1 / (4 π(cid:15) ) and R ij is the displacement betweenmolecules i and j . Referring to Fig. 2, we select the dou-blet |↓i = | , i and |↑i = | , i , which are energeticallyresolved from all other rotational states, to play the roleof a “spin” [18]. We let S µ denote the usual spin op-erators in this subspace, but note, that unlike S = 1 / k y + N y * k x E SU(2) XY (cid:1)(cid:2) | , i| , i | , i | , i B Ising | dE | B | E | ( d/B ) FIG. 2. The XXZ anisotropy ∆ is controlled by the electricfield strength, E , measured relative to the rotational splittingdivided by the dipole moment, B/d . Top left inset - the rota-tional states used as the two-level pseudo-spin. Bottom rightinset - molecules reside in the XY plane and the electric fieldis oriented along ˆ z . derive the effective Hamiltonian, we project H dd onto thetwo-level subspace and drop S z non-conserving terms asthey are strongly off-resonant. This projection is physi-cally justified by the separation of energy scales betweenthe dipolar interaction and the rotational level-splittings: gd /R (cid:28) B, ( d E ) /B .When the electric field is aligned perpendicular to thelattice plane (Θ = 0, inset Fig. 2), we find [18] H eff = g X i,j R ij (cid:2) d ( S xi S xj + S yj S yi ) + ( µ − d ) S zi S zj (cid:3) (3)where d = h , | d z | , i is the transition dipole mo-ment and d = h , | d z | , i , µ = h , | d z | , i are theelectric field induced “permanent” dipole moments. Thesign of the couplings shows that the interaction is anti-ferromagnetic along all spin-axes.As depicted in Fig. 2, the ratio ∆ = ( µ − d ) d , betweenthe Ising and XY interactions [Eqn. 3] is controlled bythe magnitude of the applied electric field. SO(3) sym-metry emerges for | dE | ≈ . B , at which point the ef-fective Hamiltonian is precisely the dipolar Heisenbergmodel. We note that H eff is in stark contrast to the typ-ical spin models analyzed for polar molecules. In partic-ular, previous works have generally considered rotationalstates that lead to ferromagnetic interactions favoringeasy-plane (XY) magnetism; frustrated phases arise onlyupon fine-tuning via microwave and optical dressing [16–20]. Ground State of the Dipolar Heisenberg Antiferromag-net —While the antiferromagnetic dipolar Heisenberg in-teraction is frustrated on any lattice, geometries with tri-angular motifs typically enhance this frustration as it isimpossible for all neighboring spins to anti-align. Here,we consider kagome and triangular lattices, both of whichhave been realized in optical lattices [23–25].The ground state of the dipolar Heisenberg antifer-
YC8-2 YC10-2 YC12-2 J1 J2 J3 J8 J11 J5 ⇥ ⇠ m ⇠ ⇥ XY Magnet Striped Neel Spin Liquid Chiral Spin Liquid
XY Magnet (a) (d) (e) z (b) -0.006 - . - . FIG. 3. a ) Triple extrapolation of the chiral order parameter χ as a function of the DMRG bond dimension ( m ), the cylindercircumference and the range of the dipolar interaction. For YC8-2 and YC10-2, all numerics have converged to a truncationerror < − while for YC12-2, we observe a truncation error ∼ . × − at bond dimension m = 5800. b , c ) The NN andNNN h S i · S j i correlations of the kagome (YC10-2) / triangle (YC8) spin liquid respectively. The magnitude of the correlationfunction for each bond is shown and is directly proportional to the linewidth of the bond (see supplementary information forfurther detail). d ) Phase transition out of the chiral spin liquid (holding ∆ = 1 . ) as characterized by thevanishing of χ and the diverging correlation length ξ (with m ). e ) Phase transition out of the triangular spin liquid (holding∆ = 1 . ) as characterized by σ z , the variance of S z across the unit cell, and the correlation length ξ . romagnet is unknown for either lattice. Even for short-range interactions, the phase diagram in these geometrieshas been an open question for more than two decades,due to delicate energetic competition between many com-peting phases. Recently, progress has been madeusingthe density matrix renormalization group (DMRG) [26–34]. As DMRG is a 1D method, it requires mapping the2D lattice to a quasi-1D geometry; here, we study bothfinite-length and infinitely long cylinders of circumfer-ence L . The dipolar interaction introduces an additionaldifficulty, as its range must be truncated for a consistentdefinition on the cylinder. Thus, our numerics requirea triple extrapolation in L , the interaction range, andthe accuracy of the DMRG as quantified by the ‘bonddimension’ m .Detecting and characterizing a quantum spin liquidphase follows a decision tree. By definition, “liquid”refers to the absence of spontaneous symmetry break-ing, specifically of spin-rotations and translation invari- ance. Any liquid phase with half integer spin in the unitcell must be exotic: the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem requires that the phase be either an ex-otic gapless spin liquid or a gapped spin liquid with frac-tionalized excitations [35, 36]. In the gapless case, theground state has a diverging correlation length as thecircumference of the cylinder is increased. In the gappedcase, the ground state will have exponentially decayingcorrelations, protected ground state degeneracy, and cer-tain characteristic signatures in its entanglement spec-trum [37, 38].There exists a zoo of gapped spin liquids distinguishedby the braiding and statistics of their fractional exci-tations. The two simplest cases are the time-reversalsymmetric (TRS) Z spin liquid and the time-reversalbreaking chiral spin liquid (CSL) [2, 39]; the spontaneousbreaking of time-reversal is detected by using a chiral or-der parameter χ = h S i · S j × S k i /
3, where i, j, k are thethree sites of a triangle. k a /⇡ semion YC10 k a /⇡ l og ( p a ) (c) (a) (b) vacuum k a /⇡ fermionic spinon FIG. 4. The entanglement spectrum { p a } as a function of k a , the momentum of the Schmidt state around the cylinder. Datapoints are colored and displaced slightly according to their S z quantum numbers. a ) The kagome YC10 model truncated at J . The momentum-resolved entanglement spectrum is consistent with the vacuum sector of a chiral spin liquid and exhibitsthe characteristic counting { , , , , · · · } predicted by the WZW edge theory. b ) Shifting the cut of the YC10 cylinder by asingle column gives the semion sector of the spin liquid, with half-integral representations of SO(3). c ) The triangular YC8model truncated at J . The spectrum is consistent with the fermionic spinon topological sector of a Z spin liquid. Let us now turn to the numerics. We refer to the cylin-der geometries using the notation of [26]; YCn is a cylin-der of circumference n lattice spacings periodized alonga Bravais vector (Fig. 3b,c). For both lattices, we define J n to be the coupling between n th nearest neighbor sites,ordered by their distance in real space, R n . We will be-gin by characterizing the ground state of each lattice atthe dipolar Heisenberg point and will subsequently mapout the full phase diagram of the molecular proposal. Kagome Model — Extensive theoretical and numericalstudies of the J − J − J kagome model reveal a richphase diagram, consisting of a honeycomb valence bondsolid, a Z spin liquid, a chiral spin liquid, and a multi-tude of ordered N´eel states [26–32, 40–42]. In contrast tothese previous studies, the long-range dipolar couplingscannot be tuned. For the kagome lattice it is necessaryto distinguish between two couplings of length R = 2 a : J (across hexagons) and J (along bow-ties). Motivatedby exchange interactions in Mott insulating materials,previous numerics have always considered J = 0. Inthe dipolar Heisenberg model, all couplings at a givendistance are equally important and a finite J in factstabilizes the CSL phase (see supplemental informationfor details). This is highlighted by the fact that keepingonly the J or J part of the dipolar interaction results inthe magnetically ordered q = (0 ,
0) phase [29–32]; onlyupon restoring the dipolar tail of the interaction does thesystem transition into the CSL.Let us now turn to the diagnostics of liquidity. Westudy cylinders of circumference L = 8 , ,
12 with dipo-lar cutoffs ranging from J to J . In addition to the YCngeometry, we also consider the so-called ‘YCn-2’ geom-etry in which cylinders are rolled up with a ‘twist’ thatidentifies sites that differ by Bravais vector n~a + ~a . Thisconvenient choice of boundary condition reduces the com-putational cost by decreasing the effective iDMRG unitcell, enabling better convergence for certain diagnostics. Crucially, neither the spin liquids nor the q = (0 ,
0) phaseare frustrated by this boundary condition; more gener-ally, for liquid phases, the resulting physics should beunaffected once the cylinder circumference is larger thanthe correlation length.We find h S µ i = 0, as required by the Mermin-Wagnertheorem in our quasi-1D geometry (note that for ∆ > ξ . . a (as calculated from theDMRG transfer matrix), consistent with a gapped para-magnet. The absence of local magnetization and long-range correlations indicates that spin rotation symmetryis preserved.To check that translational symmetry is also preserved(i.e. to rule out valence-bond order), we verify that thebond correlations are translation invariant (Fig. 3b) andalso calculate the overlap of the ground state, | Ψ i , witha translated version of itself, h Ψ | ˆ T y | Ψ i = (1 − (cid:15) ) V . Thisoverlap scales with the volume of the system, V , witherror (cid:15) < . J , but similar results arefound when truncating to J or extending to J , as wellas on the smaller YC8 geometry and the larger YC12-2geometry (see supplementary information for details).A key indication of the CSL phase is the spontaneousbreaking of time-reversal symmetry. To this end, thechiral order parameter | χ | is shown in Fig. 3a as a func-tion of the size of the cylinder, the truncation cutoff, andthe DMRG accuracy; | χ | increases weakly with cylindercircumference, converges with bond dimension, and sat-urates for large dipolar cutoff.In addition to spontaneous TRS breaking, the mostspectacular signature of a CSL is a chiral edge state.Quantum entanglement provides a way to probe theseedge states given only the ground state. The reduceddensity matrix ρ L for half of the cylinder can be viewedas a thermal density matrix of a semi-infinite cylinder,introducing a single ‘edge’. The spectrum p a of ρ L (the‘entanglement spectrum’) is known to mimic the energyspectrum of the physical edge; plotting this spectrumversus the momentum around the edge, k a , should re-veal a chiral dispersion relation [37, 43, 44]. As shownin Fig. 4a,b the momentum-resolved entanglement spec-trum of a YC10 cylinder indeed displays characteristiclevel counting { , , , , · · · } organized into SO(3) mul-tiplets consistent with the SU(2) Wess-Zumino-Wittenedge theory [45].
Triangular Model —Truncating the dipolar Heisenbergmodel at short range leads to N´eel order: for J only,a 120 o degree N´eel phase [46], and for J , J , a two-sublattice collinear N´eel phase [33, 34]. However, addingin the dipolar J coupling directly penalizes the order ofthe collinear state and appears to drive the system into aliquid; this is evidenced by a drastic change in the h S i · S j i correlation function as the long-range tail of the interac-tion is restored (see supplementary information). Withcouplings through J , the YC8 ground state has an XYcorrelation length of ξ . . a and is translationally sym-metric with (cid:15) < × − . Similar results are found whentruncating to J or extending to J , as well as on thesmaller YC6 geometry and the larger YC10 geometry.The phenomenology of the observed spin liquid phaseis equivalent to the J − J spin liquid reported in[33, 34]. The lowest energy state is time-reversal sym-metric and has an entanglement spectrum consistent withthe fermionic spinon topological sector of a Z spin liq-uid; it exhibits a four-fold degeneracy and a half-integralrepresentation of SO(3) as shown in Fig. 4c [38]. Whilethe bond correlations are translation invariant (Fig. 3c),they exhibit a noticeable striping consistent with nematicordering (note that this nematicity may be an artifact ofthe cylindrical geometry which breaks C symmetry) [33]. Phase Diagram —The above results (for both triangu-lar and kagome) were presented for the SO(3) symmetricHeisenberg anti-ferromagnet (∆ = 1) at | dE | ≈ . B .For both stronger (∆ = 1 .
6) and weaker (∆ = 0 . z -axis and zero net magnetization. As one tilts the electric field into the lattice plane, thespin liquids we observe begin to compete with magneti-cally ordered phases. The tilt generates angular depen-dence in the effective Hamiltonian, H eff = g X i,j R ij [1 − (Φ − Φ ) sin Θ ] × [2 d ( S xi S xj + S yj S yi ) + ( µ − d ) S zi S zj ] (4)where Φ, Φ are the polar angles of ~R ij and the elec-tric field orientation, respectively (inset of Fig. 2). Fornonzero Θ , full frustration is lost as dipoles begin topoint head-to-tail along the field direction, thereby ex-hibiting ferromagnetic interactions. For large Θ , a vari-ety of ordered phases appear as shown in Fig. 1a,b (forfull details, see supplementary information). Here, werestrict our interest to the phase boundaries of the spinliquid states.In Fig. 3d,e, we present two representative verticalcuts: 1) out of the kagome CSL at ∆ = 1 . .
6. In the kagome case,we identify the transition out of the chiral spin liquid viathe vanishing of the chiral order parameter (Fig. 3d). Inthe triangular case, we diagnose the phase transition byexamining the correlation length and the variance of the S z -magnetization (Fig. 3e). This reveals two phases, anXY magnet directly proximate to the spin liquid and theexpected striped N´eel phase for larger Θ . In additionto showing that the spin liquid phases persist to mod-erate electric field tilts, understanding the nature of theordered phases surrounding the spin liquids may enablethe preparation of these topological states [47].In summary, our proposal provides a new route towardstudying frustrated quantum magnetism in an ultracoldlattice gas. The dipolar Heisenberg antiferromagnet ex-hibits promising signs of spin liquid behavior on both thekagome and triangular lattices, distinct from models ofnearest-neighbor exchange. Looking forward, it is im-portant to consider the effects of lattice vacancies anddipolar relaxation as well as to identify unique signals offrustration in quench dynamics.We gratefully acknowledge the insights of and discus-sions with B. Lev, A. Gorshkov, A. M. Rey, M. Lukin,C. Laumann, J. Moore, M. Zwierlein and J. Ye. Thiswork was supported by the AFOSR MURI grant FA9550-14-1-0035 and the Miller Institute for Basic Research inScience. [1] P. Anderson, Materials Research Bulletin , 153 (1973).[2] V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. ,2095 (1987).[3] L. Balents, Nature , 199 (2010).[4] Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, andG. Saito, Phys. Rev. Lett. , 107001 (2003).[5] T. Itou, A. Oyamada, S. Maegawa, M. Tamura, and R. Kato, Phys. Rev. B , 104413 (2008).[6] M. Rigol and R. R. P. Singh, Phys. Rev. Lett. , 207204(2007).[7] J. S. Helton, K. Matan, M. P. Shores, E. A. Nytko, B. M.Bartlett, Y. Yoshida, Y. Takano, A. Suslov, Y. Qiu, J.-H.Chung, D. G. Nocera, and Y. S. Lee, Phys. Rev. Lett. , 107204 (2007). [8] K.-K. Ni, S. Ospelkaus, M. De Miranda, A. Pe’er,B. Neyenhuis, J. Zirbel, S. Kotochigova, P. Julienne,D. Jin, and J. Ye, science , 231 (2008).[9] A. Chotia, B. Neyenhuis, S. A. Moses, B. Yan, J. P.Covey, M. Foss-Feig, A. M. Rey, D. S. Jin, and J. Ye,Physical review letters , 080405 (2012).[10] J. Deiglmayr, A. Grochola, M. Repp, K. M¨ortlbauer,C. Gl¨uck, J. Lange, O. Dulieu, R. Wester, and M. Wei-dem¨uller, Phys. Rev. Lett. , 133004 (2008).[11] J. W. Park, S. A. Will, and M. W. Zwierlein, Phys. Rev.Lett. , 205302 (2015).[12] B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R.Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Nature ,521 (2013).[13] K. R. Hazzard, B. Gadway, M. Foss-Feig, B. Yan, S. A.Moses, J. P. Covey, N. Y. Yao, M. D. Lukin, J. Ye, D. S.Jin, et al. , Physical review letters , 195302 (2014).[14] M. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller,Chemical Reviews , 5012 (2012).[15] M. Lewenstein, Nature Physics , 309 (2006).[16] A. Micheli, G. Brennen, and P. Zoller, Nature Physics , 341 (2006).[17] A. V. Gorshkov, S. R. Manmana, G. Chen, J. Ye,E. Demler, M. D. Lukin, and A. M. Rey, Physical reviewletters , 115301 (2011).[18] A. V. Gorshkov, S. R. Manmana, G. Chen, E. Demler,M. D. Lukin, and A. M. Rey, Physical Review A ,033619 (2011).[19] N. Y. Yao, A. V. Gorshkov, C. R. Laumann, A. M.L¨auchli, J. Ye, and M. D. Lukin, Physical review let-ters , 185302 (2013).[20] S. R. Manmana, E. Stoudenmire, K. R. Hazzard, A. M.Rey, and A. V. Gorshkov, Physical Review B , 081106(2013).[21] S. R. White, Phys. Rev. Lett. , 2863 (1992).[22] I. P. McCulloch, (2008), arXiv:arXiv:0804.2509.[23] G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vish-wanath, and D. M. Stamper-Kurn, Physical review let-ters , 045305 (2012).[24] C. Becker, P. Soltan-Panahi, J. Kronj¨ager, S. D¨orscher,K. Bongs, and K. Sengstock, New Journal of Physics ,065025 (2010).[25] J. Struck, C. ¨Olschl¨ager, R. Le Targat, P. Soltan-Panahi,A. Eckardt, M. Lewenstein, P. Windpassinger, andK. Sengstock, Science , 996 (2011).[26] S. Yan, D. A. Huse, and S. R. White, Science , 1173(2011).[27] S. Depenbrock, I. P. McCulloch, and U. Schollw¨ock,Phys. Rev. Lett. , 067201 (2012).[28] H.-C. Jiang, Z. Wang, and L. Balents, Nature Physics , 902 (2012).[29] S.-S. Gong, W. Zhu, L. Balents, and D. N. Sheng, Phys.Rev. B , 075112 (2015).[30] Y. C. He, D. Sheng, and Y. Chen, Phys. Rev. Lett. ,137202 (2014).[31] S.-S. Gong, W. Zhu, and D. N. Sheng, Sci. Rep. (2014),10.1038/srep06317.[32] Y.-C. He and Y. Chen, Phys. Rev. Lett. , 037201(2015).[33] Z. Zhu and S. R. White, ArXiv e-prints (2015),arXiv:1502.04831.[34] W.-J. Hu, S.-S. Gong, W. Zhu, and D. N. Sheng, ArXive-prints (2015), arXiv:1504.00654. [35] M. Oshikawa, Phys. Rev. Lett. , 1535 (2000).[36] Hastings, M. B., Europhysics Letters , 824 (2005).[37] A. Kitaev and J. Preskill, Phys. Rev. Lett. , 110404(2006).[38] M. Zaletel, Y.-M. Lu, and A. Vishwanath, arXiv preprintarXiv:1501.01395 (2015).[39] X. G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B ,11413 (1989).[40] R. R. Singh and D. A. Huse, Physical Review B ,180407 (2007).[41] A. M. L¨auchli, J. Sudan, and E. S. Sørensen, PhysicalReview B , 212401 (2011).[42] B. Bauer, L. Cincio, B. P. Keller, M. Dolfi, G. Vidal,S. Trebst, and A. W. Ludwig, Nature communications (2014).[43] H. Li and F. Haldane, Phys. Rev. Lett. , 010504(2008).[44] X.-L. Qi, H. Katsura, and A. W. W. Ludwig, Phys. Rev.Lett. , 196402 (2012).[45] J. Wess and B. Zumino, Physics Letters B , 95 (1971).[46] T. Jolicoeur, E. Dagotto, E. Gagliano, and S. Bacci,Phys. Rev. B , 4800 (1990).[47] M. Barkeshli, N. Y. Yao, and C. R. Laumann, Phys.Rev. Lett. , 026802 (2015). Supplementary Information for A Quantum Dipolar Spin Liquid
N. Y. Yao, M. P. Zaletel, D. M. Stamper-Kurn, A. VishwanathHere, we present additional numerics for both the kagome and triangular models. We also provide a description of the orderedphases that appear at large electric field tilts Θ and examples of phase transitions among them. Kagome lattice —We computed the ground state on geometries YC8, YC10, YC12 and YC8-2, YC10-2, YC12-2 (‘shifted’).The topologically degenerate ground states of a spin-liquid on an un-shifted even circumference cylinder (e.g. YC8) are of twotypes: ‘odd’ sectors, in which Schmidt states carry half-integral representations of SO(3), and the ‘even’ sectors, in which theycarry integer representations. For YC8, we find ground states in both the even and odd sectors. For odd circumference andshifted cylinders, there is no such distinction. Calculations were repeated for a dipolar cutoff at J , J , J and J . There is atradeoff between the number of couplings kept and how well we can converge the DMRG in the bond dimension m . In the maintext, we presented data for the shifted geometries for cutoffs from J to J . Here, since we discuss the more difficult YC8 and - . -0.006 - . . . -0.01 . - . . - . . - . -0.0076 (a) (b) (c) (d) (e) (f) (g) FIG. S1: a, b ) The NN and NNN h S i · S j i correlation function of the YC10-2 kagome J − J model. Data is taken at m = 6000 . In panel (a) we show h S i · S j i , with blue negative and red positive; magnitudes of two typical bonds are labeled. Inpanel b), we plot the correlations after subtracting off the mean value for the bond type, revealing deviations. There isapparently a slight anisotropy, which is expected from the combination of a chiral order and the ‘twisted’ nature of the YC10-2cylinder. c-d ) same as in (a,b) for the YC10 kagome model. Unfortunately, we cannot fully converge the YC10 geometry( m = 3600 ), leading to a mottled pattern in the correlations at the level of 10%. Within this error results are consistent withYC10-2. e ) The h S · S i i correlation function of the YC10-2 model; site is indicated with an × . f ) The h S · S i i correlationfunction of the YC10 mode. g ) For contrast, the h S · S i i correlation function of the dipolar YC8 model truncated at J − J ,which is known to have N´eel order. Data is plotted on the same scale as (e), (f). a r X i v : . [ c ond - m a t . s t r- e l ] O c t S e n t ⇠ E S e n t E (a) (b) (c) m mm S e n t (d) Kagome YC10, YC10-2 Kagome YC10, YC10-2 Kagome YC8, YC8-2 YC12, YC12-2 Triangle YC8 FIG. S2: ( a - b ) Convergence of the energy ( E ), entanglement entropy ( S ent ), correlation length ( ξ ) and chiral order ( χ ) versusthe MPS bond dimension m , for the kagome YC10 and YC10-2 models with couplings up to J . c ) Same for the triangularYC8 model with couplings up to J d ) Convergence of ξ and S ent on the smaller YC8 (square markers) and larger YC12(triangle markers) geometry for the kagome J model.YC10 geometries, we focus on the J cutoff. We address two questions: 1) does the system have spontaneous chiral order? 2)is the state a liquid? a. Chiral order. All DMRG simulations were done with complex wavefunctions in order to allow for spontaneous chiralorder. For every sample except one - the even sector of YC8 - the state breaks time reversal with chiral order parameter χ ∼ . . Data for all the shifted samples was summarized in the main text, and strongly suggests that chiral order persists inthe thermodynamic limit. The YC10 chiral order is also χ ∼ . , consistent with the shifted result. For the exceptional case,the YC8 model truncated at J , the energies are E YC8:E = − . (with χ = 0 ) and E YC8:O = − . (with χ = 0 . )at m = 4200 . The YC8 odd sector has a chiral entanglement spectrum characteristic of the CSL. To interpret the discrepancybetween the odd and even cases, note the splitting of the topological degeneracy in the TR-symmetric nearest neighbor YC8model is ∆ E ∼ . , with the even sector lowest in energy. One interpretation is that the energies of the CSL and a TRsymmetric SL are split by about ∆ E ∼ . per site; for the YC8 cylinder, the splitting of the topological degeneracy iscomparable to this competition, and we find one state from each phase. For the other cylinders, there is no odd-even distinction,and we always find a CSL phase. It would be useful to study the odd / even sectors of the YC12 cylinder to verify this hypothesis.We are unable to converge YC12 for the J and J models, so must leave this to future SU(2) DMRG studies. b. Evidence for a liquid. To assess if the state is a liquid, we focus here on the YC10-2 and YC10 cylinders. While we canreasonably converge YC10-2 ( m = 6000 ), we cannot fully converge YC10 ( m = 4000 ) leading to artifacts in the correlations.Fig. S1(a-d) depicts the NN and NNN valence-bond correlations. Note that these geometries are expected to have a very slightdoubling of the unit cell, which should decrease exponentially with the cylinder circumference; indeed we find the bond energiesdiffer on the order of ∆ E ∼ . between the two sublattices, which is not observable on the scale of the figure. Fig. S1(e-f)shows the S · S i correlation function, both for YC10 and YC10-2. For contrast, we also show the S · S i correlation functionwhen the dipolar model is truncated at J [Fig. S1(g)], which exhibits q = (0 , order. As expected, the correlations in the q = (0 , phase are far larger, and longer ranged, than those of the putative CSL.Finally, to illustrate the convergence of the DMRG for YC10 and YC10-2 , we plot [Fig. S2(a,b)] the energy, correlationlength, chiral order and entanglement entropy versus the MPS bond dimension m . Fig. S2(d) provides a representative exampleof data at the smaller YC8 geometry and the larger YC12 geometry, both of which are consistent with the numerics for YC10. Triangular lattice —We first review the phenomenology found in Refs. 1 and 2, because equivalent results (and open ques-tions) are found in the dipolar model. For YC8, the odd sector has the lowest energy, and is significantly easier to converge withbond dimensions than the even sector. . . -0.012 - . . (a) (b) (c) (d) (e) FIG. S3: a ) The NN and NNN h S i · S j i correlation function of the YC8 triangular J − J model (odd sector). Data is taken at m = 6000 . In panel a we show h S i · S j i , with blue negative and red positive; magnitudes of two typical bonds are labeled. Inpanel b ), we plot the correlations after subtracting off the mean value for the bond type, revealing deviations. c ) The h S · S i i correlation function of the YC8 triangular J − J model (odd sector). Site is indicated with an × . d , e ) Correlation functionsof the triangular YC8 J − J dipolar model, which is known to be magnetically ordered. In d) we show the NN and NNN h S · S i i correlation function.The odd sector has 1) a short correlation length, 2) no sign of chiral ordering, and 3) a four-fold degenerate entanglementspectrum, which was argued in Ref. 3 to be consistent with the ‘fermionic spinon’ topological sector of a Z spin-liquid. Butconfusion arises when considering the even sector. In Ref. 2, it was found that the YC8 even sector has chiral-correlations out tolong distances, and on YC10 it spontaneously breaks time-reversal. This is problematic if both sectors are to be interpreted as thedegenerate ground states of a single spin-liquid. One possibility is that there are two distinct spin liquids, one which preservesTR and one which does does not. On these cylinders the splitting of their topological degeneracy may be comparable to thesplitting between the two spin liquids, and hence the DMRG finds one state from each. Larger system sizes will be required toresolve this issue, which are currently beyond the reach of DMRG.In light of this situation, our goal is simply to convince the reader that the lowest energy state of the truncated dipolarinteraction on a YC8 cylinder is essentially equivalent to the fermionic spinon state observed in previous studies. In our U(1)DMRG we are unable to study the even sector beyond bond dimensions m > , as the simulation tunnels into the odd sector. ⇠ XY AF Ising AF z ⇠ S e n t z S e n t XY AF Ising AF (a) (b) FIG. S4: a) Transition (at Θ = π/ ) between the XY antiferromagnet and the Ising antiferromagnet on the YC6 triangulargeometry with couplings up to J . b) Transition (at Θ = π/ ) between the XY antiferromagnet and the Ising antiferromagneton the YC8 kagome geometry with couplings up to J . In both cases, the phase transition is diagnosed by the diverging of ξ inthe XY phase and the onset of N´eel order in the Ising phase.The entanglement spectrum of the odd sector is shown in Fig. 4c of the main text; it can be compared with Fig. 5d) of Ref. 2,which has the same 4-fold degeneracy.In Fig. S3(a-b) we plot the nearest and next-nearest neighbor correlations h S i · S j i , both in absolute value and relative to themean value of the bond-type. Most importantly, translation symmetry is preserved, which rules out valence bond crystal orderat this circumference. There is a noticeable stripe pattern; the stripes preserve translation, but on the plane would imply nematicorder. Nematic order does not invalidate the Hastings-Lieb-Schultz-Mattis theorem. The cylinder geometry breaks the π/ rotational symmetry, so the nematic order could be a finite size artifact. The same stripe was found in Refs. [1, 2]. In Fig. S3(c)we plot the set of h S · S i i correlations with site ‘ ’. We compare both types of correlations with the J − J only model,Fig. S3(d-e), which is expected to be a magnetically ordered N´eel state. By contrast, the dipolar case shows no sign of N´eelorder. Note that in the presence of power law interactions, the true ground state of the plane will generically have power lawcorrelations, even if it is gapped, with a power law related to the fall-off of the Hamiltonian. Finally, to demonstrate convergenceof the DMRG for this geometry, in Fig. S2 we plot the energy, correlation length, and entanglement entropy versus the MPSbond dimension m . Order states at Θ > and Associated Phase Transitions —The nature of the ordered phases away from Θ = 0 and theirphase boundaries are likely to be sensitive to the tails of the dipolar interaction. In this study, we restrict to dipolar interactionscut-off at J (kagome) and J (triangular) [see Fig. 1 in the maintext].The nature of the ordered phases depends strongly on the XXZ anisotropy ∆ . Consider first the triangular case with ∆ > .As large tilts, ferromagnetism begins to develop along the field direction yielding two different striped Z -ordered Neel phases(maintext, Fig. 1b). Because of the Ising anisotropy ∆ > , this state does not break any continuous symmetries, resultingin a gapped state with low entanglement. In contrast, for ∆ < , similar striped ordering occurs but in the XY-plane, whichbreaks a continuous symmetry and leads to a Goldstone boson. In the cylinder geometry studied here, the ordering is onlyalgebraic (Mermin-Wagner theorem), consistent with the diverging correlation length found in our simulations. This distinctionis exemplified by observing the phase transition across the ∆ = 1 [Fig. S4(a)].The kagome model also hosts a similar array of ordered phases: for ∆ < we find algebraically-ordered XY magnetism,while for ∆ > we observe a number of different Neel states. A similar phase transition between these ordered states is shownin Fig. S4(b). XY Antiferromagnetism for Molecular Rotational State | , i —Working with the pseudo-spin defined by |↓i = | , i and |↑i = | , i , we will demonstrate that the effective Hamiltonian naturally favors XY Neel order rather than spin liquidity.Let us assume that the state |↑i = | , − i is energetically split away, so that the only non-zero matrix elements arise from T ( d ( i ) , d ( j ) ) . It is natural to define a hardcore bosonic operator a † i = |↑ih↓| i .The only non-zero hopping matrix element for the rotational states is: h↑ i ↓ j | T |↓ i ↑ j i = − d √ , (S1)where d = h , ± | d ± | , i is the transition dipole moment. This yields a hardcore bosonic hopping term, − d √ ( a † i a j + a † j a i ) .The coefficient of the interaction term n i n j depends on the induced permanent dipole moment of the rotational states, h↓ i ↓ j | d z d z + 12 ( d + d − + d − d + ) |↓ i ↓ j i = d , (S2) h↑ i ↓ j | d z d z + 12 ( d + d − + d − d + ) |↑ i ↓ j i = d d , (S3) h↓ i ↑ j | d z d z + 12 ( d + d − + d − d + ) |↓ i ↑ j i = d d , (S4) h↑ i ↑ j | d z d z + 12 ( d + d − + d − d + ) |↑ i ↑ j i = d , (S5)where d = h , | d z | , i and d = h , ± | d z | , ± i are the (electric field) induced dipole moments [4]. The effective density-density interaction strength is given by, V ij = h↑ i ↑ j | H dd |↑ i ↑ j i + h↓ i ↓ j | H dd |↓ i ↓ j i − h↑ i ↓ j | H dd |↑ i ↓ j i − h↓ i ↑ j | H dd |↓ i ↑ j i =( d − d ) . Thus, the effective molecular Hamiltonian is given by, H = g X i,j R [1 − (Φ − Φ ) sin Θ ][ − d a † i a j + a † j a i ) + ( d − d ) n i n j ] (S6)Assuming that the electric field is pointed along Z (perpendicular to the plane where the molecules are sitting), Θ = 0 , meaningthat the Hamiltonian simplifies to H = g X i,j R [ − d a † i a j + a † j a i ) + ( d − d ) n i n j ] (S7)Thus, the natural Hamiltonian that emerges is a /R , U(1) conserving hardcore bosonic Hamiltonian. Note that the ratio of theinteractions to the hopping, ( d − d ) d can be tuned by varying the strength of the electric field (see Fig. 2, maintext). In the limit, | E | → , ( d − d ) → , leaving only H = R [ − d ( a † i a j + a † j a i ) ]. It is useful to rewrite the Hamiltonian in terms of spindegrees of freedom: S + i = b † i , S − i = b † i , S zi = n i − / , wherein (upon dropping constant shifts) [5], H = g X i,j R [ − d S + i S − j + S + j S − i ) + ( d − d ) S zi S zj ] . (S8)While interesting, the fact that the hopping term of the above Hamiltonian is unfrustrated implies that there are two naturalphases. In the limit where the hopping dominates, we expect an easy-plane XY antiferromagnet, while in the limit whereinteractions dominate, we expect a crystal. [1] Z. Zhu and S. R. White, ArXiv e-prints (2015), arXiv:1502.04831.[2] W.-J. Hu, S.-S. Gong, W. Zhu, and D. N. Sheng, ArXiv e-prints (2015), arXiv:1504.00654.[3] M. Zaletel, Y.-M. Lu, and A. Vishwanath, arXiv preprint arXiv:1501.01395 (2015).[4] N. Y. Yao, A. V. Gorshkov, C. R. Laumann, A. M. L¨auchli, J. Ye, and M. D. Lukin, Physical review letters , 185302 (2013).[5] A. V. Gorshkov, S. R. Manmana, G. Chen, E. Demler, M. D. Lukin, and A. M. Rey, Physical Review A84