Abstract
The fractional quantum Hall effect (FQHE) realized in two-dimensional electron systems under a magnetic field is one of the most remarkable discoveries in condensed matter physics. Interestingly, it has been proposed that FQHE can also emerge in time-reversal invariant spin systems, known as the chiral spin liquid (CSL) characterized by the topological order and the emerging of the fractionalized quasiparticles. A CSL can naturally lead to the exotic superconductivity originating from the condense of anyonic quasiparticles. Although CSL was highly sought after for more than twenty years, it had never been found in a spin isotropic Heisenberg model or related materials. By developing a density-matrix renormalization group based method for adiabatically inserting flux, we discover a FQHE in a isotropic kagome Heisenberg model. We identify this FQHE state as the long-sought CSL with a uniform chiral order spontaneously breaking time reversal symmetry, which is uniquely characterized by the half-integer quantized topological Chern number protected by a robust excitation gap. The CSL is found to be at the neighbor of the previously identified Z2 spin liquid, which may lead to an exotic quantum phase transition between two gapped topological spin liquids.
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Introduction
The experimentally discovered fractional quantum Hall effect (FQHE)1,2,3 is the first demonstration of topological order and fractional (anyonic) statistics4,5,6,7,8 realized in two-dimensional electronic systems under a magnetic field breaking time-reversal symmetry (TRS). A related new state of matter with fractionalized quasiparticle excitations is the topological quantum spin liquid (QSL) emerging in frustrated magnetic systems9,10,11,12,13,14,15,16,17,18. Such spin systems, related to strongly correlated Mott materials and holding the clue to the unconventional superconductivity in doped systems, are of fundamental importance to the condensed matter field9,10,18,19,20. To understand the emergent physics of frustrated magnetic systems, where spins escape from the conventional fate of developing symmetry broken ordering, the concept of QSL with the fractionalized quasiparticles was established8,10,14. Experimental candidates for such a new state of matter are identified including kagome antiferromagnets21,22,23 and triangular organic compounds24,25,26. The simplest QSL with TRS is the gapped Z2 spin liquid, which possesses the Z2 topological order and fractionalized spinon and vison quasiparticle excitations8,14. The Z2 QSL is identified as an example of the resonating valence-bond liquid state, which was first proposed by Anderson10. Although explicitly demonstrated in contrived theoretical systems11,13,16,17, the searching of the gapped QSL in realistic Heisenberg models has always attracted much attention over the last twenty years. The primary example is the recent discovered gapped Z2 QSL for kagome Heisenberg model (KHM) with the dominant nearest neighbor (NN) interactions based on the density-matrix renormalization group (DMRG) simulations27,28,29,30.
Another class of QSL with fractionalized quasiparticles obeying fractional (anyonic) statistics is chiral spin liquid (CSL)31,32,33,34,35, which breaks TRS and parity symmetry while preserves other lattice and spin rotational symmetries. Kalmeyer and Laughlin31 first proposed that, in a time-reversal invariant spin system with geometry frustration, one can realize a ν = 1/2 FQHE as a CSL state32 through mapping the frustrated in-plane exchange interactions to the uniform magnetic field. A CSL is also considered to be a simple way in which frustrated spin systems develop topological order through statistics transformation to cancel out the frustration32,33. The CSL may also lead to the exotic anyon superconductivity with doping holes into such systems32,33. The existence of CSL through spontaneously TRS breaking has been demonstrated in a Kitaev model on a decorated honeycomb lattice with contrived anisotropic spin interactions36 and most recently in a spin anisotropic kagome model37. Interestingly, based on the classical and Schwinger boson mean-field analyses, QSLs with different chiral spin orders have been suggested for extended KHM38,39. Other theoretical studies show that one can also induce a CSL state through adding multi-spin TRS breaking chiral interactions42,43,40,41. Although CSL has been explored for more than twenty years31,32,33,34,35,38,44,43, the accurate DMRG27,28,29,30 and variational Monte Carlo45 studies on various frustrated Heisenberg models often lead to the conventional ordered phases or TRS preserving Z2 and U(1) QSLs. The simple concept of realizing CSLs through spontaneously breaking TRS and statistics transformation32,33 remains illusive in realistic frustrated magnetic systems.
In this article, we report a new theoretical discovery of the CSL in an extended KHM based on the state of art DMRG simulations46,47. As illustrated in the inset of Fig. 1(a), the system has the NN coupling J = 1 as energy scale, as well as the second and third NN couplings Jâ² inside each hexagon of the kagome lattice, described by the following Hamiltonian16,38:
We perform the numerical flux insertion simulations on cylinder systems based on the newly developed adiabatical DMRG to detect the topological Chern number, which uniquely characterizes the chiral spin liquid. We have fully established a robust ν = 1/2 FQHE state for by observing the half-integer quantized topological Chern number protected by a robust excitation gap, the degenerate ground states and the uniform chiral order spontaneously breaking TRS.
Results
Phase diagram
Our main findings are summarized in the phase diagram Fig. 1(a). With the turn on of a positive Jâ², we find a robust CSL phase in the region of . We design and perform the Laughlin flux insertion numerical experiment through developing an adiabatic DMRG, which inserts flux and obtains the ground state for each flux. The adiabatic DMRG allows us to obtain the topological Chern number3,34, which characterizes the topological nature of the quantum phase. Our simulation experiment shows that the CSL is characterized by a fractionally quantized Chern number C = 1/2, which is a âsmoking gunâ evidence of the emergent ν = 1/2 Laughlin FQHE state31 in the frustrated KHM. The CSL phase is also characterized by a four-fold degeneracy in two topological sectors. In each sector, there is a double degeneracy representing the two sets of CSL states with opposite chiralities. The near uniform chiral order measured for a state spontaneously breaking TRS is illustrated in Fig. 1(b). We also establish that the CSL is neighboring with the Z2 QSL previously found27,28,29,30 at Jâ² = 0, while the transition region appears to be under strong influence of the nonuniform Berry curvature resulting from gauge field, which may provide new insights to many puzzles regarding theoretical27,28,29,30 and experimental findings18,21,22,23 for kagome antiferromagnets.
Fractional quantization of topological number
To uncover the full topological nature of the phase at large system scale, we perform the flux inserting simulation based on the adiabatic DMRG. For conventional FQHE systems, a quantized net charge transfer would appear as ÎN = C from one edge of the sample to the other edge after inserting one period of flux θ = 0 â 2Ï, corresponding to a nonzero fractionally quantized topological invariant Chern number C34, which is C = 1/2 for the ν = 1/2 bosonic Laughlin state.
By adiabatically inserting the flux θ in our DMRG experiment, we study the evolution of the local magnetization , which is the spin-z average of the ground state at a local lattice site Ri = (x, y). With the increase of θ, we measure the corresponding spin accumulations of each ground state at θ = jÏ/2 (j is an integer). One example with θ = 2Ï is shown in Fig. 2(a). We find nonzero magnetization starting to build up at the left and right edges of cylinder, which grows monotonically with the growing of θ as shown in Fig. 2(b). Since our system has total spin conservation, the net spin-z transfer ÎSz|edge (which is the total magnetization around the right edge of the system) is equivalent to the pumping of the hardcore bosons from the left edge to the right edge without going through the bulk. In Fig. 2(c), we show the net spin transfer ÎSz|edge as a function of θ. A near linear spin pump is being realized in this chiral spin state, which is exactly quantized as ÎSz|edge = 0.5 at θ = 2Ï. From the fundamental correspondence between edge spin transfer and bulk Chern number48, we identify the bulk Chern number of the system as C = 1/2, fully characterizing the state as the Kalmeyer-Laughlin CSL31 of ν = 1/2 FQHE. Physically, the pumping in FQHE system is achieved through the adiabatical rotation of the basis states of the many-body wavefunction, which can be viewed as a non-local operation by developing a âspinonâ line in the cylinder. We find the entanglement spectrum of the spinon sector obtained here by inserting 2Ï flux is identical to the one of the S-sector shown below in Fig. 3(b) obtained through pinning. With further increasing the flux to θ = 4Ï, the net spin transfer ÎSz|edge = 1.0, where the system evolves back to the vacuum sector. These observations fully establish the bosonic ν = 1/2 FQHE emerging in the J â Jâ² KHM. While the Chern number simulations characterize the ground state as the long-sought CSL, we will further measure the topological degeneracy, chiral correlations, topological entanglement entropy and modular matrix to demonstrate the full nature of the topological state in our time-reversal invariant system.
Low-energy spectrum and topological degeneracy
The Kalmeyer-Laughlin CSL has two-fold topological ground-state degeneracy and the spontaneously TRS breaking for such a time-reversal invariant system must have an additional double degeneracy in each topological sector. On cylinder geometry, one can control the boundary condition near the cylinder edges to target into different topological sectors27,49, which we denote as the vacuum and S-sectors, respectively. By using this technique in DMRG, we find the two lowest-energy states in each sector whose energy differences and drop to small values for . One example is shown in Fig. 1(a) for a cylinder with Lx = 24 and Ly = 4. Importantly, the degenerating states in each topological sector also have near identical entanglement spectra. The double degeneracy of entanglement spectrum for the ground states is explicitly shown using two different symbols (line and circle) in Figs. 3(a) and 3(b). These observations are consistent with the spontaneously TRS breaking double degeneracy. We also find the ground-state energies between the two sectors are degenerate ( for Jâ² = 0.5 at Ly = 4), which, combined with the distinct entanglement spectra50 as shown in Figs. 3(a) and 3(b) of the two sectors, establish the topological degeneracy for these two sectors in the intermediate phase. By searching for other low energy excited states from both DMRG and exact diagonalization (ED), we exclude that there are other distinct topological degenerating sectors for the intermediate region, while a lot more lower energy states appear near Jâ² = 0.
The energy and entanglement spectra doubling are signatures of finding the maximally entangled states in each sector, which is forced by the TRS of the system Hamiltonian (here we used a real number initial wavefunction in DMRG calculations which forbids any spontaneous TRS breaking). To demonstrate the nature of the new quantum phase, we first find the minimum entangled states (MESs) in each topological sector28,52,51, which represent the eigenstates of the Wilson-loop (string-like) operators encircling the cylinder and are the simplest states of the quasiparticles. In Fig. 3(c), we show two MESs emerging (labeled by two red dots) in the vacuum sector: , which are equal magnitude superposition of the real states with a phase difference ±Ï/2. The MES breaks the TRS spontaneously and demonstrates a uniform nonzero chirality order for each triangle as illustrated in Fig. 1(b). The chiral order reaches a value around 0.08 comparable to its classical value 1/8. The conjugate state as another MES has the opposite sign of chirality. The doubling of the entanglement spectra for the maximum entropy state simply results from the superposition of the MESs with the same entanglement spectra. Consequently, one finds an entanglement entropy difference ln 2 comparing to the MESs as illustrated in Fig. 3(c). Near identical results and two MESs are also found in the topological degenerating S-sector. Furthermore, if we initiate the DMRG state with a random complex number state, we automatically find such a MES, which spontaneously breaks TRS.
By obtaining the MES, we find the topological entanglement entropy γ consistent with the result ln 2/2 of the ν = 1/2 Laughlin state53,54. The ED calculations further confirm this state on a N = 3 à 4 à 3 cluster by extracting modular transformation matrix51,52 from the MESs of two noncontractable cuts (see Supplementary Information for more details).
Quantum phase transitions
We use both the chiral-chiral correlation functions and the topological Chern number obtained from inserting flux to identify the quantum phase diagram and transitions in the J â Jâ² model. In Fig. 4(a), we compare the chiral correlations ãÏiÏjã for the states from the two topological sectors with different system widths at Jâ² = 0.5. We find long-range correlations for the states from both topological sectors, which are further enhanced with increasing system width Ly. To reveal the quantum phase transitions, we show the chiral correlation functions calculated from the ground state of the vacuum sector for different Jâ² in Fig. 4(b). ãÏiÏjã is positive everywhere and has the long-range order for 0.1 ⤠JⲠ⤠0.7, while transitions to other phases are detected at Jâ² = 0.05 and 0.8 by identifying the exponential decaying chiral correlations.
In the flux insertion simulations, we find that the Chern number remains to be quantized at C = 1/2 for the same parameter range 0.1 ⤠JⲠ⤠0.7, thus we establish the quantum phase diagram as shown in Fig. 1(a). The quantum phase transition around Jâ² ~ 0.7â0.8 is charaterized by an excitation gap closing in the bulk of system, where we detect a strong bulk magnetization (boson density) response to the inserted flux. Between Jâ² = 0 and 0.1, we detect a strong nonuniform Berry curvature resulting from the gauge field in the inserting flux simulations, possibly indicating the forming of new quasiparticles and the emerging of Z2 QSL. We also study the stability of the CSL when the second and third neighbor couplings are different. We find the CSL phase in a region around the line with J2 = J3. For example, when J2 = 0.1, the CSL is robust for . Physically, the J3 coupling suppresses the magnetic order formed in the J1 â J2 (J2 ~ 0.2) kagome model, thus substantially enlarges the non-magnetic region. Meanwhile, classically the J3 term will enhance a noncoplanar spin chiral order38, which may induce a CSL in the quantum J1 â J2 â J3 model as demonstrated here.
Discussion
In the past twenty years, the gapped QSL in realistic magnetic systems have attracted intensive attention. While the NN or J1 â J2 KHM27,28,29,30 is the primary candidate of a possible Z2 QSL, there are still many puzzles left unresolved. The frustrated kagome antiferromagnets Herbertsmithite Cu3(Zn,Mg)(OH)6Cl2 and Kapellasite Cu3Zn(OH)6Cl218,21,22,23 are possible candidates of QSL; however, they appear to be more consistent with gapless or critical states. At theoretical side, redundant low-energy excitations are found for the NN KHM from ED simulations55, variational studies find that U(1) gapless QSL45,56 has relatively low energy and DMRG studies have not been able to identify all the four topological sectors for Z2 QSL49. Our finding of the robust CSL at the neighbor of the NN KHM indicates that the latter is not a fully developed Z2 QSL yet and the nature of states for the experimental relevant kagome systems may be strongly affected by a new quantum critical point between two gapped QSLs, the Z2 and the CSL. In a parallel work, a CSL has also been uncovered in an anisotropic kagome spin system37 with only spin-z interactions for further neighbors. We believe that our numerical findings will stimulate new theoretical and experimental researches in this field to resolve the nature of the quantum phases for different frustrated magnetic systems. An exciting next step will be identifying theoretical models and experimental materials which can host exotic topological superconductivity by doping different CSLs.
Methods
DMRG is a powerful tool to study the low-lying states of strongly correlated electron systems46. The accuracy of DMRG is well controlled by the number of kept states M, which denotes the M eigenstates of the reduced density matrix with the largest eigenvalues. The highly efficiency of DMRG for one-dimensional systems or two dimensional cylinder systems have been shown for different systems27,47. An improvement in DMRG calculations is to implement symmetry to reduce the Hilbert space. The spin-z or total particle U(1) symmetry is commonly used in DMRG, which is preserved in many model systems. For some systems with spin rotational SU(2) symmetry such as the Heisenberg spin model, the more efficient choice is to apply the SU(2) symmetry57, from which we can obtain more accurate results for wider systems. This algorithm has been applied to study various frustrated Heisenberg systems successfully29,58,59.
Details of the SU(2) DMRG calculation
We study the frustrated KHM without flux using SU(2) DMRG. We study the cylinder system with open boundaries in the x direction and periodic boundary condition in the y direction. For Ly = 4 (Ly = 6) systems, we keep up to 3000 (4600) SU(2) states with the DMRG truncation error for most calculations. To find the ground states in both vacuum and S- topological sectors on cylinders in the DMRG calculations, we take pinning sites in the open boundaries or insert flux to target the two different sectors49.
Adiabatic DMRG and fractionally quantized Chern number
For the first time, we develop the numerical flux insertion experiment for cylinder systems based on the adiabatical DMRG simulation to detect the topological Chern number34 of the bulk system, which uniquely characterizes the CSL as a ν = 1/2 FQHE state emergent from the J â Jâ² Heisenberg model on kagome lattice. In this simulation, we impose the twist boundary conditions along the y direction by replacing terms for all neighboring (i, j) bonds with interactions crossing the y-boundary in the Hamiltonian. Starting from a small θ ~ 0, a state with the definite chirality and sign of Chern number will be randomly selected, which remains the same through out the whole adiabatical process of θ = 0 â 4Ï. We find states with the opposite Chern numbers (C = ±1/2) in different runs of the simulations due to spontaneously TRS breaking. A robust excitation gap Î ~ 0.24 is obtained for Jâ² = 0.5 after we create two spinons (at θ = 2Ï) at the opposite edges of the cylinder (see Fig. 2(a)), which protects the CSL state. This method can be applied to study different interacting systems and characterize different topological states.
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Acknowledgements
We thank Y. C. He for extensive discussions. We also thank Leon Balents, Matthew P. A. Fisher, Olexei I. Motrunich and F. Duncan M. Haldane for stimulating discussions and explanations of spin liquid as well as topological physics. This research is supported by the National Science Foundation through grants DMR-1205734 (S.S.G.), DMR-0906816 and DMR-1408560 (D.N.S.), the U.S. Department of Energy, Office of Basic Energy Sciences under grant No. DE-FG02-06ER46305 (W.Z.).
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S.S.G. and W.Z. performed main calculations based on different numerical programs they developed. S.S.G., W.Z. and D.N.S. made significant contributions from the design of the project to the finish of the manuscript.
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Gong, SS., Zhu, W. & Sheng, D. Emergent Chiral Spin Liquid: Fractional Quantum Hall Effect in a Kagome Heisenberg Model. Sci Rep 4, 6317 (2014). https://doi.org/10.1038/srep06317
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DOI: https://doi.org/10.1038/srep06317
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