Signatures of Dirac Cones in a DMRG Study of the Kagome Heisenberg Model

The antiferromagnetic spin-$1/2$ Heisenberg model on a kagome lattice is one of the most paradigmatic models in the context of spin liquids, yet the precise nature of its ground state is not understood. We use large-scale density matrix renormalization group simulations (DMRG) on infinitely long cylinders and find indications for the formation of a gapless Dirac spin liquid. First, we use adiabatic flux insertion to demonstrate that the spin gap is much smaller than estimated from previous DMRG simulation. Second, we find that the momentum-dependent excitation spectrum, as extracted from the DMRG transfer matrix, exhibits Dirac cones that match those of a $\pi$-flux free-fermion model [the parton mean-field ansatz of a $U(1)$ Dirac spin liquid].

Popular Summary

A quantum spin liquid is an exotic phase of matter of highly entangled spins that hosts fractionalized quasiparticles. Realizing such a phase of matter has been a long-standing quest in the field of condensed-matter physics. However, a persistent, fundamental problem in this context is to understand the nature of the spin-liquid ground state realized in the spin-$1/2$ kagome Heisenberg model, which serves as a minimal model to describe various spin-liquid materials. Despite more than two decades of work, the solution still remains elusive. Here, we present unbiased numerical evidence that the kagome spin liquid is a Dirac spin liquid. This liquid is described by a strongly interacting $2+1$ dimension conformal field theory that has Dirac fermions coupled to the gauge field.

We use large-scale density matrix normalization group simulations to theoretically study the kagome spin liquid. First, we show that its spin gap has a strong dependence on the boundary conditions and is much smaller than estimated based on previous simulations. Second, we find that the momentum-dependent excitation spectrum shows signatures of relativistic Dirac particles, consistent with theoretical predictions of a Dirac spin liquid.

Our results shed light on the long-standing kagome spin-liquid problem and will hopefully motivate future experiments in this field. Furthermore, realizing an interacting Dirac spin liquid (i.e., a $2+1$ dimension conformal field theory) without fine-tuning may have applications in other fields such as high-energy physics.

Figure 1

(a) Illustration of the kagome lattice with a width of $L_y=4$ unit cells. The different geometries, YC8-0, YC8-2, YC8-4, and YC8-6, correspond to identifying the site $x$ with the site $a$, $b$, $c$, or $d$, respectively. (b) The $\pi$-flux state, where each hexagon has $\pi$ flux. We chose a specific gauge, in which the red bond has $s_{ij}=-1$, and the black bond has $s_{ij}=1$. The different geometries [here (c) YC8-0 and (d) YC8-2] correspond to different ways of cutting the Brillouin zone (shown as blue and red lines). The $M$ points are [labeled by $(k_1,k_2)$], $M_1=(\pi ,0)$, $M_2=(0,\pi )$, and $M_3=(\pi ,\pi )$. The DSL has a halved magnetic Brillouin zone (dashed line) because it is a $\pi$-flux ansatz. The two Dirac points of the DSL are at $\pm Q$, and we refer to this as $Q$, $Q^{\prime }$ in the text. Note the blue and red lines represent cuts in the original Brillouin zone; the allowed momenta of the fractionalized spinons form similar cuts through the magnetic Brillouin zone, but the cuts are displaced according to the spin flux $\theta$ and the emergent gauge flux $\varphi$, as we explain in Sec. 3c.

Figure 2

Diagram for a spin liquid under the twist. (a) The spinon feels half-twist boundary or flux due to the fractionalization. (b) The response of the energy under the twist. (c) Two topological sectors of a DSL, with one gapped sector and one gapless sector. (d) The collapse of the twist simulation will have a quasiparticle emerging as the domain wall.

Figure 3

(a) Upper bound on the spin gap ${\mathrm{\Delta }}_{S=1}$ under the twist boundary condition $\theta$ for $J_2=0$; the estimate is obtained using DMRG bond dimension $m=4000$ and using the lowest-energy topological sector. Data end at the failure of adiabaticity. The qualitative behavior depends on the type of cylinder (I or II), which in a DSL would be gapless at $\theta =2\pi$, $\pi$, respectively. (b) Dependence of the spin gap ${\mathrm{\Delta }}_{S=1}$ (solid line) and $S^z=1$ correlation length $\xi$ (dashed line) on the spin flux $\theta$ and DMRG bond dimension $m$. Data are taken with $J_2=0.05$ on the type II YC8-2 cylinder. Generally, the estimated gap decreases with the bond dimension $m$; the larger the system size is, the more likely we are to overestimate the spin gap.

Figure 4

The transfer matrix spectrum in the $S^z=1$ sector (roughly the triplet channel) for the kagome Heisenberg model (a,c,e) and free-fermion model (b,d,f). The vertical axis is the inverse of correlation length $1/\xi$, which can be considered as the gap $\mathrm{\Delta }$ of the excitations, up to a prefactor $\mathrm{\Delta }=v_s/\xi$. The horizontal axis denotes (a,b) the twist angle $\theta$; (c,d) momentum $2k_1$; and (e,f) momentum $k_2$. In panels (c)–(f), we plot only the lowest level of each excitation type, cf. panel (g). The cylinder we show here is the YC8-2 cylinder for both cases, and the truncation error of DMRG is around $2\times {10}^{-6}$, which corresponds to bond dimension $m=6000$ for the kagome Heisenberg model and $m=250$ for the free-fermion model. For the kagome Heisenberg model, we also include a small $J_2=0.05$. The finite “gap” at the Dirac point appears to be due to the truncation error of DMRG simulations (see Fig. 15 in Appendix pp2-s2e). (g) Three different types of particle-hole excitations. The arrows represent the direction of the movement of the discretized momentum under the twist boundary conditions.

Figure 5

(a) The transfer matrix spectrum of the kagome Heisenberg model in the $S^z=0$ sector versus the twist angle $\theta$. Here, we have the YC8-2 cylinder, $J_2=0.05$, and bond dimension $m=6000$. (b) Correlation function of the scalar chirality $⟨\chi (0)\chi (r)⟩$ at $\theta =\pi$ and $17\pi /18$. The correlation lengths from fitting $⟨\chi (0)\chi (r)⟩$ are $\xi =119$ ($\theta =\pi$) and $\xi =38.5$ ($\theta =17\pi /18$), which are virtually identical to the correlation lengths obtained from the transfer matrix $\xi =121$ and $\xi =39$.

Figure 6

The graphical representation of (a) the MPS and (b) the transfer matrix. The quantum number of the eigenvalue is determined by $q=Q_r-Q_{r^{\prime }}$. (c) The mixed transfer matrix to calculate the momentum $k_1$ of each Schmidt basis. (d) The $-1$ geometry for the square lattice (it is analogous to the $-2$ geometry for the kagome lattice). The periodic condition is found by identifying the sites labeled by the same number. (e) The mixed transfer matrix for the fermionic excitation.

Figure 7

(a) iDMRG–finite DMRG combined algorithm. (b) Finite DMRG algorithm.

Figure 8

Transfer matrix spectrum for free fermions [Eq. (3)] on the YC8-2 cylinder with bond dimension $m=3000$. Here, we compare the free Dirac fermion model and a Chern insulator: the inverse of correlation lengths $1/\xi$ versus (a,b) twist angle, (c,d) momentum $2k_1$, and (e,f) momentum $k_2$. The Chern insulator we show here is obtained by assigning $\pi /25$ flux in the up and down triangles (of the kagome), and $23\pi /25$ flux in the hexagon.

Figure 9

The energy and entanglement entropy of the KSL under flux insertion. Here, the bond dimension is $m=6000$.

Figure 10

The $S^z=1$ transfer matrix spectrum of $J_2=0.05$ for the YC6-2, YC8-6, YC10-2, and YC12-2 cylinders. Here, the bond dimension is $m=6000$.

Figure 11

The $S^z=1$ transfer matrix spectrum of $J_2=0.1$ for the YC6-2, YC8-2, YC8-6, YC10-2, and YC12-2 cylinders. Here, the bond dimension is $m=6000$.

Figure 12

The $S^z=1$ transfer matrix spectrum of $J_2=0$ for the YC6-2, YC8-2, YC8-6, YC10-2, and YC12-2 cylinders. Here, the bond dimension is $m=6000$.

Figure 13

(a) The singlet excitation spectrum for the (a)-i YC10-2 and (a)-ii YC12-2 cylinders; here, $J_2=0.05$, and bond dimension $m=6000$. (b) The dependence on the bond dimension $m$ of the lowest singlet excitation spectrum for the (b)-i YC10-2 and (b)-ii YC12-2 cylinders. The singlet excitation keeps decreasing as the bond dimension $m$ increases.

Figure 14

The $S^z=1$ transfer matrix spectrum of YC8-0; here, $J_2=0$, and the bond dimension is $m=6000$.

Figure 15

The dependence of the inverse of correlation ($1/\xi$) at the Dirac point versus the bond dimension $m$. We show both the kagome Heisenberg model (YC8-2, $J_2=0.05$) and the free-fermion model.

Figure 16

Spin gap versus the truncation error. The larger the bond dimension is, the smaller the truncation error. For the comparable truncation error, the larger system size has a smaller spin gap. Here, we show $J_2=0$, and the twist angle $\theta =5\pi /6$. The bond dimensions shown are $m=1000$, 2000, 3000, 4000. (For YC6-2, they are $m=2000$, 3000, 4000.)

Figure 17

The transfer matrix spectrum of a magnetic order. Here, we show a YC8-2 cylinder, with $J_2=0.3$, and the bond dimension $m=2000$.

Figure 18

The kagome spin liquid on the YC8-2 cylinder is unstable to an ordered state when $\theta =\pi$. The ordered state spontaneously breaks the spin-flip symmetry $P_x=\prod (2S^x)$ (hence, the time-reversal symmetry) by forming a staggered Ising magnetization ($⟨S_i^z⟩\approx \pm 0.03$) as represented by arrows. It breaks lattice symmetry, with bond correlations $⟨{\overrightarrow{S}}_i·{\overrightarrow{S}}_j⟩$, to be $-0.26$ (green solid bonds), $-0.22$ (red hollow bonds), and $-0.17$ (black thin bonds).