Dynamical and Topological Properties of the Kitaev Model in a [111] Magnetic Field

Matthias Gohlke,1 Roderich Moessner,1 and Frank Pollmann2

1Max-Planck-Institut fur Physik komplexer Systeme, 01187 Dresden, Germany2Technische Universitat Munchen, 85747 Garching, Germany

(Dated: December 6, 2018)

The Kitaev model exhibits a quantum spin liquid hosting emergent fractionalized excitations.We study the Kitaev model on the honeycomb lattice coupled to a magnetic field along the [111]axis. Utilizing large scale matrix product based numerical models, we confirm three phases withtransitions at different field strengths depending on the sign of the Kitaev exchange: a non-abeliantopological phase at low fields, an enigmatic intermediate regime only present for antiferromagneticKitaev exchange, and a field-polarized phase. For the topological phase, we numerically observe theexpected cubic scaling of the gap and extract the quantum dimension of the non-abelian anyons.Furthermore, we investigate dynamical signatures of the topological and the field-polarized phaseusing a matrix product operator based time evolution method.

I. INTRODUCTION

Quantum spin liquids (QSLs)1,2 are realized in cer-tain spin systems where the interplay of frustration andquantum fluctuations suppresses long range order. Theseexotic phases of matter cannot be understood in terms ofspontaneous symmetry breaking, but are instead charac-terized by their long range entanglement and emergentfractionalized excitations. The lack of local order param-eters makes it difficult to experimentally detect QSLs bytheir static properties–except for showing the absence ofconventional order. Instead it appears more promising tostudy dynamical properties of QSLs (e.g., the dynamicalspin structure factor) which encode characteristic finger-prints of topological order3–7.

On the theory side, significant insight into the physicsof QSLs comes from the study of exactly solvable mod-els. A prominent example is the Kitaev model on thehoneycomb lattice8, which exhibits a QSL phase fea-turing fractionalization of spin-1/2 degrees of freedominto fluxes and Majorana excitations. The Kitaev in-teraction, a strongly anisotropic Ising exchange appearsto be realized approximately in compounds with strongspin-orbit interaction9–13, such as the iridates Na2IrO3,Li2IrO3

14, and α-RuCl315–17. It may also be real-

ized in metal-organic frameworks18. In such materi-als, additional interactions are important and typicallylead to long-range magnetic order, nonetheless signa-tures of being in the proximity to the Kitaev QSL arediscussed17,19,20. Recent attention has shifted to apply-ing a magnetic field19,21–27, in particular experiments onthe Kitaev compound α-RuCl3 (with an in-plane mag-netic field) reveal a single transition into quantum para-magnetic phase with spin-excitation gap28–34.

In this article, we consider the Kitaev model in a mag-netic field along [111], such that the field couples to thespins in a symmetry-equivalent way and the field does notprefer any bond in particular. While the magnetic fieldbreaks integrability, Kitaev has identified two three-spinexchange terms within perturbation theory, that breaktime-reversal symmetry and open a gap in the spectrum.

One of the terms retains integrability and upon adding tothe Kitaev model, leads to a topologically ordered phasehosting non-abelian anyons8. However, numerical simu-lations35 reveal that the same topological phase occursfor small magnetic fields and ferromagnetic Kitaev cou-pling. The topological phase turns out to be more stable,by one order of magnitude in the critical field strength, ifan antiferromagnetic coupling is considered36. Remark-ably, an additional regime, possibly gapless, between thelow-field topological and the high-field polarized phaseappears to exist36.

In this work, we employ large scale infinite density ma-trix renormalisation group (iDMRG) methods37–40 to in-vestigate the ground state phase diagram of the Kitaevmodel in a magnetic field along [111] and simulate itsdynamics using a matrix-product operator (MPO) basedtime-evolution41.

The topologically ordered phase is characterized by itsfinite topological entanglement entropy (TEE)42,43. Bysubtracting contributions of the Majorana fermions andthe Z2-gauge field from the numerically obtained entan-glement entropy of a bipartition, we extract a remainderwhich is identical to the TEE in the integrable case. Indoing so, we obtain a clear signature of non-abelian any-onic quasiparticles in the topological phase. In a mag-netic field, this remainder is still consistent with the ex-istence of non-abelian anyons.

Furthermore, within the topological phase the correla-tion length decreases with magnetic field in a way thatis consistent with a cubic opening of the gap as foundfor the three-spin exchange8. However, the dynamicalspin-structure factor in presence of a field behaves verydifferently compared to what is known for the three-spinexchange44. The magnetic field causes the flux degreesof freedom to become mobile. As a consequence the low-energy spectrum contains more structure and the gap inthe dynamical spin-structure factor is reduced.

Approaching the intermediate regime from the po-larized phase, the magnon modes reduce in frequencyand simultaneously flatten. This resembles the phe-nomenology within linear spin wave theory (LSWT)21,45,but the transition is significantly renormalised to lower

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tr-e

l] 5

Dec

201

8

2

(a)

z

xy

Syi

Syj

(b)

Sxi Sy

k

Szj

FIG. 1. (a) Bonds labeled with x,y, and z and an exemplaricSyi S

yj Kitaev-exchange (orange), (b) a single three-spin term

Sxi S

zj S

yk of the three-spin interaction in HK3 .

fields. Close to the transition, a broad continuum ex-ists that, within our reachable resolution in frequency,reaches down to zero frequency and merges with the sin-gle magnon branches. At the transition, the spectrumappears to be (nearly) gapless in the entire reciprocalspace. We do not observe an opening of a gap in theintermediate regime.

The remainder of this paper is structured as follows:In Sec. II we introduce the model consisting of Kitaevterm, Zeeman coupling to a magnetic field along [111],and three-spin exchange. In Sec. III, the ground statephase diagram is discussed for both signs of the Kitaevcoupling. We then focus on the antiferromagnetic Kitaevcoupling in Sec. IV and study its dynamical signatureswithin the low-field topological as well as the high-fieldpolarized phases. We conclude with a summary and dis-cussion in Sec. V.

II. MODEL

The Hamiltonian describing the Kitaev model in amagnetic field along [111] direction reads

H =∑〈i,j〉γ

KγSγi S

γj − h

∑i

(Sxi + Syi + Szi ) , (1)

where the first term is the pure Kitaev model exhibitingstrongly anisotropic spin exchange coupling8. Neighbor-ing spins couple depending on the direction of their bondγ with SxSx, SySy or SzSz, cf. Fig. 1(a). The secondterm is the Zeeman-coupling of the spins to a magneticfield applied in [111] direction.

In the zero field limit, the Kitaev model exhibits aquantum spin liquid ground state with fractionalizedexcitations8. Depending on Kγ , the spectrum of thefermions is either gapped (A-phase) or gapless (B-phase).Let the Kγ be sorted as Kα ≥ Kβ ≥ Kγ , then the gaplessB-phase occurs if |Kα| ≤ |Kβ |+ |Kγ | and the A-phase if|Kα| > |Kβ | + |Kγ |. In the remainder, we consider theisotropic case Kγ = K = ±1.

Flux degrees are defined by the plaquette operator

Wp =∏i∈P σ

γ(i)i , where γ(i) = {x, y, z} equals the bond,

that is not part of the loop P around the plaquette. TheWp commute with the Hamiltonian (in the h = 0 limit)and have eigenvalues ±1. Thus, the Wp’s are quantum

numbers separating the full Hilbert space into subspaces,for each of which a free fermion problem remains to besolved. The ground state lies in the flux-free sector, thatis ∀i : Wp,i = +1.

For later use, we comment on placing the Kitaevmodel on a cylinder. A second flux operator of a non-contractable loop C going around the cylinder can be de-

fined: Wl =∏i∈C σ

γ(i)i . Similarly to Wp, Wl commutes

with the Hamiltonian, has eigenvalues ±1, and separatesthe full Hilbert space in two subspaces. With respectto the free fermions, Wl = −1 (flux-free) correspondsto periodic and +1 to antiperiodic boundary conditionsalong the circumference of the cylinder. The ground statewithin each of the two sectors are separated in energy by∆E, which depends on the circumference Lcirc and van-ishes in the limit Lcirc →∞.

Applying a magnetic field h along [111], as in Eq.(1), breaks time-reversal symmetry and opens a gap inthe fermionic spectrum. The lowest order terms break-ing time-reversal and not changing the flux configurationare the three-spin exchanges Sxi S

yj S

zk . Two such terms

exist8. The one illustrated in Fig. 1(b) plus symmetricvariants results in a quadratic Hamiltonian for the Majo-rana fermions and thus preserves the integrability of theoriginal model. The corresponding Hamiltonian reads

HK3=∑〈i,j〉γ

KγSγi S

γj +K3

∑〈〈i,j,k〉〉

Sxi Syj S

zk , (2)

where 〈〈.〉〉 denotes an ordered tuple (i, j, k) of neighbor-ing sites such that the Sx, Sy, and Sz at the outer twosites coincide with the label of the bond connecting to thecentral site. The flux operators Wp and Wl still commutewith HK3 and separate the Hilbert space. The remainingfermionic Hamiltonian is quadratic with the correspond-ing bands having non-zero Chern number±1 and yieldingcomposite excitations with anyonic exchange statistics8.

III. GROUND STATE PHASES

The ground state is obtained using the matrix prod-uct state (MPS) based infinite density matrix renormal-isation group (iDMRG) method37–40. Being a standardtechnique for one-dimensional systems, it has been usedin two dimensions by wrapping the lattice on a cylinderand mapping the cylinder to a chain with longer rangeinteractions.

We employ a rhombic-2 geometry with a circumfer-ence of Lcirc = 10 sites and a rhombic geometry withLcirc = 6 as illustrated in Fig. 2. Both geometries capturethe K−points in reciprocal space and hence are gaplessfor pure Kitaev-coupling (h = 0). A main advantage ofthe rhombic-2 geometry is its translational invariance ofthe chain winding around the cylinder. While the map-ping to a cylinder for the rhombic geometry requires aniDMRG unit cell of at least Lcirc sites, a single funda-mental unit cell with two sites is sufficient to simulate an

3

(a)

n2

n1

I II

I II

y x

z

(b)

K

K′

Γ′

M

Γ

k1

k2

(c)

n2

n1

II III

I II

y xz

(d)

K

K′

Γ′

M

Γ

k1

k2

FIG. 2. Geometries used for iDMRG and their correspondingaccessible momenta (orange lines) in reciprocal space withrespect to the first Brillouin zone (inner hexagon). The secondBrillouin zone is shown partially. The roman numbers labellinks across the boundary. (a) rhombic geometry with threeunit cells, Lcirc = 6 sites, along the circumference and (b)its corresponding reciprocal space. (c,d) rhombic-2 geometrywith five unit cells circumference, Lcirc = 10 sites.

infinite cylinder using the rhombic-2 geometry. DifferentiDMRG cells have been used to test for possible break-ing of translational symmetry and corresponding resultswill be presented when of relevance. We use bond dimen-sions of up to χ = 1600 for the computation of the phasediagram.

We confirm the existence of two phases and a singletransition for ferromagnetic Kitaev coupling35,36 (FMK,K < 0), and of at least three phases for antiferromagneticKitaev coupling36 (AFK, K > 0). For both, FMK andAFK, we find a topological phase at low field and a field-polarized phase at high field. Only for AFK, we identifyan intermediate, seemingly gapless, phase.

A. Topological Phase

For small h, the system forms a non-abelian topolog-ical phase8. Its stability upon applying h vastly differsdepending on the sign of the Kitaev interaction. Employ-ing a rhombic-2 geometry with Lcirc = 10, we find, incase of AFK, that this phase ranges up to hc1,AF ≈ 0.22,whereas for FMK it ranges only up to hc,FM ≈ 0.014.Both values are based on the peaks in the second deriva-

0.0

8

−d2E

/dh

2

χ = 400

χ = 400

χ = 800

χ = 800

χ = 1200

χ = 1200

χ = 1600

0.2

0.6

SE

/L

y

100

101

ξ

0

0.5

1

⟨Wp⟩

0.0 0.1 0.2 0.3 0.4 0.5

h

−1

1

Wl

0.00 0.005 0.01 0.015 0.02 0.025

h

Wl = −1 :

Wl = +1 :

FIG. 3. Several observables of the Kitaev model with an-tiferromagnetic coupling, K > 0, in a magnetic field along[111]. From top to bottom: Second derivative −d2E/dh2 ofthe energy with respect to the field h, entanglement entropySE of a bipartition of the cylinder divided by the numberLy of bonds cut, correlation length ξ, average of plaquettefluxes Wp, and flux Wl of a non-contractible loop aroundthe cylinder. At least three phases are observed: topologicalphase for h < hc1,AF ≈ 0.22, intermediate possibly gaplesshc1,AF < h < hc2,AF ≈ 0.36, and a subsequent field-polarisedphase. Solid blue lines are for the Wl = −1 sector, dashedblue lines for Wl = +1, and its intensity encodes the bonddimension χ used, where dark blue refers to a large χ. Thindashed black lines depict the phase transitions obtained fromthe peaks in −d2E/dh2.

tive −d2E/dh2 of the energy with respect to the mag-netic field. However, subtle features are present for AFKat slightly lower h ≈ 0.2, which become less pronouncedwith larger bond dimension χ. In comparison to valuesreported earlier35,36 we find a nearly 30% lower value forthe FMK transition hc,FM . This is due to the fact thatfor rather small circumferences, the ground state energywithin the topological phase is strongly sensitive to theboundary condition as has already been noted in Ref. [8].The rhombic-2 geometry we employ has the same twistedboundary condition as the (Ln1, Ln2+n1) geometry em-ployed in [8], which is shown to converge better in energywhen increasing L or Lcirc, respectively. The transitionfield hc,FM may still decrease slightly upon further in-creasing Lcirc and approaching the two-dimensional limit

4

Lcirc →∞.

For small h, the total magnetisation, |〈S〉| (not shownhere), grows proportionally with h. The two sectorsfound on the cylindrical geometry and determined byWl = +1 or Wl = −1 are distinguished by their be-haviour of the entanglement entropy SE and the cor-relation length ξ. The Wl = +1 sector is character-ized by finite ξ and SE due to being gapped by impos-ing antiperiodic boundary conditions on the Majoranafermions46. In contrast, the Wl = −1 sector has diver-gent ξ and SE when h = 0, where it is gapless. In thelatter, encoding the wave function as MPS with a finiteχ induces an effective gap that limits ξ and SE . In fact,the growth of ξ and SE with increasing χ is connectedvia SE,χ = c/6 log ξχ + const47,48, where c is the univer-sal central charge. This has been named finite entangle-ment scaling and allows to confirm c = 1 (for h = 0,Wl = −1) as has been checked previously on a differentcylinder geometry46. As a side remark, the notion of acentral charge is applicable due to using a cylinder ge-ometry and effectively mapping the model in question toa one-dimensional system.

In a magnetic field, 〈Wp〉 as well as the cylinder flux Wl

begin to slowly deviate from ±1 until they vanish closeto the transition. The plaquette fluxes Wp, as defined inthe integrable limit, are not conserved anymore for finiteh as the application of a single Sγi creates a flux each onthe two plaquettes adjacent to bond γ at site i. However,an adiabatically connected operator Wl of Wl is expectedto exist, such that Wl ≈ ±149. Such a dressed Wilsonloop Wl separates the two sectors found on the cylinderfor any h within the topological phase.

Numerical convergence, that is ξ and SE become χ-independent, is achieved for 0.1 < h < 0.18. In thatrange ξ reflects the physical excitation gap50 via ∆E ∝1/ξ.

Figure 4, where the x-axis has been rescaled h→ 32h3,enables a direct comparison with the three-spin exchangeK3 in HK3

. Both exhibit a very similar decrease of ξwith a ξ ∝ 1/x scaling, where x is either 32h3 or K3. ξreaches a plateau at x = 0.2 with a low ξ ≈ 1. If h isapplied, a small χ-dependent dip and the phase transi-tion into the intermediate regime follows, whereas for K3

the plateau ranges up to K3 = 1, from where ξ increasesagain51. The entanglement entropy SE reaches, in thecase of a magnetic field, a plateau already at 32h3 ≈ 0.06(h ≈ 0.12) beyond which it raises again until the tran-sition field hc1,AF is reached. At all fields the entan-glement remains larger than for the corresponding K3.A more detailed discussion about the entanglement inthe context of topological excitations and topological en-tanglement entropy follows below. The Wl = +1 sectorhas χ-independent ξ and SE up to h ≈ 0.18. Beforethe transition (0.18 < h < hc1,AF ) both sectors exhibita χ-dependents which suggests a closing of the gap atthe transition and, thence, indicates that the transitionmight be continuous.

We now focus on the characterization of the topologi-

10−2 10−1 100

32h3

100

101

102

ξ

χ = 400

χ = 800

χ = 1200

χ = 1600

ξ ∝ h−3

10−2 10−1 100

K3

100

101

102

ξ

χ = 400

χ = 800

χ = 1200

ξ ∝ K−13

(a) ξ – 32h3

(b) ξ – K3

K3, 32h3

FIG. 4. Comparison of correlation length ξ between (a)the rescaled magnetic field h → 32h3 and (b) the three-spin exchange K3. The solid red line is a guide-to-the-eye corresponding to a 1/K3 or 1/(32h3) scaling. Within0.05 < K3, 32h3 < 0.2, that is where numerical convergenceis achieved, the behaviour of ξ is consistent with a predictedopening of the gap as ∆E ∝ h3 or as ∆E ∝ K3, respectively.

cal order occurring at low magnetic fields h or when non-zeroK3 is considered. First, let us recall some facts abouttopologically ordered systems on an infinite cylinder52,53.Generally, topological order leads to a ground state de-generacy with a number of degenerate states being equalto the number of emergent quasiparticle species. Theseground states are conveniently represented as minimallyentangled states (MES)53,54, say |ψi0〉, where i denotesthe particular quasiparticle. By utilizing iDMRG, suchMES are selected naturally, and the obtained MPS cor-responds to one of the quasiparticles52,55.

Upon cutting a cylinder into two semi-infinite halves,the entanglement entropy grows proportional with thecircumference Lcirc as43

SE = αLcirc − γi , (3)

where γi denotes the topological entanglement entropy(TEE)42,43. A non-zero TEE γi = log(D/di) revealstopological order and is connected to the total quantumdimension D, which itself is a sum of the quantum di-mension di of each quasiparticle

D =

√∑i

d2i . (4)

The quantum dimension is related to the fusion vectorspace, which is spanned by all the different ways anyons

5

0.0

8

−d2E

/dh

2χ = 400

χ = 400

χ = 800

χ = 800

χ = 1200

χ = 1200

χ = 1600

0.2

0.6

SE

/L

y

100

101

ξ

0

0.5

1

⟨Wp⟩

0.0 0.1 0.2 0.3 0.4 0.5

h

−1

1

Wl

0.00 0.005 0.01 0.015 0.02 0.025

h

Wl = −1 :

Wl = +1 :

10−2 10−1 100

32h3

−1.0

− log 2

− log 2

2

0.0

0.5

∆S

E

(a) ∆SE – 32h3

10−2 10−1 100

K3

−1.0

− log 2

− log 2

2

0.0

∆S

E

(b) ∆SE – K3

K3, 32h3

FIG. 5. Remainder ∆SE of the entanglement entropy of abipartition of the cylinder after subtracting a fermionic anda gauge field contribution following Eq. (7). The magneticfield has been rescaled, h → 32h3, based on the behaviourof the correlation length in Fig. 4. The vertical dashed linessignal the transitions in a magnetic field. The horizontal linescorrespond to log(D/da) as discussed in the main text.

can fuse to yield a trivial total charge43,56. The quan-tum dimension of abelian anyons is di = 1, whereas fornon-abelian anyons di is generally larger than one. Thegapped phase of the Kitaev model upon applying K3 isknown to exhibit topological order hosting non-abelianIsing anyons8. The following quasiparticles exist: 1 (vac-uum), ε (fermion), and σ (vortex), of which σ has a quan-

tum dimension dσ =√

2 and the other two d1 = dε = 1.From (4) follows a total quantum dimension of D = 2.

The Kitaev model has two separate contributions57 tothe entanglement entropy

SE = SG + SF . (5)

The first contribution, SG, originates from the static Z2-gauge field and is stated to be57,58

SG =

(Ny2− 1

)log 2 , (6)

where Ny is the number of unit cells along the circumfer-ence and equals the number of bonds cut by the bipar-tition, thus Ny = Lcirc/2. The second contribution, SF ,describes the entanglement of the matter fermions57. Bycomparison with Eq. (3), the constant part in (6) resem-bles the TEE γi = log 2.

We turn to our iDMRG results now, where the entan-glement entropy is readily available from the MPS rep-resentation of the ground state wave function. As will

become clear later, we consider the following quantity

∆SE = SE − SF −Ny2

log 2 ≈ γi , (7)

where SE is the entanglement entropy extracted numer-ically using iDMRG. SF can be computed exactly viathe eigenvectors of the fermion hopping matrix if HK3 isconsidered57,59. We compute SF on a torus with one di-mension equalling Lcirc and the second dimension beingmuch larger. Note that a bipartition of a torus leavestwo cuts of length Lcirc, whereas on the cylinder there isonly one such cut. Thence, only half of SF of a torus isconsidered in Eq. (7).

In the exactly solvable case of HK3, ∆SE reproduces

the TEE, such that ∆SE,K3= γi for all K3, except when

iDMRG is not converged with respect to χ. From Fig. 5,we recover the following TEE

γi =

{log 2 (Wl = +1),

log 2√2

(Wl = −1),(8)

which depends on the sector Wl = ±1. In the gaplesslimit of the Wl = −1 sector (K3 = 0), SF is divergent.Thus, at small K3 the MPS improves with increasing χsimilar to the behaviour of ξ discussed before. Nonethe-less, from (8) a total quantum dimension of D = 2 cansimply be read off. The Wl = −1 sector contains anon-abelian anyon, a vortex σ, with quantum dimensiondi =

√2. The ground state of the Wl = +1 sector is

doubly degenerate with di = 1 for both states. Thus, theexpected degeneracy is recovered.

Upon applying the magnetic field, the integrability ofHK3

in Eq. (2) is lost and the fermionic contributionSF cannot be computed exactly. Based on the fact thatwe observe a similar opening of the gap in the fermionicspectrum forK3 and h when the magnetic field is rescaledas h → 32h3, we assume that SF as a function of therescaled magnetic field SF (32h3) is similar to SF (K3) as afunction ofK3. This assumption is at least justified in thelimit of small h. Figure 5(a) displays ∆SE in a magneticfield, where it approaches the same values of γi for smallh. At elevated fields, ∆SE begins to deviate from γ =log 2 or γσ = log

√2. ∆SE increases monotonically until

the transition into the intermediate phase is reached.In a magnetic field, the separability of fluxes and

fermions is lost and generically additional entanglementbetween fluxes and fermions is created. Such entan-glement generates an additional contribution SF⊗G tothe entanglement entropy, which is not accounted for inEqs. (5) and (7). As this deviation occurs continuously,we like to argue that the topological phase in a low mag-netic field is adiabatically connected to the topologicalphase of HK3

at non-zero K3.As a remark, the difference of ∆SE between the Wl =±1 sectors is not constant. This is due to the correlationlength of the fermions being enhanced in the −1 sector,particularly near the gapless limit (h = 0), where it di-verges. Thus, the fermions may build up entanglement

6

with the fluxes in an increased area near the cut resultingin an enhanced SF⊗G.

We like to conclude that we find numerical evidencefor a total quantum dimension D = 2 with non-abeliananyons having quantum dimension di =

√2 in the ex-

actly solvable model using the three-spin term. The re-sults using the magnetic field, breaking integrability ofthe original model, are still consistent with the resultsabove. However, a significant contribution to the entan-glement entropy arises at increased magnetic fields.

B. Intermediate Regime

Only for AFK, an intermediate region exists rangingfrom hc1,AF < h < hc2,AF , where hc1,AF ≈ 0.22 (forrhombic-2, Lcirc = 10) marks the transition from thetopological phase and hc2,AF ≈ 0.36 the transition intothe field-polarised phase.

The ground state within the intermediate regimerequires to go to comparably large bond dimensionsχ ≈ 1000. Using smaller χ, the ground state is very sen-sitive to the cylindrical geometry as well as the size of theiDMRG cell. However, based on the 1/χ-extrapolation ofthe ground state energy, that is presented in Appendix A,we find evidence for a translationally invariant groundstate. In particular, when using a larger iDMRG cell,we observe a restoration of translational symmetry uponreaching a sufficiently large χ.

This motivates the use of the rhombic-2 geometry withan iDMRG cell equivalent to a single fundamental unitcell, which on the one hand suppresses ground states withenlarged unit cells due to broken translational symmetry,but on the other hand saves computational resources bet-ter spent in reaching larger χ.

Returning to its physical properties, the intermediateregion exhibits a behaviour typical for a gapless phase.Both correlation length ξ and entanglement entropy SEare not converged with respect to χ, where ξ increasesslowly with χ, while SE increases somewhat faster than inthe gapless Kitaev limit. As we are studying effectively aone-dimensional system due to the cylindrical geometry,the finite-χ scaling48 extracting a central charge may beapplicable60. In that context, the behaviour of SE andξ indicate a larger central charge c, than found in theB-phase of the bare Kitaev model. However, the finite-χscaling, see also Appendix A, does not reveal a conclusivec. Furthermore, the behaviour ξ for larger χ ≥ 800 sug-gests a separation of the intermediate region into threephases, of which the middle one grows in extent withlarger χ. Given the large entanglement and the sensitiv-ity to boundary conditions, our iDMRG results can onlybe suggestive for the nature of the ground state in thetwo-dimensional limit.

The flux expectation values Wp and Wl approachzero continuously. Interestingly, the coexistence of bothsectors found in the topological phase, Wl|h=0 = ±1,persists beyond the transition hc1,AF . The peak in

−d2E/dh2 signaling this transition is independent of theparticular sector.

C. Polarized Phase

A transition to the large-h field-polarized phase oc-curs at hc2,AF ≈ 0.36 (AFK), or hc,FM ≈ 0.014 (FMK),respectively. The polarized phase is gapped, which issignaled by the DMRG simulations by a finite correla-tion length ξ and finite entanglement entropy SE . Theentanglement SE decreases with increasing field h andvanishes once the magnetic moments approach satura-tion, where the ground state is a simple product state.At the transition both, FMK and AFK, exhibit a longitu-dinal magnetic moment of ≈ 55% of saturation along the[111] direction without any transverse component. Thelongitudinal moment grows with h reaching 90% satu-ration near h ≈ 0.6 (AFK) and h ≈ 0.2 (FMK). Largemagnetic moments motivate perturbative methods likespin wave-theory45. In comparison to linear spin wavetheory (LSWT)21, the transition gets renormalized sig-

nificantly from hLSW,AF = 1/√

3 ≈ 0.58 down to hc2,AF .For FMK, LSWT predicts a transition at exactly zero21,whereas in iDMRG it occurs at small, non-zero field.

IV. DYNAMICAL SPIN-STRUCTURE FACTOR

The dynamical spin-structure factor S(k, ω) containsinformation about the excitation spectrum and is experi-mentally accessible via scattering experiments, in partic-ular inelastic neutron scattering. We consider S(k, ω) =∑γ={x,y,z} Sγγ(k, ω) with Sγγ(k, ω) being the spatio-

temporal Fourier transform of the dynamical correlations

Sγγ(k, ω) = N

∫dt eiωt

∑a,b

ei(rb−ra)·k Cγγab (t) , (9)

where γ = {x, y, z} is the spin component, ra and rbare the spatial positions of the spins, and diagonal ele-ments Sxx, Syy, and Szz are considered. N is definedby normalizing Sγγ(k, ω) as

∫dω∫dk Sγγ(k, ω) =

∫dk.

Cγγab (t) denotes the dynamical spin-spin correlation

Cγγab (t) = 〈ψ0|Sγa (t)Sγb (0)|ψ0〉= 〈ψ0|U(−t)SγaU(t)Sγb |ψ0〉= 〈ψ0|SγaU(t)Sγb |ψ0〉 , (10)

where the unitary time-evolution operator U(t) =e−i(H−E0)t is modified by subtracting the ground stateenergy E0. Thus, the time-evolution U(−t) acts triv-ially on the ground state 〈ψ0|U(−t) = 〈ψ0|. FollowingRef. [41], we express U(t) into a matrix product operator(MPO) with discretized time steps.

Equation (10) provides the numerical protocol we em-ploy: (i) Obtain the ground state wave function |ψ0〉 us-ing iDMRG and enlarge the iDMRG cell along the cylin-drical axis to make room for the excitation to spread

7

Γ M Γ′0

1

2

3

ω10−2 10−1

Γ M Γ′0

1

2

3

ω

10−2 10−1

Γ M Γ′0

1

2

3

ω

10−2 10−1

0.0

1.5

ω

0.0

1.5

ω

0.0

1.5

ω0.01 Sxx(k, ω) 0.1

(a) h = 0.0

(b) h = 0.1

(c) h = 0.2

Γ M Γ′0

1

2

3

ω

10−2 10−1 100

Γ M Γ′0

1

2

3

ω

10−2 10−1 100

Γ M Γ′0

1

2

3

ω

10−2 10−1 100

0.01 Sxx(k, ω) 1.0

(d) h = 0.0,K3 = −0.25

(e) h = 0.1,K3 = −0.25

(f) h = 0.175,K3 = −0.25

0.0

0.4

Sxx(K

,ω) h = 0.0

h = 0.1

h = 0.15

h = 0.2

0.0

0.4

Sxx(M

,ω)

0.0 0.5 1.0 1.5ω

0.0

0.4

Sxx(Γ

′ ,ω)

(g) K3 = 0.0

0.0

0.4

Sxx(K

,ω) h = 0.0

h = 0.1

h = 0.15

h = 0.175

0.0

0.4

Sxx(M

,ω)

0.0 0.5 1.0 1.5ω

0.0

0.4Sx

x(Γ

′ ,ω)

(h) K3 = −0.25

FIG. 6. Dynamical spin-structure factor Sxx(k, ω) along Γ–M–Γ′ at: (a) h = 0.0, (b) h = 0.1, (c) h = 0.2, (d) h = 0.0,K3 =−0.25, (e) h = 0.1,K3 = −0.25, (f) h = 0.175,K3 = −0.25 within the topological phase. (a)-(f) have a logarithmic color scaleranging over two decades. (g,h) Sxx(k, ω) at high-symmetry points Γ, K, M , and Γ′ for different h and K3. An vertical offsetis used for better visibility. In all plots, Sxx(k, ω) is normalized as given in the main text.

spatially, (ii) apply spin operator Sγi at site i, (iii) time-evolve the MPS by U(t), (iv) apply Sγj at j, and (v)compute the overlap.

On the technical side, we first compute the spatialFourier transform of Cγγab (t), extend the time-signal usinglinear predictive coding61, and multiply with a gaussianto suppress ringing due to the finite-time window. Theextension of the time-signal allows for much wider finite-time windows keeping a significant part of the simulatedreal-time dynamics. All spectra shown in the remainderhave a broadening of σω = 0.018 due to multiplying thereal-time data with a Gaussian of width σt = 55.8. Thereal-time data is obtained for times up to T = 120 oncylinders with rhombic geometry and Lcirc = 6.

In the following, we discuss S(k, ω) within the topo-logical phase and the polarized phase. Simulating thedynamics within the intermediate regime is left for fu-ture work as the necessary bond dimension for encodingthe ground state is to large to achieve appreciably longtimes in the time-evolution.

A. Topological Phase

Near h = 0, see Fig. 6(a), the numerically obtainedS(k, ω) exhibits the features of the analytic solution44,62

with some adjustments due to the cylindrical geometry46.Firstly, this involves a low-energy peak at ω ≈ 0.03 ofwhich its spectral weight is shifted towards Γ′ due to theantiferromagnetic nearest-neighbor spin-spin correlationcaused by the antiferromagnetic Kitaev exchange. Whenusing a cylindrical geometry, an additional δ-peak withfinite spectral weight occurs at the two-flux energy. Thisδ-peak, together with the finite-time evolution and subse-quent broadening in frequency space, hides the two-fluxgap. Nevertheless, the δ-peak position coincides with thetwo-flux gap63, ∆2 ≈ 0.03.

Secondly, a broad continuum exists, that is cut off nearω ≈ 1.5. Increasing h to 0.1 and 0.2, cf. Fig. 6(b) and(c), only leads to minor changes of the spectrum. Mostnotably, the low-energy peak develops a shoulder towardsslightly elevated energies, and the cut-off at ω ≈ 1.5 is

8

Γ M Γ′0

1

2

3

ω10−2 10−1 100

Γ M Γ′0

1

2

3

ω

10−2 10−1 100

0.01 S(k, ω) 1.0

0.0

1.5

ω

0.0

1.5

ω

(a) h = 0.58

(b) h = 0.5 Γ M Γ′0

1

2

3

ω

10−2 10−1 100

Γ M Γ′0

1

2

3

ω

10−2 10−1 100

0.01 S(k, ω) 1.0

(c) h = 0.425

(d) h = 0.375

0

2

S(K

,ω)

h = 0.375

h = 0.40

h = 0.425

h = 0.5

h = 0.58

0

2

S(M

,ω)

0.00 0.25 0.50 0.75 1.00 1.25 1.50ω

0

2

S(Γ

′ ,ω)

(e)

FIG. 7. Dynamical spin-structure factor S(k, ω) in the field-polarized phase along Γ–M–Γ′ at: (a) h = 0.58, (b) h = 0.5, (c)h = 0.425, and (d) h = 0.375. (e) S(k, ω) at high-symmetry points K, M , and Γ′ for different h. An vertical offset is used forbetter visibility. In all plots, S(k, ω) is normalized as given in the main text.

blurred out. Both features are more prominent in the lineplots, Fig. 6(g). Any changes to the low-energy spectrumnear or even below the two-flux gap are hidden in theenergy resolution caused by the finite time-evolution.

In order to get a qualitative view on how the magneticfield affects the spectrum, we investigate the effect ofboth, K3 and h.

For K3 = −0.25 and h = 0.0, Fig. 6(d), the low-energypeak gets elevated to ω ≈ 0.2. This peak originates froma single fermion bound to a pair of fluxes44 and its shiftis caused by K3 increasing the two-flux gap. The fermioncontinuum starts at ω ≈ 0.4, and the upper cut-off of thecontinuum remains near ω ≈ 1.5. Both edges are sharp.Note that, K3 = −0.25 has a similar correlation length ash = 0.2 as discussed above in relation to Fig. 4. Yet, thecorresponding spectra, Fig. 6(c) and (d), are qualitativelydifferent, in that for h = 0.2 the spectral weight is shiftedsignificantly towards zero with no observable gap.

Upon increasing h to 0.1, the low-energy peak splitsinto at least three peaks, two of them develop a disper-sion. Due to the field, the fluxes acquire a finite hoppingamplitude and become mobile. The fluxes are thenceno longer required to lie on neighboring plaquettes, butinstead may separate. Hence, the mode describing afermion bound to the two-flux pair generally attains morestructure64. Moreover, interaction between fluxes mayinduce further dispersion65,66. At further elevated fields,cf. Fig. 6(f) at h = 0.175, somewhat before the phasetransition into the intermediate regime67, the splittingincreases with lots of the spectral weight shifting to thepeak that is lowest in energy. The spectral gap reduces

significantly with h and has its minimum at the Γ and Γ′

high-symmetry points.

B. Polarized Phase

From linear spin-wave theory (LSWT) it is known thatthe magnons are topological. Their bands carry a ±1Chern number and chiral edge modes have been observedon a slab geometry45,68. But LSWT is only applicablefor fields above the classical critical field hclas = 1/

√3 ≈

0.58. Here, we focus on the bulk excitation spectrum atfields between the numerically observed, hc2,AF ≈ 0.36,and the classical critical field. Results for larger fields arepresented in Ref. [45] using the same method.

Beginning our discussion at the classical critical fieldh = 0.58 shown in Fig. 7(a), we observe two magnon-bands with a minimum of ω ≈ 0.15 at the high-symmetrypoints Γ and Γ′. The two-magnon continuum has someoverlap with the upper magnon band. With loweringthe field, the magnon bands move down in energy andflatten in the sense that their bandwidth decreases. Ath = 0.5, cf. Fig. 7(b), the continuum already overlapswith major parts of the upper magnon band. This opensdecay channels, limiting its lifetime, and consequentlybroadening the mode.

Approaching the transition, cf. Fig. 7(d) at h = 0.375and (c) at h = 0.425, S(ω,k) shows a very broad contin-uum ranging down to almost zero energy, where also mostof the spectral weight is observed. The upper magnonband is completely obscured by the multi-magnon con-

9

tinuum and lots of the spectral weight is distributed overa wide range in energy. The lower edge of the spectrumflattens towards the transition, which is even more evi-dent in the line plots shown in Fig. 7(e). In particularat h = 0.375 the low-energy peaks shift down to almostzero energy simultaneously at the K, M , and Γ′ high-symmetry points, with most of the spectral weight stillappearing above the Γ′-point.

This reproduces to some extent the phenomenology ofLSWT, namely that the lower magnon band flattens com-pletely while decreasing to zero energy21,45, yet it occursat lower fields than in LSWT. On the other hand, a clearremnant of the single magnon branch cannot be observedclose to the transition as it overlaps and merges with themulti-magnon continuum. It may be possible that thesingle magnon branch is still dispersive, even though witha significantly reduced bandwidth.

A feature in the spectrum not mentioned so far,emerges at around ω ≈ 1 at magnetic fields near the tran-sition. Initially this high-energy feature is very broad inenergy, but sharpens and moves to higher energy uponincreasing the field. At h = 0.5 (h = 0.58) it appearsaround ω ≈ 1.25 (ω ≈ 1.5) and exhibits a slight dis-persion. At even larger fields, beyond what is presentedhere, the high-energy feature moves up in energy with alinear dependence on the field and twice the slope com-pared to the single-magnon excitations. Furthermore,the high-energy feature is situated at the upper edge ofthe two magnon continuum. Its intensity first increases,but starts to decrease at higher fields. It would be inter-esting to investigate, if this may be due to the appearanceof an anti-bound state69 of two magnons experiencinga repulsive interaction on account of the antiferromag-netic Kitaev exchange interaction between two adjacentflipped spins.

V. CONCLUSION

We confirm the vastly different phenomenology be-tween ferromagnetic and antiferromagnetic Kitaev inter-action, if a magnetic field along [111] direction is applied.In case of ferromagnetic Kitaev coupling, only a singlemagnetic transition is observed, that separates a low-htopological phase from the large-h field-polarized phase.Whereas for antiferromagnetic Kitaev coupling, the topo-logical phase is more stable and an intermediate regimeexists, that is possibly gapless. The topological order ofthe low-h phase and its non-abelian anyonic excitationsare verified by extracting the topological entanglemententropy. In addition to Ref. [35], the topological orderobtained with a finite three-spin term or when applyinga weak magnetic field is the same also for antiferromag-netic Kitaev coupling.

Upon applying the magnetic field, the spectral gapin the dynamical spin-structure factor remains withinthe frequency resolution and the overall spectrum ex-hibits only minor changes. However, the dynamical spin-

structure factor is remarkably different when applyingthe three-spin term lifting the spectral gap, both due tothe flux gap increasing and the fermions gapping out.When a combination of magnetic field and three-spinterm is applied, we observe a drastic reduction of thespectral gap with increased field and more structure inthe low-energy peak corresponding to a bound state of aflux pair and a fermion. This additional structure is dueto the fluxes becoming mobile and the flux-pair may sep-arate providing a richer energy manifold for that boundstate. Upon approaching the intermediate regime, thespectral gap reduces with a minimum at the Γ′ high-symmetry point. We can conclude, that even though theenergy gap opens in a similar manner when either themagnetic field or three-spin term is varied, the dynami-cal spin-structure factor exhibits a remarkably differentlow-energy structure. Thus, additional terms in pertur-bation theory, other then the three-spin term preservingintegrability, are relevant to describe the dynamical spin-structure factor in the topological phase.

When approaching the intermediate region from high-fields, we observe a strong reduction in frequency witha simultaneous flattening of the lower magnon band.A broad continuum develops, that ranges down to thelowest frequencies and merges with the single magnonbranch. It remains an open question, whether this flat-tening could be attributed to the collapse of the lowermagnon branch, as observed within LSWT, or rather tomulti-magnon excitations obscuring any dispersion of thevery same magnon branch. Nonetheless, the flat gap clos-ing as such is interesting in various aspects as it mayindicate exotic spin states like a quantum spin liquid.

VI. ACKNOWLEDGEMENTS

We are grateful to Balasz Dora, Lucas Janssen, PaulMcClarty, Karlo Penc, Jeffrey G. Rau and Ruben Verre-sen for stimulating discussions. This work was supportedin part by DFG via SFB 1143. FP acknowledges the sup-port of the DFG Research Unit FOR 1807 through grantsno. PO 1370/2- 1, TRR80, and the Nanosystems Initia-tive Munich (NIM) by the German Excellence Initiative.

Appendix A: Finite-size dependents withinintermediate phase

Here, we investigate the intermediate regime with re-spect to possible finite-size effects as well as finite bonddimension of the matrix product state (MPS). Figure 8provides a comparison of the ground state energy EGSfor two different geometries, rhombic with Lcirc = 6or rhombic-2 with Lcirc = 10, as well as several differ-ent sizes of the iDMRG cluster at a magnetic field ofh = 0.275. Similar checks are done at different h.

In case of rhombic-2 with Lcirc = 10 (green symbols),the smallest cluster is similar to a single fundamental

10

0.000 0.001 0.002 0.003 0.004 0.005

1/χ

−0.465

−0.464

−0.463EGS

3x1 rho

3x3 rho

5x1 rho-2

5x1 rho-2 n = 2

5x1 rho-2 n = 5

5x2 rho-2 n = 10

5x3 rho-2 n = 15

FIG. 8. Comparison of the ground state energy EGS vs bond-dimension χ for different geometries and sizes of the iDMRGcluster at a single field strength of h = 0.275. For small χ,larger iDMRG cluster have a smaller EGS . However, in thelimit 1/χ→ 0, EGS is of very similar value for all geometriesused. In fact, large iDMRG cluster show a phase transitionfrom an ordered ground state at small χ to a translationalinvariant ground state at large χ captured by the smallestiDMRG cluster.

unit cell with two sites (green circles), that is repeatedalong a chain winding around the cylinder. Next largerclusters are: four sites (n = 2 fundamental unit cells,green ’x’), 10 sites (n = 5, green ’+’), 20 sites (n = 10,green lower triangle), and 30 sites (n = 15, green uppertriangle). When using small bond-dimensions χ < 500,larger iDMRG clusters result in lower ground state en-ergies EGS . Upon increasing χ, the different energiesapproach each other until eventually a transition to theground state of a smaller cluster occurs, e.g., at χ ≥ 512the 10 site cluster (’x’) has the same ground state prop-erties as the fundamental unit cell (circles). Such a χ-transition is unphysical and a mere property of truncat-ing the MPS.

In case of rhombic with Lcirc = 6 (black symbols)in Fig. 8, the smallest iDMRG cluster is a single ringwith three fundamental unit cell along the circumference(’3x1’, black circles). Larger clusters of three repetitionsalong the cylinder (’3x3’, black triangles) and six repeti-tions (not shown, but equivalent to ’3x3’) are checked. Asabove, a similar χ-transition at χ ≈ 800 is found, wherefor smaller χ the ’3x3’ has a lower EGS , but transitions

to the same state as ’3x1’ for larger χ.In conclusion, the ground states for larger χ are not

exhibiting any broken translational symmetry and mayresemble the physical ground state. Thus, the use ofiDMRG cell composed of a single fundamental unit cellis justified for computing the phase diagram shown inFig. 3.

The previous results signify, that large bond dimen-sions are necessary to resemble the physical ground state.We cannot say for sure, that the χ we are able to achieveare already sufficient, thus any statement regarding theintermediate region has to be taken with care. Nonethe-

1006× 10−1 2× 100 3× 1004× 100

ξ

2.0

2.5

3.0

S

5x1 rho-2

h = 0.35

h = 0.325

h = 0.30

h = 0.275

h = 0.25

FIG. 9. Entanglement entropy S of a bipartition of the cylin-der over correlation length ξ for various bond dimension χ tocheck for possible finite-χ scaling. Data is shown for differentmagnetic field strength h = 0.25, 0.275, 0.3, 0.325, 0.35 acrossthe intermediate regime. Lines are fits to the five points withlargest ξ at each field.

less, let us assume the MPS do reflect physical proper-ties of the underlying phase and apply a finite-χ scal-ing. For h = 0.275, 0.3, 0.325, and 0.35 we obtain aSE,χ = c/6 log ξχ + const scaling typical for a gaplessphase47,48, see Fig. 9. Linear regression of the fivepoints with largest χ reveal slopes corresponding to cen-tral charges of c = 3.49 at h = 0.275, c = 3.31 ath = 0.30, c = 4.54 at h = 0.325, and c = 4.01 forh = 0.35, all for rhombic-2 with Lcirc = 10. We wantto remark, that not all of the c represent physical centralcharges. h = 0.3 has a very similar behaviour in terms ofSE vs ξ, but a slightly smaller c, which may converge to3.5 for larger χ. For h = 0.25, χ does not yet suffice toenter a linear SE ∝ log ξ regime.

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12

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