K. S. Tikhonov and M. V. Feigel’man- Quantum spin metal state on a decorated honeycomb lattice

8/3/2019 K. S. Tikhonov and M. V. Feigel’man- Quantum spin metal state on a decorated honeycomb lattice

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Quantum spin metal state on a decorated honeycomb lattice

K. S. Tikhonov 1,2 and M. V. Feigel’man 1,21 L. D. Landau Institute for Theoretical Physics, Kosygin str.2, Moscow 119334, Russia and

2 Moscow Institute of Physics and Technology, Moscow 141700, Russia (Dated: March 4, 2010)

We present a modication of exactly solvable spin- 12 Kitaev model on the decorated honeycomb

lattice, with a ground state of ”spin metal” type. The model is diagonalized in terms of Majoranafermions; the latter form a 2D gapless state with a Fermi-circle those size depends on the ratio of exchange couplings. Low-temperature heat capacity C (T ) and dynamic spin susceptibility χ (ω, T )are calculated in the case of small Fermi-circle. Whereas C (T ) T at low temperatures as it isexpected for a Fermi-liquid, spin excitations are gapful and χ (ω, T ) demonstrate unusual behaviourwith a power-law peak near the resonance frequency. The corresponding exponent as well as thepeak shape are calculated.

Quantum spin liquids [1–5] present very interesting ex-amples of strongly correlated phases of matter which donot follow the classical Landau route: no local order pa-rameter is formed while the entropy vanishes at zero tem-perature. Although quite a number of different proposalsfor the realization of spin liquid states are available, andmany interesting results were obtained numerically, seee.g. [4], the progress in analitical theory was hindered fora long time due to the absense of an appropriate quan-tum spin model exactly solvable in more than one spa-tial dimension. Seminal results due to A. Kitaev [ 6] maypave the way to ll this gap. Kitaev proposed spin- 1

2model with anisotropic nearest-neighbour spin interac-tions J α σα

i σαj (where α = x,y,z ) on the honeycomb lat-

tice. The model allows for the two-dimensional general-ization of the Jordan-Wigner spin-to-fermion transforma-tion and can be thus exactly diagonalized. Kitaev modelrealizes different phases as its ground-states, the most in-teresting of them corresponds to the region around sym-metric point J x = J y = J z = J ′. The excitations abovethis ground-state constitute a single branch of masslessDirac fermions (Kitaev presents them in terms of Ma- jorana fermions those spectrum contains two symmet-ric conical points). Thus the Kitaev honeycomb modelpresents exactly solvable case of the critical QSL. Al-though long-range spin correlations vanish in this modelpresisely due to its integrability [7], it presents valuablestarting point for the construction of analitically control-lable theories possessing long-range spin correlations.

Still, here we discuss another extension of themodel [6]: it is interesting to nd exact realizations of other types of QSLs, the one with gapful excitation spec-trum, and the one with a whole surface of gapless exci-tations [ 8]. Gapful QSL was recently found by Yao andKivelson [9]. They proposed specic generalization of theKitaev model, where each site of the honeycomb latticeis replaced by a triangle (we will call it, for brevity, ”3-12 lattice”), with internal coupling strengths equal to J and inter-triangle couplings equal to J ′. Topologicallyequivalent structure of such a lattice is shown in Fig. 1.Yao-Kivelson model is exactly solvable and contains a

FIG. 1: (Color online) ”Brick-wall” representation of the dec-orated honeycomb lattice. Elementary cell for the MajoranaHamiltonian contains 12 sites and is bounded by the blue line.

critical point at g ≡ J ′/J = √3. At g = √3 the ex-citation spectrum has single low-energy branch of Diracfermions, like the original Kitaev model, whereas at anyother g the excitation spectrum is gapful. Yao and Kivel-son have shown that the ground state at g < √3 is a topo-logically nontrivial chiral spin liquid with Chern numberC = ±1, whereas g > √3 phase is topologically trivial,with C = 0. Exactly solvable QSL model of spin-metaltype with spins- 3

2 was proposed by Yao, Zhang and Kivel-son [10]. In the present Letter we take a different route:we show that slight modication of the Hamiltonian of spin- 1

2 Yao-Kivelson model on the 3-12 lattice leads tothe spin-metal QSL with a pseudo-Fermi-circle. Similarapproach was proposed recently in Ref. [11] where QSLwith pseudo-Fermi-line was found in the spin- 1

2 modelon a decorated square lattice. Apart from another lat-tice studied, our study differs from [11] in two respects: i)our Fermi-liquid-like ground state is the result of sponta-neous symmetry breaking leading to a ”chiral antiferro-magnet” ordering dened in terms of 3-spin products, ii)we present analitic results for heat capacity and dynamicspin susceptibility in the limit of small Fermi-surface size.We consider the Hamiltonian:

http://arxiv.org/abs/1003.0992v1













































2

FIG. 2: (Color online) Flux congurations in G1,

2,

3,

4 states

correspondingly. Empty/lled plaquette corresponds to theux +1 , − 1 for dodecagons and + i, − i for triangles.

H=l= ij

J l (σ i n l ) ( σ j n l ) + λx ′ ,y ′ ,z ′ −links

T pT p′ . (1)

Here T p = σxia σy

jb σzkc is the three-spin ”exchange” op-

erator corresponding to the p-th triangle. We takeJ x,y,z = J, J x ′ ,y ′ ,z ′ = J ′ and without loss of general-ity assume J, J ′ > 0. Unit vectors n l are parallel tox, y and z axis for the corresponding links x, x′, etc.Eq. (1) reduces to the original Hamiltonian of Ref. [ 9]at λ = 0. This spin Hamiltonian is rather special, sinceit posesses large number of independent integrals of mo-tion, so called uxes dened as W p = n

s =1 σαj s

σαj s − 1

,where j 0 , j1 , ..., j n = j 0 denes a minimal close loopon the lattice. All W p commute with Hamiltonian andwith each other and divide total Hilbert space into sec-tors, corresponding to different sets of W p eigenvalues.Hamiltonian ( 1) can be diagonalized by means of Ki-taev representation of spins via 4 Majorana fermions:σα

i = ici cαi , where four Majorana operators ci , cx

i , cyi , cz

iare dened on each site of the lattice and satisfy anticom-mutation relations cα

i , cβj = 2 δij δαβ . This way, each

spin (2-dim Hilbert space) is represented by four Majo-ranas (4-dim Hilbert space). This is a representation inthe extended Hilbert space; all physical states should sat-isfy the constraint: D i |Ψ phys = |Ψ phys for any latticesite i, where D i = ci cx

i cyi cz

i . Operators D i is the gaugetransformation operator for the group Z 2 . Hamiltonian(1), extended to the Hilbert space of Majorana fermions,reads ( u ij = icα

i cαj ):

H= −il= ij

J l ci u ij cj + λx ′ ,y ′ ,z ′ −links

t pt p′ (2)

with t p = uab ubcuca . Note, that [ u ij , H ] = 0 a n du2

ij = 1 . These integrals of motion are gauge-dependent;they are related to the gauge-invariant uxes W p =n

s =1 −iu j s j s − 1 . There are two types of W p : 1) uxes,corresponding to the triangular loops W (3)

p = ±i, and 2)uxes W (12)

p = ±1 corresponding to the dodecagon loops.They respond differently to the time reversal transfor-mation: ˆT W (12)

p = W (12) p , but ˆT W (3)

p = −W (3) p . It was

shown in Ref. [12] that the ground state of Hamiltonian(1) with λ = 0 corresponds to all W (12)

p = −1 and allW (3)

p are equal (either to i, or to −i) ; these two global

eigenstates are related by the ˆT inversion. We show nowthat in some range of couplings J, J ′ even very small λstabilizes another type of the ground state, with vari-ables W (3)

p = ±i ordered alternatively (like in the AFMIsing model on honeycomb lattice), and with Fermi-lineof gapless excitations.

Hamiltonian ( 2) can be diagonalized for any periodicconguration of the gauge eld u ij . However, we restrictour consideration to the states with the same ux pe-riodicity as the original lattice. Thus we are left with 4gauge-nonequivalent states G1,2,3,4 shown in Fig. 2 , plustheir time-reversal partners G i ′ = ˆT G i . Since the gaugeeld u ij which correspond to the ux congurations G2,4does not t into 6-site unit cell, in order to describe allstates G1.. 4 we use elementary cell containing 12 sites,shown in Fig 1, with e 1 = e x and e 2 = e y (hereafterlengths are measured in units of the lattice spacing).

After the gauge is xed, we are left with the Hamil-tonian H, restricted to the Majorana space, and de-noted by H . This Hamiltonian can be diagonalized

in terms of Fourier-transformed Majorana elds ψα, k =1√2N r e−ikr cα, r , where subscript α = 1 ..12 enumeratesfermionic components inside each of N elementary cells.In the Fourier representation the Hamiltonian reads:

H =k K +

ψ+k H k ψk (3)

where summation is going over the half of the Bril-luen zone K + = (0 ≤kx ≤π, −π ≤ky ≤π). Fourier-transformed Majorana elds, restricted to K + , denecomplex fermions. Hamiltonian H is diagonal in thenumber of this fermions, which results from the trans-

lational invariance of the system. H k i s a 12 × 12gauge-dependent Hermitian matrix. Spectral equationdet H k − = 0 determines twelve bands with disper-sions α, k . The Fermi-sea energy can then be calculatedas E = f ( α, k ) α, k , with f ( ) being the fermion popu-lation numbers (below we imply periodic boundary con-ditions). When the ground state of Majorana system insome xed gauge is found, the true ground state of theoriginal spin Hamiltonian should be found applying theprojection operator P = i

1+ D i2 . However, if we are in-

terested in calculation of gauge-invariant quantities (likeground state energy or spin susceptibility), there is no

need for explicit implementation of this projection, andcalculation can be done in any particular gauge.Below we consider vicinity of the point g = gc = √2

where the state G2 becomes critical (see below). Groundstate energies of G1.. 4 states per unit cell at g = gc areat λ = 0: E (0)

1 = −10.758J, E (0)2 = −10.681J, E (0)

3 =

−10.664J, E (0)4 = −10.610J . That conrms that G1

has the lowest global energy at λ = 0. However, atnite λ this energies are simply shifted by ±6λ andAFM orderdering of t p realized by the G2 state becomes



3

favourable at λ > λ c = E (0)2 −E (0)

1 / 12 ≈6.4×10−3J .

The G2 state breaks both ˆT -symmetry and the symme-try ˆP of inversion between sublattices of the honeycomblattice, and it can be called ”chiral AFM” state. How-ever, it is invariant with respect to the combined inver-sion ˆT ˆP . As temperature raises above some critical valueT c , a phase transition leading to a ”chiral-disordered”

state obeying bothˆ

T - andˆ

P -inversions should occur.We assume below that T T c , and neglect excitationswhich ip chiralities t p.

Eigenstates of ( 3) are found via matrix diagonaliza-tion: H k = S k ˆH k S +k . Solving the equation det H k = 0at g = gc we nd that zero-energy excitations are locatedat two inequivalent points: K 1 and K 2 . So, there is a sin-gle gapless band 1,k containing low-energy excitations,whereas all other 11 bands 2.. 12 ,k have the gap of theorder of J . Low-energy excitations are given by φ1,k =

α S +α ψα, K 1 + k and φ2,k = α S +α ψα, K 2 + k (hereafterfor brevity we write S α = S α, K 1 and S α = S 1α, K 2 ). Per-turbation expansion up to the second order in k and upto the rst order in Γ = gc −g 1 leads to the effectiveHamiltonian of the low-energy excitations:

H eff =

|k | 1

φ+2,k φ2,k −φ+

1,k φ1,k k (4)

where k = 12√3 J 3k2

x + k2y −µ , with µ = 8/ 3 J Γ.

Density of states is dened by d 2 k(2 π ) 2 = ν d and is

equal to ν = (2 πJ )−1 . Hamiltonian (4) determines low-energy properties of the spin system ( 1) at small Γ andunder condition λ > λ c . The spectrum is gapful at Γ < 0.Positive Γ corresponds to the spin metal state. At T

µ J the heat capacity of spin liquid (per unit cell)is C (T ) = π

3 T/J , demonstrating standard Fermi-liquidbehaviour at low temperatures. However, these gaplessexcitations do not carry spin, while spin excitations aregapped in this model, as we discuss below.

We note that similar analysis could be developed forthe G3 state in the vicinity of the point g = √3, which isknown to be critical for the G1 state as well [9]. However,within the model dened by the Hamiltonian ( 1), thestate G3 always has energy higher than that of G2 state;the latter, however, has large gap near g = √3, thus wework near the point g = √2 where G2 becomes critical.

Now we turn to the calculation of frequency-dependentspin susceptibility χ (ω, T ) of the spin metal state. Linearsusceptibility tensor is proportional to the unit matrixdue to cubic symmetry of the Hamiltonian ( 1) in thespin space. We choose external homogenious eld h(t) inthe z direction which adds the term −h(t) r α σz

α (r ) tothe spin Hamiltonian and calculate σz . Susceptibilityreads χ (ω) = r , r ′ αβ χ αβ (r

′

−r , ω) with

χ αβ (r , ω) = i ∞0eiωt [σz

α (r , t ), σ zβ (0 , 0)] dt (5)

according to Kubo formula ( r enumerates cells and α, βstay for sites within the same cell, average hereafteris taken over the nonperturbed ground state). Wend, following [7], that correlation function Gαβ (r , t ) =

σzα (r , t )σz

β (0 , 0)T

is non-zero either for coinciding spinsor for spins which are connected by z or z ′ link. Thismeans, that Gαβ (r , t ) δ (r ) (since different elemen-

tary cells are connected by x′, y′ links) and in whatfollows we do not write spatial coordinates explicitly.Spin operator creates two Z 2 vortices in the neigbour-ing plaquettes, which have an excess energy Ω αβ J ,thus correlation function Gαβ (t) oscillates with a fre-quency Ωαβ . Therefore dynamic susceptibility χ (ω) =N l χ αβ + χ ββ + χ αβ + χ βα , where summation goesover z and z′ links l = αβ in the unit cell, containstwo resonances at frequencies ω = Ω z , Ωz ′ . Our goal isto nd lineshapes of these resonances.

The sum in parenthesis in the above expression forχ (ω) can be written as χ l = 4 i ∞0 eiωt (Gl (t) −Gl (t))with G

l(t) = eiHt ψ

le−

iH ′

l t ψ+

l, where ψ

l=

12 (cα + iu αβ cβ ) is a complex fermion dened on a linkl. In this expression H ′l stays for the Hamiltonian whichis different from H by inversion of the sign of u l : H ′l =H + V l , whereas V l = 4 J l ψ+

l ψ l − 12 . After standard

algebra we nd:

Gl (t) = ψ l (t)T exp −i t

0V l (τ )dτ ψ+

l (0) . (6)

The problem of calculation of Gl (t) seems to be simi-lar to the Fermi Edge Singularity (FES) problem with

a separable scatterer. The latter was solved exactly (inthe infrared limit µt 1) in [13] by summation of theperturbation theory series via the solution of particu-lar integral equation. However, our problem is, strictlyspeaking, different, since initial Hamiltonian H is not di-agonal in ψ+

l ψ l and hence correlation function F 0l (t) =T ψl (t)ψl (0) is not equal to zero identically. However,

unlike G0l (t) = T ψl (t) ψ+l (0) which has long-time tail

1/t , the function F 0l (t µ−1) decays very fast with tdue to exact cancellation between Fermi-surface contri-butions coming from different valleys. This means thatcorresponding pairings in the series expansion of (6) donot lead to any singulary at the threshold and can be ne-

glected. In this case, the solution is similar to the one pre-sented in Ref. [13] and can be written in terms of the long-time ( |tµ| 1) asymptotics of G0l → −iν a l

t + Λ lδ (t)(here the term with δ(t) is necessary to preserve the cor-rect weight G0l (t)dt, see [13] for details). The 1 /t tailin the Green function reects the presense of a band of gapless excitations φ1,2 which form Fermi-sea. Reshuf-ing of the Fermi-sea by the scattering off the local re-pulsive potential V l leads to the shift of the energy and tothe change in the power-law exponent in the exact Green



4

1Ω ’ T

Χ Ω ’

FIG. 3: Real (full line) and imaginary (dashed line) partsof χ (ω′ , T ) at µ/J = 0 .05, that corresponds to the exponentλ = 0 .35.

function, compared to the bare one:

G l (t) = −iνa l

t(iξ0t)λ l e−i Ωl t , (7)

where λ l = 2 ( δl /π )−(δl /π )2 and δl = −arctan 4πνa l J l1+4 ν Λl J l .

Frequency Ω l J > 0 is the shift in the ground stateenergy due to creation of two uxes. Obviously, Ω l isdifferent for z and z′ links; nally, ξ0 µ is the high-energy cutoff. Whereas a l is determined by the vicin-ity of Fermi-energy only, the parameter Λ l characterizesshort-time behaviour of the Green function, and thus isdetermined by the whole spectrum of all 12 fermionicbands: a l = 1

Nν p ,γ |S αγ, p |2 + |S βγ, p |

2 δ ξγ, p and

Λl = 2u αβ

Nν p ,γ ξ−1γ, p Im S αγ, p S βγ, p . The parameters

a l and Λ l are gauge-independent constants, which de-pend on the type of the link only. Explicite calcula-tion leads to a l = |S α |

2 + |S β |2 + S α

2 + S β2 and

Λl = cl log J µ + ηl , where cl = 2 uαβ Im S α S β −S α S β

and ηl is some number of the order of unity which canbe found only by numerical integration over K + (it isdetermined by the whole band). Evaluation leads to thefollowing result:

tan δz = −11.04+ 0 .18ln J

µ, tan δz ′ =

10.40+ 0 .26ln J

µ.

(8)Eq.( 8) determines phase shifts modulo π only. This ambi-guity is xed by the continuity condition: δl (µ →0) = 0.Since λ z < 0 and λ z ′ > 0 for any J/µ , only χ z ′ divergesat the corresponding threshold (while χ z still has a cusp);therefore below we concentrate on the contribution of thez′ links only, λ z ′ ≡λ. Using the results ( 7,8), we calcu-late spin susceptibility close to the Ω z ′ resonanse, withω′ = ω −Ωz ′ . We nd χ T =0 (ω′) = χ 0Γ (λ) (−ξ0 /ω ′)λ ,where χ 0 = 4N

3π J −1 . Note, that χ l = 0 below the thresh-old (ω′ < 0) as it should be at T = 0. These results can

be easily generalized to the nite temperature T µ.As was shown in [14], nite-temperature correlation func-tion in the FES problem can be obtained from the zero-temperature one by substitution t →sinh πT t

πT . For thesusceptibility, that gives:

χ (ω′) = χ 0iξ0

2πT

λ Γ (λ) Γ 1−λ2 − iω

2πT

Γ 1+ λ

2 −iω

2πT

. (9)

This function is plotted in Fig. 3. The ma- jor effect of nonzero temperature is the appearenceof absorption below threshold: χ (−ω′ T ) ≈χ T =0 (ω′) 1 + i sin πλe −|ω

′

|/T ; in addition, the reso-nant peak appears to be smeared out: χ (|ω′| T ) ≈χ 0eiπλ ξ 0

2πT

λ.

In conclusions, we have shown that certain (numer-ically, very weak) modication of the Yao-Kivelson ver-sion of Kitaev spin lattice leads to the ground-state of theFermi-liquid type, with a Fermi energy µ √2

−J ′/J .

We have studied the model in the continuum limit of small Fermi-circle µ J and at low temperatures T µ. Gapless excitations of the Fermi-sea do not carry spinthemselves, but they determine the shape ( 9) of the res-onance peak in the dynamic spin susceptibility.

We are grateful to A. Yu. Kitaev for numerous im-portant discussions and advises. This research was sup-ported by the RFBR grant # 10-02-00554 and by theRAS Program ”Quantum physics of condensed matter”.

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K. S. Tikhonov and M. V. Feigel’man- Quantum spin metal state on a decorated honeycomb lattice