Frustrated honeycomb-lattice bilayer quantum antiferromagnet in a magnetic field: Unconventional phase transitions in a two-dimensional isotropic Heisenberg model

0.88 MB·

by  Taras Krokhmalskii

Frustrated honeycomb-lattice bilayer quantum antiferromagnet in a magnetic field:
Unconventional phase transitions in a two-dimensional isotropic Heisenberg model
Taras Krokhmalskii,1,2 Vasyl Baliha,1 Oleg Derzhko,1,3,2,4 J¨org Schulenburg,5 and Johannes Richter3
1Institute forCondensed Matter Physics, National Academyof Sciences ofUkraine, Svientsitskii Street 1, 79011 L’viv,Ukraine
2Department for Theoretical Physics, Ivan Franko National University of L’viv, Drahomanov Street 12, 79005 L’viv, Ukraine
3Institut fu¨r theoretische Physik, Otto-von-Guericke-Universit¨at Magdeburg, P.O. Box 4120, 39016 Magdeburg, Germany
4Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
5Universita¨tsrechenzentrum, Otto-von-Guericke-Universit¨at Magdeburg, P.O. Box 4120, 39016 Magdeburg, Germany
(Dated: January 30, 2017)
7
1 We consider the spin-1/2 antiferromagnetic Heisenberg model on a bilayer honeycomb lattice
0 including interlayer frustration in the presence of an external magnetic field. In the vicinity of the
2 saturationfield,wemapthelow-energystatesofthisquantumsystemontothespatialconfigurations
of hard hexagons on a honeycomb lattice. As a result, we can construct effective classical models
n
(lattice-gas as well as Ising models) on the honeycomb lattice to calculate the properties of the
a
J frustrated quantum Heisenberg spin system in the low-temperature regime. We perform classical
Monte Carlo simulations for a hard-hexagon model and adopt known results for an Ising model to
6
discuss the finite-temperature order-disorder phase transition that is driven by a magnetic field at
2
low temperatures. We also discuss an effective-model description around the ideal frustration case

and find indications for a spin-flop liketransition in the considered isotropic spin model.
]
l
e
PACSnumbers: 75.10.-b,75.10.Jm
-
r Keywords: quantum Heisenbergantiferromagnet,frustratedhoneycomb-lattice bilayer,localizedmagnon
t
s
.
t
a I. INTRODUCTION conventional physical properties was found, see Refs. 9–
m 11 and references therein. For the flat-band systems
at hand, the local nature of the one-magnon states al-
-
d An important class of quantum Heisenberg antiferro- lowsto constructalsolocalizedmany-magnonstatesand
n magnetsconsists ofthe so-calledtwo-dimensionaldimer- to calculate their degeneracy by mapping the problem
o izedquantumantiferromagnets. Theycanbeobtainedby
onto a classical hard-core-object lattice gas; the case of
c placing strongly antiferromagnetically interacting pairs
the frustrated bilayer was discussed in Refs. 12,13. In
[
of spins 1/2 (dimers) on a regular two-dimensional lat- the strong-fieldlow-temperatureregime the independent

1 tice and assuming weak antiferromagnetic interactions localized-magnon states are the lowest-energy ones and
v between dimers. Among such models one may mention
therefore they dominate the thermodynamics. The ther-
6 the J J′ model with the staggeredarrangement of the
7 strong−J′ bonds(definingdimersandfavoringsingletfor- modynamic properties in this regime can be efficiently
8 calculated using classical Monte Carlo simulations for a
mationondimers)onasquarelattice1 (seealsoRef.2for
7 lattice-gasproblem. Evenincaseofsmalldeviationsfrom
related dimerized square-lattice models). Other exam-
0 theidealflat-bandgeometryadescriptionwhichisbased
. plesarethebilayermodels: Theyconsistoftwoantiferro- on the strong-coupling approach14 can be elaborated.15
1
magnets in each layer with a dominant nearest-neighbor
0 Again the effective theory is much simpler than that for
interlayer coupling which defines dimers.3 By consider-
7 the initial problem.
ingadditionalfrustratinginterlayercouplingsthebilayer
1
: model can be pushed in the parameter space to a point From the theoretical side, frustrated bilayer systems
v
which admits a rather comprehensive analysis of the have been studied by several authors. Thus, the frus-
i
X energy spectrum.4 For this special set of coupling pa- trated square-lattice bilayer quantum Heisenberg anti-
r rameters, the frustrated bilayer is a system with local ferromagnet was studied in Refs. 12,13,16–20, whereas
a conservation laws (the square of the total spin of each the honeycomb-lattice bilayer with frustrationwas stud-
dimer is a good quantum number) that explains why iedinRefs.21–23(intralayerfrustration)andRefs.24,25
it is much easier to examine this specific case. On the (interlayer frustration). For the system to be examined
other hand, the frustrated bilayer belongs to the class of inourpaper, i.e.,the spin-1/2antiferromagneticHeisen-
so-called localized-magnon spin systems,5 which exhibit berg model on a bilayer honeycomb lattice including in-
someprominentfeaturesaroundthesaturationfield,such terlayer frustration, H. Zhang et al.25 have determined
as a ground-state magnetization jump at the saturation the quantum phase diagram at zero magnetic field for a
field,afinite residualentropyatthesaturationfield,and rather general case of an arbitrary relation between the
anunconventionallow-temperaturethermodynamics,for nearest-neighbor intralayer coupling and the frustrating
a review see Refs. 6–8. The singlet state of the dimer interlayer coupling. Another recent study reported in
is the localized-magnon state which belongs to a com- Ref. 24 concerns the antiferromagnetic classical Heisen-
pletely dispersionless (flat) one-magnon band. Over the berg model on a bilayer honeycomb lattice in a highly
last decade a large variety of flat-band systems with un- frustrated regime in the presence of a magnetic field. Its
2
mainresultisthephasediagramofthemodelintheplane
“magnetic field – temperature”. However, this analy-
sis cannot contain any hallmarks caused by the local-
ized magnons, since localized-magnon features represent
a pure quantum effect which disappears in the classical
limit. m a + 1 , m b +1,1
ma , m b +1,1
ma , m b ,2
From the experimental side, one may mention sev-
erallayeredmaterials,whichcanbe viewedasfrustrated
b y
bilayer quantum Heisenberg antiferromagnets. Thus, a
the compound Ba CoSi O Cl could be described as ma , m b ,1
2 2 6 2 x
a two-dimensionally antiferromagnetically coupled spin-
1/2XY-like spindimer systeminwhichCo2+ sites form
thefrustratedsquare-latticebilayer.26Theinterestinthe
ma , m b ,3
frustrated honeycomb-lattice bilayers stems from exper-
iments on Bi Mn O (NO ).27 In this compound, the
3 4 12 3
ionsMn4+ formafrustratedspin-3/2bilayerhoneycomb
FIG. 1: (Color online) Frustrated honeycomb-lattice bilayer.
lattice.28 Finally, let us mention that a bilayer honey-
It can be considered as a triangular lattice with four sites in
comb lattice can be realized using ultracold atoms.29 the unit cell. a and b are the basis vectors for the triangu-
larlatticeandtheintegernumbersma andmb determinethe
The presentstudy has severalgoals. Motivatedby the position of the unit cell. The vertical (red) bonds have the
recentpaperofH.Zhangetal.,25 wewishtoextenditto strength J2. The nearest-neighbor intralayer (black) bonds
the case of nonzero magnetic field. On the other hand, have the strength J1. The frustrating interlayer (brown)
bonds have the strength Jx. The main focus of our study
with our study we complement the analysisof the classi-
cal case24 to the pure quantum case of s=1/2. Finally, is the case Jx = J1, J2 > 3J1 (ideal frustration case). The
case Jx =J1, J1 Jx /J2 1 is considered in Sec. IVB.
the present study can be viewed as an extension of our 6 | − | ≪
previouscalculations12,13tothehoneycomb-latticegeom-
etry. Although we do not intend to provide a theoretical
description of Bi3Mn4O12(NO3), our results may be rel- II. INDEPENDENT LOCALIZED-MAGNON
evant for the discussion of the localized-magnon effects STATES
in this and in similar materials.
In the presentpaper,we considerthe spin-1/2Heisen-
The outline of the paper is as follows. Sec- berg antiferromagnetwith the Hamiltonian
tionII contains the spectroscopicstudy ofthe frustrated
honeycomb-lattice bilayer spin-1/2 Heisenberg antiferro- H = Jijsi·sj −hSz, Jij >0, Sz = szi (2.1)
magnet: Byexactdiagonalizationforfinitequantumsys- Xhiji Xi
tems and direct calculations for finite hard-core lattice-
definedonthehoneycomb-latticebilayershowninFig.1.
gas systems we show the correspondence between the
ThefirstsuminEq.(2.1)runsoverallbondsofthelattice
ground states in the large-Sz subspaces and the spatial
andhenceJ acquiresthreevalues: J (dimerbonds),J
configurations of hard hexagons on an auxiliary honey- ij 2 1
(nearest-neighbor intralayer bonds), and Jx (frustrating
comb lattice. Based on the established correspondence,
interlayerbonds),seeFig.1. Inwhatfollowsweconsider
in Section III we report results of classical Monte Carlo
simulations for hard hexagons on the honeycomb lattice the case Jx = J1 and call it the “ideal frustration case”
(or “ideal flat-band case”). Only in Sec. IVB we discuss
and use them to predict the properties of the frustrated
honeycomb-lattice bilayer spin-1/2 Heisenberg antiferro- Sdienvciaetitohneszfrocmomtphoeniednetaloffrtuhsetratotitoanl cspasine,Si.ze.,coJmxm6=uJte1s.
magnet in the strong-field low-temperature regime. The
withtheHamiltonianwecanconsiderthesubspaceswith
mostintriguingoutcome isanorder-disorderphasetran-
different values of Sz separately.
sition which is expected at low temperatures just below
In the strong-field regime the subspaces with large
the saturation field. This transition is related to the
Sz are relevant. The only state with Sz = N/2 is the
ordering of the localized magnons on the two-sublattice
fully polarized state ... ... with the energy E =
honeycomblatticeasthedensityofthelocalizedmagnons N(J /8+3J /4). In| the↑subsipace with Sz = N/F2M 1
increases. Section IV deals with some generalization of 2 1 −
(one-magnon subspace) N eigenstates of H (2.1) belong
theindependentlocalized-magnonpicture: Weshowhow
to four one-magnon bands, E + Λ(α), α = 1,2,3,4,
to take into account the contribution of a low-lying set FM k
with the dispersion relations:
of other localized states as well as discuss the effect of
deviations from the ideal frustration case. We end with
a summarizing discussionin Section V. Severaltechnical Λ(k1) =Λ(k2) =−J2−3J1, Λ(k3,4) =−3J1∓J1|γk|,
details are put to the appendixes. γ(k) = 3+2[cosk +cosk +cos(k +k )].(2.2)
a b a b
| |
p
3
FIG. 3: (Color online) Finite lattices used in our exact-
diagonalization studies. = 12, 16, 18, 24, i.e. N =
N
24, 32, 36, 48 (from left to right). Periodic boundary con-
ditions are implied.
FIG. 2: (Color online) Independent localized-magnon states 13. Each overlapping pair of hexagons (i.e., occupation
(corresponding to the shaded vertical dimers in Fig. 1) and of neighboring dimers by localized magnons) increases
hard-hexagon configurations on an auxiliary honeycomb lat- the energy by J . If J /J is sufficiently large, the over-
1 2 1
tice. lapping hexagon states are the lowest excited states in
the subspaces with Sz =N/2 2,...,N/4, but they are
the ground states in the subsp−aces with lower Sz. From
exact-diagonalizationdata for N =24, 32, 36, 48 we de-
Here k = (k ,k ), k = √3a k , k = 3a k /2
x y a 0 x b 0 y termined the required values of J /J as 3.687, 3.781,
− 2 1
√3a0kx/2, where a0 is the hexagon side length, and k 3.813, 3.874, respectively: For these values the first ex-
acquires /2valuesfromthefirstBrillouinzone,seeAp- cited state in the subspace with Sz = N/2 2 are the
pendix AN. The states from the two flat bands α = 1 overlapping hexagon states. −
N
and α = 2 can be chosen as a set of localized states
We check our statements on the character of the
where the spin flip is located on one of the vertical
groundstates andthe excited states by comparisonwith
N
dimers, see Fig. 1. The remaining states (i.e., from
exact-diagonalization data. Clearly, exact diagonaliza-
N
the two dispersive bands α=3 and α=4) are extended
tions are restricted to finite lattices, which are shown in
over the whole lattice. As can be seen from Eq. (2.2), Fig.3. Weusethespinpackpackage30andexploitthelo-
the two-fold degenerate dispersionless (flat) one-magnon
cal symmetries to perform numerical exact calculations
band becomes the lowest-energy one, if J > 3J , i.e., if
2 1 for large sizes of the Hamiltonian matrix. The ground-
the strength of the vertical bond J is sufficiently large.
2 state degeneracy coincides with the number of spatial
In what follows, we assume that this inequality holds.
configurations of hard hexagons on the honeycomb lat-
From the one-magnon spectra (2.2) we can also get the
tice for all considered cases, see Table I. In Table I we
value of the saturation field: h =J +3J .
sat 2 1 also report the energy gap ∆ to the first excited state
We pass to the many-magnon ground states. Be- and the degeneracy of the first excited state. While in
cause the localized one-magnon states have the lowest the one-magnon subspace we have ∆ = J 3J , see
2 1
energy in the one-magnon subspace, the ground states Eq.(2.2),theenergygapinthesubspacesSz =−N/2 n,
in the subspaces with Sz = N/2 n, n = 2,...,nmax, n = 2,..., /2 1 agrees with the conjecture that−for
− N −
nmax = N/2 = N/4 can be obtained by populating large enough J2/J1 &4 the first excited states are other
the dimers. However, for the ground-state manifold a localized-magnon states for which two of the localized
hard-core constraint is valid, i.e., the neighboring verti- magnons are neighbors (two hard hexagons overlap),see
caldimerscannotbepopulatedsimultaneously,sincethe above. Further evidence for this picture is provided by
occupation of nearest neighbors leads to an increase of the value ∆=2J for Sz =N/4: The first excited state
1
the energy. Thus, we arrive at the mapping onto a clas- with respect to the localized-magnon-crystalstate corre-
sical lattice-gas model of hard hexagons on an auxiliary sponds to three overlapping hard hexagons resulting in
honeycomb lattice: Each ground state of the quantum an increase of energy by 2J [see also Eq. (B4) in Ap-
1
spin model can be visualized as a spatial configuration pendix B].
ofthehardhexagonsonthehoneycomblatticeexcluding The zero-temperature magnetization curve is shown
the population of neighboring sites (hard-core rule), see by the thick solid red curve in Fig. 4. The magneti-
Fig. 2. zation curve probes the ground-state manifold and it is
The occupation of neighboring sites, excluded for the in a perfect agreement with the above described pic-
ground-state manifold at Sz = N/2 2,...,N/4, pro- ture. There are two characteristic fields, h = J and
2 2
−
vides another class of localized states which canbe visu- h = J +3J , at which the ground-state magnetiza-
sat 2 1
alizedasoverlappinghexagonsonthehoneycomblattice: tioncurvehasajump. Todemonstratetherobustnessof
ThesestateswerecompletelycharacterizedinRefs.4and the mainfeatures of the magnetizationcurve againstde-
4
TABLEI: =12, 16, 18, 24: exactdiagonalizations[J1 =1,
N
J2 =5(for =12, 24)andJ2 =10(for =16, 18)]versus
N N
countingofthenumberofhard-hexagonconfigurations. DGS
is the degeneracy of the ground state, ∆ is the energy gap,
D1ES is the degeneracy of the first excited state, # HHS is
thenumberof configurations of hard hexagons.
z
N S DGS ∆ D1ES # HHS
24 11 12 2 1 12
10 48 1 18 48
9 76 1 108 76
8 45 1 168 45
7 12 1 48 12
6 2 2 42 2
5 12 1 48 —
32 15 16 7 1 16
14 96 1 24 96 FIG. 4: (Color online) Exact-diagonalization results for the
13 272 1 240 272 zero-temperature magnetization curve of the honeycomb-
12 376 1 816 376 lattice bilayer spin-1/2 Heisenberg antiferromagnet. The
11 240 1 1104 240 thick solid red line is for theideal frustration case at J1 =1,
J2 = 5. Although the data refer to the lattice of N = 32
10 — 1 — 72
sites they do not show finite-size effects. The magnetiza-
9 — 1 — 16 tion curve has two jumps: at h = h2 = J2 = 5 and
8 — 2 — 2 h=hsat =J2+3J1 =8. Thethinsolidblackcurveshowsthe
36 17 18 7 1 18 zero-temperaturemagnetization curveforJ1 =1.1,Jx =0.9,
J2 = 5 for N = 32, i.e., slightly away from the ideal frustra-
16 126 1 27 126
tion case, see also Sec. IVB.
15 438 1 324 438
14 801 1 1404 801
13 — 1 — 756
regime.
12 — 1 — 348
11 — 1 — 90
10 — 1 — 18
III. HARD HEXAGONS ON THE HONEYCOMB
9 — 2 — 2 LATTICE
48 23 24 2 1 24
22 240 1 36 240 The lowest eigenstates in the subspaces with large Sz
21 1304 1 648 1304 become ground states for strong magnetic fields. Thus,
20 — — — 4212 the energy of these lowest eigenstates in the subspaces
with Sz =N/2 n, n=0,1,...,n in the presence of
19 — — — 8328 max
−
the field h is
18 — — — 10036
17 — — — 7176 N
Elm(h)=E h (ǫ h)n, ǫ =J +3J .(3.1)
16 — — — 2964 n FM− 2 − 1− 1 2 1
15 — — — 752
At the saturation field, i.e., at h = h = ǫ , all these
sat 1
14 — — — 156 energies become independent of n, Elm(h ) = E
n sat FM −
13 — — — 24 ǫ N/2. Therefore, the system exhibits a huge ground-
1
12 — — — 2 state degeneracy at h which grows exponentially with
sat
thesystemsizeN:7,31 = N/2g (n) exp(0.218N),
W n=0 N ≈
see Eq. (3.8) below. Here g (n) denotes the degeneracy
N
P
ofthegroundstateforthe2 -sitefrustratedhoneycomb-
viationsfromthe idealfrustrationcase,wealsoshowthe
lattice bilayer in the subspaNce with Sz =N/2 n.
curve when Jx slightly differs from J1. A more detailed −
Furthermore, following Refs. 32 and 7, the contribu-
discussion of this issue is then provided in Sec. IVB.
tion of the independent localized-magnon states to the
In the next section we use the established correspon-
partition function is given by the following formula:
dence between the spin model and the hard-hexagon
model to calculate the thermodynamic properties of the N
2 Elm(h)
frustrated honeycomb-lattice bilayer quantum Heisen- Z (T,h,N)= g (n)exp n . (3.2)
lm N
bergantiferromagnetinthestrong-fieldlow-temperature − T
n=0 (cid:20) (cid:21)
X
5
Since g (n)=Z (n, ) is the canonicalpartitionfunc-
N hc
N
tionofnhardhexagonsonthe -sitehoneycomblattice,
N
Eq. (3.2) can be rewritten as
E hN
Z (T,h,N)=exp FM− 2 Ξ (T,µ, ),
lm hc
− T ! N
N
2 µn
Ξ (T,µ, )= Z (n, )exp ,µ=ǫ h.(3.3)
hc hc 1
N N T −
nX=0 (cid:16) (cid:17)
As a result, we get the following relations:
F (T,h,N) E h 1Ω (T,µ, )
lm FM hc
= + N ,
N N − 2 2
N
Ω (T,µ, )= T lnΞ (T,µ, ) (3.4)
hc hc
N − N
for the free energy per site f(T,h),
M (T,h,N) 1 1 ∂ Ω (T,µ, )
lm hc
= + N (3.5)
N 2 2∂µ
N
for the magnetization per site m(T,h),
S (T,h,N) 1S (T,µ, )
lm hc
= N (3.6)
N 2
N
for the entropy per site s(T,h),
C (T,h,N) 1C (T,µ, )
lm hc
= N (3.7)
N 2
N
for the specific heat per site c(T,h). Note that h and µ
FIG.5: (Coloronline)Specificheatversusfieldatlowtemper-
arerelatedbyµ=h h. Thehard-hexagonquantities
sat− atures. Upper panel: Exact-diagonalization data for J1 =1,
in the r.h.s. of these equations depend on the tempera- J2 =5,N =24( =12). TheresultsforT =0.01, 0.02, 0.1
ture and the chemical potential only through the activ- are almost indistNinguishable. The results for T = 0.2 and
ity z = exp(µ/T). That means that for the frustrated T = 0.5 start to deviate from the universal dependence on
quantum spin system at hand all thermodynamic quan- (h hsat)/T. Lower panel: Exact-diagonalization data for
−
titiesdependontemperatureandmagneticfieldonlyvia = 12 (empty diamonds) and classical Monte Carlo data
x=(h h)/T =lnz,i.e.,auniversalbehavioremerges Nfor = 2 with = 288, 576, 1152. Empty circles cor-
sat N L L
− respond to the direct calculation for hard hexagons on the
in this regime.
honeycomb lattice with =64.
To check the formulas for thermodynamic quanti- N
ties given in Eqs. (3.4) – (3.7) we compare the exact-
diagonalization data with the predictions based on the
hard-hexagon picture. We set J = 1, J = 5 and
1 2 h=8.05arewelldescribedbythehard-coremodelagain
performexact-diagonalizationcalculationsforthermody-
up to about T =0.1.
namicsforthefrustratedquantumspinsystemofN =24
Usingthecorrespondencebetweenthefrustratedquan-
sites,30 see Fig. 3, where the total size of the Hamilto-
tumspinmodelandtheclassicalhard-core-objectlattice-
nian matrix is already 16777216 16777216. We also
× gas model, we can give a number of predictions for the
perform the simpler calculations for the corresponding
formermodelbasedontheanalysisofthelatterone. For
hard-hexagonsystems, see Appendix B.
example, we can calculate the ground-state entropy at
Ourresultsfortemperaturedependencesofthespecific
the saturation field:
heataroundthe saturationarecollectedinFigs.5and6.
As can be seen from these plots, the hard-hexagon pic- S(T 0,h=h ,N) lnΞ (z =1, )
sat hc
tureperfectlyreproducesthelow-temperaturefeaturesof → = N 0.218(.3.8)
2 2 ≈
thefrustratedquantumspinmodelaroundthesaturation N N
field. Deviationsfromthehard-core-modelpredictionsin This number follows by direct calculations for finite lat-
the upper panelofFig.5become visibleonlyatT =0.2. tices up to = 64 sites. On the other hand, for the
N
From the middle panel of Fig. 6 one can conclude that problem of hard hexagons on a honeycomb lattice κ =
thetemperatureprofilesforspecificheatath=7.95and exp[lnΞ (z = 1, )/ ] = 1.546... plays the same role
hc
N N
6
as the hard-squareentropy constant κ=1.503048082...
forhardsquaresonthesquarelatticeorthehard-hexagon
entropy constant κ = 1.395485972...for hard hexagons
onthe triangularlattice.33 Suchconstantsdetermine the
asymptotic growth and are also of interest to combina-
torialists. A more precise value of this constant for hard
hexagonsonahoneycomblatticecanbefoundinRef.34.
The most interesting consequence of the correspon-
dence between the frustrated quantum bilayer and the
hard-corelattice gasis the existenceofanorder-disorder
phase transition. It is generally known that for the
lattice-gas model on the honeycomb lattice with first
neighbor exclusion the hard hexagons spontaneously oc-
cupy one of two sublattices of the honeycomb lattice as
the activity z exceeds the critical value z = 7.92...,
c
see Ref. 34. In the spin language, this corresponds to
the ordering of the localized magnons as their density
increases. This occurs at low temperatures just below
the saturationfield. For the fixed (small) deviationfrom
the saturation field, h h, the formula for the critical
sat
−
temperature reads:
h h
sat
T = − 0.48(h h). (3.9)
c sat
lnz ≈ −
c
Furthermore, the critical behavior falls into the univer-
sality class of the two-dimensional-Ising-model.35 That
means, the specific heat at T (3.9) shows a logarithmic
c
singularity. Of course, the calculated T (3.9) must be
c
small, otherwise the elaborated effective low-energy the-
ory fails, see Figs. 5 and 6.
IV. BEYOND INDEPENDENT
LOCALIZED-MAGNON STATES
A. Other localized-magnon states
Following Ref. 13, in addition to the independent
localized-magnon states (which obey the hard-hexagon
rule) we may also take into account another class of
localized-magnonstateswhichcorrespondtooverlapping
hexagonstates(i.e., theyviolatethe hard-hexagonrule),
see also our discussion in Sec. II. The corresponding
lattice-gas Hamiltonian has the form:
N
FIG.6: (Coloronline)Temperaturedependenceofthespecific
( n )= µ n +V n n . (4.1)
heat C(T,h,N)/ for J1 = 1, J2 = 5. Upper panel: h = 0 H { m} − m m n
(red), h = 4.95 (Nblue), and h = 5.05 (black). Middle panel: mX=1 hXmni
h = 6.5 (magenta), h = 7.95 (green), and h= 8.05 (brown).
Lowerpanel: h=5.05. Exact-diagonalizationdata(symbols) Here nm = 0,1 is the occupation number attached to
were obtained for the lattice of N = 24 sites. Hard-hexagon each site m = 1,..., of the auxiliary honeycomb lat-
N
predictions(3.7),(3.3)areshown bythinsolidlines. Lattice- tice, the first (second) sum runs over all sites (nearest-
gas-model predictions (4.2) are shown by thick dashed lines. neighborbonds)ofthisauxiliarylattice,andµ=h h,
sat
In the lower panel the Monte Carlo data for the lattice-gas V =J . The interaction describes the energy increas−e if
1
model of 288 288 sites are shown by the thick dashed gray
× two neighboring sites are occupied by hexagons. In the
line.
limit V the hard-core rule is restored.
→∞
7
The partition function is given by
E hN
Z (T,h,N)=exp FM− 2 Ξ (T,µ, ),
LM lg
− T ! N
( n )
m
Ξ (T,µ, )= ... exp H { } .(4.2)
lg
N − T
n1X=0,1 nNX=0,1 (cid:20) (cid:21)
Since Z contains not only the contribution frominde-
LM
pendent localized-magnon states, but also from overlap-
ping localized-magnonstates,itis validinasignificantly
wider region of magnetic fields and temperatures.
Evidently, new Ising variables σ = 2n 1 may be
m m
−
introducedinEqs.(4.1)and(4.2)andasaresultweface
the antiferromagnetic honeycomb-lattice Ising model in
a uniform magnetic field:
FIG. 7: (Color online) Phase diagram of the frustrated
N honeycomb-lattice bilayer spin-1/2 Heisenberg antiferromag-
µ 3V
= + Γ σ + σ σ , netintheplane“magneticfield –temperature”. Thecoordi-
m m n
H N (cid:18)−2 8 (cid:19)− mX=1 J hXmni annadtesTco/fJt1he=h1ig/h[2eslnt(p2o+int√o3f)]the0d.o3m80e.aTrehhe/dJo1m=eJto2u/cJh1e+s3th/2e
µ 3V V ≈
Γ= , = >0.(4.3) horizontal axis at h2/J1 = J2/J1 and hsat/J1 = J2/J1 +3.
2 − 4 J 4 The black doted and the green dashed lines correspond to
the (approximate but very accurate) closed-form expressions
The Ising variable σ acquires two values 1, the
m ± suggestedinRefs.37,38(theycannotbedistinguishedinthis
nearest-neighbor interaction = J /4 > 0 is an-
1 graphic representation).
J
tiferromagnetic, and the effective magnetic field Γ =
(h h)/2 3J /4 = (J +3J /2 h)/2 is zero when
sat 1 2 1
− − −
h = J + 3J /2. The zero-field case (i.e., Γ = 0)
2 1
is exactly solvable, see Ref. 36 and references therein.
For example, the critical temperature is known to be
B. Deviation from the ideal flat-band geometry
T /J =1/[2ln(2+√3)] 0.380. Theground-stateanti-
c 1
≈
ferromagneticorder in the model (4.3) survivesat T =0
at small fields Γ <3 , i.e., for h <h<h , h =J ,
| | J 2 sat 2 2 Following Ref. 15, we can consider an effective low-
h = J + 3J . The antiferromagnetic honeycomb-
sat 2 1 energy description when the flat-band conditions are
lattice Ising model in a uniform magnetic field was a
slightlyviolatedandtheformerflatbandacquiresasmall
subject of several studies in the past.37–40 In particu-
dispersion. To this end, we assume that the intralayer
lar, several closed-form expressions for the critical line
nearest-neighbor interaction J and the interlayer frus-
1
in the plane “magnetic field – temperature” which are
trating interaction Jx are different, but the difference is
ingoodagreementwithnumericalresults wereobtained,
small J1 Jx /J2 1. Then in the strong-field low-
seeRefs.37,38andalsoRefs.39,40. Onthebasisofthese | − | ≪
temperature regimethere aretwo relevantstates ateach
studies we can construct the phase diagram, see Fig. 7. dimer: u = and d = ( )/√2. Their
Here we have used the two closed-form expressions for | i | ↑↑i | i | ↑↓i−| ↓↑i
energies, ǫ = J /4 h and ǫ = 3J /4, coincide at
u 2 d 2
the criticalline ofthe antiferromagneticIsingmodel ina h = h = J . Now th−e 2N-fold dege−nerate ground-state
0 2
magnetic field suggested in Refs. 37 and 38, where both
manifold is splitted by the perturbation, which consists
areindistinguishableinthescaleusedinFig.7. Although of the Zeeman term (h h ) sz and the interdimer
the two-dimensional Ising model in a field has not been − − 0 i i
interactions with the coupling constants J1 and Jx. The
solved analytically, the results of Refs. 37,38 are known P
effectiveHamiltonianactingintheground-statemanifold
to be very accurate.38–40 can be found perturbatively:41,42
In Fig. 6 the temperature profiles for the specific heat
in a wide range of magnetic fields are shown. The
comparison with the exact-diagonalization data demon-
H =PHP +..., (4.4)
stratesaclearimprovementofthehard-hexagondescrip- eff
tionafterusingthelattice-gasmodel(4.2). Furthermore,
in the lower panel of Fig. 6 we report classical Monte
Carlo data for h = 5.05 [lattice-gas model (4.2)] which where P = ϕ0 ϕ0 is the projector onto the ground-
| ih |
shows how the temperature profile C(T,h,N)/ modi- state manifold, ϕ = N v , where v is either the
N | 0i m=1| i | i
fiesanddevelopsasingularityasthesystemsizeincreases state u or the state d . After some straightforward
[see C(T,h,N)/ for = 2882 in the lower panel of calcula|tiions and introdQu|ciing the (pseudo)spin-1/2 oper-
Fig. 6]. N N ators Tz = (u u d d)/2, T+ = u d, T− = d u
| ih |−| ih | | ih | | ih |
8
at each vertical bond we arrive at the following result:
N
h J 3J
H = 2 + h Tz
eff N −2 − 4 8 − m
(cid:18) (cid:19) m=1
X
+ [JzTzTz+J(TxTx+TyTy)],
m n m n m n
hXmni
h=h J2 3J,J = J1+Jx,Jz =J,J=J1 Jx.(4.5)
− − 2 2 −
The secondsum in Eq.(4.5) runs overall 3 /2 nearest-
N
neighborbondsofthe auxiliaryhoneycomblattice. Note
thatthesignofthecouplingconstantJisnotimportant,
sincethe auxiliary-latticemodel(4.5)isbipartite. Again
the effective Hamiltonian (4.5) which corresponds to the
spin-1/2 XXZ Heisenberg model in a z-aligned field on
the honeycomb lattice is much simpler than the initial
modelanditcanbestudiedfurtherby,e.g.,thequantum FIG. 8: (Color online) Zero-temperature magnetization
Monte Carlo method.43 curves for thehoneycomb-lattice bilayer spin-1/2 Heisenberg
For the ideal flat-band geometry (ideal frustration antiferromagnet: Exact diagonalization (N = 32) and quan-
case) the effective Hamiltonian (4.5) transforms into the tum Monte Carlo simulations. The thick solid red line refers
abovediscussedlattice-gasorIsingmodels. Tomakethis to the ideal frustration case Jx = J1 = 1, J2 = 5. Thin
evident we have to take into account that J = J = V, lines[exact diagonalization forthefull modelofN =32sites
1
h = h h +3J/2 = µ+3V/2, Jz = J = V, J = 0, (solid) and for the corresponding effective XXZ model of
sat 1
− − = 16 sites (dashed)] and very thin lines (quantum Monte
and replace Tz by σ/2: N
− Carlo for = 2304) correspond to deviation from the ideal
N
h J2 3V frustration case, J1 = 1.2, Jx = 0.8 and J1 = 1.5, Jx = 0.5,
Heff =N −2 − 4 + 8 whereas J2 =5. Note that for J1 =1.2, Jx =0.8 the exact-
(cid:18) (cid:19) diagonalizationdataforthefullmodelandforthecorrespond-
µ 3V N V ing effective model almost coincide. In the inset we show
σm+ σmσn separately more quantum Monte Carlo data ( =2304) for
− 2 − 4 4 N
(cid:18) (cid:19)mX=1 hXmni J1−Jx =0.2 (black), J1−Jx =0.4 (brown), J1−Jx =0.6
µ 3V (green),J1−Jx =0.8(darkblue),andJ1−Jx =1(magenta).
=E h + +
FM
− N N −2 8
(cid:18) (cid:19)
N
µ 3V V
σ + σ σ , (4.6)
− 2 − 4 m 4 m n effectivemodel(4.5): Itisaspin-floptransition,whichis
(cid:18) (cid:19)mX=1 hXmni presentina two-dimensionalIsing-likeXXZ Heisenberg
cf. Eqs. (4.2) and (4.3). antiferromagnet in an external field along the easy axis,
Toillustratethequalityoftheeffectivedescription,we see, e.g., Refs. 44–46. Note that according to Eq. (4.5)
compare the results for the ground-state magnetization theeffectiveeasy-axisXXZ modelbecomesisotropicfor
curve obtained by exact diagonalization for the initial J1+Jx =2(J1 Jx), i.e., the spin-flop transition disap-
−
modelofN =32sites(thinsolidcurvesinFig.8)andfor pears as increasing the deviation from the ideal frustra-
the effective model of = 16 sites (thin dashed curves tion case J1 =Jx. Although, we are not aware of previ-
in Fig. 8). N ousstudiesofthespin-floptransitionforthehoneycomb-
It is worthnoting the symmetry presentin the Hamil- latticespin-1/2XXZ model(andsuchastudyisbeyond
tonian (4.5): If one replaces h = J + 3J + δh to the scope of the present paper), we may mention here
2
h = J δh and all Tz to Tz the Hamiltonian (4.5) that the square-lattice case was examined in Ref. 44. In
(up to2t−he constant) rmemain−s tmhe same. This symme- particular, one may find there the dependences of the
try of the effective model is also present in the exact- height of the magnetization jump and of the transition
diagonalization data for the initial model, if deviations field on the anisotropy. Furthermore, for temperature
from the flat-band geometry are small, see the thin solid effects, see Ref. 45. Supposing, that for the honeycomb-
blackcurveandthethinsolidbrowncurveinFigs.4and lattice case the same scenario as for the square-lattice
8. Moreover,itisalsoobviousinthelattice-gasHamilto- case is valid, we may expect that the spin-flop like tran-
nian(4.1): Afterthereplacementµ=δµtoµ=3J δµ sitioninourmodelonlydisappearsattheisotropicpoint
1
aconndstaallntn)mretmoa1in−s tnhme stahmeeH.amiltonian (4.1) (up to−the sJh1o+wnJxin=th2e(Jin1s−etJoxf)F.igO.u8rsquupapnotrutmthMisocnotnecCluasriolon.data
Ascanbe seeninFig.8,themagnetizationjumpssur- Let us complete this section with a general remark
vive evenfor moderate deviationsfromthe ideal frustra- on effective models around the ideal flat-band geome-
tion case. The nature of the jump is evident from the try (the ideal frustration case). Recalling the findings of
9
Ref. 15, where several localized-magnon systems includ- there is a line of phase transitions which occur below
ing the square-kagome model were examined, we con- T /J = 1/[2ln(2 + √3)] 0.380 for h in the region
c 1
≈
clude that the effective model aroundthe ideal flat-band between h = J and h = J +3J . Finally, for devi-
2 2 sat 2 1
geometry essentially depends on the universality class of ations from the ideal frustration case we observe for the
the localized-magnon system. (For a comprehensive dis- isotropicHeisenbergmodelathandmagnetizationjumps
cussion of the various universality classes of localized- which can be understood as spin-flop like transitions.
magnon systems, see Ref. 47.) While for the square- There might be some relevance of our study for the
kagomemodelfallingintothemonomeruniversalityclass magnetic compound Bi Mn O (NO ). The most in-
3 4 12 3
we obtained the (pseudo)spin-1/2 XXZ models with triguing question is: Can the phase diagram from Fig. 7
easy-plane anisotropy,15 for the considered frustrated be observed experimentally? First, the exchange cou-
honeycomb-lattice bilayer model, which belongs to a plings for Bi Mn O (NO ) are still under debate28 but
3 4 12 3
hard-hexagonuniversalityclass,wegetthe(pseudo)spin- the relation J /J 2 looks plausible. In this case the
2 1
1/2XXZ modelswitheasy-axisanisotropy. Clearly,the flat band is not th≈e lowest-energy one, see Eq. (2.2).
magnitudeoftheIsingtermsintheeffectiveHamiltonian Second, the spin value is s = 3/2 for this compound
are related to the specific hard-core rules. (eachMn4+ ioncarriesaspins=3/2)andthelocalized-
magnon effects are less pronounced in comparison with
the s = 1/2 case. For example, the magnitude ground-
V. CONCLUSIONS state magnetization jump at the saturation is still /2,
N
butthismagnitudeisonly1/6ofthesaturationvalue(in
Inthis paper weexamine the low-temperatureproper- contrast to 1/2 of the saturation value for the s = 1/2
ties of the frustrated honeycomb-lattice bilayer spin-1/2 case). Thus,furtherstudiesonthiscompoundareneeded
Heisenberg antiferromagnet in a magnetic field. For the to clarify the relation to the localized-magnon scenario
considered model, when the system has local conserva- presented in our paper.
tionlaws,itispossibletoconstructasubsetof2N eigen-
states ( = N/2) of the Hamiltonian and to calculate
N
their contribution to thermodynamics. For sufficiently
Acknowledgments
strong interlayer coupling, these states are low-energy
onesforstrongandintermediatefieldsandthereforethey
dominate the thermodynamic properties. The present study was supported by the Deutsche
Themostinterestingfeaturesofthe studiedfrustrated Forschungsgemeinschaft (project RI615/21-2). O. D.
quantum spin model are: The magnetization jumps acknowledges the kind hospitality of the University of
as well as wide plateaus, the residual ground-state en- Magdeburg in October-December of 2016. The work
tropy, the extra low-temperature peak in the tempera- of T. K. and O. D. was partially supported by Project
ture dependence of the specific heat around the satu- FF-30F(No. 0116U001539)from the Ministry of Educa-
ration, and the finite-temperature order-disorder phase tion and Science of Ukraine. O. D. would like to thank
transition of the two-dimensional Ising-model universal- the Abdus Salam International Centre for Theoretical
ity class. The phase transition occurs just below the Physics (Trieste, Italy) for partial support of these stud-
saturation field h . However, for large enough J /J , ies through the Senior Associate award.
sat 2 1
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