Effects of Two Energy Scales in Weakly Dimerized Antiferromagnetic Quantum Spin Chains

0.24 MB·English

by  A. Brühl

Effects of Two Energy Scales in Weakly Dimerized Antiferromagnetic Quantum Spin
Chains
A. Bru¨hl1, B. Wolf1, V. Pashchenko1, M. Anton1, C. Gross1, W. Assmus1, R.
Valenti2, S. Glocke3, A. Klu¨mper3,T. Saha-Dasgupta4, B. Rahaman4, and M. Lang1
1Physikalisches Institut, Universit¨at Frankfurt, D-60438 Frankfurt, Germany
2Institut fu¨r Theoretische Physik, Universit¨at Frankfurt, D-60438 Frankfurt(M), Germany
8 3Theoretische Physik, Universit¨at Wuppertal, 42097 Wuppertal, Germany and
0 4S.N. Bose National Centre for Basic Sciences, Salt Lake City, Kolkata 700098, India
0 (Dated: February 4, 2008)
2
By means of thermal expansion and specific heat measurements on the high-pressure phase of
n
(VO)2P2O7, the effects of two energy scales of the weakly dimerized antiferromagnetic S = 1/2
a
Heisenbergchainareexplored. Thelow energyscale, givenbythespin gap ∆,isfound tomanifest
J
itself in a pronounced thermal expansion anomaly. A quantitative analysis, employing T-DMRG
8
calculations, shows that this feature originates from changes in the magnetic entropy with respect
1 to ∆, ∂Sm/∂∆. This term, inaccessible by specific heat, is visible only in the weak-dimerization

limit whereit reflectspeculiarities of theexcitation spectrum and itssensitivity to variations in ∆.
]
l
e PACSnumbers: 75.50.Ee,75.40.Cx,75.30Et,75.80.+q,71.15.Mb,05.10.Cc
-
r
t
.s In recent years, many interesting and unexpected ef- al.[9], comprises one kind of S = 1/2 (V4+) chains with
t
a fects have been discovered in low-dimensional antiferro- alternating afm Heisenberg interactions [9, 10], as op-
m magnetic (afm) quantum-spin systems. Prime examples posed to the ambient-pressure (AP) variant, where two
- include the Haldane gap for spin S = 1 chains, mag- slightly different chain types were identified [10, 11]. No
d
netization plateaux in coupled-dimer systems, and the magnetic ordering has been found in HP-VOPO down
n
o different ground states for even- and odd-leg ladder sys- to 1.5K ∼0.01J1/kB, the lowesttemperature studied so
c tems, see, e.g.[1, 2, 3, 4]. These phenomena reflect the far, indicating that interchain interactions are weak.
[ intricate many-body quantum character of the systems
Toverifythatthe modelofanalternatingafmHeisen-
which renders advanced mathematical methods neces-
2 berg chain is appropriate for HP-VOPO, we carried
v sary to gain a quantitative understanding of the experi- out ab initio density functional (DFT) calculations in
1 mental observations. In this communication, we want to
the generalized gradient approximation using an LMTO
3 addressanothernon-trivialcollectivephenomenon, over-
basis. Our analysis of the DFT results in terms of
4
looked in the past, of the effects of two different energy
3 the Nth-ordermuffin-tinorbital-baseddownfolding tech-
scalescausedbyaweakalternationinthespin-spininter-
0 nique [12], shows that the dominant V-V effective hop-
7 actionofanS =1/2afmHeisenbergchain. Thesescales ping integrals are the intrachain hopping via the PO4
0 determine the physical properties of the interacting sys-
unit,t1 =0.114eV,followedbythenearestneighborV-V
t/ tematdifferenttemperatures. IfJ1andJ2(<J1)denote intrachainhoppingt2 =0.10eV.Thenextnon-negligible
a the alternating afm coupling constants between nearest
m V-V hopping is along the b-axis, tb = 0.02eV. Since all
neighboring spins along the chain, we define J1 as the other possible V-V interaction paths are more than one
- large scale and the spin gap ∆, induced by any amount
d orderofmagnitudesmallerthant1,2,thisanalysisclearly
n of alternation [5] α˜ < 1 with α˜ ≡ J2/J1, as the small confirms the alternating-chainnature of HP-VOPO.
o one. The dominant feature, governing the magnetic and
Single crystals of HP-VOPO were prepared by trans-
c thermodynamic properties at elevated temperatures, is
: formingpreviouslygrownAP-VOPOcrystalsundersuit-
v connected with J1. In the magnetic susceptibility χ, for ◦
ablepressureandtemperatureconditions(3GPa/800 C)
i example, the maximum due to short-range order is lo-
X cated at kBTχmax = 0.64J1, practically independent of [13]. Structurestudies revealphase purity withinthe ex-
r perimentalresolutionof2-3%andgiveparametersinac-
a α˜ [6]. However as will be shown below, the small en-
cordance with published data [14]. ESR measurements
ergyscaleintroducesaseconddistinctanomaly[7],which
yieldag-factorofg = 1.977forB⊥c. Themagnetic sus-
can be extraordinarily large in the coefficient of thermal
ceptibility, measured by employing a Quantum Design
expansion. The feature observed here is no longer pro-
SQUID magnetometer, is very similar to the results in
portional to that in the specific heat, viz., the so-called
ref.[10]. A good description of our χ(T) data for B⊥c
Gru¨neisen-scaling [8] does not apply.
at 2-350K is obtained by using eq.(13a) in ref.[10], the
For the study of the effects of two energy scales in the above g-factor, and the following parameters: J1/kB =
interesting weak-dimerization regime, the high-pressure 127K,J2/kB=111K,i.e.,α˜=0.873and∆/kB =31.8K,
variant of (VO)2P2O7 (HP-VOPO) appears particularly andaconcentrationof3.9%ofparamagneticS=1/2mo-
well suited. HP-VOPO, first synthesized by Azuma et mentspermoleV4+ withanafmWeisstemperatureΘ
W
2

20
a
15 0.3 experimental data
T-DMRG + Debye
10
2 K]
-6 [10/K]i 05 Pocly T[J/mol 0.2
C/
-5 0.1
-10
-15 0.0
0 50 100 150 200 0 5 10 15 20
T[K] T[K]
FIG.1: Expansioncoefficientsofsingle-crystallineHP-VOPO FIG. 2: Specific heat as C/T of single crystalline HP-VOPO
along the inter-chain a- (open circles), and intra-chain c-axis (⋄) of mass m = 0.74mg. Solid line corresponds to model
(open squares) and for polycrystalline material α (open calculations described in thetext.
poly
triangles). Solid lines are fits discussed in the text.
For the description of the magnetic contribution Cm
= (2 ± 0.6)K. These parameters, being in good agree- to the specific heat of HP-VOPO in Fig.2, the density
ment with those given in ref.[10], also provide a consis- matrix renormalization group approach for transfer ma-
tent description of the specific heat and thermal expan- trices (T-DMRG) has been used where the free energy
sion data, see below, and indicate that impurity effects of a one-dimensional quantum system can be calculated
are of no relevance in the T range discussed here. by use ofthe Trotter-Suzukimapping [17]. For allcalcu-
The thermal expansion was measured at tempera- lations we have retained between 64 and 100 states and
tures 1.5-200K employing a high-resolution capacitive usedtheTrotterparameterǫ=0.025sothattheerrorof
dilatometer [15]. For the specific heat measurements at theTrotter-SuzukimappingisoftheorderO(ǫ2)≈10−4,
2-30K, a home-built AC calorimeter [16] was used. whereas the truncation error is of the order 10−7. The
Figure 1 depicts the uniaxial expansivities α (T) = phonon background Cph was approximated by a Debye
i
l−1dl/dT measuredonanHP-VOPOsinglecrystalalong modelwithaDebyetemperatureΘD =426K,whichpro-
(i = c) and perpendicular (i = a) to the spin-chain vides a reasonable description of the lattice contribution
direction, together with data for polycrystalline mate- to the thermal expansion, see below. Instead of fitting
rial αpoly. In accordance with the crystal structure, the the Cm data with J1 and J2 as adjustable parameters,
α ’s are strongly anisotropic. Particularly striking is the requiring extensive computing time, T-DMRG calcula-
i
anomalousin-chainc-axisexpansivity,αc,yieldingahuge tions for J1/kB = 127K and J2/kB = 111K were used.
negativepeakanomalyaround13Kfollowedbyachange As the solid line in Fig.2 demonstrates, this ansatz de-
ofsignandamaximumat50K.Atelevatedtemperatures scribes the C(T) data reasonably well. Most notably, it
T >∼ 150K, αc becomes very small, indicating an unusu- proves the hump anomaly in C(T) at low temperatures
ally smalllattice contributionto α . In contrast,α and to be an intrinsic feature of the weakly-dimerized afm
c a
α showamuchlargerlatticeexpansivitywithasmall S = 1/2 Heisenberg chain. The departure of the solid
poly
minimum at 13K and no obvious anomaly at 50K. line from the data at higher temperatures is assigned to
As will be argued below, it is these two distinct fea- the inadequacy of the simple Debye model in accurately
turesat50Kand13K,dominatingtheintra-chainexpan- describing the phonons in this temperature range.
sivity α , in which the two energy scales become mani- Togainmoreinsightintothenatureofthedistinctlow-
c
fested in HP-VOPO. An indication for the low-energy temperatureanomaliesinHP-VOPO,asimplifiedmodel,
scale, though much less strongly pronounced, is also vis- denoted independent-magnon approximation (IMA) in
ible in the specific heat of HP-VOPO, shown in Fig.2. the following, will be introduced, see also ref.[6]. The
The data reveal a weak variation at low temperatures, significanceoftheIMAisverifiedbycomparisonwiththe
consistent with a spin gap, and a shoulder-like anomaly accurateT-DMRG results. The thermodynamic calcula-
around 13K. The latter feature can be discerned also in tions performed within the IMA are based on a singlet
the C(T) data on HP-VOPO by Azuma et al.[9], where groundstate separatedby an energy gap ∆ from a band
it has been assigned to a Schottky-type anomaly. of spin-1 excitations, cf. inset to Fig.3. To restrict the
3

T-DMRG
IMA
n]1.0
kT/J) [k/SpiB1B0.5 ~E(,k)/J111..05 ~~==00..8573 m /S*[a.u.]0 m /SJ[a.u.]01
m /( 0.5
C
0.0 *= /J1 T-DMRG
0.0 -0.4 -0.2 0.0 0.2 0.4 IMA, E(~=0.873,k)/J1 from [6]
ka/ IMA, E(~=0.873,k)/J1=1-(1- *)cos(2ka)
0.0 0.1 0.2 0.3 0.4 0.5 0.6
kBT/J1 0 50 100
T[K]
FfoIrGt.h3e: aMfmagnSet=ics1p/e2ciafiltcehrneaattidnigv-iedxecdhbayngkeBHT/eiJs1envbs.erkgBcTh/aJin1 FIG.4: Temperaturedependenceof(i)∂S0m/∂∆∗ (leftscale)
with J1/kB = 127K and J2/kB = 111K calculated by T- and(ii)∂S0m/∂J1(rightscale)forthealternatingafmS=1/2
DMRG (solid line) and IMA (broken line). Inset shows one- HeisenbergchainwithJ1/kB =127KandJ2/kB =111K,i.e.
magnon dispersion relations for the alternation parameter α˜ α˜=0.873and∆/kB =31.8K.Whilefor(i)T-DMRGresults
= 0.873 and 0.5, calculated by using eq.(23a) in ref.[6]. are compared with IMA results for different dispersion rela-
tions given in thefigure, (ii) is calculated within T-DMRG.
maximum number of excited states to 2N, with N the
numberofS =1/2sites,theoccupancyofaone-magnon
a sum of separate contributions S = Sj, the vol-
state with energy E(k) and wave vector k is allowed to j
ume expansion coefficient can be writPten as [8] β =
be either 0 or 1. In zero field, where the one-magnon βj = κ (∂Sj/∂V) , with κ =−V−1(∂V/∂P)
band is threefolddegenerate,we obtainfor the magnetic j T j T T T
tPhe isothermPal compressibility. This holds true also for
contribution to the free energy per spin:
anisotropic systems as long as the volume changes occur
underhydrostaticconditions[8]. ForHP-VOPO,thesum
k Td π/2d E(α˜,k) jrunsoverthemagnetic(j =m)andphonon(ph)contri-
Fm =− B ln 1+3exp − dk. butions. TheuseoftheDebyemodelandthescaleinvari-
2π Z−π/2d (cid:18) (cid:18) kBT (cid:19)(cid:19) ance ofSm imply thatS =Sph(T/ΘD)+Sm(T/J1,∆∗),
(1)
Here d is the average spin-spin distance along the where ∆∗ = ∆/J1. With C0ph = −ΘD∂S0ph/∂ΘD and
chains. The one-magnon dispersion E(α˜,k) was calcu- C0m = −J1∂S0m/∂J1, the phonon and magnetic molar
specific heat contributions, respectively, one finds
latedtohighaccuracybyBarneset al.[19]. Afairlysim-
ple parametrization, given by eq.(23a) in ref.[6], is used
herefornumericalcomputationsofFm,themagneticen- β = κT(−∆∗Γ∆∗∂S0m +ΓJ1Cm+ΓphCph). (2)
tropySm =−(∂Fm/∂T)V andCm =−T(∂2Fm/∂T2)V. V0 ∂∆∗ 0 0
Figure 3 compares the so-derived Cm with results from
T-DMRG calculations. A representation C/T is chosen Here Γ∆∗ = −∂ln∆∗/∂lnV, ΓJ1 = −∂lnJ1/∂lnV and
to make the low-temperature anomaly clearer. As ex- Γph =−∂lnΘD/∂lnV definedifferentGru¨neisenparame-
pected, the IMA nicely traces the T-DMRG results at terswhichmeasurethevolumedependenceoftherespec-
low temperatures, where the magnon density is small. tive characteristic temperature or energy. Note that as
In particular, it shows the peak anomaly in Cm/T at a consequence of Sm being a function of the two scaling
∗
tkeBmTp/eJr1at∼ure0s.10,.a06si≤gnaktBuTre/Jo1f ≤the0.l5o,wh-eonweervgeyr,sctahlee.IMFoAr v∝ar∂iaSb0mle/s∂T∆/∗J,1wahnicdh∆ha,sβno(ecqo.u2)ntceornptaaritnsinatchoenmtraibgunteitoinc
tends to overestimate Cm by about 10% at maximum. specific heat, i.e., βm is no longer proportional to Cm,
The accompanied extra entropy is balanced at higher viz. Gru¨neisen-scaling does not apply.
mteomdpeelsrarteuvreeasl,asbootuhta9t7at%koBfTth=e f2uJll1s(pninotenshtroowpny)., both paEnsqiuvaittiieosnb2y cαain =be γgi∆en∗∂erSa0mliz/e∂d∆t∗o+thγeiJ1u∂nSi0max/i∂aJl1ex+-
With these models at hand, a detailed description of γiph∂S0ph/∂ΘD, i = a,b,c. The prefactors γi∆∗, γiJ1 and
the anomalous thermal expansion of HP-VOPO is pos- γiph contain the respective strain dependencies of ∆∗, J1
sible. If the entropy of a system can be expressed as and Θ , some combinations of elastic compliances and
D
4
themolarvolumeV0,see[8]. Sincethesequantities,along estimate for the bulk modulus of cB = κ−T1 = (47 ±
withtheGru¨neisenparametersΓj ineq.(2),generallyre- 6)GPafromΘ =426K[22], thefityieldsthefollowing
D
veal only a very weak variation with temperature, they Gru¨neisen parameters Γ∆∗ = -45 ± 6, ΓJ1 = 1.3 ± 0.2
can be treated as constants [20]. In leading order, the T and Γph = 0.3 ± 0.1. Particularly striking is the large
dependence of α and β is thus given by the derivatives Γ∆ = Γ∆∗ + ΓJ1 ≃ -43 ± 6, reflecting a huge pressure
i
oftheentropywithrespecttothedifferentenergyscales. dependenceofthespingap∂∆/∂P =∆·κ ·Γ∆=-(28±
T
The two magnetic terms of interest here, computed 8)K/GPa. A linear extrapolation to finite pressure im-
numerically within the IMA (eq.(1)) and T-DMRG for plies that hydrostatic pressure of about (1.1 ± 0.3)GPa
J1/kB = 127K and J2/kB = 111K, are displayed in is sufficient to close the spin gap in HP-VOPO, i.e. to
Fig.4 as a function of temperature. For ∂S0m/∂J1 (= drive the system to the quantum critical uniform limit.
−C0m/J1), a broad anomaly is revealed around 50K, re- In summary, the high-pressure variant of (VO)2P2O7
flecting the short-range afm correlations associated with has been used to study the effects of two distinct energy
J1. Inaddition,∂S0m/∂J1 disclosesatinyfeaturearound scales of the afm S = 1/2 weakly-dimerized Heisenberg
13K,correspondingtothelow-temperaturemaximumin chain. We show that ∂S0m/∂J1 accounts for the signa-
Cm/T =−J1/T·∂S0m/∂J1inFig.3,asaresultofthegap turesinthespecificheatassociatedwithshort-rangeafm
formation. Since this anomaly is much smaller and has correlationsand spin-gap formation. However,this term
thesamesignastheoneat50K,itcannotaccountforthe cannot explain the large low-temperature anomaly ob-
changeinsignandthelargenegativepeakanomalyinαc served in the intra-chain expansivity. The latter feature
at 13K. However, as shown in Fig.4, such an anomaly iswelldescribedby∂Sm/∂∆∗,whichprobestheentropy
is revealed by ∂S0m/∂∆∗. This feature, constituting the contours with respect 0to changes in the spin gap. This
central result of this paper, reflects the sensitivity of the term, inaccessible by specific heat, is observable only in
magnetic entropy to changes in the spin gap ∆, probed theweak-dimerizationlimitandreflectsthehighsensitiv-
in a most sensitive differential way by the expansivity. ity of the excitation spectrum, as regards small changes
Theeffectcanbeobservedonlyintheweak-dimerization in ∆. Our results, which are based on the scale invari-
limit,wheretheenergyscalesareclearlyseparated. Here ance of the entropy Sm(J1,J2,T) = Sm(T/J1,J2/J1),
the minimum in the one-magnondispersion is narrow so can be generalized to cover the large family of magnetic
that the magnetic excitation spectrum is very suscepti- multi-scalesystemsdescribedby Heisenberg-likemodels.
ble tosmallchangesin∆, cf.insettoFig.3. To simulate
The authors acknowledge fruitful discussions with B.
theeffectofsuchchangesontheexpansivity,comparative
Lu¨thi. The T-DMRG calculations have been supported
calculationsof∂Sm/∂∆∗havebeenperformedwithinthe
0 bytheDFGundercontractsNo.KL645/4-2andGK1052.
IMA using different dispersion relations E(α˜,k) delimit-
ing the spectrum to low energies, cf. Fig.4. While for
E(α˜ =0.873,k)shownin the inset to Fig.3, the calcula-
tionswithinthe IMA(brokenline inFig.4)nicelyrepro-
ducetheT-DMRGresults(solidline)uptotheminimum [1] F.D.M. Haldane, Phys. Rev.Lett. 50, 1153 (1983).
position,agappedcos(2ka)dispersionwiththesamegap [2] I. Affleck,J. Condens. Matter 1, 3047 (1989).
value givesriseto ananomalyin∂Sm/∂∆∗ (dottedline) [3] E. Dagotto and T.M. Rice, Science 271, 618 (1996).
0
whichmarkedlydeviatesfromtheT-DMRGdata. These [4] M. Oshikawa et al.,Phys. Rev.Lett. 78, 1984 (1997).
[5] J.C. Bonner et al.,J. Appl. Phys.50,1810 (1979).
results highlight the extraordinarily high sensitivity of
[6] D.C. Johnston et al.,Phys. Rev.B 61, 9558 (2000).
the expansivity to peculiarities of the magnetic excita-
[7] An anomaly can also bediscerned in χ(T)calculated by
tion spectrum associated with the low energy scale. various techniques, provided α˜>∼ 0.8 [6].
Employing the prefactors γiJ1, γi∆∗ and γiph and ΘD [8] T.H. Barron et al., Adv.in Phys. 29, 609 (1980).
as adjustable parameters,the expressionfor the uniaxial [9] A. Azumaet al.,Phys.Rev.B 60, 101454 (1999).
expansivities can be used to fit the α and α data in [10] D.C. Johnston et al.,Phys. Rev.B 64, 134403 (2001).
a c [11] A.W. Garrett et al.,Phys.Rev. Lett.79, 745 (1997).
Fig.1 (solid lines). The good quality of the fit to α for
T >∼70K,wherethephononcontributiondominates,aval- [12] O.K.AndersenandT.Saha-Dasgupta,Phys.Rev.B62,
R16219 (2000), and references therein.
idates the use of the Debye model with ΘD = 426K for [13] C. Gross et al.,High Pressure Research 22,581 (2002).
satisfactorily describing the phonons in HP-VOPO. The [14] T.Saitoetal.,J.SolidStateChemistry153,124(2000).
lattice contribution, however, plays a minor role for αc. [15] R. Pott et al.,J. Phys.E 16, 444 (1983).
Here an excellent fit is obtained essentially by the su- [16] J. Mu¨ller et al.,Phys.Rev. B65, R140509 (2002).
perposition of the two magnetic contributions depicted [17] J. Sirker,A. Klu¨mper, Europhys.Lett. 60, 262 (2002).
in Fig.4, properly scaled by γJ1 and γ∆∗. After having [18] T. Barnes, Phys.Rev.B 67, 024412 (2003).
c c [19] T. Barnes et al.,Phys.Rev. B 59, 11384 (1999).
assured the validity of our ansatz through these fits to
[20] Thisisnolongervalidclosetothequantumcriticalpoint
αc and αa, a modeling of the volume expansivity β = (∆→0), where theGru¨neisen parameter diverges [21].
αa + αb + αc is possible by means of eq.(2), cf. solid [21] L. Zhu et al., Phys.Rev.Lett. 91, 066404 (2003).
line through αpoly = β/3 in Fig.1. Employing a rough [22] This is possible within an isotropic approximation, see
5
G.A.AlersinPhysicalAcoustics,AcademicPress(1965).