Disorder effects in the quantum Heisenberg model: An Extended Dynamical mean-field theory analysis

0.32 MB·English

by  S. Burdin

Disorder effects in the quantum Heisenberg model: An Extended Dynamical
mean-field theory analysis
S. Burdin1, D. R. Grempel2 , and M. Grilli3
1 Max-Planck-Institut fu¨r Physik komplexer Systeme. N¨ot∗hnitzer Straβe 38, 01187 Dresden, Germany
2 CEA-Saclay/DSM/DRECAM/SPCSI, 91191 Gif-sur-Yvette Cedex, France and
3INFM-CNR SMC Center, and Dipartimento di Fisica
7
Universit`a di Roma ”La Sapienza” piazzale Aldo Moro 5, I-00185 Roma, Italy
0
(Dated: February 3, 2008)
0
2
WeinvestigateaquantumHeisenbergmodelwithbothantiferromagneticanddisorderednearest-

n neighbor couplings. We use an extended dynamical mean-field approach, which reduces the lattice
a problem toaself-consistent local impurity problem thatwe solvebyusing aquantumMonteCarlo
J algorithm. We consider both two- and three-dimensional antiferromagnetic spin fluctuations and

0 systematicallyanalyzetheeffectofdisorder. Wefindthatinthreedimensionsforanysmallamount
1 ofdisorderaspin-glassphaseisrealized. Intwodimensions,whilecleansystemsdisplaytheproper-
tiesofahighlycorrelatedspin-liquid(wherethelocalspinsusceptibilityhasanon-integerpower-low

] frequencyand/ortemperaturedependence),inthepresentcasethisbehaviorismoreelusiveunless
n
disorderisverysmall. Thisisbecausethespin-glasstransitiontemperatureleavesonlyaninterme-
n
diatetemperatureregimewherethesystemcandisplaythespin-liquidbehavior,whichturnsoutto
-
s bemore apparent in the static than in thedynamical susceptibility.
i
d
PACSnumbers: 75.10.Jm,75.10.Nr,75.50.Ee,75.40.Gb
.
t
a
m
I. INTRODUCTION tween disorder and fluctuations8,9,10, analyzing the role
- ofquantumspinfluctuationsindrivingaspin-glass(SG)
d
phase into a paramagnetic (or spin-liquid) phase upon
n
o Quantum magnetism is an important topic of mod- varying spin size and/or temperature. Within the same
c ern solid state physics: Not only do magnetic phases frameworkweconsiderinthispapera“simple”quantum
[ take place in the phase diagram of many correlated AF Heisenberg model for S = 1/2 spins and we intro-

electronsystems, like high-temperature superconducting ducearandommagneticcoupling. BysolvingthisDisor-
1
v cuprates1 and heavy fermions2, but many different ma- deredQuantumAntiferromagneticHeisenberg(DQAFH)
1 terials display magnetic properties, which change upon model within the Extended Dynamical Mean-Field The-
1 varying parameters such as doping, temperature, disor- ory (EDMFT)11,12,13, we aim to clarify the interplay be-
2 der, or pressure. A few examples may (very partially) tween AF and random magnetic couplings as well as the
1 illustrate the enormous variety of behaviors that one role of dimensionality in the instability from a paramag-
0 encounters in this field: Upon increasing doping the netictoaSGphaseandtherelevanceofquantumfluctu-
7
/0 lmigahgtnlyetidcop(AedF)Lato2−axSsrpxiCnu-gOla4sspa(SssGes) fprhomasea1;ntahnetiifnertrroo-- aotuitonthseinpsruespepnrteswsoinrgktwheewcriiltliacalwlateyms rpeemraatiunrein. Tthheropuagrha--
at ductionofnon-magneticimpuritiesinquantumparamag- magnetic phase with neither AF- nor replica-symmetry
m nets like SrCu2O3, KCuCl3, CuGeO3, gives rise to local- breaking and we will determine the conditions for these
moment formation3; the glassy phase of the disordered (second-order) instabilities to take place.
-
d dipolar-coupled magnet LiHoxY1 xF4 may be driven to
n other magnetic phases by means−of a magnetic field4; The EDMFT technique11,12,13 is an extension of the
o SrCr Ga O is a S = 3/2 magnet with a Kagom´e standardDynamicalMean-FieldTheory14, wherethe lo-
9p 12 9p 19
c latticestruc−ture,whichabovethe(low)SGfreezingtem- cal (spin or charge) degrees of freedom are coupled to a
:
v perature displays the spin-liquid behavior5,6 typical of self-consistent bosonic bath. In our case the Heisenberg
i strongly correlated spin systems7. From this large va- model is mapped onto a single quantum spin embedded
X
riety we extract three essential ingredients, which are in a bosonic bath which represents the surrounding spin
r
a commonly and physically relevant in quantum magnetic excitationsasafluctuatingmagneticfield. Thisapproach
materials: disorder, frustration, and fluctuations (both hassomedefinite limitations: Itisnotsuitedto keepfull
thermalandquantum). Theseparateroleofthesephysi- account of the spin character of the surrounding degrees
calmechanismsandtheirinterplayindeterminingdiffer- of freedom (for instance it misses the possibility of sin-
entphysicalpropertiesisobviouslyarelevantissueinthe gletformationbetweenthesinglespinandthe surround-
field of quantum magnetism. Since one is facing a huge ing ones) and it neglects the momentum dependence of
variety of systems with different structures, properties the spinself-energythereby preventingthe occurrenceof
and phase diagrams,the search of common fundamental spatial anomalous dimensions. This latter limitation is
mechanisms mostly relies on the analysis of schematic more severe at low temperatures, where spatial correla-
models to highlight the deep physical effects. For in- tions between the spins become relevant. As also dis-
stance some work has been devoted to the interplay be- cussed in Section VI the consequence of this is the ap-
2
pearanceofwell-known13,15 low-temperatureinstabilities 1. Disorder and replicas
occurring as an unphysical sign in the spin self-energy.
Also related to this poor treatment of spatial correla-
The free energy is averaged with respect to the prob-
tions (and the related lack of anomalous dimensions) is ability P(JD). We use the well-known replica trick17,
ij
the appearance of spurious first-order transitions, where
which relies on the fact that the free energy can be
second-order are instead expected16. Nevertheless, it is F
calculated from the partition function using the rela-
commonly recognized16 that the EDMFT approach is a Z
tion
valuabletooltoinvestigatethe dynamicsofspinsystems
anditmakesanimportantfirststeptowardtheinclusion 1
β = log = lim log n . (2)
of spatial correlations between the spins. F h Zidis n 0+ n hZ idis
OuranalysisoftheDQAFHnotonlyhasabroadvalid- →
ity and addressesgeneraltheoreticalissues,but it is also In order to calculate n we introduce n replicas of
dis
hZ i
of pertinence for those numerous AF materials, where the system, each one being represented by an index α
disorder can induce competition between antiferromag-
netism (possibly having a correlated anomalous charac- + n
n = ∞dJDP(JD)Tr exp β H ,
ter) and a spin glass (SG) phase, or can induce local hZ idis ij ij " − α#
moment formation with low-temperature Curie-like de- Yi<jZ−∞ αX=1
(3)
pendencies. The role of low dimensionality in determin-
where H is the disordered Heisenberg Hamiltonian Eq.
ing the relevance of magnetic fluctuations is also investi- α
(1)actingonthereplicaα. InEq.(3),thetraceisapplied
gated.
toalltheoperatorsSαofthereplicatedHilbertspace(see
The paper is organized as follows. In Section II we i
Appendix A).
present the model and the EDMFT technique. Section
IIIpresentsthecriterianecessarytodetecttheestablish-
ment of broken-symmetry phases or the spin-liquid be-
havior. The numericalresults forstatic propertiesin the 2. Extended Dynamical Mean Field
three- and two-dimensional case are reported in Section
IV, while the dynamical behavior is discussed in Sect. In the standardDMFT technique14, the electronic de-
V. In Section VI we discuss our results and present our grees of freedom of a specific single site are coupled to
conclusions. a fermionic effective bath. On the other hand, in the
EDMFT spin(or charge)degreesof freedomarecoupled
to an external bosonic bath simulating the spin/charge
II. THE MODEL AND THE TECHNIQUE degrees of freedom of the rest of the lattice11,12,13. The
condition that the single specific site must be equivalent
A. The model to all other sites of the lattice is then implemented by
means ofselfconsistency equations. We arenow goingto
We consider the Disordered Quantum Antiferromag- suitably adaptthis procedure to the disorderedspin sys-
neticHeisenberg(DQAFH)model,givenbythefollowing tem represented in Eq. (1). The first step is standard:
Hamiltonian: We single out a specific site of the system (arbitrarily
chosen as the origin); the disorder (JD couplings) will
H = J˜ijSiSj . (1) be formally averaged and the trace wiijll be taken with
i<j respect to the operators Sα with i=0. For any value of
X i 6
z,oneobtainswithinthisprocedureaneffectivelocalac-
Here S is a spin 1/2 operator at the i th site of a lat-
i − − tionforthe localspins Sα. The localactionis calculated
tice of coordination z. The nearest-neighbor magnetic 0
couplings J˜ J˜AF +J˜D are the sum of a translation- withacumulantexpansion18: Inthelarge-zlimit,allthe
ij ≡ ij ij cumulants vanish, except the first two terms. One de-
atrllaynisnfovramrisanJt(pqa)r]taJ˜niAjdFa≡dJisiAojFrd/e√rezd≡pJar/t√Jz˜D[withJFDo/u√riezr. pfieelndd.sTlihneeasrelcyonondiSsαiquanaddrcaotricreasnpdonrdeflsetcotsthtehestdaytnicammeicaanl
ij ≡ ij
The translationally invariant couplings are related to fluctuations. Here, we study only the paramagnetic and
the spin-wave spectral density ρ(ǫ) δ(ǫ J(q)). replica-symmetricsolutionwith Sα =0: Thetransition
≡ q − h ii
The disordered parts JD are random couplings with a to an AF phase or to a SG phase will only be analyzed
ij P
GaussianprobabilitydistributionP(JD),withzeromean fromtheparamagneticphase,consideringthecriteriafor
ij
a second-order transition (see Sect. III).
value JD = 0 and JDJD = δ δ J2 [
h ijidis h ij klidis i,l j,k D h···idis Thelocaleffectiveactionisthuscastintothefollowing
represents the averaging over the disorder distribution].
form:
Inthefollowing,wewillperformalarge-zexpansion,in
ordertostudythespin-wavecorrelationswithinasingle-
site effective model. This derivation has been described = 1 βdτ βdτ Sα(τ)Sα′(τ ) αα′(τ τ ) ,
in Ref. 11 for a clean Heisenberg model. Here, the pres- A −2Z0 Z0 ′αα′ 0 0 ′ K − ′
X
enceofdisorderrequiresaspecificpreliminarytreatment. (4)
3
with where the explicit Matsubara frequency dependency
has been dropped for clarity. Using standard
Kαα′(τ −τ′)= 31z hJ0iJj0hSαi(τ)Sαj′(τ′)icavityidis . considerations11,14,19 of power counting estimates of the
ij correlationfunction dependence on 1/z one can cast the
X
(5) part proportional to J2δ in a fully local form
D ij
InthisexpressionfortheKernel,asinthefollowingones,
1 1
thesummationwithrespecttoiandj concernsthenear- = JAFJAF χ χ χ /χ + J2χ
K z 0i j0 ij − i0 0j 00 z D ii
estneighborsofthesite0,whileα,α arereplicaindexes.
′ ij i
Thecavitycorrelationfunctions cavity correspondto X (cid:0) (cid:1) X
+ . (13)
h···i AF D
thecorrelationfunctionscalculatedwithacavityinstead ≡ K K
ofthesite0(seeAppendicesAandB).Thecouplingfac- Althoughthe disorder obviouslyenters the averagedχ’s,
tors J J in Eq. (5) precisely invoke the site 0 (of the itisimportanttonoticethatinEq. (13)thekernelsplits
0i j0
cavity). in a part only involving ordered AF couplings and the
UsingtheargumentsofAppendixB,theaveragingover generic χij −χi0χ0j/χ00 combination, and a term pro-
the disorder can be factorized portionaltoJ2 andtothepurelylocalsusceptibilityχ .
D ii
ThisdecompositionofthekernelintoanAFterminvolv-
J J Sα(τ)Sα′(τ ) cavity =
h 0i j0h i j ′ i idis ing non-local correlations and a D term with only local
hJ0iJj0idishhSαi(τ)Sαj′(τ′)icavityidis . (6) cTohrerreelafotiroen,sasisfaornaesodfistohredenroitsicceoanbcleernreesdu,latsnyoflatthtiiscewtoyrpke.
Since we are concerned with (the instabilities of) the becomes equivalent to a Bethe lattice: In the latter case
paramagneticphasewe onlyconsideraverages(with and the topology of the lattice allows for retraceable paths
without cavities) leading to diagonal quantities both in only, while in the former the statistical properties of the
the spin components and in the replica indexes. All the JD’sthemselvesselectthesametypeofpaths. Therefore
followingargumentscouldbecarriedoutkeepingthema- it is not surprising that, as in Bethe lattices, in the part
trix character of the spin correlation functions, but we of the kernel proportional to the disordered interaction
adopt this simplified presentation to put more clearly in S (τ)S (τ ) cavity = S (τ)S (τ ) for any lattice type.
i i ′ i i ′
evidence the role ofdisorder without unnecessaryformal h i h i
complications. In this way all thermally averaged quan-
tities are scalar quantities and accordingly 3. Spin self-energy and self-consistent procedure
(τ τ′) αα(τ τ′) , (7)
K − ≡K − Inthelarge-zlimitandintheparamagnetic(spin-and
and one can introduce the (diagonal, i.e., scalar) spin replica-symmetric) phase the local effective action (4) is
susceptibility cast in the simpler form
χ (τ τ )= Sα(τ)Sα(τ ) /3= Sαµ(τ)Sαµ(τ ) . (8) 1 β β
ij − ′ h i j ′ i h i j ′ i = dτ dτ′S(τ)S(τ′) (τ τ′) . (14)
A −2 K −
[µisaspin-componentindex]. Wealsodefinethedisorder Z0 Z0
The standard EDMFT procedure is to solve this impu-
averagedsusceptibilities
rity problem with a kernel self-consistently depend-
K
χ χ , (9) ing on the spin S Sα via the local susceptibility
ij ≡h ijidis ≡ 0
χ χ and an (approximate) evaluation of χ enter-
with a similar definition for the average cavity suscepti- loc ≡ ii ij
ing Eq. (13). Here we emphasize again that, while the
bilities χcavity. Now, invoking Eq. (6) in Eq. (5), fixing
ij functionaldependence of AF(ωn)fromthe spinsuscep-
the unnecessary replica index and dropping it, we find K
tibilities (cf. Eq. (13)) depends on the dimensionality of
for the paramagnetic phase
the AF spin fluctuations, the disorder part alwaysyields
(τ)= 1 J J χcavity(τ) . (10) KD(ωn)=JD2χloc(ωn).
K z h 0i j0idis ij To solve the impurity problem of Eq. (14) is usu-
Xij ally the difficult step of the EDMFT procedure. In the
In the large z limit, the averaged cavity susceptibility presentpaperweadoptaQuantumMonteCarlo(QMC)
canbeexpre−ssedintermsofthefullphysical(i.e. without technique(seesubsectionII.B)toobtainthelocalsuscep-
cavity) averagedsusceptibility (see Appendix B) tibility χloc(ωn). The next step is then to suitably mod-
ify the kernel of Eq. (14) to iterate the self-consistency
χcavity(ω )=χ (ω ) χ (ω )χ (ω )/χ (ω ) . (11) procedure. To this purpose we customarily introduce a
ij n ij n − i0 n 0j n 00 n spin-fluctuationself-energyΠ−01(ωn),which,owingtothe
where ω is a Matsubara frequency. The Kernel (10) is
n large coordination of the original lattice, is taken to be
then given by the self-consistent relation
momentum independent. Then in momentum space the
K= z1 J0AiFJjA0F +JD2δij χij −χi0χ0j/χ00 , npoennd-lioxcaBl)susceptibility χij can be written as (see Ap-
ij
X(cid:0) (cid:1)(cid:2) (cid:3)(12) χ(q,ωn)= Π−01(ωn)+J(q) −1 . (15)
(cid:2) (cid:3)
4
The calculations reported in Appendix C of Ref. 11 can III. TRANSITION CRITERIA AND PHASE
easily be generalized (see Appendix B) to the present DIAGRAMS
disorderedcase,toobtaintheself-energyΠ−01(ω)interms
of the local susceptibility and the Fourier-transformed A. AF instability criterium
kernel
1 The establishment of an AF long-range order through
Π−01(ωn)=KAF(ωn)+ χ (ω ) . (16) a second-order transition is naturally described within
loc n
the EDMFT formalism via a divergent momentum-
The relation for the averagedlocal susceptibility
dependent spin susceptibility
1
χ (ω ) χ(q,ω )= (17)
loc n ≡Xq n Xq Π−01(ωn)+J(q) χ (q QAF,ω =0;TN)
→
pthreovciadlecsulaantoetdhleorceaqlusautsicoenptaiblliolwityingantdotrheelabteoscoonmicplbeatethly, = Π−01(ω =0;TN)+J(QAF) −1 →∞ (19)
allowing to determine from χloc(ω) a new (ωn). for some spe(cid:2)cific AF wavevectorand a N(cid:3)´eel temperature
K
The procedure is then iterated until self-consistency is T 11,22. Thiswouldoccurwhenthe Fouriertransformed
N
reached. magneticcouplingreachesanextremalvalue,which,e.g.,
on a hypercubic lattice in D = z/2 dimensions, yields
J(Q )=2 J D cos(Q )= J√z.
B. The quantum Monte Carlo approach AF √z µ=1 AF −
However, the cumulant expansion and the related
P
rescaling of J˜ =J /√z, discussed in the previous Sec-
Within our EDMFT scheme, we are faced with the ij ij
tion and needed to keep the dynamics of the quantum
problem of solving the model in Eq. (14) with an im-
spins, would locate the energy of this ground state very
purity quantum spin coupled to itself via a retarded in-
far (E √zJ) from the other states forming the
teractionkernelrepresentingthe surroundingmedium of 0 ∼ −
“body” of the spectrum. Therefore,having in mind that
the other spins. As announced above, we here adopt
after all real systems occur only up to three dimensions,
a numerical QMC technique, starting from a Hubbard-
one is naturally led to use a bounded band (with semiel-
Stratonovich transformation of the local partition func-
liptic and rectangular densities of states to mimic the
tion
three-dimensionaland the two-dimensionalcases respec-
β tively). In this case the divergence of the spin suscepti-
Z = ηTr exp dτη(τ) S(τ)
loc bilityisnolongerrelatedtosomespecificwavevectorcor-
Z D T "Z0 · # responding to an easily recognizable space arrangement
1 β β of the spins, but is rather reached when the border of
exp dτdτ 1(τ,τ )η(τ) η(τ ) ,
′ − ′ ′ the band reaches some specific value. Here, after a suit-
× "−2Z0 Z0 K · # able rescaling of the energy units, we set this condition
(18) to J(Q ) J = ε (all the z factors have been
AF min
− ≡ −
eliminated via the rescaling). Therefore, the condition
where is the imaginary time ordering and Tr denotes
T for AF order reads
a pathintegral,takenwith respectto the spin degreesof
freedom S(τ). This represents the partition function of
a spin coupled to an effective imaginary-time-dependent Π−01(ω =0;TN)=J . (20)
randommagneticfieldwithaGaussiandistribution. No-
tice that within the functional-integral formalism, the
B. Spin-glass instability criterium
Hubbard-Stratonovich field η is a bosonic field. This
relates previous analyses of Bose-Kondo models20,21 to
our EDMFT analysis of the magnetic model, where the On the other hand, one can show that the condition
spin degrees of freedom of surrounding medium are rep- for the instability of the paramagnetic phase toward the
resented by a simpler bosonic bath. formation of a SG phase is given by
Following Ref.10, to implement the QMC algorithm,
1
we discretize the imaginary-time axis into L small time 1 = J2 χ2(q,ω =0)=J2
slicessothatthetime-orderedexponentialinthetraceof DXq n DXq Π−01(0)+J(q) 2
Eq. (18) can be written as the product of L 2 2 matri-
ces using Trotter’s formula. In our calculation×s we take = J2 J dε ρ(ε) =(cid:2) J2 ∂χloc(0)(cid:3),
the inverse temperature J0β ≤ 120 and L ≤ 256 while DZ−J Π−01(0,TSG)+ε 2 − D ∂Π−01
keeping J0∆τ = J0β/L ≤ 0.25, where J0 is a natural (cid:2) (cid:3) (21)
magnetic energy scale: In the 3D case we identify this
scale with J (see Sect. IV.A), while in the 2D case we where the spin-excitation density of states ρ(ε) δ(ε
use the nota×tion Jˆ(see Sect. IV.B). J(q)) has been introduced to calculate the mom≡entum−
5
sum. In the particular case when the ordered AF cou-
plingisabsenttheactionismadelocalbyaveragingover
the disorder irrespective of the EDMFT expansion [cf.
SectionII andEq. (6)in Ref. 23]andthe condition(21)
reduces to 1 = J2χ2 = J χ 24 This condition was
D loc D loc
established long ago23 for the determination of the SG
instabilityforaquantumSherrington-Kirkpatrickmodel.
C. Dimensional analysis
We now aim to establish which one of conditions (20)
and(21)isrealizedfirstdependingonthedimensionality
D. This latter determines the specific form of the bare
density of states of the spins, which, near the (lower)
band edge ε = J behaves like
min
−
D−2
ρ(ε) (J +ε) 2 . (22)
∝
To proceed we assume that the condition (20) is real-
ized first and we replace Π−01 by J inside Eq.(21)
J ρ(ǫ)
1=J2 dǫ . (23)
DZ−J (ǫ+J)2
The convergence of the integral is determined by the FIG.1: SchematicphasediagramsofthedisorderedAFquan-
band-edge behavior at (ǫ J). For D 4 the in- tum Heisenberg model in (a) D = 2, (b) D = 3, and (c)
≈ − ≤ D>4. ThedashedlineT isacrossovertemperaturebelow
tegral diverges indicating that, for any finite value of SL
which spin-liquid correlation appear. T and T are the
J , the conditionEq.(21) is satisfiedbeforeT is lowered SG AF
D spin-glass and the AF transition temperatures respectively.
to its N´eel value when Π−01(TN) = J. In this case the ThedotschematicallyindicatesthepositionatwhichtheSG
system always becomes a SG before forming any static
temperature becomes larger than the AF one. The dotted
AF order. We expect that, at least for small J ’s, the
D lineindicatesthe“causalinstability”lineoccurringin2D(see
AForderandareplica-symmetrybreakingordercoexist. text).
It should also be considered that first-order transitions
might occur driving the system AF or SG ordered with-
out passing through the instabilities determined by the
also allow for the quantitative determination of the SG
above conditions25. To investigate and settle all these
transition temperature T and of the spin-liquid in-
issues, however, an EDMFT analysis in the presence of SG
stability temperature T . This latter marks the tem-
finite AF and/or SG order parameters should be car- ∗
ried out. This is technically challenging and beyond the perature below which the spin self-energy Π−01(ω) as a
functionofrealfrequencieshasapositiveimaginarypart
scope of our work. On the other hand, for D > 4, the
integral in Eq. (23) converges. In this case a minimum Π−01(ω)′′ > 0 thereby violating the causality condition
value of J has to be present, above which a SG order ωΠ−01(ω)′′ <0.13,15
D
is established before an AF order parameter is formed. In three dimensions Fig. 1(b) shows that, as soon as
Therefore, while for D 4 even a small amount of dis- a disordered three-dimensional magnetic coupling JD is
order is enough to prod≤uce a SG phase, for D > 4 the turned on, the system undergoes a second-order transi-
pure AF order is stable and a sufficiently large amount tion to a SG state. This occurs below a transition tem-
of disorder is required to break the replica symmetry.26 perature TSG, which, according to our criterion above,
Fig. 1 schematically depicts the phase diagrams for is always larger than the N´eel temperature TN (in par-
our model in two, three, and above four dimensions. ticular it is also larger than the AF temperature in the
In two dimensions [Fig. 1(a)], the AF long-range order absence of disorder TN0. In the opposite limit of JD ≫J
never occurs28 and AF correlations only manifest them- onenaturallyrecoversthe transitiontemperatureforthe
selveswiththeformationofaspin-liquidphasewithare- “pure” quantum Sherrington-Kirkpatrick model10,23,29
ducedspinsusceptibility (with respectto the Curie form and the TSG/J curve smoothly approaches the straight
valid at temperatures above J) and with critical spin line TSG =0.142JD.
dynamics. This behavior arises below a crossover tem- We finally notice that the qualitative picture reported
perature T , which has been numerically determined in Fig. 1(c) for the case above four dimensions, is in
SL
via the QMC analysis reported in Section V. This will qualitativeagreementwiththemean-fieldcalculationsof
6
Refs. 30, where the competition between a SG and a Using the relation
ferromagnetic phase was studied.
1 √1 x2 π
− dx= y y2 1 , (29)
IV. STATIC PROPERTIES Z−1 [y+x]2 y2−1h −p − i
p
the SG criterion can be written as
A. The three-dimensional case
Π−01 2 1=2JD2 Π−01 Π−01 2 1 .
When the AF coupling is assumed to act in three di- s J − J2  J −s J − 
(cid:18) (cid:19) (cid:18) (cid:19)
mensions, the spin-wave spectral density ρ(ǫ) takes a
 (30)
semielliptic form
Finally, this criterion can be cast into the form
2
ρ3D(ǫ)= πJ2 J2−ǫ2 . (24) Π−01 = 1+2(JD/J)2 . (31)
p (cid:12) J (cid:12) 1+4(JD/J)2
In this case the integral in Eq. (17) can be analytically (cid:12) (cid:12)
(cid:12) (cid:12) p
performedto show thatthis equationis identically satis- It is worth em(cid:12)phasi(cid:12)zing again that the self-consistent
fied for valueofΠ−01(TSG)tobeusedintheaboverelationsisthe
same as in the non-disordered system. Therefore, to de-
Π−01(ωn)=J2χloc(ωn) , (25) termine the 3D phase diagram as a function of tempera-
× tureanddisorderedcouplingJ ,weself-consistentlycal-
D
with culated with QMC the spin self-energy at different tem-
peratures for a unit value of J = J = 1. The results
J2 J2/4+J2. (26) are reported in the inset of Fig.×2. Then, for each value
× ≡ D
of J /J, the SG instability temperature is obtained by
D
Therefore in three dimensions the problem with mixed equatingΠ−01(T =TSG)withther.h.s. ofEq.(31). While
AFanddisorderedcouplingbecomesformallyequivalent Fig. 1 is a sketch comparing the phase diagrams of our
to a model with a rescaled coupling J only31. modelinvariousdimensions,Fig. 2showsthephasedia-
×
graminthreedimensionsasnumericallydeterminedfrom
the procedure outlined above.
1. Phase diagram It is worth noticing that in the three-dimensionalcase
the regime with strong spin correlationsreducing the lo-
Starting from the free-spin high-temperature regime, calsusceptibilityisseverelyreducedbytheoccurrenceof
upondecreasingT,the magneticcorrelationsstarttore- theSGphase. Thisistobe contrastedwiththe substan-
duce the magnetic susceptibility. We therefore introduce tialspin-liquidregime foundin twodimensions (see next
a crossover temperature T , which is arbitrarily defined Section)
asthe temperatureatwhic×hthe localmagneticsuscepti-
bility is reduced by ten per cent with respect to the free
moment Curie law 4T χ (T )=0.9. From the numer- 3. Local-moment formation
loc
ical calculations,we fi×nd T ×0.75J . Since we decided
to use the AF magnetic co×u≈pling J×as the energy unit, The temperature dependence of the static local sus-
we recast this relation as ceptibility is shown in the inset of Fig. 3 for various val-
ues of the disorder J /J at a fixed value of J . Again,
D
T 1 J 2 by exploiting the fact that in the three-dimens×ional case
D
J× ≈0.75s4 + J , (27) the disordered coupling can be “hidden” inside the ef-
(cid:18) (cid:19) fective coupling J , we first worked in units of J , and
where the disordered part of the magnetic interaction is calculated once th×e universal curve J χloc(T,JD×,J) =
made explicit. f(T/J )). Then we expressed J i×n terms of JD/J
to find×the different curves at var×ious values of disor-
derinthe (meta)stable paramagneticregion,both above
and below T . It is quite evident that the suscepti-
2. Spin-glass temperature SG
bility curvesdisplay both a high-temperature and a low-
temperatureCurie-likebehaviorwithaconstant(unityin
Usingthethree-dimensionaldensityofstatesEq.(24),
log-logplot)universalslope. Thisisanindicationoflocal
the SG criterion Eq. (21) is
momentformation,asshowninthemainframeofFig.3.
Here the temperature dependence of the square local
2J2 J √1 ǫ2
1= D − dǫ . (28) moments µ2(T) = Tχloc(T) is displayed, showing that
π J2 Z−J Π−01(ω =0,TSG)+ǫ 2 µ(T)2 ≈ 3/4 at high temperature, while µ(T)2 ≈ 1/4
(cid:2) (cid:3)
7
at low temperature. It can also be noticed that the dis-
ordered coupling strongly favors the formation of static
10 low-T local moments. However, as shown by the dia-
monds in Fig. 3, the SG transition already occurs at
7.0
much higher temperatures when the local moments are
J
5.0−1/ still close to their high-temperature value 3/4.
3.0 Π0
1.0
T/J 1 0.01 0.1T/J 1 B. The two-dimensional case
T
X We also consider here the case of two-dimensional
T AF fluctuations having a spectral density of the form
N
T ρ (ǫ)= 1 for J ǫ J. Contrary to the above 3D
SG 2D 2J − ≤ ≤
case, the effect of disorder can no longer be included in
0.1 0.01 0.1 1 10 aneffective coupling J together with the AF J. There-
fore, for any value of J× one has to recalculate the self-
J /J D
d consistent quantities χ , . These quantities are re-
loc
K
lated by the selfconsistency Eq.(17), which in two di-
FIG.2: PhasediagramoftheDQAFHmodelinthreedimen-
mensions, reads
sions as obtained from the QMC calculations at fixed (unit)
pvaelruaetuorfeJT×. TahsensuolmiderliicnaellryeporbetsaeinntesdthfreoSmGthtreancrsiitteiorniumtemo-f Π−01(ω,T)=Jcoth[Jχloc(ω,T)] . (32)
SG
Eqs. (21) and Eq. (31). The dashed line is the crossover
T (see text). Inset: zero-disorder spin self-energy Π−1 as
× 0
1. Phase diagram
a function of temperature. The antiferromagnetic instability
occurs at T ≈0.28J, when Π−1=J.
0
The results of our EDMFT calculations are summa-
rized in the numerical phase diagram of Fig. 5. Here
the presence of various regions is apparent, where the
system displays different physical properties. First of all
oneshouldnoticeacrossoverlineseparatingthefree-spin
high temperature region from a region at intermediate
temperatures and low disorder, where the spin system
0.75
has non-trivial correlations. This crossover temperature
hasbeendeterminedfromahigh-temperatureexpansion
10 of Eq. (32), where the explicit form of Π−01 in 2D has
been inserted. Specifically, by assuming that Jχ is
loc
1χloc small at high temperatures, one obtains that
0.1
2µ 0.5 T JD 2
× =A 1+9 . (33)
0.010.1 1 10 J s J
T/J (cid:18) (cid:19)
The prefactor A is then numerically determined, by im-
posing that the spin susceptibility is reduced by ten
percent from its high-energy Curie law. This yields
A 0.25, for which one obtains the dashed line in Fig.
0.25 ≈
0.001 0.01 0.1 1 10 4. Below this line and before the SG phase takes place
at low temperature, the spins form a correlated “liquid”
T/J
with the static and dynamic local susceptibilities show-
ing anomalous power-law behavior. This behavior, at
FIG. 3: Temperature dependence of the square effective
moment µ2(T) = Tχ (T) in 3D at fixed (unit) value of intermediate temperatures where the effects of JD are
loc
not significant, is quite similar to the case of the pure
J and various values of disorder: J /J = 0, 0.5, 1., 5, 10
× D
from left to right. The high-temperature limiting value is two-dimensional system described in Ref. 13. Specifi-
S(S+1)=3/4, while the low-temperature limit is 1/4. The cally, one finds over an order of magnitude in T, where
diamondsshowthesquaremomentatthecorrespondingT . the static local spin susceptibility displays the quantum-
SG
Inset: Temperaturedependenceofthelocalspinsusceptibility critical power-law behavior χ (T) J 1(J/T)2/3. Re-
loc −
≈
for JD/J =0, 0.5, 1, 5, 10 from top to bottom. latedcriticaldynamicalpropertiesarefoundinthe same
region (see below). However, upon lowering the temper-
ature, the paramagnetic phase is characterized by the
8
10 0.75
10
1
TX 1χloc
J
T/
0.1 2µ 0.5
T* T 0.01 0.1 1 100.1
SG T/J
0.01
0.001 0.01 0.1 1 10
J/J
d
0.25
0.01 0.1 1 10
FIG. 4: Phase diagram of the DQAFH model in two dimen- T/J
sions as obtained from the QMC calculations. The dashed
line represents the analytic estimation of the crossover tem-
FIG. 5: Temperature dependence of the square effective mo-
perature T the high-temperature free-spin regime to the
× ment µ2(T) = Tχ (T) in 2D at various values of disorder:
intermediate-temperaturespin-liquidregimeasobtainedfrom loc
J /J =0, 0.10, 0.58, 0.98, 2.06 from left to right. The high-
the high-temperature expansion (see text). The filled dia- D
temperature limiting value is S(S+1)=3/4, while the low-
monds represent the SG transition temperature T as nu-
SG
temperature limit is 1/4. The circle in the J /J = 0 curve
mericallyobtainedfromthecriteriumofEq. (21). Theheavy D
showsthesquaremomentatT =T∗≈0.03J. Thediamonds
dashedlinerepresentstheasymptotic(J ≫J transitionline
D
show the square moment at the corresponding T . Inset:
for the quantum Sherrington-Kirkpatrick model. The dot- SG
dashedlinecorrespondstothetemperatureT∗ of thecausal- Temperature dependence of the local spin susceptibility for
J /J = 0, 0.58, 0.98, 2.06 from top to bottom. The dashed
ity violation occurring at zero disorder, while the thin solid D
line is a guide for the eyes corresponding to a power-law be-
lineisobtained from thelow-temperature estimation ofT .
SG
havior with an exponent 2/3 to illustrate the intermediate
spin-liquid behavior.
gradual formation of local moments, whose temperature
dependence for various values of the disorder is reported
in Fig. 5. SG instability as determined by Eq. (21). In the present
two-dimensional case, the relation Eq. (21) characteriz-
ing the SG temperature can be simplified to
2. Local-moment formation
The fact that disorder favors the formation of local Π−01(ω =0,TSG)= JD2 +J2 ≡Jˆ, (34)
moments at progressively larger temperatures, is made
q
apparent by the flattening of the curves to the classical
value1/4. Itisworthnoticingthatthetendency toform with Π−01(ω,T) given by Eq. (32) In the weak disorder
a finite local moment at low temperature is also present limit J <<J, we approximate χ by its zero disorder
D loc
in the absence of disorder (J = 0). This tendency was expression Jχ (ω = 0,T << J) 0.338(J/T)2/3 (see
D loc
also just visible in our previous work13, in our QMC cal- Fig. 1(b)ofRef. 13). Thisgivesthe≈asymptoticbehavior
culations at T = J/100. However, it was impossible to in the limit J /J 0
D
→
confirm this numerical hint by well-convergedQMC cal-
culations at lower temperatures32. On the other hand,
3/2
the static moment formation becomes quite clear in the 0.338
T J , (35)
presence of disorder. SG ≈ "ln J #
JD
3. Spin-glass temperature showing that a significant T can be obtained even for
SG
a small disorder. The numerical data seem indeed to be
So far we have discussed the properties of the param- smoothlyextendedtothislow-disorderbehavior(seeFig.
agnetic phase disregarding the possible occurrence of a 4).
9
withasinglecharacteristicrelaxationfrequencyω given
L
by ω2 = 2J 2S(S + 1)/2[1 S(S + 1)(βJ )2/18] In
imagiLnary-tim×e this correspon−ds to ×
0.25
χL (τ) = S(S+1)e−21“βω2L”2h1−(1−2βτ)2i
0.2 loc 3
) S(S+1)
τ(oc ≡ 3 ΦL(τ) . (37)
χl 0.15
On the other hand, at low temperatures the curves in
Fig. 6 clearly display two regimes: At long times the
0.1
curve flattens indicating the formation of nearly static
moments, while at short times a rapid variation (de-
0 0.5 1
crease) of χ (τ) indicates that some spectral weight is
τ/β loc
shifted at high frequencies. From an analysis similar to
FIG.6: Imaginary-timedependenceofthelocalsusceptibility the one of Ref. 10 one can see that this leads to dissipa-
in threedimensions at different valuesof theinverse temper- tion from transverse spin excitations
ature βJ =1, 3, 5, 10, 30, 60, 120.
×
χT (ω) S β 3/2 ωtanh(βω/2)
loc =
V. DYNAMICAL PROPERTIES πω 2 (cid:20)2ωT(cid:21) √2π(1+βωT/2)
β(ω ω )2 β(ω+ω )2
T T
exp − +exp ,
A. Dynamical properties in 3D × (cid:20) (cid:20)− 4ωT (cid:21) (cid:20)− 4ωT (cid:21)(cid:21)
(38)
From our QMC analysis we also obtained dynami-
cal quantities in imaginary time. Fig. 6 displays the atfrequenciesωT J 2/T. Inimaginarytime thisgives
∼ ×
imaginary-time dependence of the local spin suscepti-
bility χloc(τ) in the paramagnetic phase.33 Indeed, al- χT (τ) = S(S+1)1+βωT/2(1−2τ/β)2
though the system rapidly passes from a paramagnetic loc 3 1+βωT/2
high temperature phase to a low-temperature SG phase βω 2τ 2
(see Fig. 2), we follow the paramagnetic phase down to exp T 1 1
low temperatures to allow a comparison with the two- × "− 4 " −(cid:18) − β (cid:19) ##
dimensional case, where the critical SG temperature is S(S+1)
Φ (τ) . (39)
much lower. In this way we can investigate the role of T
≡ 3
dimensionality in determining the spin dynamics.
In χ (τ) one can recognize the different temperature The longitudinal fluctuations instead still produce a
loc
regimesfoundinthestaticquantitiesandexploredinde- quasi-elasticabsorptionpeakoftheformofEq. (36),but
tailinthepreviousstudyofthepurelydisorderedcasein with a reduced frequency ωL T anda prefactor βS2/3
∝
Ref. 10. This isquite obvioussince,asalreadydiscussed replacing βS(S +1)/3. This reduction of the effective
for the static properties, our system in three dimensions Curie constant is natural since the formation of static
is equivalent to a purely disordered Heisenberg model moments must coexist with the high-energy absorption.
with an effective magnetic coupling [see Eq.(25)]. This Theknownbehaviorinthehigh-andlow-temperature
equivalence turns out to be quite important, since the regimes allows to construct an interpolation function
dynamical analysis involves a delicate procedure of ex-
S(S+1)
tractingreal-frequencyinformationfromdataobtainedin
χ = [pΦ (τ)+(1 p)Φ (τ)] . (40)
loc L T
imaginary time or frequencies. This is usually a difficult 3 −
task,butitisgreatlysimplifiedbytheknowledgeofwell-
Obviously the temperature variation of the relaxation
established limits at low and high temperatures for the
function F(ω) χ (ω)/(πχ (T)ω) is quite similar to
dissipativepartofthelocalresponseχloc(ω). Thisallows the one shown≡in F′i′g. 2 of lRocef. 10 provided that J is
the construction of an interpolating function that suc-
replaced by J .
cessfully matches the numerical QMC data. Specifically ×
one finds a high-temperature regime, where the spin dy-
namicabsorptionproducesaGaussianquasi-elasticpeak
B. Dynamical properties in 2D
χ (ω) βS(S+1) (βJ )2
′l′oc = 1 S(S+1) × Fig. 7displaystheresultsofourQMCcalculationsfor
πω 3 − 18
(cid:20) (cid:21) the imaginary-time dependent spin susceptibility. The
e−1/2(ω/ωL)2 framesfromtoptobottom arefordifferentvalues ofdis-
, (36)
× 2πωL2 order JD = 0.1J, JD = J, and JD = 10J. The various
p
10
curves are calculated at different values of the inverse
temperature βJˆ = 1, 3, 5, 10, 30, 60. Frames (b) and
(c) also contain the curves for βJˆ = 120, which could
0.25
not be obtained for the zero disorder case due to non-
convergenceof the QMC code. Similarly to the 3D case,
thelow-temperaturecurvesclearlydisplayaflatteningat
largetime markingthetendency toformstaticlocalmo- )
ments. Thisformationisclearlyfavoredbydisorder. The τ(oc 0.15
sameconclusioncouldqualitativelybedrawnfromthelo- χl
cal susceptibility as a function of Matsubara imaginary
frequenciesforvariousvaluesofthedisorder: Atlowtem-
perature the more disordered systems display a visible
0.25
tendency to form an additional contribution at zero fre-
quency,whichcannotbeobtainedfromasmoothextrap-
olation from the finite-frequency data. This is because
)
thelocalmomentsprovideasubstantialdelta-likecontri- τ( 0.15
butiontothezero-frequencysusceptibility(however,due oc
χl
tothedifficultyinreliablyextractingthisadditionalcon-
tribution, we refrain from showing curves for χ (ω )).
loc n
For the curves of χ as a function of imaginary-time,
loc
we performed an analysis similar to the one carried out 0.25
in Subsection V.A for the three-dimensional case.
Upon inspecting the two-dimensional phase diagram
of Fig. 4 one sees that a substantial region is present
)
between T and T∗ or TSG, where strong spin corre- τ( 0.15
lations sho×uld give rise to a non-trivial spin dynam- oc
χl
ics. From our previous work on the non-disordered
Heisenberg model in two dimensions13, we learned that
a spin-liquid phase arises below T . At long imaginary 0.05
times this phase has a critical susc×eptibility of the form 0.0 0.5 1.0
χloc(τ) [sin(πτ/β]−1/3,whichcorrespondstoascaling τ/β
∼
of the imaginary local susceptibility13
FIG. 7: Imaginary-time dependence of the local susceptibil-
χ (ω) J 1(ω/J) δ (ω/T) , (41)
′l′oc ∼ − − Fδ ity in D = 2 for different values of the inverse temperature
βJˆ=1, 3, 5, 10, 30, 60, 120andthreevaluesofdisorder: (a)
with (x) = xδ Γ(1 δ +i x )2sinh x and δ = 2/3.
Fδ | −2 2π | 2 JD = 0.1J, (b) JD = J, and (c) JD = 10J. For the low-
Eq. (41) gives a power-law divergence at low energy, disorder case (a) the lowest-temperature data βJˆ= 120 are
χ′l′oc(T = 0,ω) ∼ J−31ω−32 and χ(cid:0)′loc((cid:1)T,ω = 0) ∼ not available (see text).
J−13T−23.
Thequestionthennaturallyarises,whetherornotthis
criticalbehaviorcanbe experimentallyobservedinmag- other physicalparametersω andω lead to substantial
L T
netic systems with strong quantum fluctuations, weak variationsinthereal-frequencysusceptibility. Therefore,
disorder, and strong anisotropy in the AF fluctuations. while our analysis is fully compatible with the possibility
In principle one could answer this question by extend- ofintermediatefrequencywindowswithanomalousspin-
ing the analysis of Subsect. V.A and generalizing Eq. liquid exponents, we cannot provide reliable bounds to
(40) to a formincluding anadditional spin-liquid contri- the observation of these regimes.
bution
S(S+1)
χ = [pΦ (τ)+qΦ (τ)(1 p q)Φ (τ)] . VI. DISCUSSION AND CONCLUSIONS
loc L SL T
3 − −
(42)
with Φ (τ) [sin(πτ/β] 1/3. Starting from the In this paper we systematically investigated the in-
SL −
∝
FouriertransformofEq. (42),wetriedtouseastandard terplay between disorder and fluctuations in a quantum
procedure based on Pad´e approximants to analytically spin-1 Heisenberg model. Without entering any broken-
2
continuetheMatsubarasusceptibilitytorealfrequencies. symmetry phase and neglecting the possible occurrence
Unfortunately, the analytic continuation procedure is so of first-order transitions, we investigated the properties
delicatethatreliableresultscanhardlybeextractedfrom and the instabilities of the paramagnetic phase. We
a form like Eq. (42). In particularwe noticed that small found that a marked difference exists between the two-
variations in the relative weights p,q as well as in the and the threedimensional cases. In 3D the system is or-