Quantum magnetism on small-world networks
QQuantum magnetism on small-world networks
Maxime Dupont
1, 2 and Nicolas Laflorencie Department of Physics, University of California, Berkeley, California 94720, USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, 31062 Toulouse, France
While classical spin systems in random networks have been intensively studied, much less is known aboutquantum magnets in random graphs. Here, we investigate interacting quantum spins on small-world networks,building on mean-field theory and extensive quantum Monte Carlo simulations. Starting from one-dimensional(1D) rings, we consider two situations: all-to-all interacting and long-range interactions randomly added. Theeffective infinite dimension of the lattice leads to a magnetic ordering at finite temperature 𝑇 c with mean-fieldcriticality. Nevertheless, in contrast to the classical case, we find two distinct power-law behaviors for 𝑇 c versusthe average strength of the extra couplings. This is controlled by a competition between a characteristic lengthscale of the random graph and the thermal correlation length of the underlying 1D system, thus challengingmean-field theories. We also investigate the fate of a gapped 1D spin chain against the small-world effect. I. INTRODUCTIONA. Complex networks and the small-world effect
Understanding complex networks is at the heart of manyscientific fields [1–11], such as computer science, mathemat-ics, physics, biology, sociology, epidemiology, etc. During thepast two decades, critical phenomena arising in such randomtopologies have emerged as a key subject of intense researchin statistical physics [7, 9].A complex network is a graph with non-trivial and randomproperties, as opposed to periodic (or quasi-periodic) latticesof finite dimension. There are two main features which contrastwith regular graphs: (i) a fluctuating connectivity (a certainproportion of the links are randomly placed) and (ii) the so-called small-world (SW) effect [12], which can dramaticallyshorten the distances across the network. More precisely, fora finite graph of 𝑁 sites, the average distance ℓ between twoarbitrary points, also called the graph diameter, grows slowerthan any power-law with 𝑁 : ℓ ∼ ln 𝑁 , resulting in an infiniteeffective dimension.The SW effect occurs in a large class of complex networks,such as Erdös-Rényi random graphs [13], scale-free [3] andSW networks [2]. For the later case, the most popular SWsystem is the Watts-Strogatz model [2] in which one ran-domly rewire with a probability 𝑝 each edge of an initialone-dimensional (1D) ring, as depicted in Fig. 1 (a). Shortlyafter, a variant was proposed in Refs. [14, 15] by simply addinglong-range bonds with probability 𝑝 , without diluting the un-derlying 1D structure, see Fig. 1 (b). This undiluted versionof the SW network, more amenable to analytical treatments,was argued [4] to bring similar physics as compared to theoriginal SW proposal of Watts and Strogatz. Another sim-plification was later proposed by Hastings in Ref. [16] witha mean-field (MF) version, see Fig. 1 (c), where all possiblelong-range links are added, but with a reduced strength ∝ / 𝑁 vanishing at large sizes. This MF variant was introduced toavoid randomness and thus facilitate analytical calculations. ( a ) Watts − Strogatz small − world ( b ) Undiluted small − world ( c ) Hastings model s0s1s2s3s4s5s6s7 s8 s9 s10 s11(a) Watts-Strogotz small-world p = 1 / p = 0 . p =
13 p = FIG. 1. Three types of small-world networks with 𝑁 =
12 sites.( a ) The Watts-Strogatz model in which one randomly rewire with aprobability 𝑝 each edge of an initial 1D ring. ( b ) Long-range bondsare added with probability 𝑝 , without diluting the underlying 1Dstructure. ( c ) All possible long-range links are added, but with areduced strength ∝ / 𝑁 vanishing at large sizes. B. Classical magnetism and small-world effect
A strong consequence of the SW effect is that for any finiteconcentration 𝑝 > 𝑑 (exemplified in Fig. 1 for 𝑑 = 𝑑 = ∞ , providedthe number of sites 𝑁 is large enough, typically exceeding acrossover size 𝑁 ★ ∼ / 𝑝 [4, 17, 18]. This drastic change in theeffective dimension of the problem has attracted a lot attentionin the context of interacting classical spin systems [4, 19–28],while much less is know for the quantum case [29–32].Classical O ( 𝑛 ) models on SW networks have been heavilyinvestigated for 𝑛 = 𝑛 = 𝑑 u =
4) was found to describe the criticalproperties. Note however that scale-free networks with power-law distributed connectivities [3] do not necessarily displayMF behavior, depending on the power-law exponent of theconnectivity distribution [23, 25, 26, 34]. Non-universal andnon-MF behaviors have also been reported in SW networkswhere the long-range interactions [35] or the branching prob-ability [36] decay as a power-law with the distance. To someextent, this reminds early renormalization-group results for 𝑛 -vector models with power-law decaying interactions [37, 38]. a r X i v : . [ c ond - m a t . s t r- e l ] F e b C. Small-world quantum magnets
In this work, we want to address the following question:is there some specificity of quantum spins as compared tothe aforementioned classical results? To this end, we willfocus on spin-1 / H and a random long-rangecontribution H LR . For the short-range piece, we choose theXXZ Hamiltonian, defined on a ring by, H = 𝐽 ∑︁ 𝑁𝑖 = ℎ Δ 𝑖,𝑖 + , with ℎ Δ 𝑖,𝑖 + = 𝑆 𝑥𝑖 𝑆 𝑥𝑖 + + 𝑆 𝑦𝑖 𝑆 𝑦𝑖 + + Δ 𝑆 𝑧𝑖 𝑆 𝑧𝑖 + , (1.1)with periodic boundary conditions. Δ is the Ising anisotropyparameter, and the long-range part, which describes interac-tions beyond nearest neighbors, takes a similar XXZ form, H LR = ∑︁ 𝑖, 𝑗 𝐽 LR 𝑖 𝑗 ℎ Δ 𝑖, 𝑗 , | 𝑖 − 𝑗 | > . (1.2)In the rest of the paper, we will focus on two emblematic cases:1) The ferromagnetic XY model with Δ =
0, and all couplingsnegative:
𝐽 < 𝐽 LR 𝑖 𝑗 = −| 𝐽 LR 𝑖 𝑗 | .2) The (staggered) antiferromagnetic Heisenberg model de-fined by Δ = 𝐽 >
0, and staggered couplings be-yond nearest-neighbor 𝐽 LR 𝑖 𝑗 = −(− ) | 𝑖 − 𝑗 | | 𝐽 LR 𝑖 𝑗 | which pre-vent magnetic frustration. This alternating exchange is well-known to enhance antiferromagnetic correlations [39, 40].
1. Undiluted small-world
Starting with a 𝑁 sites ring, the undiluted SW model,Fig. 1 (b), is controlled by a branching parameter 0 < 𝑝 ≤ . (cid:98) 𝑝𝑁 (cid:99) long-ranged links ( 𝑖, 𝑗 ) having a coupling strength | 𝐽 LR 𝑖 𝑗 | = 𝐽 (cid:48) , while 𝐽 LR 𝑖 𝑗 = 𝑧 = + 𝑝 ,and the average strength of extra long-range couplings is, 𝐽 (cid:48) ( 𝑝 ) = 𝑝𝐽 (cid:48) . (1.3)
2. Hastings model
As shown in Fig 1 (c), the MF version of the SW net-works [16] is built by distributing the long-range couplingsover all sites with | 𝐽 LR 𝑖 𝑗 | = 𝑝𝐽 (cid:48) / 𝑁 for all pairs | 𝑖 − 𝑗 | > 𝐽 (cid:48) ( 𝑝 ) = 𝑝𝐽 (cid:48) 𝑁 − 𝑁 −→ 𝑝𝐽 (cid:48) ( 𝑁 → +∞) , (1.4)thus making this model equivalent to the undiluted small-worldfrom an energetic point of view, while the connectivity of theHastings model is extensive 𝑧 = 𝑁 . D. Structure of the paper
The rest of the paper is organized as follows. In Sec. II, wereview previous results on classical spin systems and discusstwo MF theories for SW networks. Because of the effectiveinfinite dimensionality of the lattice, one expects a temperaturephase transition on this geometry. Interestingly, the two ap-proaches lead to different qualitative behaviors of the criticaltemperature 𝑇 c with the average strength of extra long-rangecouplings 𝐽 (cid:48) ( 𝑝 ) , as defined in Eqs. (1.3) and (1.4). The firstmethod is based on a comparison between the thermal corre-lation length of the system without the extra couplings and apurely geometric quantity: the average distance between twoshortcuts. The other approach is based on the random phaseapproximation (RPA). In Sec. III, we consider these approxi-mate MF treatments for SW graphs built on top of 1D quan-tum spin chains for the classical Ising chain and the quantum 𝑆 = / 𝑝 . In order to go beyond gapless XXZ physics, we alsoexplore the fate of a gapped 1D dimerized chain against theSW effect. To conclude, we present a summary of our findingsand discuss a few perspectives of our study in Sec. V. II. MEAN-FIELD THEORYA. Ising and XY models: discussion of previous results
As expected from the infinite-dimensional nature of SWnetworks, several authors agreed on the MF nature of the finitetemperature ordering transition for both classical Ising [4, 19,21, 22, 27] and XY [20] models. In the limit of small branchingprobability 𝑝 (cid:28)
1, a simple MF argument predicts a criticaltemperature when the correlation length of the underlying 𝑑 -dimensional lattice 𝜉 ( 𝑇 ) ∼ | 𝑇 − 𝑇 c ( )| − 𝜈 (with 𝑇 c ( ) the 𝑝 = 𝜈 the associated critical exponent)becomes of the order of the average distance between twoshortcuts 𝜁 𝑝 ∼ 𝑝 − / 𝑑 . This simple argument gives, 𝑇 MFc ( 𝑝 ) − 𝑇 c ( ) ∝ 𝐽 𝑝 / 𝑑𝜈 . (2.1)For the 𝑑 = 𝑇 c ( ) = 𝜈 = ∞ since 𝜉 ( 𝑇 ) ∼ exp ( 𝐽 / 𝑇 ) , the above MF argument yields, 𝑇 MFc , Ising ∝ 𝐽 (cid:14) ln (cid:0) / 𝑝 (cid:1) , (2.2)in good agreement with the literature [4, 19, 27].However, when the very same MF reasoning is applied to theclassical XY chain, for which 𝜉 ( 𝑇 ) ∼ 𝐽 / 𝑇 at low temperature,we get 𝑇 MFc , XY ∝ 𝐽 𝑝 , a result in disagreement with Monte Carlosimulations where a surprising 𝑎 ln 𝑝 + 𝑏 (with 𝑎, 𝑏 ∈ R )scaling has been found [20]. The MF prediction for 𝑇 MFc inEq. (2.1) has been critically analyzed by Hastings in Ref. [16]where a different scaling with the branching probability wasfound, ˜ 𝑇 MFc ( 𝑝 ) − 𝑇 c ( 𝑝 = ) ∝ 𝐽 𝑝 / 𝛾 , (2.3)with 𝛾 the critical exponent controlling the susceptibility (asso-ciated to the order parameter) of the underlying 𝑑 -dimensionalmodel: 𝜒 ( 𝑇 ) ∼ | 𝑇 − 𝑇 c ( )| − 𝛾 when 𝑇 → 𝑇 c ( ) + . As we willdiscuss in more details below, the expression of Eq. (2.3) isa direct consequence of a random phase approximation treat-ment of the problem.When comparing Eq. (2.1) and Eq. (2.3) with numericalresults obtained for the Ising model by Herrero [22], Hastingsargued in favor of Eq. (2.3) since 𝑝 / 𝛾 < 𝑝 / 𝑑𝜈 in the 𝑝 → 𝛾 < 𝑑𝜈 , orequivalently, using Fisher’s identity 𝛾 = ( − 𝜂 ) 𝜈 [41], 𝑑 + 𝜂 > , (2.4)with 𝜂 the anomalous dimension. This condition is fulfilled forclassical phase transitions in spin systems, but as we will seebelow, low-dimensional quantum magnets provide a uniqueexample where the critical temperature 𝑇 c ( 𝑝 ) can cross-overfrom Eq. (2.3) to Eq. (2.1) when 𝑝 → B. Random phase approximation
The random phase approximation [42, 43] gives a self-consistent MF estimate for the ordering transition temperatureof weakly coupled 𝑑 -dimensional systems, using, 𝑇 RPAc = 𝜒 − 𝑑 (cid:18) 𝐽 ⊥ (cid:19) , (2.5)where 𝜒 − 𝑑 is the inverse-susceptibility function of the underly-ing 𝑑 -dimensional system, and 𝐽 ⊥ is the (weak) MF couplingbetween the 𝑑 -dimensional units, see App. A.The RPA expression for the critical temperature of Eq. (2.5)has proven to be very useful in the context of weakly coupledlow-dimensional systems [44] such as coupled spin chains andladders [45–47], or layered magnets [48–53]. Interestingly,a direct quantitative comparison between exact QMC simula-tions for various 𝑑 = 𝐽 ⊥ by anon-universal factor 𝐽 ⊥ → 𝛼𝐽 ⊥ with 𝛼 (cid:39) . 𝑑 -dimensional systems induces extra-couplings of av-erage strength 𝐽 (cid:48) ( 𝑝 ) , as given by Eqs. (1.3) and (1.4). Usingthe susceptibility divergence of the bare system at 𝑝 = 𝐽 𝜒 ( 𝑇 ) ∝ (cid:18) 𝑇 − 𝑇 c ( ) 𝐽 (cid:19) − 𝛾 , (2.6)the above RPA formula of Eq. (2.5) yields 𝑇 RPAc ( 𝑝 ) − 𝑇 c ( ) ∝ 𝐽 (cid:18) 𝑝 𝐽 (cid:48) 𝐽 (cid:19) / 𝛾 , (2.7) which recovers Hastings’ expression [16], as given above inEq. (2.3). Here, we notice that the RPA estimate explicitlydepends on the shortcut coupling strength 𝐽 (cid:48) , while the simplerMF expression of Eq. (2.1) does not.In the absence of finite temperature transition 𝑇 c ( ) = 𝑑 =
1, or 𝑑 = III. THE SPECIAL CASE OF 𝑑 = A. Ising chain
We shall start with a brief discussion of the 𝑑 = 𝑝 (cid:28)
1, yields 𝜉 ( 𝑇 c ) ∼ e 𝐽 / 𝑇 c ( 𝑝 ) ∼ / 𝑝 ,which leads to the well-know form of Eq. (2.2) [4, 19, 27].However, one can also invoque an RPA treatment of this prob-lem, using the exponential divergence of the susceptibility 𝜒
1d Ising = 𝑇 exp (cid:18) 𝐽𝑇 (cid:19) , (3.1)which gives in the limit 𝑝 (cid:28) 𝑇 RPAc = 𝐽 ln (cid:16) 𝑇 RPAc 𝑝𝐽 (cid:48) (cid:17) ≈ 𝐽 ln (cid:16) 𝐽𝑝𝐽 (cid:48) (cid:17) . (3.2)One sees that if shortcut and nearest-neighbor couplings haveequal strengths 𝐽 = 𝐽 (cid:48) , the RPA of Eq. (3.2) becomes equiva-lent to the simple MF expression of Eq. (2.2).If 𝐽 (cid:48) < 𝐽 , the ordering will be controlled by 𝑇 RPAc < 𝑇
MFc . Inthe opposite case 𝐽 (cid:48) > 𝐽 , the MF temperature 𝑇 MFc < 𝑇
RPAc willtake over because the 1D correlation length at 𝑇 RPAc has notreached the average distance between two shortcuts 𝜁 𝑝 ∼ / 𝑝 ,and one would need to further cool down the system to reachthis threshold. We therefore expect from this simple examplethat the transition temperature will be given by the minimumof the two estimates: 𝑇 c = min (cid:16) 𝑇 RPAc , 𝑇
MFc (cid:17) . (3.3) B. XXZ chain: the case of Tomonaga-Luttinger liquids
1. Analytical results
The spin-1 / H inEq. (1.1) is a well-known example of Tomonaga-Luttingerliquid (TLL) in the regime − < Δ ≤
1. Among the vastamount of knowledge available for this class of systems [57],let us briefly summarize a few of them, in particular the onesuseful in the context of an RPA treatment of 𝑑 = H : the velocity of excitations 𝑢 and the so-called Luttinger exponent 𝐾 . Their dependence onthe Ising anistropy Δ are well-known [58], 𝑢 = 𝜋 √ − Δ Δ , 𝐾 = 𝜋 (− Δ ) . (3.4)In the easy-plane regime | Δ | <
1, the dominant correlations aretransverse with respect to the Ising anistropy and power-lawdecaying at 𝑇 = (cid:10) 𝑆 𝑥𝑚 𝑆 𝑥𝑛 (cid:11) = 𝐴 𝑥𝑥 | 𝑚 − 𝑛 | 𝐾 e − 𝑖𝑞 | 𝑚 − 𝑛 | + · · · , (3.5)with 𝑞 = 𝑞 = 𝜋 ) for ferrromagnetic (antiferromagnetic)interactions. The amplitude 𝐴 𝑥𝑥 in Eq. (3.5) is also knownexactly [59]. This quasi-long range (algebraic) order doesnot survive at finite temperature where all correlations decayexponentially with a finite correlation length, diverging at lowtemperature, 𝜉 ( 𝑇 ) ∝ 𝑢𝐽 (cid:14) 𝑇 𝜈 with 𝜈 = . (3.6)In the regime | Δ | <
1, the transverse susceptibility, associatedto the dominant correlation of Eq. (3.5), has the followinglow- 𝑇 behavior [46, 57], 𝜒 𝑥𝑥 (cid:0) 𝑇 (cid:1) = 𝐴 𝑥𝑥 sin (cid:0) 𝜋 𝐾 (cid:1) 𝐵 (cid:16) 𝐾 , − 𝐾 (cid:17) 𝑢𝐽 (cid:18) 𝜋𝑇𝑢𝐽 (cid:19) − + 𝐾 , (3.7)with 𝐵 ( 𝑥, 𝑦 ) = Γ ( 𝑥 ) Γ ( 𝑦 )/ Γ ( 𝑥 + 𝑦 ) , making Eq. (3.7) aparameter-free expression.
2. Consequences for the critical temperature
From the above expression of the transverse susceptibilityEq. (3.7), one can identify the susceptibility exponent to be 𝛾 = − 𝐾 . Quite interestingly, we see that the above conditionEq. (2.4) is not fulfilled for TLL with 𝐾 = ( 𝜂 ) − > /
2, whichapplies to the entire easy-axis regime ( − ≤ Δ < 𝑇 RPAc ( 𝑝 ) = 𝑢𝐽 𝑓 (cid:0) 𝐾, 𝐴 𝑥𝑥 (cid:1) (cid:18) 𝑝𝐽 (cid:48) 𝑢𝐽 (cid:19) 𝐾 𝐾 − , (3.8)with the dimensionless prefactor, 𝑓 (cid:0) 𝐾, 𝐴 𝑥𝑥 (cid:1) = 𝜋 (cid:20) 𝐴 𝑥𝑥 sin (cid:16) 𝜋 𝐾 (cid:17) 𝐵 (cid:18) 𝐾 , − 𝐾 (cid:19)(cid:21) 𝐾 𝐾 − . (3.9)When comparing the RPA prediction with the simple MF ex-pression of Eq. (2.1) using the temperature dependence of thecorrelation length of Eq. (3.6), 𝑇 MFc ( 𝑝 ) = 𝑢𝐽 𝑝, (3.10)we anticipate a crossover at low branching probability 𝑝 ★ from an RPA regime of Eq. (3.8) to the linear MF regime .
01 0 . ? ¢ = . = . = . ? FIG. 2. The crossover probability 𝑝 ★ , defined in Eq. (3.11), is plottedversus the long-range coupling 𝐽 (cid:48) / 𝐽 for different values of Isinganisotropy Δ . The inset shows the behavior of the 𝐽 (cid:48) / 𝐽 -independentprefactor of 𝑝 ★ versus Δ . Note its singular behavior as | Δ | → of Eq. (3.10) (provided that 𝐾 > / 𝑇 RPAc ( 𝑝 ★ ) = 𝑇 MFc ( 𝑝 ★ ) , meaning that, 𝑝 ★ = 𝑝 ( Δ ) (cid:18) 𝐽 (cid:48) 𝐽 (cid:19) 𝜇 Δ , (3.11)where 𝑝 ( Δ ) = 𝑢 − 𝜇 Δ [ 𝑓 ( 𝐾, 𝐴 𝑥𝑥 )] + 𝜇 Δ is plotted inFig. 2 (inset) as a function of the Ising anisotropy, and theexponent 𝜇 Δ = 𝜋 / arccos ( Δ ) varies between 1 for Δ = − +∞ when Δ →
1. Eq. (3.11) is plotted against 𝐽 (cid:48) / 𝐽 in Fig. 2for various anisotropies Δ . This defines the range of validityof the RPA expression for the critical temperature Eq. (3.8) for 𝑝 > 𝑝 ★ . Below 𝑝 ★ , the simpler linear MF argument Eq. (3.10)is expected.The antiferromagnetic Heisenberg case ( Δ =
1) is moresubtle since the TLL parameter 𝐾 = / C. Quantum Monte Carlo results for the 𝑑 = susceptibilities We simulate the 𝑆 = / 𝑇 with QMC, using the stochastic seriesexpansion with directed loop updates [68–70].Noting ℎ sb a symmetry-breaking field coupled to the orderparameter (cid:104) 𝑚 (cid:105) , the linear response function (susceptibility 𝜒 )takes the form, 𝜒 = 𝜕 (cid:10) 𝑚 (cid:0) ℎ sb (cid:1)(cid:11) 𝜕ℎ sb (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℎ sb = = ∫ / 𝑇 d 𝜏 (cid:10) 𝑚 † ( 𝜏 ) 𝑚 ( ) (cid:11) , (3.12)with 𝑚 ( 𝜏 ) = e − 𝜏 H 𝑚 e 𝜏 H in the Heisenberg picture where 𝜏 is the imaginary time. The right-hand side of Eq. (3.12) is Tomonaga Luttinger − liquid∝ T −3/2 (XY model) .
001 0 .
01 0 . ) / . . j = = = = = FIG. 3. QMC transverse susceptibility of the XY chain (
Δ = 𝑝 =
0) as a function of the temperature 𝑇 for different system sizesfrom 𝑁 =
128 to 𝑁 = 𝑇 / 𝐽 (cid:46) .
1, when the Tomonaga-Luttinger liquiddescription becomes valid. derived from the Kubo formula [71]. In the ferromagnetic XYmodel, 𝑚 = (cid:205) 𝑗 ( 𝑆 𝑥𝑗 + 𝑖𝑆 𝑦𝑗 )/ 𝑁 , while in the antiferromagneticHeisenberg model, one has 𝑚 = (cid:205) 𝑗 (− ) 𝑗 𝑆 𝑧𝑗 / 𝑁 .
1. The spin- / ferromagnetic XY chain Eq. (3.7) provides a parameter-free expression which givesfor
Δ =
𝐽 𝜒 Δ= 𝑥𝑥 ( 𝑇 ) = . ... (cid:18) 𝐽𝑇 (cid:19) / . (3.13)This expression is plotted in Fig. 3, together with QMC resultswhere one sees a very good agreement at low temperature.
2. The spin- / antiferromagnetic Heisenberg chain The Heisenberg spin-1 / 𝜒 Δ= 𝜋 (cid:0) 𝑇 (cid:1) = 𝜒 𝑇 √︃ ln (cid:0) 𝐽 Λ / 𝑇 (cid:1) . (3.14)In order to apply the RPA analysis, it appears very impor-tant to have a correct description for 𝜒 Δ= 𝜋 ( 𝑇 ) . In Fig. 4 weshow our QMC results for large spin chains, up to 𝐿 = 𝜒 = . ( ) and Λ = . ( ) , in the temperature range0 . 𝐽 ≤ 𝑇 ≤ . 𝐽 , These parameters differ from the ones re-ported in Refs. [75–77] where QMC was performed at highertemperature. Classical .
001 0 .
01 0 . ) / . . j = = = = = .
01 10 ) / . . . . . ) j FIG. 4. QMC results for the staggered susceptibility of the Heisen-berg chain ( 𝑝 =
0) as a function of the temperature 𝑇 . Differentsymbols show system sizes from 𝑁 =
128 to 𝑁 = (cid:0) 𝑇 𝜒 (cid:1) is plottedagainst (cid:0) 𝑇 / 𝐽 (cid:1) − . The solid line is a fit to the form 𝜒 ( ln Λ − ln 𝑇 ) ,according to Eq. (3.14), with 𝜒 = . ( ) and Λ = . ( ) . IV. QUANTUM MONTE CARLO RESULTS FOR THESMALL-WORLD
We now turn to SW networks. In the undiluted case shownin Fig. 1 (b), we average the QMC results over different latticeswith 𝑝 >
A. Observables
To characterize the finite-temperature transition, we con-sider the square of the order parameter (cid:10) 𝑚 (cid:11) , directly accessi-ble from the normalized structure factor for both the staggeredantiferromagnetic Heisenberg and ferromagnetic XY models(see also Sec. III C). It can also be evaluated by looking at thespin-spin correlation at long distance, (cid:10) 𝑚 (cid:11) = lim | 𝑚 − 𝑛 |→+∞ (cid:40) (cid:12)(cid:12)(cid:12)(cid:10) 𝑆 𝑧𝑚 𝑆 𝑧𝑛 (cid:11)(cid:12)(cid:12)(cid:12) XXX case , (cid:10) 𝑆 𝑥𝑚 𝑆 𝑥𝑛 + 𝑆 𝑦𝑚 𝑆 𝑦𝑛 (cid:11) XY case . (4.1)On a finite-size system, the longest distance is taken along the1D ring with | 𝑚 − 𝑛 | = 𝑁 /
2. One can average over the 𝑁 / 𝑄 = (cid:10) 𝑚 (cid:11)(cid:46)(cid:10) 𝑚 (cid:11) , (4.2)which takes a system-size independent value at the transitionand is therefore useful to detect it. B. Mean-field behavior
1. Critical exponents
In infinitely coordinated systems, where each site is coupledto all others (e.g., the Hastings model), the concepts of dimen-sionality and length, involved in the standard finite-size scalinghypothesis, are not well-defined. Botet, Jullien, and Pfeuty ex-tended the hypothesis to such systems [79, 80] by substitutingthe correlation length 𝜉 with a coherence number N , indepen-dent of the dimensionality. Similarly to 𝜉 , it diverges at thetransition N ∼ | 𝑇 − 𝑇 c | − ˜ 𝜈 with ˜ 𝜈 a critical exponent dependingon the system but not its dimension. The authors found that,˜ 𝜈 = 𝜈 MF 𝑑 u , (4.3)with 𝑑 u the upper critical dimension and 𝜈 MF the correla-tion length exponent of the MF theory. Eq. (4.3) has beenverified for various physical systems and found to apply tomore generic infinite-dimensional geometries such as SW net-works [20, 29, 79, 80]. For the XY and Heisenberg universalityclasses considered in this work, one has 𝑑 u = 𝜈 MF = / 𝜈 =
2. As a result, close to the critical temperature 𝑇 c , the square of the order parameter follows, (cid:10) 𝑚 (cid:11) = 𝑁 − 𝛽 MF / ˜ 𝜈 F 𝑚 (cid:16) 𝑡𝑁 / ˜ 𝜈 (cid:17) , (4.4)with 𝑁 the number of lattice sites, 𝛽 MF = / F 𝑚 a universal scaling function and 𝑡 = ( 𝑇 − 𝑇 c )/ 𝑇 c the reduced temperature. The Binder ratio ofEq. (4.2) equally follows, 𝑄 = F 𝑄 (cid:16) 𝑡𝑁 / ˜ 𝜈 (cid:17) , (4.5)with F 𝑄 the corresponding scaling function.
2. Corrections to scaling
Note that irrelevant corrections to the above scaling lawsshould also be considered with a modified scaling function,
F → (cid:0) + 𝑏𝑁 − 𝜔 (cid:1) F (cid:16) 𝑡𝑁 / ˜ 𝜈 + 𝑐𝑁 − 𝜙 / ˜ 𝜈 (cid:17) , (4.6)where 𝑏, 𝑐 are non-universal parameters, and 𝜔, 𝜙 are cor-rections to scaling exponents [81, 82]. In practice, we makea fourth order Taylor expansion of the scaling function, i.e., F ( 𝑥 ) (cid:39) (cid:205) 𝑛 = 𝑎 𝑛 𝑥 𝑛 with 𝑥 = 𝑡𝑁 / ˜ 𝜈 + 𝑐𝑁 − 𝜙 / ˜ 𝜈 according toEq. (4.6). At this stage, 𝑇 c , 𝑏, 𝑐, 𝜔, 𝜙, 𝑎 , 𝑎 , 𝑎 , 𝑎 , 𝑎 areall parameters obtained by non-linear least squares fitting. Us-ing the values of the exponents 𝛽 MF and ˜ 𝜈 give very good datacollapses, see Fig. 5.We use a standard least squares fitting method to obtain theparameters. For each dataset, the fitting procedure is repeated ≈ times where each data point is generated from a normaldistribution of mean and standard deviation corresponding tothe statistical QMC average and error, respectively [83]. . . . . . & + l = = = = = = = . . . ⌦ < ↵ c o rr V M F / ˜ a + l C / ˜ a + q / ˜ a . . . . ⌦ < ↵ s u m V M F / ˜ a + l .
12 0 .
15 0 . ) / . . . & .
12 0 .
15 0 . ) / . . ⌦ < ↵ c o rr V M F / ˜ a .
12 0 .
15 0 . ) / . . ⌦ < ↵ s u m V M F / ˜ a ( a )( b )( c ) FIG. 5. Hastings model. Data collapse for the staggered XXX anti-ferromagnet with 𝐽 (cid:48) / 𝐽 = .
25. Three estimates are considered: ( a )the binder cumulant, ( b ) the square of the order parameter evaluatedfrom the spin-spin correlation at long distance and ( c ) the normalizedstructure factor. Setting ˜ 𝜈 = 𝛽 MF = /
2, we find that 𝑇 c (cid:39) .
3. Hastings model
We first discuss the infinitely connected Hastings modelwhere the long-range couplings take the form | 𝐽 LR 𝑖 𝑗 | = 𝐽 (cid:48) / 𝑁 , ∀| 𝑖 − 𝑗 | >
1. Fig. 5 shows QMC results for the staggeredantiferromagnetic Heisenberg model with 𝐽 (cid:48) / 𝐽 = .
25. Thethree panels display the different observables used to extract theordering transition, which all agree perfectly with an estimate 𝑇 𝑐 (cid:39) . 𝐽 (cid:48) / 𝐽 , for both the XY ferromagnet and the staggered XXXantiferromagnet. QMC results for the critical temperature as RPA (Analytical) .
01 0 . ( ? )/ . . ) c / Hastings (QMC)RPA (QMC)Ferromagnetic XYStaggered AF Heisenberg
FIG. 6. Critical temperature of the Hastings model, plotted as afunction of the average long-range coupling strength 𝐽 (cid:48) ( 𝑝 )/ 𝐽 forthe ferromagnetic XY and staggered antiferromagnetic Heisenbergmodels. The RPA estimates (analytical and QMC) are also displayedfor each model. a function of 𝐽 (cid:48) / 𝐽 are reported in Fig. 6 where a direct com-parison to the RPA prediction is provided. Here, we clearlyobserve that not only the critical exponents obey MF predic-tions (Fig. 5), but the chain-MF theory provides through theRPA prediction of Eq. (2.5) a perfectly quantitative estimate for 𝑇 c of the Hastings model, and this remains true up to 𝐽 (cid:48) / 𝐽 = 𝑇 RPAc is also shown in Fig. 6, together with QMC data. Forthe XY ferromagnet, Eq. (3.8) yields 𝑇 RPAc , XY (cid:14) 𝐽 = . ... (cid:18) 𝐽 (cid:48) 𝐽 (cid:19) / , (4.7)which clearly agrees very well with QMC below 𝐽 (cid:48) / 𝐽 ∼ . 𝜒 𝑇 c √︁ ln ( 𝐽 Λ / 𝑇 c ) = 𝐽 (cid:48) (4.8)Using the Lambert 𝑊 -function [84], it gives for 𝐽 (cid:48) (cid:28) 𝐽 , 𝑇 RPAc , XXX ≈ 𝜒 𝐽 (cid:48) (cid:118)(cid:117)(cid:116) ln (cid:18) 𝐽𝐽 (cid:48) (cid:19) + 𝐴 − ln √︄ (cid:18) 𝐽𝐽 (cid:48) (cid:19) + 𝐴, (4.9)with 𝜒 = . ( ) and 𝐴 = ln (cid:16) Λ √ (cid:17) − ln 𝜒 = . ( ) .This RPA analytical estimate for 𝑇 c compares very well toQMC data, as shown in Fig. 6.In this toy-model with infinite connectivity, the chain-MFtheory provides with the RPA expression an exact estimate for the critical temperature measured by QMC. As anticipatedby Hastings in Ref. [16], the simple MF expression Eq. (2.1)obtained for SW geometries in the diluted limit does not applyin this case. In the following we address the disordered casewith low connectivity for both the XY ferromagnet and theXXX antiferromagnet. C. The Undiluted small-world geometry
We now turn to the disordered case with a finite branch-ing probability 𝑝 ≤ . 𝐽 (cid:48) = 𝐽 . Incontrast to the disorder-free Hastings model, here we have toperform disorder averaging, typically over a few hundreds ofindependent samples. The average distance between short-cuts being 𝜁 𝑝 ≈ ( 𝑝 ) − , QMC simulations have to be ideallyachieved over systems of length 𝑁 (cid:29) 𝜁 𝑝 . This natural scalefixes a limit to the accessible concentrations 𝑝 (cid:38) − in oursimulations.Despite the very low connectivity 𝑧 = + 𝑝 , a finite tem-perature transition is clearly detected in our QMC simulations,see Fig. 7. We obtain MF critical exponents, as expected fromthe 𝑑 = ∞ nature of the SW network, even in the vanishing 𝑝 limit. Nevertheless, there are notable differences with theinfinitely connected Hastings model, as we discuss now.Fig. 7 (c) shows the concentration 𝑝 dependence of the crit-ical temperature for both ferromagnetic (XY) and antiferro-magnetic (XXX) ordering transitions. QMC estimates for 𝑇 c are compared to the RPA result. We first discuss the staggeredXXX antiferromagnet (orange colors). In this case, QMC re-sults and RPA estimates are not equal, but they seemingly getcloser when 𝐽 (cid:48) ( 𝑝 ) →
0. Following similar ideas developed inRefs. [44, 45, 54–56], we introduce a renormalization param-eter 𝛼 , such that the exact critical temperature follows fromthe RPA formula, with a 𝑝 -dependent renormalization of theaverage long-range coupling 𝐽 (cid:48) ( 𝑝 ) , 𝑇 c = 𝜒 − (cid:32) 𝛼𝐽 (cid:48) ( 𝑝 ) (cid:33) . (4.10)Remarkably, we observe in the inset of Fig. 7 (c) that 𝛼 in-creases towards unity when 𝑝 →
0, thus making the RPAresult asymptotically exact in this extreme limit.The XY model shows a strikingly different trend. Indeed,while the RPA behavior 𝑇 c ∼ ( 𝐽 (cid:48) ( 𝑝 )/ 𝐽 ) / gives a reasonabledescription of the exact QMC data at intermediate couplingstrengths, this is no longer the case when 𝐽 (cid:48) ( 𝑝 )/ 𝐽 (cid:46) . Δ = 𝑝 ★ ≈ .
11 below whichthe average distance between two shortcuts 𝜁 𝑝 becomes largerthan the 1D correlation length value at the RPA temperature 𝜉 ( 𝑇 RPAc ) . As predicted, we clearly observe a downturn for 𝑇 c towards the linear behavior Eq. (3.10) shown by a dashed linein Fig. 7 (c). This crossover is a direct consequence of theunderlying Luttinger liquid behavior which allows to break the
10 0 100 . . . . . ⌦ < ↵ c o rr V M F / ˜ a + l = = = = =
10 0 10 C / ˜ a + q / ˜ a . . . . ⌦ < ↵ s u m V M F / ˜ a + l = = = = = .
04 0 .
06 0 . ) / . . ⌦ < ↵ c o rr V M F / ˜ a .
04 0 .
06 0 . ) / ⌦ < ↵ s u m V M F / ˜ a RPA (Analytical) .
01 0 . ( ? )/ . . ) c / QMCFerromagnetic XYStaggered AF Heisenberg∝ J ′ ( p ) / J .
01 0 . ( ? )/ . . . U ( a )( b ) ( c ) FIG. 7. Data collapse for the XY ferromagnet on the SW networkwith 𝑝 = . a ) the normalized structure factor and ( b ) the spin-spin correlation atlong distance. Setting ˜ 𝜈 = 𝛽 MF = /
2, we find that 𝑇 c (cid:39) . c ) Criticaltemperature of the ferromagnetic XY and staggered antiferromagneticHeisenberg models versus the average strength of the extra couplings.In each case, the analytical RPA estimate is also displayed. In the stag-gered antiferromagnetic Heisenberg model, the two estimates agreeas 𝐽 (cid:48) ( 𝑝 ) →
0. We plot in the inset the renormalization parameter 𝛼 (see text). As 𝐽 (cid:48) ( 𝑝 ) →
0, one sees that 𝛼 →
1. In the ferromagneticXY case, the RPA and QMC estimates deviate below 𝐽 (cid:48) ( 𝑝 )/ 𝐽 ∼ . ∝ 𝐽 (cid:48) ( 𝑝 )/ 𝐽 . condition of Eq. (2.4): here 𝜂 >
1, with 𝜂 = arccos (− Δ )/ 𝜋 ≤ 𝑑 = 𝑝 →
0. Therefore one should expect, in principle, to observe asimilar crossover towards the MF expression Eq. (3.10) for thestaggered XXX antiferromagnet. However, this effect is clearlyout of reach since it would theoretically occur for 𝑝 ★ ≈ − . D. Influence of a spin gap
We finally investigate a dimerized antiferromagnetic chain,governed by the following Heisenberg Hamiltonian, H = 𝐽 ∑︁ 𝑁𝑖 = (cid:2) + 𝛿 (− ) 𝑖 (cid:3) 𝑺 𝑖 · 𝑺 𝑖 + . (4.11)In contrast with the previous study, here the ring has a gappedground-state, with a finite 𝑇 = 𝜒 𝜋 saturatesat low temperature to a finite value, 𝜒 𝜋 , as visible in Fig. 8(inset) for a dimerization parameter 𝛿 = . 𝑇 divergence for 𝜒 𝜋 should imply a critical coupling 𝐽 (cid:48) c = / 𝜒 𝜋 , below which 𝑇 c =
0. This is well known for instance inthe case of coupled Haldane chains [86, 87]. Here we performeQMC simulations of the Hastings model with extra couplingsof varying strength 𝐽 (cid:48) / 𝑁 for a dimerization parameter 𝛿 = . 𝑇 c estimates are reported in Fig. 8, together with theRPA result, obtained using 𝑇 RPAc = 𝜒 − 𝜋 ( / 𝐽 (cid:48) ) when a solutionexists, and 𝑇 c = 𝐽 (cid:48) / 𝐽 >
1. The 𝑇 = 𝐽 (cid:48) c ≈ .
53 is alsoperfectly captured by the RPA treatment.
RPA (QMC)Hastings (QMC) . . . . . ( ? )/ . . . . . ) c / .
01 0 . ) / j c = = = = = FIG. 8. Critical temperature of the dimerized antiferromagneticmodel Eq. (4.11) with 𝛿 = .
25 and additional long-range couplingsof the Hastings form of strength 𝐽 (cid:48) ( 𝑝 ) . The RPA estimate comparesperfectly to the QMC results. Inset: Temperature dependence of thestaggered susceptibility of the 1D dimerized system. V. SUMMARY AND CONCLUSION
In this work, building on mean-field theory and extensivequantum Monte Carlo simulations, we investigated interactingquantum spins on small-world networks. Starting from 1Drings, we considered two situations: all-to-all interacting andlong-range interactions randomly added. The effective infinitedimension of the lattice leads to a magnetic ordering at finitetemperature 𝑇 c with mean-field criticality.First, we showed that different mean-field treatments led todifferent power-law behaviors for the scaling of 𝑇 c versus theaverage strength 𝐽 (cid:48) of the extra couplings. The first approachis controlled by a competition between a characteristic lengthscale of the small-world network and the thermal correlationlength of the underlying 1D system, and leads to 𝑇 c ∝ 𝐽 (cid:48) . Theother approach is based on the random phase approximation.For a critical 1D system with anomalous exponent 𝜂 , it gives 𝑇 c ∝ 𝐽 (cid:48) /( − 𝜂 ) .Before confronting these approximate treatments with un-biased quantum Monte Carlo simulations of the problem,we compared analytical RPA based on low-energy physicswith numerical RPA in order to quantitatively define the low-temperature limit of the analytical approaches. By computingthe transverse susceptibility of the XY chain and the staggeredsusceptibility of the Heisenberg chain with exchange coupling 𝐽 , we found that the analytical low-energy approaches becomeasymptotically exact for 𝑇 (cid:46) 𝐽 .Starting with the all-to-all interacting system (Hastingsmodel), we first checked that the transition belonged to themean-field universality class. Because of the effective infi-nite dimensionality of the system, the correlation length expo-nent is rescaled as ˜ 𝜈 → 𝜈 MF 𝑑 u with 𝜈 MF = / 𝑑 u = 𝑇 c scales according to the random phaseapproximation versus the average strength 𝐽 (cid:48) of the extra cou-plings.We then considered the system with long-range interactions,randomly added with a finite probability 𝑝 . Similarly to theHastings model, we checked that its criticality belongs to themean-field universality class. However, we found in this casethat the critical temperature shows both the 𝑇 c ∝ 𝐽 (cid:48) /( − 𝜂 ) and 𝑇 c ∝ 𝐽 (cid:48) scalings, with a crossover from one to the other. As 𝐽 (cid:48) is reduced, the linear scaling takes over the RPA behavior whenthe characteristic length scale of the network becomes largerthan the 1D thermal correlation length at the RPA temperature.Finally, we investigated the fate of a gapped 1D spin chainagainst the small-world effect by considering the dimerizedspin-half Heisenberg chain. We found that the gap of the1D system leads to a critical value 𝐽 (cid:48) c for magnetic ordering.Beyond 𝐽 (cid:48) c , the critical temperature behavior behavior is well-captured by the RPA estimate.For future work, it would be interesting to investigate howthe order parameter at zero temperature responds to the small-world effect. For instance, a TLL-based approach predicts that (cid:104) 𝑚 (cid:105) ∝ 𝐽 (cid:48) 𝜂 /( − 𝜂 ) [46, 47, 57, 88, 89], but as for the scaling of the critical temperature, one might expect a crossover towardsanother MF regime as a function of 𝐽 (cid:48) .Besides, it is very stimulating to think of the small-worldeffect in the presence of disorder [90]. It has been found tohost unusual physics for non-interacting fermions [91, 92],and it would be interesting to study the problem for interact-ing quantum systems, similarly to what has been done on theCayley tree [93] for bosons in a random potential. Randomexchange spin systems also offer a very promising platform, inparticular to explore the issue of infinite randomness critical-ity [94] against the small-world effect [95]. We further notethat magnetic frustration, occurring in the long-range interac-tions across the ring [96] for instance, is another fascinatingroute where one could find more exotic quantum phases ofmatter. Finally, it is fascinating to observe that small-worldquantum magnets are now available in experiments, with forinstance all-to-all spin models, or tree-like tunable Heisenberg-type systems which can be realized in cold atom setups coupledto an optical cavity [97–99]. ACKNOWLEDGMENTS
We are grateful to G. Lemarié for very fruitful discussions.M.D. was supported by the U.S. Department of Energy, Of-fice of Science, Office of Basic Energy Sciences, MaterialsSciences and Engineering Division under Contract No. DE-AC02-05-CH11231 through the Scientific Discovery throughAdvanced Computing (SciDAC) program (KC23DAC Topo-logical and Correlated Matter via Tensor Networks and Quan-tum Monte Carlo). N. L. acknowledges the French NationalResearch Agency (ANR) under Projects THERMOLOC ANR-16-CE30-0023-02, and GLADYS ANR-19-CE30-0013. Thisresearch used the Lawrencium computational cluster resourceprovided by the IT Division at the Lawrence Berkeley Na-tional Laboratory (supported by the Director, Office of Sci-ence, Office of Basic Energy Sciences, of the U.S. Departmentof Energy under Contract No. DE-AC02-05CH11231). Thisresearch also used resources of the National Energy ResearchScientific Computing Center (NERSC), a U.S. Department ofEnergy Office of Science User Facility operated under Con-tract No. DE-AC02-05CH11231. We also acknowledge theuse of HPC resources from CALMIP (grants 2019-P0677 and2020-P0677) and GENCI (grant x2020050225).
Appendix A: Chain mean-field theory and random phaseapproximation1. Chain mean-field theory
We recall the basic idea of treating the long-range part inmean-field. One looks at the fluctuations around the averagevalue of the operators in the long-range part by making thesubstitution, 𝑆 𝑥,𝑦,𝑧𝑖 = (cid:10) 𝑆 𝑥,𝑦,𝑧𝑖 (cid:11) + (cid:16) 𝑆 𝑥,𝑦,𝑧𝑖 − (cid:10) 𝑆 𝑥,𝑦,𝑧𝑖 (cid:11)(cid:17) . (A1)0Neglecting quadratic terms and up to an irrelevant constant,one gets the following effective 1D Hamiltonian, H XXXeff = 𝐽 ∑︁ 𝑖 𝑺 𝑖 · 𝑺 𝑖 + + ∑︁ 𝑖, 𝑗 𝐽 LR 𝑖 𝑗 (cid:10) 𝑆 𝑧𝑗 (cid:11) 𝑆 𝑧𝑖 , (A2)in the staggered antiferromagnetic Heisenberg case ( Δ = 𝑧 spin component. For the ferromag-netic XY model ( Δ = 𝑥𝑦 plane. We suppose it is along the 𝑥 spin component and obtainthe effective 1D model as follows, H XYeff = 𝐽 ∑︁ 𝑖 (cid:16) 𝑆 𝑥𝑖 𝑆 𝑥𝑖 + + 𝑆 𝑦𝑖 𝑆 𝑦𝑖 + (cid:17) + ∑︁ 𝑖, 𝑗 𝐽 LR 𝑖 𝑗 (cid:10) 𝑆 𝑥𝑗 (cid:11) 𝑆 𝑥𝑖 . (A3)In both cases, the idea is to assume symmetry breaking. Con-sidering an homogeneous system, one can replace (cid:10) 𝑆 𝑥,𝑧𝑗 (cid:11) bythe corresponding order parameter (cid:104) 𝑚 (cid:105) . In the staggered anti-ferromagnetic Heisenberg case, H XXXeff = 𝐽 ∑︁ 𝑖 𝑺 𝑖 · 𝑺 𝑖 + + (cid:104) 𝑚 (cid:105) ∑︁ 𝑖, 𝑗 𝐽 LR 𝑖 𝑗 (− ) 𝑗 𝑆 𝑧𝑖 , (A4)where the factor (− ) 𝑗 comes from the fact that (cid:104) 𝑚 (cid:105) = (− ) 𝑗 (cid:10) 𝑆 𝑧𝑗 (cid:11) . In the XY case, the order is ferromagnetic, H XYeff = 𝐽 ∑︁ 𝑖 (cid:16) 𝑆 𝑥𝑖 𝑆 𝑥𝑖 + + 𝑆 𝑦𝑖 𝑆 𝑦𝑖 + (cid:17) + (cid:104) 𝑚 (cid:105) ∑︁ 𝑖, 𝑗 𝐽 LR 𝑖 𝑗 𝑆 𝑥𝑖 . (A5)
2. Random phase approxmation
The linear response to a tiny symmetry-breaking field ℎ sb coupled to the order parameter operator 𝑚 takes the form, (cid:104) 𝑚 (cid:105) = 𝜒 (cid:0) 𝑇 (cid:1) ℎ sb , (A6)where 𝜒 ( 𝑇 ) is the susceptibility. Neglecting possible inho-mogeneities in the SW branching by using the fact that the aver-age strength of extra-couplings across the ring is 𝐽 (cid:48) ( 𝑝 ) = 𝑝𝐽 (cid:48) and including explicitely the symmetry-breaking field for theantiferromagnetic XXX model of Eq. (A4), one gets, H XXXeff = 𝐽 ∑︁ 𝑖 𝑺 𝑖 · 𝑺 𝑖 + + (cid:104) 𝑚 (cid:105) 𝐽 (cid:48) ( 𝑝 ) ∑︁ 𝑖 (− ) 𝑖 𝑆 𝑧𝑖 + ℎ sb ∑︁ 𝑖 (− ) 𝑖 𝑆 𝑧𝑖 , (A7) and similarly for the XY case of Eq. (A5), H XYeff = 𝐽 ∑︁ 𝑖 𝑺 𝑖 · 𝑺 𝑖 + + (cid:104) 𝑚 (cid:105) 𝐽 (cid:48) ( 𝑝 ) ∑︁ 𝑖 𝑆 𝑥𝑖 + ℎ sb ∑︁ 𝑖 𝑆 𝑥𝑖 . (A8)Within the chain mean-field approach, the total effective sym-metry breaking field is ℎ sb + 𝐽 (cid:48) ( 𝑝 )(cid:104) 𝑚 (cid:105) . Therefore, (cid:104) 𝑚 (cid:105) = 𝜒 ( 𝑇 ) (cid:16) ℎ sb + 𝐽 (cid:48) ( 𝑝 )(cid:104) 𝑚 (cid:105) (cid:17) . (A9)Isolating the order parameter from Eq. (A9), one gets, (cid:104) 𝑚 (cid:105) = 𝜒 ( 𝑇 ) − 𝐽 (cid:48) ( 𝑝 ) 𝜒 ( 𝑇 ) ℎ sb = 𝜒 RPA ( 𝑇 ) ℎ sb . (A10)Because magnetic ordering occurs at 𝑇 = 𝑇 RPAc with a diver-gence of the susceptibility, one finds the condition, 𝜒 (cid:16) 𝑇 RPAc (cid:17) = (cid:46) 𝐽 (cid:48) ( 𝑝 ) . (A11)By inverting it, one can get the RPA estimate of the criticaltemperature 𝑇 RPAc . Appendix B: Fitting parameters for the scaling functions
Following the scaling analysis including corrections to thesccaling (see Sec. IV B 2), the fitting parameters for the datacollapse of Figs. 5 and 7 are reported in Tab. I.
Quantity 𝑇 c 𝜔 𝜙 𝑏 𝑐 Staggered AF Heisenberg (Hastings, 𝐽 (cid:48) / 𝐽 = . 𝑄 (Binder) 0 . ( ) . ( ) . ( ) − ( ) ( )(cid:104) 𝑚 (cid:105) corr . ( ) . ( ) . ( ) − . ( ) . ( )(cid:104) 𝑚 (cid:105) sum . ( ) . ( ) . ( ) ( ) ( ) XY Ferromagnet (Small-world, 𝑝 = . (cid:104) 𝑚 (cid:105) corr . ( ) . ( ) . ( ) ( ) − ( )(cid:104) 𝑚 (cid:105) sum . ( ) . ( ) . ( ) ( ) ( ) TABLE I. Fitting parameters for the data collapse of Figs. 5 and 7.See Sec. IV B 2 for a definition of the different parameters.[1] J. Scott, Sociology , 109 (1988).[2] D. J. Watts and S. H. Strogatz, Nature , 440 (1998).[3] A.-L. Barabási and R. Albert, Science , 509 (1999).[4] A. Barrat and M. Weigt, Eur. Phys. J. B , 547 (2000).[5] S. H. Strogatz, Nature , 268 (2001).[6] M. Girvan and M. E. J. Newman, PNAS , 7821 (2002).[7] R. Albert and A.-L. Barabási, Rev. Mod. Phys. , 47 (2002).[8] A. Barrat, M. Barthélemy, R. Pastor-Satorras, and A. Vespig-nani, PNAS , 3747 (2004).[9] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. 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