Abstract
The MerminWagner theorem states that spontaneous continuous symmetry breaking is prohibited in systems with shortrange interactions at spatial dimension D ≤ 2. For longrange interactions with a powerlaw form (1/r^{α}), the theorem further forbids ferromagnetic or antiferromagnetic order at finite temperature when α ≥ 2D. However, the situation for α ∈ (2, 4) at D = 2 is not covered by the theorem. To address this, we conduct largescale quantum Monte Carlo simulations and field theoretical analysis. Our findings show spontaneous breaking of SU(2) symmetry in the ferromagnetic Heisenberg model with 1/r^{α}form longrange interactions at D = 2. We determine critical exponents through finitesize analysis for α < 3 (above the upper critical dimension with Gaussian fixed point) and 3 ≤ α < 4 (below the upper critical dimension with nonGaussian fixed point). These results reveal new critical behaviors in 2D longrange Heisenberg models, encouraging further experimental studies of quantum materials with longrange interactions beyond the MerminWagner theorem’s scope.
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Introduction
In recent years, the importance of the studies on longrange (LR) lattice models have been gradually noticed, due to the fact that they exhibit intrinsically different properties from their shortranged (SR) counterparts. For example, LR Heisenberg models at spatial dimension D = 2 acquires anomalous magnon dispersion different from the linear and quadratic spinwaves in the SR antiferromagnetic and ferromagnetic models^{1,2}. In addition, the violation of MerminWagner theorem and unconventional critical properties in LR systems also attracted much attention in investigations of both quantum spin models and interacting fermionic models^{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23}.
These phenomena also have immediate experimental relevance. Due to the fast development in the Rydberg atom arrays^{24,25,26,27,28}, the magic angle twisted bilayer Graphene and other 2D quantum moiré materials^{29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66} and the programmable quantum simulators^{67,68} such as quantum gases coupled to optical cavities^{69}. LR interactions in the forms of van der Waals, dipoledipole and Coulomb have given rise to a plethora of correlated topological and quantum phases of matter beyond the semiclassical or meanfield type descriptions, and new theoretical paradigm that could cope with these fast emergent experimental facts are critically called for.
One particularly interesting direction is to explore the critical properties of phase transitions with continuous symmetry breaking, outside the realm of the established MerminWagner theorem. For 1D LR antiferromagnetic Heisenberg chain^{23} and Heisenberg ladders^{14} with 1/r^{α}form LR interactions, the phase diagram as well as the critical exponents have been addressed and it has been found that there is an upper critical value α_{c} above which there is no phase transitions for these systems. Below α_{c}, the transition exists and the critical exponents are dependent on α, as identified by both field theory analysis and numerical evidence. However, for 2D LR Heisenberg models with finitetemperature transitions, it was only known that, for D = 2 Heisenberg model with ferromagnetic LR interaction 1/r^{α}, a finitetemperature ferromagnetic phase will not exist when α ≥ 4 which has been proved analytically in Ref. ^{70}, and for α ≤ 2 the system is gapped due to the generalized Higgs mechanism^{1,2} and the finitetemperature ferromagnetic order should be allowed. However, the situation in α ∈ (2, 4) is not well understood. Although there are classical field theory predictions and renormalization group analysis on this issue^{3,4,7}, which state there is a Gaussian fixpoint for 2 < α < 3 and a nonGaussian fixedpoint for 3 ≤ α < 4, a thorough numerical treatment on the 2D quantum Heisenberg model has not been performed to date. Such unbiased numerical analysis of this model is crucial not only because the fieldtheory scenario needs to be impartially examined on the realistic lattice models, but also due to the fact that the Heisenberg model is one of the most central toy models in condensed matter and statistic physics and a complete clarification of the critical properties of this model will serve as the cornerstone of further studies on LR quantum many body systems.
Here we bridge these gaps by largescale QMC simulations and field theory analysis. We find clear evidence of the breakdown of the MerminWagner theorem with finitetemperature phase transitions in α ∈ (2, 4), as shown in Fig. 1. By performing the stateoftheart finitesize scaling analysis, as illustrated in Fig. 2, we obtain the accurate critical exponents of the phase transition as a function of α as shown in Fig. 3, and demonstrate these results nicely satisfy the fieldtheory predictions both for α < 3 where the system is above the upper critical dimension with Gaussian fixed point and for 3 ≤ α < 4 where the system is below the upper critical dimension with nonGaussian fixed point. Our results explicitly show the critical behaviors for α ∈ (2, 4) in LR Heisenberg model at D = 2 and will intrigue further theoretical and experimental physics and even mathematics studies of systems with LR interactions beyond the realm of the MerminWagner theorem^{3,4,5,6,7,8,71}.
Results
Model
The Hamiltonian of the LR ferromagnetic Heisenberg model is
where \({J}_{ij}=\frac{1}{{r}_{ij}^{\alpha }}\) denotes the LR coupling and r_{ij} is the nearest distance between site i and site j under the periodic boundary condition. In order to alleviate the strong finitesize effects in systems with LR interactions arising from the cutoff of LR interactions under the periodic boundary condition, we replace J_{ij} with the Ewaldcorrected coupling \({\tilde{J}}_{ij}\)^{19,72} which takes the form of
This modified coupling parameter \({\tilde{J}}_{ij}\) counts all the possible distances between two sites under the periodic boundary condition, so that the effect of cutting off the tail of LR interactions is minimized, and this trick has been shown to be very useful in the simulation of many LR systems^{14,18,19,72}. For 2D there is no closed form for Eq. (2), so we truncate the summation at ∣m∣_{max}, ∣n∣_{max} = 1000 for α < 3 which is large enough to have the wellconverged finitesize scaling behavior, as shown in Fig. 2. For α ≥ 3 the finitesize effects are mainly from crossovers to SR case, and we find the original coupling J_{ij} is fine to obtain converged results.
When α ≥ 2D the system reduces to the SR case where there is no spontaneously continuous symmetry breaking phase at finitetemperature. When α ≤ D, the Hamiltonian is no longer extensive and there is no welldefined thermodynamic limit. Between α ∈ (2, 4) we carry out the QMC simulations^{73,74,75} up to the linear system size of L = 256, as shown in Fig. 2, to determine the precise phase boundary as well as the critical exponents ν, β, and η. Note that because of strong finitesize effects, we only compute the region of α ∈ [2.3, 3.7] where our QMC simulations can obtain wellconverged results. The origins of finitesize effects as α approaches the two boundaries, α = 2 and α = 4, exhibit inherent distinctions. When α → 2, the finitesize effect arises from the escalating intensity of LR (longrange) interactions, which fundamentally reduces the efficiency of the Ewaldcorrected scheme. Conversely, as α → 4, the system approaches the regime where finitetemperature phase transitions do not exist. Consequently, near this boundary, the convergence of data points becomes exceedingly slow to be overcome. The results are shown in Figs. 1 and 3 and will be discussed in the critical exponents section. The QMC implementation is explained in the Supplementary Note 1.
Note that when α ≤ D, the Hamiltonian defined in Eq. (1) can actually be Kacnormalized^{10,76} to be extensive with the addition of a factor \(\frac{N1}{{\sum }_{i < j}{J}_{ij}}\) to the Hamiltonian. Although this is not the focus of our paper, we examine the Kacnormalized Hamiltonian and the results are shown in Supplementary Note 2.
Critical exponents
Figure 2 shows our results at α = 2.5. We first use the crossing points of the Binder ratios to locate the critical temperature T_{c}. The crossing points of U(T, L) with U(T, 2L) are denoted as T^{*}(L), and through fitting to Eq. (9) the precise value of T_{c} can be obtained. We then use the value of T_{c} to perform data collapse according to Eq. (10) and Eq. (13) separately for 3 ≤ α < 4 and α < 3, to obtain the critical exponents \({\nu }^{{\prime} }\) and β. To obtain the anomalous dimension η_{Q}, we measure the correlation function G(L/2) at the obtained critical temperature T_{c} and obtain the anomalous dimension separately by fitting to Eq. (11) for 3 ≤ α < 4 and Eq. (14) for α < 3.
According to the conventions defined in Eq. (16) and field theory results of the meanfield critical exponents in Eq. (7), we can extract the expression for the three critical exponents in the Gaussian region which are \({\nu }^{{\prime} }=1\), \(\beta =\frac{1}{2}\) and η_{Q} = 1. Outside the Gaussian region, we have η = 4 − α and γ defined in Eq. (8), and the value of β and ν can be obtained via solving the scaling relations between the critical exponents with \(\nu =\frac{\gamma }{2\eta }\) and \(\beta =\frac{\gamma \eta }{2(2\eta )}\).
The critical exponents we have obtained are shown in Fig. 3. We find that within the region we simulated, our QMCobtained critical exponents \({\nu }^{{\prime} }(\alpha ),\beta (\alpha )\), and η_{Q}(α) match nicely with the prediction of both LR Gaussian theory (for α < 3) and the twoloop perturbative RG (for 3 ≤ α < 4), although there is a sign of deviating from twoloop RG predictions when α approaches 4. The possible deviation might be explained by the increasing finitesize effects near the boundary or the inefficiency of twoloop perturbative RG predictions when α is away from α = 3. The results can be further improved by either considering higherorder RG corrections or by pushing the QMC simulations to larger system sizes. Notably, the predicted form of anomalous dimension η receives no corrections at any α ∈ (2, 4)^{3} and our results confirm this argument with η matching with η = 4−α well in the whole region.
Discussions
Our investigation reveals a finitetemperature phase transition point in the 2D LR Heisenberg model, occurring for values of α within the range of α ∈ (2, 4), which separates the ferromagnetic phase from the paramagnetic phase. We observe that the phase transition point exhibits distinct behaviors: a Gaussian fixed point characterizes the transition for α ≤ 3, while a nonGaussian fixed point emerges for 3 < α < 4. Similar phenomena have been observed in various LR systems^{6,7,8,9,10,14,18,19,23}. However, it is important to note that LR Isinglike systems differ intrinsically from LR Heisenberglike systems. The former does not adhere to the MerminWagner theorem, guaranteeing a finitetemperature transition for all α > 0, while the latter exhibits an upper critical value α_{c} beyond which the MerminWagner theorem precludes the existence of phase transitions. In conclusion, our results clearly point out the LR quantum manybody system exhibit unconventional critical properties beyond the realm of the MerminWagner theorem, which are also worthwhile to pursue in future experimentalrealizations, such as the quantum simulators.
Methods
Field theory analysis
We review here the field theory description of the model at the thermodynamic limit dating back to Ref. ^{3}. The action can be written as
to match the lattice model, we need α = d + σ. Under the scaling symmetry
the kinetic term remains unchanged when \({{{\Delta }}}_{\phi }=\frac{D\sigma }{2}\). The coupling constant of ϕ^{4} interaction, on the other hand, scales as
When \(\alpha \,< \frac{3D}{2}\), the coupling constant decays at larger length scale, which means the λϕ^{4} term is an irrelevant operator. The Gaussian fixed point at λ = 0 is a stable fixed point. Notice when λ = 0, the action is in a purely quadratic form, hence named “Gaussian" fixed point. This was established mathematically in Ref. ^{6}. When \(\alpha \,> \frac{3D}{2}\), the λϕ^{4} term becomes relevant, which triggers a renormalization group towards a different nonGaussian fixed point^{3}. One can perform standard renormalization technique to calculate the scaling dimension of various operators, by evaluating Feynman diagrams with nonconventional propagators. Such a calculation was first performed in^{3}. Since the kinetic term in Eq. (3) is nolocal, which can not receive corrections from any local counter terms, the scaling dimension of ϕ will not be renormalized (This can be easily seen by analyzing the CallanSymanzik equation for the twopoint function 〈ϕ(x)ϕ(y)〉, see for example, Ref. ^{77}). Equivalently, we have η = 2Δ_{ϕ} − D + 2. Our numerical result clearly confirms such a theoretical prediction. For a fixed σ in Eq. (3), we can define the upper critical dimension as the spacetime dimension at which the ϕ^{4} term is marginal. The \({{{\Delta }}}_{{\phi }^{4}}=4{{{\Delta }}}_{\phi }={D}_{uc}\) gives us
We now focus on the D = 2 case. When α < 3, the critical behavior is controlled by the λ = 0 Gaussian fixed point. The critical behavior is similar to the usual Ising model at D > 4, due to the effect of dangerously irrelevant operators^{78}, the critical exponents are given by
For example, the β = 1/2 exponent can be seen from the following argument. Deform the action (3) by a mass term ∫dx^{D}tϕ(x)^{2} with negative t and minimize the potential, we get 〈ϕ〉 ∝ (−t/λ)^{β}, with β = 1/2. The other exponents can be calculated by similar mean field theory analysis. The critical exponent η controls the two point function 〈ϕ(x)ϕ(y)〉 only at the strict thermodynamic limit. At finite sizes, the power law behavior will be modified to (14), which follows from analysing the effect of dangerously irrelevant operators carefully^{79}.
When α > 3, on the other hand, the second term in Eq. (3) becomes relevant, and renormalization group flows towards a different nonGaussian fixed point^{3}. The critical exponent η will remain at its mean field theory value^{3} as in Eq. (7). The other exponents, on the other hand receives correction at \({{{\mathcal{O}}}}\left({(\alpha 3)}^{2}\right)\). The twoloop perturbation results for γ is
with \(Q(\sigma )=\sigma \left[\psi (1)2\psi \left(\frac{1}{2}\sigma \right)+\psi (\sigma )\right]\) where ψ(z) is the logarithmic derivative of the gamma function. The other critical exponents can be obtained by scaling relations between them.
When α > 4, the longrange model becomes equivalent to shortrange models, due to the MerminWagner theorem^{80,81,82}, the system will be gapped at finitetemperature. In the fieldtheory language, the value of α at which such a longrange to shortrange crossover happens when the scaling dimension of ϕ equals to the scaling dimension of the short range model. In two dimensions, this gives α = 4^{3,4}.
Finitesize scaling analysis
To identify the phase transitions and obtain the critical exponents, we compute the square magnetization 〈m^{2}〉, the correlation function G(r), and the Binder ratio \(U(T,L)=\frac{5}{2}(1\frac{1}{3}\frac{\langle {m}^{4}\rangle }{{\langle {m}^{2}\rangle }^{2}})\) in the QMC simulation. The crossing point of U(T, L) with U(T, 2L) is denoted as T^{*}(L) and it is expected to converge to the thermodynamic limit critical temperature T_{c} following the scaling relation:
Given the values of T*(L) with sufficiently small errors and large enough system sizes L, the critical point T_{c} can be precisely located as shown in Fig. 2. To obtain the critical exponents ν, β and η, when D ≤ D_{uc}, the standard finitesize scaling behavior (FSS)^{3,83} allows us to perform a data collapse near the critical points with the relation
The anomalous dimension can also be obtained by fitting to the correlation function at the critical point T_{c}
However, when D > D_{uc}, which is our case when α < 3, the system enters the meanfield region where the hyperscaling relation breaks down, famously due to the effect of dangerously irrelevant operator^{79,84,85}. The scaling of the correlation length in this region shall follow the relation \({\xi }_{L} \sim {L}^{\frac{{D}_{{{\rm{uc}}}}}{D}}\) instead of ξ_{L} ~ L^{10,18,19,79,84,85,86}, and this leads to the modification of hyperscaling relation with
where \({\nu }^{{\prime} }=\frac{{D}_{{{\rm{uc}}}}}{D}\nu\) and α_{H} is the critical exponent associated with the specific heat. For our system Eq. (1), the upper critical dimension is D_{uc} = 2(α − D), which we will explain later in the field theory analysis section. Accordingly, Eq. (10) also needs to be modified and the correct relation for data collapse in mean field region is^{10,18,19,79,84}
The scaling of correlation function for α < 3 is also modified with
where
By fitting to Eq. (14), the modified anomalous dimension η_{Q} as well as η can be obtained.
To unify the conventions, we define
and
Then \({\nu }^{{\prime} }\), β and η_{Q} will be obtained with the same scaling functions for both α < 3 and 3 ≤ α < 4.
Data availability
The data that support the findings of this study are available from the corresponding author upon request to the authors.
Code availability
All numerical codes in this paper are available upon request to the authors.
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Acknowledgements
We thank Subir Sachdev, Fabien Alet, Fakher Assaad, Kai Sun, Michael Scherer and Lukas Janssen for valuable discussions on the related topic. J.R.Z. thanks Mr. Tianyu Wu and Ms. Zhenzhi Qin for valuable discussions. J.R.Z., M.H.S., and Z.Y.M. acknowledge the support from the Research Grants Council (RGC) of Hong Kong SAR of China (Project Nos. 17301420, 17301721, AoE/P701/20, 17309822, HKU C703722G), the ANR/RGC Joint Research Scheme sponsored by RGC of Hong Kong and French National Research Agency (Project No. A_HKU703/22), the K.C. Wong Education Foundation (Grant No. GJTD202001) and the Seed Fund “QuantumInspired explainableAI” at the HKUTCL Joint Research Centre for Artificial Intelligence. The authors also acknowledge the TianheII platform at the National Supercomputer Center in Guangzhou, the HPC2021 system under the Information Technology Services and the Blackbody HPC system at the Department of Physics, University of Hong Kong for their technical support and generous allocation of CPU time.
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J.Z., Z.Y.M., and J.R. initiated the work. J.Z. carried out the Quantum Monte Carlo simulations. Juncheng Rong conducted the field theory analysis. All authors contributed to the analysis of the results and the preparation and revision of the draft.
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Zhao, J., Song, M., Qi, Y. et al. Finitetemperature critical behaviors in 2D longrange quantum Heisenberg model. npj Quantum Mater. 8, 59 (2023). https://doi.org/10.1038/s41535023005916
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DOI: https://doi.org/10.1038/s41535023005916
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