Spectral damping without quasiparticle decay: The dynamic structure factor of two-dimensional quantum Heisenberg antiferromagnets
SSpectral Damping without Quasiparticle Decay: The Dynamic Structure Factorof Two-Dimensional Quantum Heisenberg Antiferromagnets
Matthew C. O’Brien
1, 2, ∗ and Oleg P. Sushkov † School of Physics, The University of New South Wales, Sydney, New South Wales 2052, Australia Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA (Dated: July 28, 2020)Two-dimensional Heisenberg antiferromagnets play a central role in quantum magnetism, and yetthe nature of dynamic correlations in these systems at finite temperature has remained poorly un-derstood for decades. We solve this problem by using a novel quantum-classical duality to calculatethe dynamic structure factor analytically and find a broad frequency spectrum despite the very longquasiparticle lifetime. The solution reveals new multi-scale physics to which the conventional renor-malization group approach is blind. We also challenge the common wisdom on static correlationsand perform Monte Carlo simulations which demonstrate excellent agreement with our theory.
Keywords: antiferromagnets; lifetimes & widths; finite temperature field theory; path-integral Monte Carlo
The dynamic structure factor encodes the fundamentalphysical processes involved in the response of a systemto an external probe, and is the most common experi-mental observable in studies of magnetic systems, deter-mined, for example, using inelastic neutron [1], or reso-nant X-ray scattering spectroscopy [2]. However, theo-retical analyses of these processes which are both quanti-tatively accurate and physically insightful can be elusive.The two-dimensional quantum Heisenberg antiferromag-net (2DQHA) plays an important role in the field of quan-tum magnetism precisely because of the theoretical chal-lenges it poses in addition to its descriptive power: First,the model describes the parent compounds of cupratehigh temperature superconductors [3]. Second, while themodel supports long-range order at zero temperature, or-der is destroyed at any finite temperature [4]. Because ofthe importance of thermal fluctuations, 2DQHAs mani-fest highly non-trivial classical and quantum long-rangedynamics which are not fully understood [5, 6]. The na-ture of quantum critical points to and from quantum spinliquid phases is also a problem of intense theoretical inter-est (see Ref. [7] for a review). Somewhat surprisingly, thephysics of thermal fluctuations in isotropic 2DQHAs isclosely related to the zero temperature quantum Lifshitzphase transition between antiferromagnetically orderedstates and a spin liquid phase in systems with long-rangefrustrated interactions (e.g., the J - J model) [8].In their seminal work, Chakravarty, Halperin and Nel-son used the O (3)-symmetric nonlinear σ model (NLSM)to describe the long wavelength physics of 2DQHAs atlow temperature and argued that the spin-spin correla-tions in the so-called “renormalized classical” regime areessentially classical in nature [9]. Crucially, their analysisrelied on a quantum-classical mapping which integratesout all dynamics of the quantum model. Consequently,this approach allowed the authors to derive a scaling formfor the static structure factor but not for the dynamicstructure factor. Later studies of the dynamic structurefactor raised surprising questions. First, a direct per- turbative calculation of the magnon decay rate due toscattering from the thermal bath predicted the dynamicstructure factor should have a very narrow linewidth [10].Similarly, a 1 /N expansion of the O ( N ) NLSM predicteda narrow quasi-Lorentzian frequency distribution [11]. Incontrast, classical time-dependent numerical simulationsshowed a broad dynamic structure factor [12], and it hasso far remained unclear how to rigorously reconcile thisapparent contradiction.We resolve the long-standing discrepancy in this paperwith a novel analytical calculation of the dynamic spinstructure factor of the isotropic O ( N ≥
3) NLSM at finitetemperature. In recent works [8, 13], we demonstratedthat infrared-divergent fluctuations—either thermal orquantum—lead to the emergence of a new quantum-classical duality: when an external probe interacts withthe system, it creates a classical field which contains aninfinite number of quanta with finite total energy. Thisconcept actually originates from particle physics where itwas first developed by Bloch and Nordsieck to solve theproblem of the radiation field of accelerating electrons[14]. Since the O (2) NLSM is exactly solvable, we wereable to rigorously show that despite the infinite quasi-particle lifetime, the dynamic structure factor at nonzerotemperature is broad and non-Lorentzian [13].The O ( N ≥
3) models are not exactly solvable, andhence, the diagrammatic expansion we derived in Ref.[13] is not applicable. However, we leverage the conceptof the infrared catastrophe to develop a new analyticaltechnique and use it to show for the first time that the dy-namic spin structure factor of the O ( N ) quantum NLSMat finite temperature is very broad and non-Lorentzian.Our analysis demonstrates that this broadening is notdue to short-lived quasiparticles but instead is due to theradiation of multiple spin waves by the external probe.With this result, we also obtain the static structure fac-tor by integrating over frequency and find similaritieswith the scaling form known in the literature [6, 9, 11].However, our static structure factor has a different tem- a r X i v : . [ c ond - m a t . s t r- e l ] J u l FIG. 1. (a)
Zero temperature spin stiffness ρ and averagestaggered magnetization n of the O (3) NLSM as a functionof g = (cid:126) c/ρ b measured using MC on a 64 lattice. (b) Thedominant contribution to the dynamic structure factor is theemission by the probe (dashed) of a quasiparticle with energy ω k (dark wavy) and a second “soft” particle with energy | ω − ω k | (cid:28) ω k (light wavy). perature dependence which originates from the under-lying quantized nature of the highly-classical radiationfield. Fortunately, unlike the dynamic factor, the staticstructure factor can be calculated numerically using pathintegral quantum Monte Carlo—referred to hereafter asMonte Carlo (MC). Therefore, to confirm our result forthe static structure factor we also perform extensive MCsimulations of the O (3)-symmetric NLSM and find excel-lent agreement. Formalism.—
The long-range dynamics of 2DQHAsat low temperature can be described by the O (3) NLSMwith Lagrangian L = ( ρ / c − ( ∂ t (cid:126)n ) − ( ∇ (cid:126)n ) ], (cid:126)n =1, where ρ and (cid:126)n are the spin stiffness and staggeredmagnetization order parameter, respectively, defined atthe ultraviolet scale Λ ∼ π/b , and b is the lattice spacing[9, 15]. Quantum fluctuations are ultraviolet-divergentas a power of the momentum scale, and at a scale Λ (cid:28) Λ corresponding to several lattice spacings, reduce theorder parameter down to n = |(cid:104) (cid:126)n (cid:105)| <
1. To describephysics at the scale Λ , the quantum fluctuations can beintegrated out, leading to the low energy Lagrangian L = 12 ρ (cid:20) c ( ∂ t (cid:126)n ) − ( ∇ (cid:126)n ) (cid:21) , (cid:126)n = 1 , (1)where ρ and (cid:126)n are the spin stiffness and order param-eter normalized at Λ . Both ρ and n as a function ofthe dimensionless coupling constant g = (cid:126) c/ρ b can becalculated numerically using MC, and those of the O (3)NLSM are shown in Fig. 1(a). In this paper, we addressthe regime g < g c ≈ .
46 which describes 2DQHAs [9].For the sake of generality, we consider from hereon the O ( N )-symmetric model in terms of (cid:126)n = ( n , n , . . . , n N ),with N ≥
3, and use units of (cid:126) = c = b = 1. At zerotemperature, the ground state of the model has long-range collinear antiferromagnetic order— (cid:126)n = const—which spontaneously breaks the O ( N ) symmetry. How- ever, the Hohenberg-Mermin-Wagner theorem guaran-tees the destruction of long-range order at any finitetemperature [4, 16]. Despite this, at sufficiently lowtemperatures T (cid:28) ρ , the system remains ordered onscales up to the exponentially-large correlation length ξ ∝ exp[2 πρ/ ( N − T ] [6, 9, 11, 17]. This separationof scales implies a notion of quasi-long-range order andallows for a perturbative treatment of the NLSM on mo-mentum scales Λ satisfying ξ − (cid:28) Λ ≤ Λ . For momen-tum scales on the order of temperature to fall within thisrange, it suffices for T (cid:28) ρ . The effects of fluctuationson scales Λ ∼ T can be determined within the leadingorder of perturbation theory. However, there are twotypes of contributions governing the physics of fluctua-tions on scales Λ < T : (i) Renormalization group (RG)“running” of physical parameters due to interactions oc-curring at the same scale. (ii) Beyond RG contributionsoriginating from multi-scale interactions.The RG contributions are well-understood [5, 6, 9], sowe only summarize the general principles. The unit vec-tor constraint of Eq. (1) generates interactions betweenthe components of (cid:126)n , leading to renormalization of thespin stiffness— ρ → ρ Λ —and fields— (cid:126)n → (cid:126)n Λ = Z / (cid:126)n ,where Z is the quasiparticle residue. To one-loop accu-racy at the momentum scale Λ < T < Λ , ρ Λ = ρ − ( N − T π log T Λ , (2a) Z Λ = 1 n (cid:18) ρρ Λ (cid:19) N − N − . (2b)The ultraviolet cutoff for the fluctuations in (2a) is thetemperature T rather than Λ due to the bosonic statis-tics of the quasiparticles; this is an important quantumcorrection to classical thermodynamics [18]. In Supple-mental Material (SM) Sec. I, we give a derivation of (2)and show that higher-loop contributions are negligiblewhen T (cid:28) ρ [19]. Dynamic structure factor.—
The dynamic structurefactor (DSF) is the Fourier transform of the order param-eter correlation function (cid:104) n i ( r , t ) n i (0) (cid:105) [20], and is inde-pendent of the polarization index i due to the absence oflong-range order at finite temperature. Expanding in aspectral representation in the basis of excited quasipar-ticle Fock states | α (cid:105) and | β (cid:105) yields [21], S ( k , ω ) = (cid:88) α,β e − ω β /T Z |(cid:104) α | n i (0) | β (cid:105)| × (2 π ) δ ( ω − ω α + ω β ) δ (2) ( k − k α + k β ) , (3)where Z is the quantum partition function, and ω α and k α are the energy and momentum of the state | α (cid:105) . Inthis paper we always work with ω >
0, since (3) impliesthat S ( k , − ω ) = e − ω/T S ( k , ω ).First, we account for RG contributions to the DSF byrenormalizing the fields in (3) at the scale of the incomingmomentum k , so that n i k = n i /Z / k ; since ξ − (cid:28) k < T ,local order exists at this scale. The absence of long-rangeorder means that the single magnon intermediate statedoes not contribute to (3) [13]. Therefore, we eliminatethe (cid:126)n = 1 constraint by writing the order parameterfield as (cid:126)n k = ( (cid:126)π k , √ − (cid:126)π k ), where (cid:126)π = ( π , . . . , π N − )are small transverse fluctuations, and use the componentof (cid:126)n in the direction of local order— n N k = √ − (cid:126)π k —tocompute the DSF. However, this approach assigns all dy-namics to the directions transverse to the local moment,and hence, does not respect the O ( N ) symmetry whichmust remain unbroken at finite temperature. To restoresymmetry, we rotationally average the DSF over all N polarizations by multiplying (3) by ( N − /N [6, 11].Therefore, suppressing Boltzmann factors and δ func-tions for notational clarity, the DSF is S ( k , ω ) = (cid:18) N − N (cid:19) Z k (cid:88) α,β |(cid:104) α | (cid:112) − (cid:126)π k | β (cid:105)| . (4)The leading contribution is then obtained by expand-ing √ − (cid:126)π k (cid:39) − (cid:126)π k / → − (cid:126)π k / ω > ω k the external probeexcites two quasiparticles [see Fig. 1(b)], and if ω < ω k one quasiparticle is emitted and a second is absorbed.When | ω − ω k | . = | ∆ | (cid:28) ω k both processes have the samecontribution to the sum over initial and final states (seeSM Sec. II [19]):˜ I ( k , ω ) = ( N − T ρ k ω k | ω − ω k | . (5)However, by examining the structure of the phasespace integral yielding (5), we find that one emittedquasiparticle will have energy ∼ ω k and the other willhave energy ∼ | ∆ | (cid:28) ω k . Hence, the two magnon in-termediate state is an inherently multi-scale process andcontributions at the “soft” scale Λ ∼ | ∆ | are not prop-erly accounted for; conventional RG is not sufficient to describe the process accurately. We understand fromour exact solution of the O (2) NLSM that the physicsof the soft scale is characterized by an interplay be-tween thermal fluctuations and the radiation of arbitrar-ily low energy quasiparticles, the “probabilities” of bothof which are logarithmically infrared-divergent; this di-vergence implies that no finite number of quasiparticlescan be excited by the probe [13]. Therefore, the “secondquasiparticle” with energy | ∆ | emitted/absorbed by theprobe is actually accompanied by a classical radiationfield containing infinitely-many quanta; the total energyof the soft magnon and the classical field is | ∆ | .To account for beyond RG contributions to the DSF, itis important to allow for the running of the quasiparticleresidue and spin stiffness down to the soft scale. Thisis simple since: (i) Soft radiation factorizes—to one-looporder—from the emission of the “hard” quasiparticle. (ii)The classical radiation field cuts off the divergent staticthermal fluctuations below the soft scale. This leads toa correction to (5)—denoted without a tilde—with onefactor of the spin stiffness modified ρ k → ρ ∆ and an ad-ditional field strength factor Z k /Z ∆ —since the runningof Z Λ is multiplicative. From this, we obtain the DSF forthe O ( N ) NLSM in the regime ξ − (cid:28) | ∆ | (cid:28) ω k (cid:28) T : S ( k , ω ) = (cid:18) N − N (cid:19) Z k I ( k , ω )= ( N − N Z ∆ T ρ k ρ ∆ ω k | ω − ω k | . (6)It is common to express the structure factor in terms ofappropriate length/time scales. In the present case, theonly length scale is λ = 1 T exp (cid:20) πρ ( N − T (cid:21) , (7)so that in terms of the one-loop expressions for Z ∆ , ρ k and ρ ∆ given by (2), the full form of (6) is S ( k , ω ) = ( N − N πρ ( N − T log( λω k ) (cid:20) ( N − T πρ log( λ | ω − ω k | ) (cid:21) N − T n ρ ω k | ω − ω k | . (8)Of course, this result assumes λ | ω − ω k | (cid:29)
1, and hence,represents a very broad frequency distribution decayingslower than 1 / | ∆ | . The limit N → N > O ( N ≥
3) NLSM. The dominant decay process foran on-shell quasiparticle with energy ω k (cid:28) T is 2 → k ω k (cid:39) ( N − T πρ k log Tω k . (9)Importantly, in our regime of interest ( ξ − (cid:28) ω k (cid:28) T (cid:28) ρ ) Γ k /ω k is an O ( T /ρ ) small quantity. It is then clearfrom the analysis in this section that when Γ k < | ∆ | ,radiative broadening of the DSF due to multiple emis-sions/absorptions dominates over 1 / | ∆ | Lorentzian life-time broadening. As a side note, since Γ k ∝ N , radiativebroadening may be hidden in a 1 /N expansion around N = ∞ . However, for N not much larger than O ( ρ/T ),the region | ∆ | < Γ k remains very narrow. Regardless,the decay of the hard particle cannot be neglected nearresonance. In particular, it serves to regularize the non-integrable singularity at ω = ω k in (8). Since the calcula-tion does not offer any new insights, it is presented in SMSec. II [19]. In Fig. 2(a) we compare the DSF calculatedin this paper [SM Eq. (S29)], to a Lorentzian lineshapewith the same ω integrated spectral weight. Clearly, theresonant response of the DSF is greatly suppressed com-pared to the Lorentzian, with significant spectral weightshifted to the tails of the frequency distribution. Equal-time correlations.—
The static structure fac-tor can be calculated directly from the DSF by integrat-ing over frequency in the interval − ω k < ∆ < ω k (seeSM Sec. III [19]): S ( k ) = (cid:18) N − N (cid:19) T n ρk (cid:20) ( N − T πρ log( λk ) (cid:21) N − . (10)The equal-time order parameter correlation function,which is N times the Fourier transform of (10), reads (cid:104) (cid:126)n ( r ) · (cid:126)n (0) (cid:105) = n (cid:20) ( N − T πρ log (cid:18) λr (cid:19)(cid:21) N − N − . (11)The static structure factor (10) has the same functional k dependence as the well-known scaling form [6, 9, 11].However, (10) contains log( λk ) instead of log( ξk ) in thoseRefs., where ξ is the correlation length ξ = ξ T (cid:20) ( N − T πρ (cid:21) N − exp (cid:20) πρ ( N − T (cid:21) , (12) ξ = ( e/ / ( N − Γ[1 + 1 / ( N − x ) is theGamma function [6, 11, 17]. The replacement ξ → λ leads to a particularly drastic difference for the case N = 3, where the pre-exponential factor of the corre-lation length is temperature-independent.To confirm our results (10) and (11), we performed MCsimulations of the O (3) NLSM and measured the equal-time order parameter correlation function. The zero tem-perature spin stiffness ρ and staggered magnetization n presented in Fig. 1(a) have been calculated on a 64 size FIG. 2. (a)
The dynamic structure factor [SM Eq. (S29)]accounting for lifetime broadening with ω k = T / ρ/
4. TheFWHM Γ k /ω k ≈ .
015 is given by (9). (b)
Order parameterequal-time correlations at fixed g = 1 ( ρ ≈ .
504 and n ≈ . N = 3, and dashed lines are theory replacing λ → ξ . lattice. To measure the correlation function we used lat-tices with L x = L y = 512 and L β = 4, 6, 8 imaginarytime slices which correspond to different temperatures T = gρ /L β . In Fig. 2(b) we present the MC correlationfunction for dimensionless coupling g = 1, correspondingto ρ/ρ ≈ .
504 and n ≈ . N = 3; note that thetheory has no adjustable fitting parameters . At r (cid:46) r (cid:38) λ replaced by ξ and disagree very clearly withthe MC simulations.To avoid misunderstanding we note the following: (i)The correlation length is defined in terms of the ex-ponential decay of correlations on large length scales (cid:104) (cid:126)n ( r ) · (cid:126)n (0) (cid:105) ∼ e − r/ξ when r (cid:29) ξ . (ii) In this work,we are operating in the opposite limit r (cid:28) ξ . We arenot claiming that the well known expression (12) for thecorrelation length is incorrect. However, we claim thatcorrelations on shorter scales are characterized by theparameter λ , and not ξ . Summary.—
We have calculated for the first time thefinite temperature dynamic structure factor of the 2D O ( N ) quantum nonlinear σ model in the regime describ-ing a Heisenberg antiferromagnet. The dynamic struc-ture factor displays a very broad frequency distributionwhich decays slower than the first power of the detuningfrom resonance. Since the quasiparticle lifetime remainsvery long, it is irrelevant to broad tails of the spectrum.Instead, the broadening is driven by the emission and ab-sorption of multiple soft excitations by the probe. To per-form this calculation, we developed a new analytical tech-nique which accounts for both conventional single scalerenormalization group contributions and “beyond RG”effects from multi-scale physics. We expect this methodto be applicable to studying the dynamics of a wide rangeof finite temperature interacting quantum field theories.Using our new result for the dynamic structure factorwe also calculated the static structure factor and foundagreement of the functional momentum dependence withthe previously known result. However, we predicted asignificant modification of the characteristic length scaleof correlations in the so-called scaling regime. This re-sult implies an important correction to the temperaturedependence of static correlations from the bosonic statis-tics of the quasiparticles. To confirm this predictionwe performed extensive path integral quantum MonteCarlo simulations and demonstrated perfect agreementbetween the numerical data and our analytical formula.To the best of our knowledge, this is also the first MonteCarlo study of correlations in the scaling regime. Acknowledgments.—
We thank Jaan Oitmaa for con-sultations on Monte Carlo simulations, and AndreyKatanin, Michael Schmidt, and G¨otz Uhrig for importantdiscussions. This research includes computations usingthe computational cluster Katana supported by ResearchTechnology Services at UNSW Sydney. We have alsoreceived support from the Australian Research CouncilCentre of Excellence in Future Low Energy ElectronicsTechnologies (CE170100039). ∗ [email protected] † [email protected][1] A. Banerjee, J. Yan, J. Knolle, C. A. Bridges, M. B.Stone, M. D. Lumsden, D. G. Mandrus, D. A. Tennant,R. Moessner, and S. E. Nagler, Neutron scattering in theproximate quantum spin liquid α -RuCl , Science ,1055 (2017).[2] G. B. Hal´asz, N. B. Perkins, and J. van den Brink, Res-onant Inelastic X-Ray Scattering Response of the KitaevHoneycomb Model, Physical Review Letters , 127203(2016).[3] M. A. Kastner, R. J. Birgeneau, G. Shirane, and Y. En-doh, Magnetic, transport, and optical properties ofmonolayer copper oxides, Reviews of Modern Physics ,897 (1998).[4] N. D. Mermin and H. Wagner, Absence of Ferro-magnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models, Physical Re-view Letters , 1133 (1966).[5] J. Zinn-Justin, Quantum Field Theory and Critical Phe-nomena (Clarendon, Oxford, 2002).[6] S. Sachdev,
Quantum Phase Transitions (Cambridge University Press, Cambridge, England, 2011).[7] L. Savary and L. Balents, Quantum spin liquids: a re-view, Reports on Progress in Physics , 016502 (2017).[8] M. C. O’Brien and O. P. Sushkov, Colossal quasiparti-cle radiation in the Lifshitz spin liquid phase of a two-dimensional quantum antiferromagnet, Physical ReviewB , 184408 (2020).[9] S. Chakravarty, B. I. Halperin, and D. R. Nelson, Two-dimensional quantum Heisenberg antiferromagnet at lowtemperatures, Physical Review B , 2344 (1989).[10] S. Tyˇc and B. I. Halperin, Damping of spin waves in atwo-dimensional Heisenberg antiferromagnet at low tem-peratures, Physical Review B , 2096 (1990).[11] A. V. Chubukov, S. Sachdev, and J. Ye, Theory of two-dimensional quantum Heisenberg antiferromagnets witha nearly critical ground state, Physical Review B ,11919 (1994).[12] S. Tyˇc, B. I. Halperin, and S. Chakravarty, DynamicProperties of a Two-Dimensional Heisenberg Antiferro-magnet at Low Temperatures, Physical Review Letters , 835 (1989).[13] M. C. O’Brien and O. P. Sushkov, Anomalous ther-mal broadening from an infrared catastrophe in two-dimensional quantum antiferromagnets, Physical ReviewB , 064431 (2020).[14] F. Bloch and A. Nordsieck, Note on the Radiation Fieldof the Electron, Physical Review , 54 (1937).[15] L. B. Ioffe and A. I. Larkin, Effective Action of a Two-Dimensional Antiferromagnet, International Journal ofModern Physics B , 203 (1988).[16] P. C. Hohenberg, Existence of Long-Range Order in Oneand Two Dimensions, Physical Review , 383 (1967).[17] P. Hasenfratz and F. Niedermayer, The exact mass gap ofthe O( N ) σ -model for arbitrary N ≥ d = 2, PhysicsLetters B , 529 (1990).[18] D. R. Nelson and R. A. Pelcovits, Momentum-shell re-cursion relations, anisotropic spins, and liquid crystals in2 + (cid:15) dimensions, Physical Review B , 2191 (1977).[19] See Supplemental Material at ?? for details of calcula-tions and Monte Carlo numerical simulations. The Sup-plemental Material includes Refs. [23–26].[20] Note that in a previous work [13], we defined the struc-ture factor as the Fourier transform of (cid:104) (cid:126)n ( r , t ) · (cid:126)n (0) (cid:105) —thetotal response from all polarizations. The present defini-tion differs by a factor of 1 /N .[21] E. M. Lifshitz and L. P. Pitaevskii, Statistical PhysicsPart 2 (Butterworth-Heinemann, Oxford, 1980).[22] Here Γ k is the full width half maximum of the quasipar-ticle Green’s function. Some works define it as the half width, leading to a spurious factor 2 difference.[23] E. Br´ezin and J. Zinn-Justin, Spontaneous breakdownof continuous symmetries near two dimensions, PhysicalReview B , 3110 (1976).[24] M. E. Peskin and D. V. Schroeder, An Introduction ToQuantum Field Theory (Westview, Boulder, CO, 1995).[25] E. Manousakis and R. Salvador, Equivalence between thenonlinear σ model and the spin-1/2 antiferromagneticHeisenberg model: Spin correlations in La CuO , Phys-ical Review B , 2205 (1989).[26] A. W. Sandvik, Computational studies of quantum spinsystems, AIP Conference Proceedings , 135 (2010). Supplemental Material
I. FORMALISMA. One-loop renormalization of dimensional NLSM
In the main text section
Formalism we discuss how quantum and thermal fluctuations are taken into account viarenormalization. Here we provide more details on how the RG equations (2) can be derived.The O ( N ) nonlinear σ model (NLSM) in 2 + 1 dimensions, normalized at the scale Λ ∼ π/b where b is the latticespacing, is given by the Lagrangian L = 12 ρ ( ∂ µ (cid:126)n ) , (cid:126)n = 1 , (S1)where ∂ µ = ( c − ∂ t , ∂ x , ∂ y ) and ρ is the spin stiffness. From hereon we set c = 1. The unit vector constraint can beeliminated explicitly by writing (cid:126)n = ( (cid:126)π , σ ) = ( (cid:126)π , (cid:112) − (cid:126)π ). The Lagrangian in terms of the transverse fluctuations (cid:126)π is L = 12 ρ (cid:20) ( ∂ µ (cid:126)π ) + ( (cid:126)π · ∂ µ (cid:126)π ) − (cid:126)π (cid:21) . (S2)Expanding around the zero temperature state of spontaneous symmetry breaking σ = 1 and π i = 0 yields L = 12 ρ (cid:2) ( ∂ µ (cid:126)π ) + ( (cid:126)π · ∂ µ (cid:126)π ) + . . . (cid:3) , (S3)where the ellipsis denotes terms of O ( π ). The interactions have the effect of renormalizing the spin stiffness— ρ → ρ Λ —and the fields— (cid:126)n → (cid:126)n Λ . = Z / (cid:126)n . To see this, we calculate the self-energy by performing a one-loopdecoupling of the quartic term ( (cid:126)π · ∂ µ (cid:126)π ) −→ N − (cid:104) (cid:126)π (cid:105) Λ ( ∂ µ (cid:126)π ) , (S4)where (cid:104) (cid:126)π (cid:105) Λ are the fluctuations of the (cid:126)π fields with momenta in the interval (Λ , Λ ). At the scale Λ (cid:28) Λ corresponding to several lattice spacings, the fluctuations reduce the effective length of the σ component down to n = |(cid:104) (cid:126)n (cid:105)| <
1. Therefore, we require that the renormalized fields satisfy (cid:104) σ Λ (cid:105) = 1; the renormalized ground stateshould have the same form σ Λ = 1, π i Λ = 0. Far away from the quantum critical point, the first order perturbativecalculation gives Z Λ = 1 n (cid:39) (cid:104) (cid:126)π (cid:105) Λ , (S5)and hence, ρ Λ (cid:39) ρ (cid:18) − N − N − (cid:104) (cid:126)π (cid:105) Λ (cid:19) . (S6)In practice, ρ and n are calculated numerically, which we do using path integral Monte Carlo (see Section IV). Byintegrating out the ultraviolet quantum fluctuations, we can obtain the low energy Lagrangian normalized at Λ L = 12 ρ ( ∂ µ (cid:126)n ) , (cid:126)n = 1 . (S7)Turning to the case of finite temperature, we note that long-range order is destroyed by thermal fluctuations and nostate of spontaneously broken symmetry exists [4, 16]. However, the correlation length ξ ∝ exp[2 πρ/ ( N − T ] remainsexponentially large in the low temperature regime T (cid:28) ρ [9, 11, 17]. Therefore, on momentum scales ξ − (cid:28) Λ < Λ ,we can apply the same analysis as above by expanding the low energy Lagrangian (S7) around a locally ordered state.The thermal fluctuations (cid:104) (cid:126)π (cid:105) Λ can be evaluated directly when Λ < T < Λ : (cid:104) (cid:126)π (cid:105) Λ = ( N − (cid:90) Λ Λ d q (2 π ) ω q ρ e ω q /T − (cid:39) ( N − T πρ log T Λ . (S8)Within the Matsubara imaginary time formalism, this same result can be obtained by noting that the dominantcontribution comes from the zero Matsubara frequency in the low temperature T (cid:28) ρ regime. This statement isequivalent to the common wisdom that the low temperature regime of the square lattice Heisenberg antiferromagnetis characterized by classical static (zero Matsubara frequency) thermal fluctuations [9]. However, we note that theultraviolet cutoff of the logarithm in (S8) is imposed by the Bose occupation factor in the momentum integral. If thethermal fluctuations were purely classical the ultraviolet cutoff would be Λ [18]. Instead, the quantization of thefield leads to a correction in 2 + 1 dimensions.The renormalization group (RG) equations governing the flow of ρ and Z from the scale Λ down to Λ then followdirectly from Eqs. (S5), (S6) and (S8) by considering infinitesimally small l = log( T /
Λ): dρdl = − ( N − T π , (S9a) d log Zdl = ( N − T πρ . (S9b)When N ≥
3, there is a non-trivial renormalization group flow. Integrating the system of equations (S9) yields ρ Λ = ρ − ( N − T π log T Λ , (S10a) Z Λ = 1 n (cid:18) ρρ Λ (cid:19) N − N − . (S10b)These results are very well known, and follow directly from expressions given in Refs. [9, 11]. B. Two-loop contributions to renormalization
In the main text, at the end of the
Formalism section, we claimed that higher-loop order corrections to the spinstiffness and order parameter could be neglected at scales ξ − (cid:28) Λ < Λ . Here, we consider the well-known RGflow equations for the 2D NLSM [5, 23], which describe the interactions of classical fluctuations in the (2 + 1)Dmodel [9]. Since all calculations should be consistent to leading order with perturbation theory, we again use thetemperature T instead of Λ as the ultraviolet normalization point of the RG flow. The two-loop equations in termsof the dimensionless temperature t = T / πρ are [23], dtdl = ( N − t + ( N − t + O ( t ) , (S11a) d log Zdl = ( N − t + O ( t ) . (S11b)Re-writing these differential equations in terms of ρ , we find the exact solution ρ Λ = − T π (cid:32) W − (cid:34) Xe X (cid:18) T Λ (cid:19) N − (cid:35)(cid:33) , where X = − (cid:18) πρT (cid:19) , (S12a) Z Λ = 1 n (cid:18) πρ + T πρ Λ + T (cid:19) N − N − , (S12b)where W ( x ) is the inverse function of W e W = x —otherwise known as the Lambert W function or productlogarithm—which has two branches for real x . The − W − ( xe x ) = x if x ≤ −
1, which guaranteesthat the initial condition of the differential equation is satisfied.In the regime we are interested in— T (cid:28) ρ and Λ (cid:29) ξ − —the argument of the W function in (S12a) is close tozero. Using the asymptotic expansion W − ( x ) = log( − x ) − log( − log( − x )) + O (1), we find ρ Λ ρ = 1 − ( N − T πρ log T Λ + O ( T /ρ ) , (S13a) Z Λ = 1 n (cid:18) ρρ Λ (cid:19) N − N − + O ( T /ρ ) . (S13b)Therefore, neither the quasiparticle residue nor the spin stiffness are modified—to a good degree of accuracy—bytwo-loop contributions in the low temperature regime.This result may surprise, given that it is well known that the correlation length ξ is heavily modified by two-loopcorrections [6, 9]. However, these differences only appear in the far infrared limit. First, observe that the one-loopspin stiffness (S10a) vanishes at the scale Λ = λ − , where λ − = T exp (cid:20) − πρ ( N − T (cid:21) . (S14)However, since W − ( − /e ) = −
1, the two-loop spin stiffness vanishes at the scale Λ = Ξ, whereΞ = T (cid:18) πρT (cid:19) N − exp (cid:20) − πρ ( N − T (cid:21) = T (cid:18) πρT (cid:19) N − exp (cid:20) − πρ ( N − T (cid:21) (cid:20) O (cid:18) Tρ (cid:19)(cid:21) . (S15)For comparison, the exact (inverse) correlation length is [6], ξ − = (8 /e ) / ( N − Γ[1 + 1 / ( N − T (cid:20) πρ ( N − T (cid:21) N − exp (cid:20) − πρ ( N − T (cid:21) (cid:20) O (cid:18) Tρ (cid:19)(cid:21) , (S16)where Γ( x ) is the Gamma function. Most importantly, for the case N = 3 both ξ − and Ξ have a temperatureindependent pre-exponential factor—to leading order in T /ρ . This is a considerable difference compared to thetemperature dependence of λ . Obviously, any perturbative calculation like RG is not valid at and beyond the strongcoupling scale. However, it is clear from the above analysis that the behavior of the spin stiffness and order parameterare heavily modified by two-loop contributions when approaching that scale.Finally, we also acknowledge that there are two-loop corrections to the speed c . However, the renormalized speedas reported in Ref. [11] is only modified at O ( T /ρ ) when Λ (cid:29) ξ − . II. DYNAMIC STRUCTURE FACTOR
In the main text section
Dynamic structure factor , we start from the definition of the dynamic structure factor(DSF) as the Fourier transform of the correlation function and derive an expression which includes both conventionalRG contributions as well as multi-scale physics. Here, we provide more of the details of this derivation. First, weconsider the radiation-dominated region, and then the lifetime-suppressed resonant peak.
A. Derivation of main text equation (8)
Given the absence of long-range order, we can write S ( k , ω ) δ ij = (cid:90) dt d r (cid:104) n i ( r , t ) n j (0) (cid:105) e i ( ωt − k · r ) , (S17)where the average is taken over the thermal ensemble. The matrix element can be expanded in a spectral representationwhich turns the integral into a sum over initial and final states [21], S ( k , ω ) = (cid:88) α,β e − ω β /T Z |(cid:104) α | n i (0) | β (cid:105)| (2 π ) δ ( ω − ω αβ ) δ (2) ( k − k αβ ) , (S18)where Z is the partition function, | α (cid:105) , | β (cid:105) are excited Fock states, ω α , ω β and k α , k β are the energy and momentumof those states, respectively, and ω αβ = ω α − ω β , and similarly for k αβ . Before proceeding, we recall that in theabsence of long-range order, the DSF will contain no elastic spectral weight; the spectrum will not have a Bragg peakat k = 0. Then, as in the main text, we account for conventional RG contributions to the spectrum by re-writing thespectral expansion in terms of the fields renormalized at the momentum transfer from the external probe k S ( k , ω ) = 1 Z k (cid:88) α,β e − ω β /T Z |(cid:104) α | n i k | β (cid:105)| (2 π ) δ ( ω − ω αβ ) δ (2) ( k − k αβ ) . (S19) FIG. S1. The transverse response of the structure factor also contains infinitely-many multiparticle states. These are obtainedfrom the different ways of cutting diagrams (vertical dashed lines). The bold line represents the exact π i propagator, normallines represent the bare π i propagator, and horizontal dashed lines represent the source. If we have ξ − (cid:28) k (cid:28) T , then local order exists on these momentum scales. This allows us to eliminate theunit vector constraint (cid:126)n = 1 by artificially breaking the O ( N ) symmetry and writing the order parameter as (cid:126)n k =( (cid:126)π k , √ − (cid:126)π k ). However, the DSF must remain rotationally invariant, and in particular, spectral sum rules must besatisfied. Therefore, we account for the fact that we have “broken” the symmetry by rotationally averaging the DSFover all N polarizations; this amounts to multiplying by a factor of ( N − /N [6, 11]: S ( k , ω ) = (cid:18) N − N (cid:19) Z k (cid:88) α,β e − ω β /T Z |(cid:104) α | (cid:112) − (cid:126)π k | β (cid:105)| (2 π ) δ ( ω − ω αβ ) δ (2) ( k − k αβ ) , (S20)Naively, this expression appears to contain a Bragg peak contribution, seemingly in contradiction to our earlier point.However, we re-emphasize that order only locally exists on momentum scales 0 < ξ − < k ; it is incorrect to use (S20)at 0 ≤ k < ξ − .On the one hand, the DSF must be rotationally invariant. On the other hand, the n N = √ − (cid:126)π componentcontains all even powers of (cid:126)π , while the transverse components are linear in π i . There is no actual contradiction here.To see this, consider Fig. S1, where we illustrate the simplest loop corrections to the interaction of the source witha transverse π i component. The first two diagrams correspond to the one- and two-loop contributions to the selfenergy of the emitted quasiparticle. However, the third diagram shows that the interactions (and self interactions)between the π i components allow the probe to create three real and on-shell particles via an intermediate virtualstate. The Feynman rules for the interaction vertex in (S3) are given in textbooks (e.g., Ref. [24]). In particular, thepropagator is G ( q ) = iρ − /q , and the amplitude for an off-shell particle with three-momentum q to decay into threeon-shell particles is A ( q ) = − iρq = 1 /G ( q ). Therefore, the total quantum amplitude for the intermediate state is G ( q ) A ( q ) = 1. Essentially, the virtual particle “contracts” the interaction to a single point. It is straightforward tosee that all higher-order interactions in the expansion of (S2) will lead to the same amplitude for similar multiparticleemissions. We emphasize that this means that the transverse components also lead to the emission of infinitely-manyquasiparticles; whether this infinity is “odd” or “even” is irrelevant, implying the preservation of O ( N ) symmetry [13].It is, however, much simpler to perform calculations using the n N longitudinal component. We obtain the firstnon-trivial contributions by expanding the square root inside the matrix element √ − (cid:126)π k (cid:39) − (cid:126)π k /
2: (i) The emissionof two particles ( ω > ω k ). (ii) The emission of one and absorption of the other ( ω < ω k ). Since these two processesare mathematically similar, we focus on emission (of any polarization), where the total contribution to the sum overinitial and final states of this form is given by the two particle phase space integral˜ I ( k , ω ) = ( N − (cid:90) d k (2 π ) d k (2 π ) |M ( k , k ) | (2 π ) δ ( ω − ω k − ω k ) δ (2) ( k − k − k ) , (S21)where |M ( k , k ) | = n ( ω k ) + 12 ω k ρ k n ( ω k ) + 12 ω k ρ k (cid:39) T ω k ρ k T ω k ρ k , (S22)0 FIG. S2. Schematic illustrating the effect of soft radiation on the spectrum: higher order diagrams set the scale of thermalfluctuations at the momentum of the soft leg. The spin stiffness is renormalized by the self energy of the emitted/absorbedparticles, and the factorization of the two internal legs [see Eq. (S24)] implies that the renormalization scales can be setindependently. The finite lifetime of the high energy particle becomes relevant near resonance ω = ω k . is the effective finite temperature two particle emission matrix element (squared) obtained after performing the Gibbsaveraging in (S20); the factor of 1 / k . The integral can be evaluated exactly after expanding the Bose occupation factors to leading orderin T /ω , ˜ I ( k , ω ) (cid:39) ( N − T ρ k ω k | ω − ω k | , when | ω − ω k | (cid:28) ω k , (S23)and it can be observed that when | ∆ | (cid:28) ω k , the dominant contribution comes from regions of phase space where oneparticle has energy ∼ ω k and the other has energy ∼ | ∆ | ; the process involving absorption which occurs for ω < ω k reduces to the same expression in the limit | ∆ | (cid:28) ω k .As discussed in the main text, this large difference in the momentum scales of the two quasiparticles means thatcontributions at the soft scale are not accounted for. First, observe that we can perform a post-hoc simplification of(S21) using our knowledge of the momentum distribution, and find that the integral factorizes as˜ I ( k , ω ) (cid:39) ( N − (cid:32)(cid:90) d k (2 π ) T ω k ρ k (2 π ) δ (2) ( k − k ) (cid:33) (cid:32)(cid:90) d k (2 π ) T ω k ρ k (2 π ) δ (∆ − ω k ) (cid:33) , (S24)into high and low energy processes. This implies that all corrections to the external particle legs will also factorize,allowing us to account for the running of the parameters of the two particles independently. First, we allow the spinstiffness to run with the momenta of the particles ρ k → ρ k i and account for the running of the quasiparticle residuefrom the normalization point k to the momenta of the particles, which gives a factor of Z k /Z k i for each particle. Wenote that there are also two-loop corrections to the source vertex which do not factorize. However, this will be ahigher-order effect, so we neglect it. Therefore, the “beyond RG” version of (S23)—denoted with no tilde—will be I ( k , ω ) = ( N − Z k (cid:90) d k (2 π ) d k (2 π ) T ω k ρ k Z k T ω k ρ k Z k (2 π ) δ ( ω − ω k − ω k ) δ (2) ( k − k − k ) (S25a) (cid:39) Z k Z ∆ ( N − T ρ k ρ ∆ ω k | ω − ω k | . (S25b)We know from the exact solution of the O (2) NLSM that the classical radiation field created by the source cuts offinfrared-divergent static thermal fluctuations with momenta smaller than that of the soft particle [13]. Therefore, allleading-order beyond RG contributions to the DSF are accounted for by the running of the quasiparticle residue ofthe soft leg. Hence, the DSF is given by S ( k , ω ) = ( N − N (cid:20) − ( N − T πρ log Tω k (cid:21) − (cid:20) − ( N − T πρ log T | ω − ω k | (cid:21) N − T n ρ ω k | ω − ω k | , (S26)1which can also be written in terms of the length scale λ = (1 /T ) exp[2 πρ/ ( N − T ], yielding Eq. (8) of the maintext. It is simple to see using this expression that the exact solution of the O (2) DSF is reproduced by taking thelimit N →
2; the first square bracket →
1, while the second square bracket exponentiates → ( | ∆ | /T ) T/ πρ . B. Lifetime broadening near resonance
At the end of the main text section
Dynamic structure factor , we point out that the finite lifetime of quasiparticlesbecomes relevant in the narrow region of the frequency spectrum | ∆ | < Γ k , where Γ k is the Raman scattering ratefor a particle with energy ω k . The leading contributions come from the imaginary part of the two-loop self energydiagram shown in Fig. S2. The well-known expression for the scattering rate is [10, 11],Γ k ω k (cid:39) ( N − T πρ k log Tω k = ( N − T πρ (cid:20) − ( N − T πρ log Tω k (cid:21) − log Tω k , (S27)where we prefer to use the convention that Γ k is the full width at half maximum of the imaginary part of the singlequasiparticle Green’s function; this gives a factor 2 difference compared to the expressions reported in Refs. [10, 11].In our regime of interest ξ − (cid:28) | ∆ | (cid:28) ω k , it is clear that the scattering rate of the soft quasiparticle is negligiblecompared to that of the higher energy particle. Therefore, the lifetime broadening of the DSF will be dominated bythe decay of the hard particle.The lifetime has the effect of “broadening” the energy conserving δ function in (S25a)(2 π ) δ ( ω − ω k − ω k ) −→ Γ( ω − ω k − ω k ) + Γ / (cid:39) Γ k (∆ − ω k ) + Γ k / . (S28)However, there is an important subtlety in accounting for contributions from different pieces of phase space. Withoutlifetime broadening, the following cases are possible:(i) If ω > ω k >
0, two particles are emitted.(ii) If ω k > ω >
0, one particle is emitted with energy ω k and one is absorbed with energy | ∆ | .(iii) If 0 > ω > − ω k , one particle is absorbed with energy ω k and one is emitted with energy | ∆ | .(iv) If 0 > − ω k > ω , two particles are absorbed.In principle, with account of lifetime broadening, any of these processes can occur for any values of energy andmomentum transfer from the source. However, since we assume | ∆ | (cid:28) ω k , we can safely assume no mixing betweenthe positive and negative frequency branches of the spectrum. However, for ω >
0, we must allow for mixing betweenprocesses (i) and (ii). Therefore, we generalize the integral (S25a) to give us the full form of the DSF S ( k , ω ) (cid:39) ( N − N T n ρ k ω k (cid:90) ω k /λ d q (2 π ) ω q ρ q Z q (cid:20) Γ k (∆ − ω q ) + Γ k / k (∆ + ω q ) + Γ k / (cid:21) , (S29)which is plotted in Fig. 2(a) of the main text. Note that we must retain an infrared momentum cutoff for thisexpression. Given that we have already established that the characteristic momentum scale of the DSF in this regimeis λ − , we use this as the cutoff. For process (i), the two emitted particles are indistinguishable bosons, but wedistinguish between them, so we must impose the ultraviolet cutoff ω k to avoid double counting states. For process(ii), the dominant contribution comes from the absorption of particles with energy < ω k . Finally, we note that in thelimit | ∆ | (cid:29) Γ k , (S29) reduces to (S26). III. EQUAL-TIME CORRELATIONS
In the main text we present the static structure factor and equal-time correlation function. Here, we simply providemore detail on how the calculations are performed. The static structure factor is defined in terms of the dynamicstructure factor by the relation S ( k ) = (cid:90) dω π S ( k , ω ) . (S30)2Since the main text Eq. (8) has a non-integrable singularity at ω = ω k , we must use (S29) to compute the staticstructure factor. Additionally, the spectral representation (S18) implies S ( k , − ω ) = e − ω/T S ( k , ω ) (cid:39) S ( k , ω ) when ω (cid:28) T , which allows us to correctly include the negative frequency portion of the spectrum. Therefore, we have S ( k ) (cid:39) ( N − N T n ρ k ω k (cid:90) ω k /λ d q (2 π ) ω q ρ q Z q × (cid:90) ∞ dω π (cid:20) Γ k (∆ − ω q ) + Γ k / k (∆ + ω q ) + Γ k / (cid:21) . (S31a)= (cid:18) N − N (cid:19) T n ρk (cid:20) ( N − T πρ log( λk ) (cid:21) N − . (S31b)We can also check the total sum rule. Since the dynamic structure factor we derived was valid for ω (cid:28) T , we shouldintegrate (S31b) up to T : (cid:90) d k (2 π ) S ( k ) (cid:39) (cid:90) T /λ d k (2 π ) S ( k ) = n N . (S32)Therefore, summing up over the N polarisations, we recover the correct normalization of the order parameter. Wenote that this sum rule is not satisfied if all parameters are normalized at the same scale—either ω k or | ∆ | . Thisobservation validates our approach to including multi-scale physics.The equal-time correlation function then follows directly from (S31b) by taking the Fourier transform. When r (cid:28) λ ,it is straightforward to find, to logarithmic accuracy, (cid:104) (cid:126)n ( r ) · (cid:126)n (0) (cid:105) = N (cid:90) d k (2 π ) S ( k ) e − i k · r (cid:39) N (cid:90) /r /λ d k (2 π ) S ( k ) = n (cid:20) ( N − T πρ log (cid:18) λr (cid:19)(cid:21) N − N − , (S33)since the complex exponential will oscillate rapidly and average to zero when k (cid:29) /r . IV. PATH INTEGRAL QUANTUM MONTE CARLO SIMULATIONS
In the main text, we present a subset of our measurements of the equal-time correlation function using pathintegral quantum Monte Carlo simulations. Here we summarize the details of our simulations—the Monte Carloupdate algorithm we implement and how we measure physical observables—and also present a larger selection ofdata.
A. Heat bath algorithm for O (3) NLSM
The quantum partition function for the O (3) NLSM in imaginary time is given by the path integral [9, 25], Z = (cid:90) D (cid:126)n ( x , τ ) δ ( (cid:126)n − e − S [ (cid:126)n ] / (cid:126) , (S34a) S [ (cid:126)n ] / (cid:126) = ρ (cid:126) (cid:90) (cid:126) /T dτ d x (cid:20) c ( ∂ τ (cid:126)n ) + ( ∇ (cid:126)n ) (cid:21) , (S34b)where ρ is the bare, un-renormalized spin stiffness defined at the lattice spacing b , (cid:126)n = ( n ( x ) , n ( y ) , n ( z ) ), and the δ function in the integration measure enforces the unit vector constraint at every point in space. Discretizing the actionover a uniform simple cubic lattice with spacing b yields [25], S [ (cid:126)n ] / (cid:126) = − g (cid:88) (cid:104) i,j (cid:105) (cid:126)n ( x i ) · (cid:126)n ( x j ) , (S35)where x i = ( cτ i , x i ), g = (cid:126) c/ρ b = L β T /ρ , L β is the size (in number of lattice spacings) of the imaginary timedimension, and the summation is over pairs of nearest neighbors. From hereon, we set (cid:126) = ρ = b = 1. In theseunits, the bare coupling constant g = c = L β T .We performed path integral quantum Monte Carlo simulations of the O (3) model by implementing a heat bathalgorithm following Ref. [25]. To summarize:3(1) Initialize the lattice in a uniformly magnetized grid.(2) To update a lattice site at position x i , calculate the local action (cid:126)ω ( x i ) = (cid:88) (cid:104) i,j (cid:105) (cid:126)n ( x i ) · (cid:126)n ( x j ) , (S36)so that the probability density for the vector (cid:126)n ( x i ) to lie in some element of solid angle is P (Ω) d Ω = C exp (cid:18) | (cid:126)ω | cos θg (cid:19) sin θdθdϕ, (S37)where C is the normalization constant of the distribution, and θ is measured from the axis directed along (cid:126)ω .(3) Generate a new configuration for (cid:126)n ( x i ) by picking θ and ϕ from this distribution, convert from local to crystalaxis coordinates, and then update.Defining a “sweep” of the lattice to be an update of every lattice site once, we allowed 2500 sweeps for the systemto thermalize before starting measurements. We then performed 50,000 sweeps, measuring once every 10 sweeps tominimize correlations between measured configurations; we estimated a correlation time from measurements of theaverage action per site to be ≈ B. Measurement methods
To measure the zero temperature staggered magnetization n , we used the standard Monte Carlo estimator n = (cid:42)(cid:32) L (cid:88) i (cid:126)n ( x i ) (cid:33) (cid:43) , (S38)where L is the total number of lattice sites (at zero temperature all dimensions are of equal size), and the ensembleaverage (cid:104) · (cid:105) is estimated by an average over measurements.The equal-time correlation function measurements were obtained using the formula (cid:104) (cid:126)n ( r ) · (cid:126)n (0) (cid:105) = (cid:28) L L β (cid:88) i,µ = x,y (cid:126)n ( x i ) · [ (cid:126)n ( x i + re µ ) + (cid:126)n ( x i − re µ )] (cid:29) , (S39)where e µ is a unit vector along the µ direction, and we averaged over positive and negative displacements along thetwo equivalent spatial dimensions to improve our measurement statistics; at finite temperature, the imaginary timedirection is not equivalent, so is not included in the sum over directions µ . Summing over all lattice sites and divisionby L β approximates the integral over the imaginary time dimension used to obtain the equal-time correlation function.To measure the zero temperature renormalized spin stiffness, we adapted the approach described in Ref. [26], whichwe summarize here. The spin stiffness measures the response of the system to a twist of the boundary conditions ofdimension µ by a relative angle Φ = QL µ . At zero temperature, ρ = 1 L x L y ∂ E ( Q ) ∂Q (cid:12)(cid:12)(cid:12)(cid:12) Q =0 , (S40)where E ( Q ) = − ( g/L β ) log Z ( Q ) is the ground state energy functional in the presence of the twist and Z is thequantum partition function. The twisted boundary conditions can be eliminated by transforming to a “rotating”frame of reference where the twist instead modifies the local interaction: S [ (cid:126)n, Q ] = − g (cid:88) (cid:104) i,j (cid:105) (cid:126)n ( x i ) · R (cid:126)n ( x j ) , (S41)4where R is a 3 × x axis in spin space, along direction µ inreal space, expanding the action to second order in Q leads to a modification of the energy E ( Q ) (cid:39) E (0) − g L β Q (cid:16) (cid:104) S ( x ) µ (cid:105) + (cid:104) ( I ( x ) µ ) (cid:105) (cid:17) , (S42)where we have defined S ( x ) µ = − g (cid:88) (cid:104) i,j (cid:105) µ (cid:104) (cid:126)n ( x i ) · (cid:126)n ( x j ) − n ( x ) ( x i ) n ( x ) ( x j ) (cid:105) , (S43a) I ( x ) µ = − g (cid:88) (cid:104) i,j (cid:105) µ [ (cid:126)n ( x i ) × (cid:126)n ( x j )] x , (S43b)where summation is over lattice bonds directed in the µ direction. However, it is necessary to account for the factthat the direction of the twist in spin space is not generally perpendicular to the local magnetization. Therefore,the spin stiffness is obtained by averaging over the other two twist axes, weighted by 3 / L x = L y = L β . = L ), so we also averaged over all bond directions to obtain a more accurate Monte Carlo estimatorof the spin stiffness: ρ = − g L (cid:32) (cid:104) S (cid:105) + 12 (cid:88) µ,a =1 (cid:104) ( I ( a ) µ ) (cid:105) (cid:33) , (S44)where (cid:104) S (cid:105) is the average of the action (S35). C. Results & further analysis
In Table SI we present a subset of measurements of the zero temperature spin stiffness and average staggeredmagnetization on a L x = L y = L β = 64 size lattice. These results are practically identical to measurements on a 32 lattice showing that finite size scaling effects are negligible (away from the O (3) quantum critical point g = g c ≈ . λ . Evidently, as thecoupling g is increased, reducing the spin stiffness ρ , the relative importance of thermal fluctuations increases.In Figs. S3 and S4 we present measurements of the equal-time correlation function on L x = L y = 512 and L β =4, 6, 8 size lattices, for a range of values for the coupling g . The solid lines show the theoretical prediction (S33) for FIG. S3. Order parameter equal-time correlation function of the O (3) NLSM measured using Monte Carlo on a 512 × L β size lattice with (a) g = 0 . (b) g = 1 .
0, and (c) g = 1 .
25. Zero temperature renormalized parameters and temperature T = gρ /L β are given in Table SI. Symbols are Monte Carlo data, solid lines are theory (S33) for N = 3, and dashed linesare theory replacing λ → ξ where ξ is the correlation length (S16). Panel (b) is the same data as Fig. 2(b) in the main text,reproduced here for comparison. All vertical scales are identical. The L β = 4 data is omitted from panel (c) since T /ρ > N = 3 with the zero temperature spin stiffness and magnetization measured on the 64 lattice. We emphasize thatthe theory has no adjustable fitting parameters . Evidently, the agreement between the data and theoretical curvesis excellent. As stated in the main text, disagreement on short length scales ( r (cid:46)
2) is to be expected due to thedominance of ultraviolet quantum fluctuations, and on larger length scales ( r (cid:38) λ , but not at all with the correlation length ξ given by (S16).Strictly speaking, our theory is valid in the regime T (cid:28) ρ . However, the exponentially-large length scales (both λ and ξ ) mean that it is still possible to study correlations at r (cid:28) λ when T ∼ . ρ . For example, consider the case g = 1 .
25 and L β = 6, shown in Fig. S3(c) in red. Here T /ρ ≈ .
692 and the theory still agrees quite well with thedata. This is because λ ≈ L β = 4, T /ρ > λ is just over twice the (linear) size of the lattice, and as expected the theory (S33) didnot agree at all with the data [omitted from Fig. S3(c) for clarity] since the temperature is outside the domain ofvalidity. FIG. S4. Order parameter equal-time correlation function of the O (3) NLSM measured using Monte Carlo on a 512 × L β sizelattice with (a) L β = 4, (b) L β = 6, and (c) L β = 8. Zero temperature renormalized parameters and temperature T = gρ /L β are given in Table SI. Symbols are Monte Carlo data, solid lines are theory (S33) for N = 3, and dashed lines are theoryreplacing λ → ξ where ξ is the correlation length (S16). All vertical scales are identical.TABLE SI. Selection of zero temperature parameters measured using Monte Carlo on a L x = L y = L β = 64 size lattice, thetemperature in units of ρ , and the length scale λ when L β = 4, 6, 8. Statistical fluctuations in measurements are O (10 − ). L β = 4 L β = 6 L β = 8 g ρ/ρ n T /ρ λ T /ρ λ T /ρ λ . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . ×10