## Describing an exotic magnetic transition

Phase transitions can be caused by temperature fluctuations or, more exotically, by quantum fluctuations at zero temperature. To describe some of these quantum phase transitions, researchers came up with a complex theory called deconfined quantum criticality.

However, subsequent numerical simulations were inconsistent with some of the predictions of the theory, leading to a debate on its validity. By using quantum Monte Carlo simulations, Shao *et al.* show that it is possible to reconcile numerics with the theory for a specific model of 2D quantum magnetism.

*Science*, this issue p. 213

## Abstract

The theory of deconfined quantum critical (DQC) points describes phase transitions at absolute temperature *T* = 0 outside the standard paradigm, predicting continuous transformations between certain ordered states where conventional theory would require discontinuities. Numerous computer simulations have offered no proof of such transitions, instead finding deviations from expected scaling relations that neither were predicted by the DQC theory nor conform to standard scenarios. Here we show that this enigma can be resolved by introducing a critical scaling form with two divergent length scales. Simulations of a quantum magnet with antiferromagnetic and dimerized ground states confirm the form, proving a continuous transition with deconfined excitations and also explaining anomalous scaling at* T* > 0. Our findings revise prevailing paradigms for quantum criticality, with potential implications for many strongly correlated materials.

In analogy with classical phase transitions driven by thermal fluctuations, condensed matter systems can undergo dramatic changes as parameters regulating quantum fluctuations are tuned at low temperatures. Some of these quantum phase transitions can be theoretically understood as straightforward generalizations of thermal phase transitions (*1*, *2*), where, in the conventional Landau-Ginzburg-Wilson (LGW) paradigm, states of matter are characterized by order parameters. Many strongly correlated quantum materials, however, seem to defy such a description.

In two-dimensional (2D) quantum magnets (*3*, *4*), a promising proposal that extends beyond LGW physics is the theory of deconfined quantum critical (DQC) points, in which the order parameters of the antiferromagnetic (Néel) state and the competing dimerized state (the valence-bond-solid or VBS state) are not fundamental variables but composites of fractional degrees of freedom that carry a spin *S* = 1/2. These spinons are condensed and confined, respectively, in the Néel and VBS states, and they become deconfined at the DQC point separating the two states. Establishing the applicability of the still-controversial DQC scenario would be of great interest in condensed matter physics, where it may play an important role in strongly correlated systems such as cuprate superconductors (*5*). There are also DQC analogs to quark confinement and other aspects of high-energy physics (e.g., an emergent gauge field and the Higgs mechanism and boson particle) (*6*).

The DQC theory represents the culmination of a large body of field-theoretic works on VBS states and quantum phase transitions out of the Néel state (*2*, *7*–*10*). The postulated SU(*N*) (special unitary groups of *N *dimensions) NCCP^{N}^{–1} (noncompact complex projective) action can be solved when *N* → ∞ (*5*, *11*, *12*), but nonperturbative numerical simulations are required to study small* N*. The most natural physical realizations of the Néel-VBS transition for electronic SU(2) spins are frustrated quantum magnets (*9*), which are notoriously difficult to study numerically (*13*, *14*). Other models were therefore pursued. In the *J*-*Q* model (*15*), the Heisenberg exchange *J* between *S* = 1/2 spins is supplemented by a VBS-inducing four-spin term *Q*, which is amenable to efficient quantum Monte Carlo (QMC) simulations (*15*–*23*). Although many results for the *J*-*Q* model support the DQC scenario, it has not been possible to draw definite conclusions because of violations of expected scaling relations that affect many properties. Similar anomalies were later observed in 3D loop (*24*) and dimer (*25*) models, which are also potential realizations of the DQC point. Simulations of the NCCP^{1} action as well have been hard to reconcile with the theory (*21*, *26*, *27*).

One interpretation of the unusual scaling behaviors is that the transitions are first order, as is generally required within the LGW framework for order-order transitions where unrelated symmetries are broken. The DQC theory then would not apply to any of the systems studied so far, thus casting doubts on the entire concept (*17*, *21*, *26*). In other interpretations, the transition is continuous, but unknown mechanisms cause strong corrections to scaling (*18*, *27*, *28*) or modify the scaling more fundamentally in some as yet unexplained way (*19*, *24*). The enigmatic current state of affairs is summarized well in (*24*).

Here we show that the DQC puzzle can be resolved based on a finite-size scaling ansatz that includes the two divergent length scales of the theory: the standard correlation length ξ, which captures the growth of both order parameters (ξ_{Néel} ∝ ξ_{VBS}), and a faster-diverging length ξ', which is associated with the thickness of VBS domain walls and spinon confinement (the size of a spinon bound state). We show that, contrary to past assumptions, ξ' can govern the finite-size scaling even of magnetic properties that are sensitive only to ξ in the thermodynamic limit. Our simulations of the *J*-*Q* model at low temperatures and in the lowest *S* = 1 (two-spinon) excited state demonstrate complete agreement with the two-length scaling hypothesis.

Consider first a system with a single divergent correlation length ξ ∝ |δ|^{–ν}, where δ = *g* – *g*_{c} is the distance to a phase transition that is driven by quantum fluctuations arising from noncommuting interactions controlled by *g* at absolute temperature *T* = 0, *g*_{c} is the critical value of the control parameter *g*, and ν is a critical exponent. In finite-size scaling theory (*29*), for a system of linear size *L* (volume *L ^{d}* in

*d*dimensions), close to δ = 0, a singular quantity

*A*takes the form (1)where the exponents κ, ν, and ω are tied to the universality class, κ also depends on

*A*, and the scaling function

*f*→ constant (up to corrections ∝

*L*

^{–ω}) when δ → 0. We assume

*T*∝

*L*

^{–}

*,*

^{z}*z*being the dynamic exponent (or, alternatively,

*T*= 0), so that scaling arguments depending on

*T*have been eliminated.

The form in Eq. 1 fails for some properties of the *J*-*Q *model (*18*, *19*, *22*) and other DQC candidate systems (*24*–*26*). A prominent example is the spin stiffness ρ_{s}, which, for an infinite 2D system in the Néel phase, should scale as ρ_{s} ∝ δ^{z}^{ν} with *z* = 1 (*1*, *3*, *4*). To eliminate the size dependence when δ ≠ 0 and *L* → ∞ in Eq. 1, it is necessary to have κ = *z*ν and *f*(*x*,* L*^{–ω}) ∝ *x ^{z}*

^{ν}for large

*x*= δ

*L*

^{1/ν}. Thus, ρ

_{s}(δ = 0,

*L*) ∝

*L*

^{–}

*and, if*

^{z}*z*= 1,

*L*ρ

_{s}

*→ constant when*

*L*→ ∞. However,

*L*ρ

_{s}(

*L*) at criticality instead appears to diverge slowly (

*17*,

*18*,

*21*). At first glance, this might suggest

*z*< 1, but other quantities (e.g., the magnetic susceptibility) instead behave as if

*z*> 1 (

*30*). Strong scaling corrections have been suggested as a way to resolve this paradox (

*18*,

*19*,

*28*). Claims of a weak first-order transition have also persisted (

*21*,

*26*,

*27*), although the continuous DQC scenario is supported by the absence of any of the usual first-order signals (e.g., the Binder cumulant does not exhibit any negative peak) (

*18*,

*24*).

To explain the scaling anomalies phenomenologically, in the presence of a second length ξ' ∝ δ^{–ν'} in the VBS state, we propose that Eq. 1 should be replaced by the form (2)where, unlike what was assumed in the past, is not necessarily the same as the exponent ν, which governs the behavior of most observables in the thermodynamic limit. Instead, we show that the criticality in the *J*-*Q *model generically demands that .

First, assume . The correct thermodynamic limit with κ = *z*ν for ρ_{s} can then be obtained from Eq. 2 if *f*(*x*,*y*,*L*^{–ω}) ∝ *x ^{z}*

^{ν}for large

*x*= δ

*L*

^{1/ν},

*y*= δ

*L*

^{1/ν'}, and, as before, ρ

_{s}(δ = 0,

*L*) ∝

*L*

^{–}

*. This can also be expressed by using a scaling function in which the second argument is the ratio of the two lengths, (δ*

^{z}*L*

^{1/ν},

*L*

^{1/ν'–1/ν},

*L*

^{–ω}). If (δ = 0) is constant when

*L*→ ∞, then

*L*

^{1/ν'–1/ν}acts like just another irrelevant field, as in the standard scenario for dangerously irrelevant perturbations in classical clock models (

*31*). Our proposal is a different large-

*L*limit of Eq. 2, controlled by

*y*= δ

*L*

^{1/ν'}, which leads to concrete predictions of scaling anomalies. In the case of the stiffness, the correct thermodynamic limit is obtained with and κ =

*z*ν if

*f*(

*x*,

*y*,

*L*

^{–ω}) ∝

*y*

^{z}^{ν}for large

*L*. Then ρ

_{s}(δ = 0) ∝

*L*

^{–}

^{z}^{ν/ν'}, which we can also obtain with and ∝

*L*

^{z}^{(1–ν/ν') }for δ → 0. A function behaving as a power of

*L*was implicitly suggested in (

*19*), though with no specific form.

This alternative scaling behavior corresponds to ξ ∝ (ξ')^{ ν/ν'} saturating at ξ ∝ *L*^{ ν/ν'} when ξ' → *L* upon approaching the critical point, in contrast to the standard scenario in which ξ grows until it also reaches *L* (*32*). The criticality at distances *r* < *L*^{ν/ν'} is conventional, whereas *r* > *L*^{ν/ν'} is governed by the unconventional power laws. Different behaviors for *r* ≪ *L* and *r* ≈ *L* were observed in a recent loop-model study (*24*), and a dangerously irrelevant field was proposed as a possible explanation, but with no quantitative predictions of the kind offered by our approach. The anomalous scaling law controlled by ν/ν', which we confirm numerically below, is an unexpected feature of DQC physics and may also apply to other systems with two divergent lengths.

The *J*-*Q *model (*15*) for spins *S* = 1/2 is defined using singlet projectors (*P _{ij}*

_{}= 1/4 –

*S*∙

_{i}*S*) as (3)where denotes nearest-neighbor sites on a periodic square lattice with

_{j}*L*

^{2}sites, and

*ij*and

*kl*in form the horizontal and vertical edges of 2 × 2 plaquettes. The Hamiltonian

*H*has all symmetries of the square lattice, and the VBS ground state for

*g*=

*J*/

*Q < g*

_{c}(with

*g*

_{c}≈ 0.045) is columnar, breaking the translational and 90° rotational symmetries spontaneously. The Néel state for

*g*>

*g*

_{c}breaks the spin rotation symmetry.

Although we have argued that the asymptotic *L* → ∞ behavior when δ ≠ 0 in Eq. 2 is controlled by the second argument of *f*, the critical finite-size scaling close to δ = 0 (when δ*L*^{1/ν} is of order 1) can still be governed by the first argument (*32*). We will demonstrate that, depending on the quantity, either δ*L*^{1/ν} or δ*L*^{1/ν'} is the relevant argument, and, therefore, ν and ν' can be extracted using single-parameter scaling. We will first consider dimensionless quantities, corresponding to κ = 0 in Eq. 2, before testing the anomalous powers of *L* in other quantities.

If the effective one-parameter scaling holds close to *g*_{c}, then Eq. 2 implies that *A*(*g*,*L*_{1}) = *A*(*g*,*L*_{2}) at some point *g* that we denote *g**(*L*_{1},*L*_{2}), and a crossing-point analysis (Fisher’s phenomenological renormalization) can be performed (*29*). For a κ = 0 quantity, if *L*_{1} = *L* and *L*_{2} = *rL* with *r* > 1 being constant, a Taylor expansion of *f* shows that the crossing points *g**(*L*) approach *g*_{c} as *g**(*L*) – *g*_{c} ∝ *L*^{–(1/ν+ω)}, if ν is the relevant exponent (which we assume here for definiteness). *A** = *A*(*g**) approaches its limit *A*_{c} as *A**(*L*) – *A*_{c} ∝ *L*^{–ω}, and it can also be shown that the quantity (4)converges to 1/ν at the rate* L*^{–ω }. In practice, simulation data can be generated on a grid of points close to the crossing values, with polynomials used for interpolation and derivatives. We present details and tests of such a scheme for the Ising model in (*32*).

In the *S* = 1 sector, spinon physics can be studied with projector QMC simulations in a basis of valence bonds (singlet pairs) and two unpaired spins (*33*, *34*). Previously, the size of the spinon bound state in the *J-Q* model was extrapolated to the thermodynamic limit (*35*), but the results were inconclusive as to the rate of divergence upon approaching the critical point. Here we consider the critical finite-size behavior. We define the size Λ of the spinon pair by using the strings connecting the unpaired spins in valence-bond simulations (Fig. 1) (*32*–*34*).

If Λ(*g*) ∝ ξ'(*g*) when* L* → ∞, then Λ(*g*_{c}) ∝ *L* follows from our proposed limit of Eq. 2. If Λ manifestly probes only the longer length scale in a finite system, which we will confirm below, then ν' is the exponent controlling the crossing points of Λ/*L*. Data and fits are presented in Fig. 2 (left side). Unlike other quantities that have been used previously to extract the critical point (*18*), the drift of *g** with *L* is monotonic in this case, and the convergence is rapid. All *L* ≥ 16 points are consistent with the expected power-law correction, with 1/ν' + ω ≈ 3.0 and *g*_{c} = 0.04468(4), where the number in parenthesis indicates the statistical uncertainty (one standard deviation) in the preceding digit. The critical point agrees with earlier estimates (*18*). The scaled crossing value Λ*/*L* also clearly converges, and a slope analysis according to Eq. 4 gives ν' = 0.585(18).

In Fig. 2 (right side), we show the analysis of a Binder ratio, defined with the *z* component of the sublattice magnetization *m*_{sz} as and computed at *T* = 1/*L* as in (*18*). In this case, the nonmonotonic behavior of the crossing points necessitates several scaling corrections, unless only the largest sizes are used. In either case, the *L* → ∞ behavior of *g** is fully consistent with the *g*_{c} obtained from Λ/*L*. *R*_{1}(*g*_{c}) has an uncertainty of over 1% because of the small value of the correction exponent, ω ≈ 0.4 to 0.5. The slope estimator (Eq. 4) of the exponent 1/ν is monotonic and requires only a single *L*^{–ω} correction, also with a small exponent ω ≈ 0.45. The extrapolated exponent ν = 0.446(8) is close to the value obtained recently for the loop model (*24*).

The above results support a nontrivial deconfinement process in which the size of the bound state diverges faster than the conventional correlation length. However, in the DQC theory, the fundamental longer length scale ξ' is the thickness of a VBS domain wall. It can be extracted from the domain-wall energy per unit length κ, which in the thermodynamic limit should scale as κ ∝ (ξξ')^{–1} (*4*). In (*32*), we re-derive this form using a two-length scaling ansatz and discuss simulations of domain walls in a 3D clock model and the *J*-*Q* model. At criticality in the conventional scenario (exemplified by the clock model), both ξ and ξ' saturate at *L* and κ ∝ *L*^{–2}. For the *J*-*Q* model with large *L*, we instead find κ ∝ *L*^{–}* ^{a}* with

*a*= 1.715(15) (Fig. 3A). Our interpretation of this unconventional scaling is that when ξ' saturates at

*L*, ξ also stops growing and remains at ξ ∝

*L*

^{ν/ν'}. Thus, κ ∝

*L*

^{–(1+ν/ν')}with ν/ν' =

*a*– 1 = 0.715(15), which agrees reasonably well with ν/ν' = 0.76(3), obtained from the quantities in Fig. 2. The large error bar on the latter ratio leaves open the possibility that the spinon confinement exponent is between ν and the domain-wall exponent ν' (

*4*).

We also calculated the critical spin stiffness ρ_{s} and the susceptibility χ(*k* = 2π/*L*) for the smallest wavenumber *k* at *T* = 1/*L*. Conventional quantum critical scaling (*2*) dictates that both quantities should decay with *L* as 1/*L*. Instead, panels B and C in Fig. 3 demonstrate slower decays, with *L*ρ_{s} and *L*χ being weakly divergent, as has also been found in earlier works (*17*–*19*, *21*, *30*). The unconventional limit of the scaling function in Eq. 2 requires *L*ρ_{s} and *L*χ to diverge with *L* as *L*^{1–ν/ν'}. The behaviors are consistent with ν/ν' ≈ 0.715, extracted from κ and a correction ∝ *L*^{–ω} with a small ω (close to the correction for *R*_{1} in Fig. 2). The mutually consistent scaling of the three quantities lends strong support to a type of criticality in which the magnetic properties are not decoupled from the longer VBS length scale ξ' for finite *L*. The results are incompatible with a first-order transition, where κ → constant, *L*ρ_{s} → *L*, and *L*χ → *L*.

We have shown that the effects of the larger divergent length scale ξ' at the Néel-VBS transition are more dramatic than those caused by standard dangerously irrelevant perturbations (*31*), and we therefore propose the term “super-dangerous” for this case. The universality class, in the sense of the normal critical exponents in the thermodynamic limit at *T* = 0, is not affected by such perturbations, but anomalous power laws of the system size appear generically in finite-size scaling. We have determined the value ν/ν' ≈ 0.72 for the exponent ratio governing the anomalous scaling in the *J*-*Q* spin model.

Loop and dimer models exhibit similar scaling anomalies (*24*, *25*), and it would be interesting to test the consistency between different quantities in these models, as we have done in this study. In simulations of the NCCP^{1} action (*21*, *26*, *27*), one might not expect any effects related to the longer DQC length scale, because the monopoles responsible for the VBS condensation are not present in the continuum theory (*3*). Nevertheless, there could be some other super-dangerous operator present (*24*), perhaps related to lattice regularization.

The consequences of our findings carry over also to *T* > 0 quantum criticality in the thermodynamic limit, because 1/*T* is the thickness of an equivalent system in the path integral formulation (*1*, *2*). Anomalous finite-*T* behaviors of the *J*-*Q* model have already been observed (*18*, *30*). For instance, the spin correlation length at* T* > 0, which should be affected by deconfined spinons, grows more quickly than the normally expected form *T*^{–1}, and the susceptibility vanishes more slowly than *T* when *T*→0. The asymptotic forms *T*^{–ν'/ν} and *T*^{ν/ν'} can account for the respective behaviors (fig. S10). Thus, we find a strong rationale to revise the experimentally most important tenet of quantum criticality: the way that *T* = 0 scaling is related to power laws in *T* at* T* > 0. Our findings may apply to a wide range of strongly correlated quantum systems with more than one length scale and may help to resolve the mysteries that still surround scaling behaviors in materials such as high-*T*_{c} cuprate superconductors.

## Supplementary Materials

## References and Notes

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- ↵Contributions originating from the regular part of the free energy are also of the same form and, in some cases, where the exponents of the irrelevant fields are large, they can cause the leading scaling corrections (
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- ↵We thank K. Damle for suggesting a definition based on the entire spinon string.

**Acknowledgments:**The research was supported by the National Natural Science Foundation of China, grant no. 11175018 (W.G.); the Fundamental Research Funds for the Central Universities (W.G.); U.S. NSF grant no. DMR-1410126 (A.W.S.); and the Simons Foundation (A.W.S.). H.S. and W.G. gratefully acknowledge support from the Condensed Matter Theory Visitors Program at Boston University, and A.W.S. is grateful to the Institute of Physics of the Chinese Academy of Sciences and to Beijing Normal University for their support. Some computations were carried out using Boston University’s Shared Computing Cluster. We thank K. Damle for suggesting a definition of spinons that uses valence-bond strings. Computer code is available from the authors upon request.