† Corresponding author. E-mail:

Project supported by the NSF under Grant No.DMR-1710170 and by a Simons Investigator Grant.

We use quantum Monte Carlo simulations to study an *S* = 1/2 spin model with competing multi-spin interactions. We find a quantum phase transition between a columnar valence-bond solid (cVBS) and a Néel antiferromagnet (AFM), as in the scenario of deconfined quantum-critical points, as well as a transition between the AFM and a staggered valence-bond solid (sVBS). By continuously varying a parameter, the sVBS–AFM and AFM–cVBS boundaries merge into a direct sVBS–cVBS transition. Unlike previous models with putative deconfined AFM–cVBS transitions, e.g., the standard *J*–*Q* model, in our extended *J*–*Q* model with competing cVBS and sVBS inducing terms the transition can be tuned from continuous to first-order. We find the expected emergent U(1) symmetry of the microscopically *Z*_{4} symmetric cVBS order parameter when the transition is continuous. In contrast, when the transition changes to first-order, the clock-like *Z*_{4} fluctuations are absent and there is no emergent higher symmetry. We argue that the confined spinons in the sVBS phase are fracton-like. We also present results for an SU(3) symmetric model with a similar phase diagram. The new family of models can serve as a useful tool for further investigating open questions related to deconfined quantum criticality and its associated emergent symmetries.

The spin *S* = 1/2 Heisenberg model with uniform nearest-neighbor interactions on the two-dimensional (2D) square lattice has a Néel antiferromagnetic (AFM) ordered ground state.^{[1]} By introducing other interactions such as frustration or multi-spin interactions, the quantum fluctuation of the AFM order parameter can be increased, eventually destroying the AFM order and leading to a different ground state. The long-wavelength behavior of Heisenberg and similar quantum magnets can be described field-theoretically by the O(3) nonlinear sigma model with a Berry phase term.^{[2,3]} The Berry phase term will vanish for any smooth spin configurations, i.e., in the AFM phase, but will likely influence the phase diagram of the system if topological defects such as “hedgehog” singularities are considered. In (2+1) dimensions, such effects can play a dominant role and may drive the system into exotic paramagnetic phases, by which we mean collective many-body states with no direct classical analogues (see Ref. [4] for a review).

One example of an interesting 2D paramagnetic phase is the so-called valence-bond solid (VBS),^{[5–7]} which preserves spin rotational symmetry but spontaneously breaks lattice symmetries through the condensation of singlets forming a regular pattern (or, more precisely, a modulation of the singlet density forms). There are several possible ways to break the lattice symmetries, and therefore VBS phases can also appear in many different incarnations. In this work we discuss two cases of dimer VBSs (i.e., the singlets form between two neighboring spins): the columnar VBS (cVBS) and staggered VBS (sVBS). As shown in Fig. *m* down to *pmm*, while in the sVBS phase the singlets are successively shifted by one lattice spacing and form a stair-pattern. This pattern belongs to the *cmm* space group. Though the symmetry groups are different, both these VBSs are four-fold degenerate on the infinite lattice or when placed on a torus with an even number of sites in both directions. We will discuss the competition between cVBS and sVBS orderings as well as the quantum phase transitions of these phases into the AFM state.

From the field theoretical perspective, the cVBS phase can be understood as a nontrivial magnetically disordered phase resulting from quantum tunneling of Skyrmions in a certain limit.^{[6–8]} One important consequence of this picture is the deconfined quantum criticality (DQC) scenario.^{[9]} By assuming the conservation of the Skyrmion number at the AFM–cVBS transition, it was proposed that this transition can be described by a CP^{1} field theory with non-compact U(1) gauge field. There are several interesting and controversial assumptions and consequences of the DQC scenario, e.g., a generically continuous transition where normally a first-order transition would be expected (since the two ordered phases break unrelated symmetries), fractionalized excitations at the critical point, and a “dangerously irrelevant” Z_{4} perturbing field with an associated emergent U(1) symmetry.^{[10]} This type of order–order phase transition is beyond the conventional Landau–Ginzberg–Wilson (LGW) paradigm and has attracted intensive scrutiny during the past several years.

As some elements of the DQC scenario are speculative, it is important to realize such phase transitions in concrete microscopic models, to fully test and explore the possible physics arising from the field theoretical proposal and beyond (i.e., features that were not part of the original DQC proposal). Since the sign-problem free *J-Q* model^{[11]} (with Heisenberg exchange *J* and a correlated multi-singlet projection *Q*) was proposed, many detailed studies of the AFM–cVBS transition and the cVBS phase itself have been carried out with quantum Monte Carlo (QMC) simulations of this model^{[11–24]} and of classical 3D models exhibiting analogous transitions.^{[25–28]} These studies solved some previously open questions, but they also posed new ones. For example, observations of emergent higher symmetries at the critical point have been reported recently, with the AFM and cVBS order parameters combining into a larger vector transforming under SO(5)^{[22,28]} or O(4)^{[29,30]} symmetry, depending on the model. The possibility of higher symmetry was pointed out already some time ago^{[31]} and the numerical observations have further spurred the interest in this phenomenon, including in the context of the “web of dualities” between different field theories.^{[29,32]} Furthermore, a description of the transition in terms of non-unitary complex conformal field theories (CFTs)^{[33–36]} was inspired by unusual scaling behaviors observed in *J–Q* and other models,^{[13–15,25–27]} which by some have been interpreted as signs of an eventually very weakly first-order transition in accessible models, with the critical point existing only in the complex plane. Alternatively, it has also been proposed that the transition is continuous, as in the original DQC scenario, but with unusual scaling behavior stemming from two divergent length scales.^{[23]} It should be noted that there are no unambiguous signs of first-order transitions in the best candidate models, and very recently further evidence has been presented in support of a truly continuous transition.^{[38]} A further intriguing fact is that, in some models where the AFM–VBS transition is clearly first-order, the coexistence state exhibits emergent O(4)^{[37,39]} or SO(5)^{[40]} symmetry, instead of forming two distinct phases separated by tunneling barriers (in analogy with conventional classical co-existence states separated by free-energy barriers).

While the phenomenology of the DQC scenario does not rely on the AFM–VBS transition being strictly continuous (requiring in practice only that the correlation length is very long, which has already been well established), it is still of fundamental interest to try to answer the following basic questions: (i)Are the candidate DQC transitions observed in simulations truly continuous, or do discontinuities and phase coexistence develop on some large length scale? (ii)Is the emergent symmetry of the putative DQC point exact in the thermodynamic limit or does it break down at a finite length scale (i.e., even if the DQC transition itself is truly continuous)? (iii)Are the emergent O(4) and SO(5) symmetries observed in the coexistence states at some first-order AFM–VBS transitions asymptotically exact, or do they break down above some length scale?

These questions are still hard to answer conclusively based on current numerical results [though regarding (i), we again note that Ref. [38] has tilted the scale further toward a continuous transition], because they appear to involve exceedingly large length scales (system sizes). In order to gain deeper understanding of the AFM–cVBS transition, it would be fruitful to construct models that enable tuning of the exotic properties and ingredients addressed in the theory of DQC points, and more broadly in scenarios of weakly first-order transitions with unusual properties. For example, suppressing the possible emergent symmetry of the transition is a promising approach, since it may give rise to a conventional strongly first-order transition that could be detectable with current numerical techniques. If the transitions can be tuned from continuous (or extremely weak first-order), to moderately weakly and strongly first-order, many of the intriguing phenomena listed above could be studied more systematically than what has been possible so far.

The emergent U(1) symmetry of the DQC scenario [and also the possible higher spherical symmetry like SO(5)] is also related to the properties of the vortices in the VBS state. In the cVBS, a nexus of four domain walls separating the four different dimer patterns must necessarily have an unpaired spin,^{[10]} i.e., the vortex is a spinon. Such spinons are bound into pairs in the cVBS phase and their deconfinement upon approaching the DQC point is directly related to the Z_{4} vortices evolving into U(1) vortices. Even though the sVBS is also four-fold degenerate, the Z_{4} vortices in the sVBS can either carry a *S* = 1/2 spinon or not, as illustrated in Fig. ^{[41]} Though in the case of the pVBS the fracton property may only be realized in practice in extreme cases where the four-spin singlets can be regarded as rigid objects, as pointed out in Ref. [40], the phenomenon may actually be more easily realizable in an sVBS.

It would be interesting to investigate whether competing sVBS interactions in a cVBS phase can suppress the fluctuation required for emergent U(1) symmetry and deconfined spinons, thus leading to a different type of AFM–cVBS phase transition. We here desire to affect the properties of the vortices by tuning suitable microscopic couplings. When the sVBS favoring interactions are sufficiently strong they may render the cVBS–AFM transition first-order, unless there is a direct cVBS–sVBS transition. Conversely, the AFM–sVBS transition may possibly become less strongly first-order in the presence of the cVBS favoring interactions.

Studying an sVBS phase and its transition to either an AFM phase or directly into a cVBS phase is interesting in its own right. There have been some discussions regarding phase transitions of the sVBS phase: Vishwanath *et al*. proposed a possible continuous phase transition in a bilayer honeycomb lattice,^{[42]} and Xu *et al*. studied certain types of transitions between the sVBS and a Z_{2} spin liquid or an AFM phase.^{[43]} Numerically, an sVBS state was studied with a *J–Q*_{3} model where the three singlet projectors in the Q_{3} terms are arranged in a staircase fashion for the square lattice.^{[44]} The AFM–sVBS transition in this case as well as in other lattices^{[45]} are very strongly first-order, and the sVBS phase itself exhibits only weak fluctuations from the maximally ordered dimer singlet configuration. Therefore, the model in its current form does not offer many insights into the more interesting case of a strongly fluctuating sVBS where topological defects can play a more prominent role. Considering the possibility of fracton-type spinons and their propensity to nucleate AFM order (in analogy with Ref. [41]), it is, however, not even clear if very strong quantum fluctuations of the sVBS can be achieved before a first-order transition takes place.

Since we are interested in the generic properties of the VBS phases, we will take the approach of “designer Hamiltonians”.^{[46]} The central idea here is to construct an easy-to-study model (e.g., without sign problems in QMC simulations) and to analyze particular phases and phase transitions. The concepts of universality and renormalizaion-group fixed points ensure that the low-energy properties of interest are still of relevance to real-world applications, even though interactions that can be realized in real materials are not faithfully represented microscopically. The following is a list of desired features for the designer Hamiltonian in our study:

harbors all of cVBS, sVBS, and AFM phases;

the cVBS–AFM transition can be tuned from continuous or weakly first-order to more strongly first-order;

realizes a direct cVBS–sVBS transition as well as cVBS and sVBS in parts of its phase diagram.

In previous studies, none of the existing models were able to connect the first-order sVBS–AFM transition and the cVBS–AFM DQC transition, nor could they realize a direct transition between the two VBS states. In this work, we introduce a model Hamiltonian with two competing terms that individually favor a cVBS and an sVBS ground state, respectively, as discussed above. The motivation here is that the presence of the sVBS favoring interactions in the cVBS phase may to some degree suppress the development of the emergent U(1) symmetry that characterizes the DQC transition between the cVBS and the AFM. We will show here that these expectations are borne out by our results for the model we have constructed, in particular as regards the evolution of the AFM–cVBS transition from continuous or very weakly first-order to clearly first-order.

The structure of the remainder of this article is as follows: In Section *S* = 1/2 [SU(2)] models we study in this work. In Section *N*) generalization of our model and present results for *N* = 3. In Section

We start from two previously studied designer Hamiltonians that host cVBS and sVBS phases, respectively. The ground state of a columnar *J–Q*_{3C} model^{[17]} undergoes a DQC type cVBS–AFM transition, and the ground state of a staggered *J–Q*_{3S} model^{[44]} exhibits a first-order sVBS–AFM transition. Here and henceforth we use the second subscript on *Q* to indicate a columnar (C) or staggered (S) spatial arrangement of the three singlet projectors used in the multi-spin interaction. A natural strategy to construct a model that harbors both cVBS and sVBS ground states would be to combine the *Q*_{3C} terms and *Q*_{3S} terms as follows:

*P*denotes a projection operator to a spin singlet state on sites

_{ij}*i*and

*j*,

*P*

_{ij}= 1/4-

*i–m*in columns or stairs, respectively. For details of the placement of the indices we refer to Figs.

In the two extreme cases where *h* would be in the cVBS and sVBS phase, respectively. However, there is not necessarily a direct transition between these two VBS phases with just the terms in the Hamiltonian defined in Eq. (

In order to achieve a direct cVBS–sVBS transition, we have to introduce another interaction. The Heisenberg exchange *J* only strengthens the AFM order and expands the AFM phase between the two VBS phases. We will therefore not further discuss the Heisenberg interaction here and instead consider another type of multi-spin *Q*-type interaction. From the field theoretical point of view, we would like to turn on some symmetry-allowed relevant terms with respect to the cVBS–AFM fixed point. An interaction achieving our aim is the

Since the

To study all three phases we define their order parameters as follows:

_{z}also taking into account that only one out of three components of the staggered magnetization is used), such that the cumulants approach

*1*with increasing

*L*if there is order of the given type and

*0*otherwise. The condition

We used the stochastic series expansion (SSE) QMC method for all the calculations presented here. For details of this algorithm we refer to the review in Ref. [47], which includes an implementation for the conventional *J–Q* model. The more complicated *Q* terms used here can be treated with simple generalizations.

To investigate the ground state phase diagram of the model, Eq. (*(g,h)*:

In Fig. *g* for several values of *h*. Here we use a rather small system, *L* = 16, but this already gives us an initial impression about the phase diagram. Figure *L* = 16 close to the transition (some of which are further discussed below), for extrapolating to the thermodynamic limit.

We observe the three different phases as expected. A key feature here is that the *h* close to 0.35, is strongly first-order, as also would be expected on the grounds that the two phases break the lattice symmetry in different ways.

The sVBS–AFM transition is always first-order, as is clear from the sharp jumps in both the order parameters and Binder cumulants in Fig. ^{[44]} The first-order transition is expected since, as discussed in Section *h*. The location of the triple point where all the phases come together still has some uncertainty, due to difficulties with the size extrapolations close to this point.

As discussed in Section *J –Q* models (the *J–*
*J–Q*_{3C} models in the notation used here) are of the DQC type, possibly with extremely weak and currently not detectable discontinuities though a strictly continuous transition appears more likely.^{[38]} The extended model used here does not reduce to the previously studied cases because of the lack of *J* terms and also because we keep *L* = 16 results shown in Fig. *h* = 0 and 0.2 (where there is still an AFM phase). However, in order to determine the true nature of the transition the system-size dependence has to be studied very carefully and on much larger lattices. Here our aim is not to draw definite conclusions, as very significant computational resources are required in order to obtain size-converged critical exponents or, alternatively, to detect weak discontinuities.

As an initial study of the AFM–cVBS transition in the extended *J–Q* model, in Figs. *h* values, *h* = 0,0.1,0.2, and 0.25, for which the transition is either continuous or weakly first order, and in each case we show results for several different system sizes between *L* = 16 and L = 80. For h = 0 and 0.1, we find that both order parameters evolve smoothly versus *g* for all system sizes, while for the larger *h* values obvious discontinuities develop with increasing *L*. For the smaller *h* values the Binder cumulants for different *L* cross each other in what appears to be a single point (though one should expect some drifts in the crossing points if the behaviors are examined in more detail with high-quality data on a denser grid), which is characteristic for continuous transitions. For the larger *h* values the cumulants are more strongly varying and also become negative in the neighborhood of the transition. The latter behavior suggests that the cumulants will eventually, for larger system sizes, develop the sharp negative peaks that are characteristic for conventional first-order transitions. Such peaks should asymptotically diverge in proportion to the system volume, as previously observed with the ^{[44]}

Recently, unconventional first-order transitions with emergent spherical symmetry of the order parameters were identified where negative cumulant values are observed, which can be traced to the lack of tunneling barriers when the two order parameters can be continuously rotated into each other.^{[37,39,40]} The results for *h* = 0 and 0.1, we can only say with certainty that the transitions are less first-order-like than those at the higher *h* values, as it is possible that discontinuities and negative cumulant peaks develop above some system size that decreases with increasing *h*.

Note again that, also at *h* = 0 there are Q_{3S} interactions present in the (*g,h*) parameterization of the model that we have used here. It is possible, in principle, that the staggered term alters the DQC transition, though we see no evidence of such behavior here (and the prospect also appears unlikely, as there is no apparent new scaling field that this operator can bring in that is originally absent). To compute the scaling dimension of the Q_{3S} coupling is possible in principle by analyzing the corresponding correlation function at the AFM–cVBS transition of the standard *J–Q* model. However, with the large number of singlet projectors involved (six in total for the correlation of the Q_{S3} terms) such a calculation would be very challenging^{[48]} and we have not attempted this.

Consider the global cVBS angle *ϕ* defined from the complex order parameter *m*_{c} in Eq. (*π*/2) can gradually shift the angle. Similarly, by replacing a pair of parallel dimers by a superposition of *x*- and *y*-oriented dimers (i.e., plaquette singlets), the angle also shifts relative to that with only static columnar dimers. Since the dimer order in the cVBS phase undergoes significant fluctuations (not only of the kind affecting short dimers as discussed above, but the actual state also contains longer bipartite valence bonds), neither the angle *ϕ* nor the magnitude

The order parameter can also be defined on some cell of size *λ* smaller than the lattice size *L*, and, in the same way as explained above, even in an cVBS state on an infinite lattice the local angle *ϕ*(** r**) thus defined in a cell centered at

**can take values also between the four columnar angles**

*r**n*over the width of the domain wall, and such intermediate angles correspond to the presence of resonating valence bonds. In the center of a

*n*values the domain wall separates). In the DQC scenario these

*π*domain walls,

^{[10]}as has been explicitly observed in studies of

*J–Q*models

^{[49]}(where an induced

*π*domain wall splits up spontaneously into a pair of

The domain-wall broadening also corresponds to a lowering of an effective potential ^{[11–13]} and the fact that the symmetry has been observed on very large length scale is one of the most important indicators of the DQC phenomenon actually being realized in these models.^{[4]}

The emergent U(1) symmetry of the cVBS is analogous to the classical 3D clock models with a potential *q*) added to the standard *XY* model with nearest-neighbor interactions ^{[10]} In the clock models, for *XY* universality class, at some critical temperature *ν*’and *ν*, respectively, and ^{[50–53]} and for which new insights were presented very recently.^{[54]} The analogous U(1) length scale has also been studied in the standard *J–Q* model,^{[17]} though not yet at the level of precision as was possible in the classical case.

Here we wish to make some observations regarding the emergent U(1) symmetry of the cVBS state as the transition changes from continuous to first-order when we increase the tuning parameter *h* (as demonstrated above in Figs. ^{[10]} As discussed above, the elementary domain walls are of the *n* of the types

It can be noted that, the classical 3D *q* = 4 clock model with potential depth *q* = 4 clock model with small *XY* universality class on account of the emergent U(1) symmetry. The exact value of ^{[21,54]}

The cVBS state in the conventional *J–Q* models also exhibits clock-like fluctuations and emergent U(1) symmetry, and is, thus, more similar to the soft *q* = 4 clock model (though the universality class is different; DQC instead of *XY*).^{[11–13]} In Fig. *h* = 0, where according to the results presented above in Subsection *L* = 32, where both the angular and amplitude fluctuations are large. The probability density is very low in the center of the distribution, reflecting a robust non-zero magnitude ^{[37,40]} would be required to draw firm conclusions, but at least it appears plausible here that the system has the emergent U(1) symmetry expected at a DQC transition.

It is now interesting to see how the cVBS order parameter distribution evolves as we increase *h* and enter the regime where the AFM–cVBS transition is clearly first-order. In Fig. *h* = 0.2. Here we no longer observe clock-like fluctuations, but instead see significant probability of fluctuations toward the center of the distribution, where in the Z_{4} case there is very little weight. If these fluctuations would be due to direct migration between the two maxima on the *x* or *y* axis, we would conclude that the *π* domain wall has replaced the *h*, we still observe the clock-like

The Binder cumulant results in Fig. *X* type, on account of Gaussian fluctuations. However, the sVBS fluctuations are not Gaussian-distributed in the cVBS phase, as we show here.

Consider the two-dimensional distribution of the complex order parameter

*N*) model and results for

*N*= 3

So far, the model we have studied was defined with the standard *n*) generalization, in which the spins on one sublattice transform under the fundamental representation of the group and the ones on the opposite sublattice transform with the conjugate of the fundamental representation.^{[6]} In the Heisenberg model with only nearest-neighbor Heisenberg exchange, a large-*N* mean-field-like theory with 1/N corrections predicts that the AFM order vanishes above ^{[7]} after which the ground state is a cVBS state. QMC simulations for integer *n* indeed found AFM order up to *N*.^{[55]} QMC simulations with loop-type algorithms can also be generalized to non-integer *N*, and a critical ^{[56]} *J–Q* models generalized to SU(*n*) have likewise been useful for testing ^{[46]} and a remarkable agreement has been found between the *N*.^{[57,58]}

Given that larger *N* drives the Heisenberg and *J–Q* systems toward VBS ordering, in the model studied here we would expect the AFM phase to shrink upon increasing *N*. Here our aim is just to demonstrate this behavior, and we leave more detailed studies to future work. Using *N* = 3, we set *h* = 0, i.e., we only have the competing

As shown in Fig. ^{[57]} We use the same definitions of the Binder cumulants as in Eq. (_{c} and U_{s} will approach 1 if there is cVBS and sVBS long-range order, respectively. If there is no such order the values will not go to 0, however, for the same reasons as discussed in Subsection _{z} will approach a value different from 1. In a non-AFM state the value should still approach 0. The results shown in Fig. *U*_{z} stays very close to 0.

We can conclude that the AFM order is always suppressed in the SU(3) Q_{3C}–Q_{3S} model and there is a direct, first-order sVBS–cVBS transition similar to the case of our SU(2) model in Fig. *h* is large. We can then achieve a similar phase diagram with an AFM phase and a triple point by adding Heisenberg exchange *J* to the SU(3) Hamiltonian. As mentioned above, for *Q* term is also used).^{[57]} In our model studied here, the Q_{3S} interactions compete against cVBS order and therefore the SU(N) *J*–Q_{3C}–Q_{3S} model should have a phase diagram similar to Fig. *h* replaced by *N*.

In this paper we have proposed a sign-free designer quantum spin model, Eq. (

Our QMC results show that the

Recently, the so-called “walking” behavior in non-unitary complex CFTs^{[32,33]} has gained attention as a way to explain the weakly first-order nature of some transitions, such as *q*-state Potts models with ^{[32,34,35]} Our model is the first in which the AFM–cVBS transition can be continuously tuned from what appears to be continuous to clearly first-order. This model should therefore constitute a good framework for studying the walking behavior in detail. The initial results presented here show that the emergent U(1) symmetry of the order parameter is absent when the transition becomes first-order. This behavior is in sharp contrast to the emergent higher symmetries, O(4) and SO(5), of the order parameters recently observed at some first-order AFM–VBS transitions in related models.^{[37,39,40]}

We also discussed a generalized model with SU(*n*) symmetry and showed some results for the SU(3) case. It is well known that the AFM order is suppressed when *n* is increased,^{[6,7,55]} and in the SU(3) *J*) term (possibly with longer-range interactions for larger *N*^{[57]}). Thus, we can in principle study phase diagrams and investigate the evolution of different phase transitions similar to what we did here for the SU(2) model.

Recently, there have been discussions on the mobility properties of spinons in a 2D VBS state different from the ones considered here — the pVBS phase. Within a scenario in which the plaquette singlets are treated as rigid objects, spinons as well as their bound “triplon” states are fractons, restricted to motion in one dimension.^{[41]} One prediction following from this picture is that spinons do not deconfine, and instead of a DQC transition (which in the DQC theory can take place in a cVBS or a pVBS) a first-order transition results from nucleation of triplons. However, the fracton argument for pVBS should only apply in extreme cases, as in generic quantum spin models the four-spin singlets can not be regarded as rigid objects but must be allowed to decay into dimer singlets.^{[40]} Here we note that some of the prerequisites for fractons outlined in Ref. [41] in the context of the pVBS have analogies in the sVBS, and the latter may be a more likely host of fractons under realistic conditions. Though we do not have quantitative results from QMC simulations, we will here argue that fracton-like excitations are generically possible in sVBS states.

The fully mobile spinons play a crucial role as vortices in the DQC theory, and it is therefore interesting to compare the nature and mobility of the spinons in cVBS and sVBS states. While the two VBS phases are both four-fold degenerate, they correspond to different kinds of symmetry breaking of the lattice, thus having different minima in the Landau free energy space. The two VBS phases also host different types of domain walls, leading to different behaviors of the vortex excitations (Fig. ^{[10]} In contrast, the sVBS phase does not necessarily require spinons-type vortices, though vortices with spinons are also possible (Fig. ^{[41]}

Figure *J*-terms in the Hamiltonian, we can see that the individual spinons can move around in two different directions; along the 45° diagonal in Fig. *π*-domain walls otherwise. In the diagrams, we indicate the *π*-domain wall.

At an AFM–cVBS DQC point, it is predicted^{[9]} and numerically observed^{[49]} that the domain wall energy vanishes, resulting in configurations such as those depicted in Fig. *π*/2- or (c)*π*-domain walls would contribute to the low-lying state, and configurations like (d) would be suppressed because of the presence also of the non-favored type of domain walls (and which one has a lower energy should depend on the specific microscopic interactions). As a consequence, the spinons are only able to move freely in one or two directions, depending on the favored domain wall type, becoming effectively fractons.

The fracton property is emergent, and in principle, the spinons can still move around in different directions in the case where the *π*/2-domain wall in Fig.

The mobility properties of the spinons in the sVBS phase we have discussed now are in sharp contrast with those in the cVBS phase. As we show in Fig. *J*-terms This mechanism only requires *π*/2-domain walls, which are indeed the one that is favored energetically. We can also see from Fig. ^{[45]} this property of allowing multiple spinons may play a role for the first-order transitions taking place, since it allows nucleation of AFM order from clusters of spinons.

The true lowest triplon excitation state should be a superposition of the type of extended triplet valence-bond basis states discussed above, each of them transforming to a different constituent of the state when a Heisenberg *J*-term is applied. The specific model discussed in the preceding sections does not have any explicit *J* terms (two-spin singlet projector) but the same fluctuation effect is also achieved by appropriately applying the *q* terms Valence-bond configurations with higher diagonal energy should have a smaller contribution to the lowest triplon excitation, and by analyzing those configurations we can argue how the spinons proliferate in space. Though we have not discussed the mechanisms in a rigorous manner here, the picture is intuitive and should capture the correct physics qualitatively.

The work we presented here illustrates the power of the designer Hamiltonian approach in engineering sign-free Hamiltonians exhibiting interesting phase transitions and enabling unbiased numerical studies of physically interesting situations. Our results suggest several possible follow-up studies, some of which we summarize here.

First, precisely determining the tricritical point in Fig. *J–Q* model^{[23]} would be helpful to confirm the universality class of the DQC point. Here we should again note that the continuous transition in the conventional *J–Q* model has not been demonstrated completely conclusively, though there are no explicit signs of first-order discontinuities. Detailed studies of the putative tricritical point, or, alternatively, demonstrating its absence (i.e., a weakly first-order transition below the blue circle indicating the putative tricritical point in Fig.

We have discussed the fluctuation patterns of the order parameter inside the VBS phases. The change in fluctuation paths where the transition changes to first order can be seen as a numerical confirmation of the graph-theoretic approach introduced in the context of the AFM–pVBS transitions in Ref. [40]. Further studying when and how the fluctuation pattern between degenerate ground states changes within a single phase could shed light on what kind of phases are allowed to exist adjacent to each other in the phase diagram, and indicate relevant and irrelevant perturbations of the DQC point.

Another interesting aspect of the sVBS phase deserving further study is the nature of the spinons and their bound states. As we show in Fig.

We would like to thank Ribhu Kaul for useful discussions. This work was supported by the NSF under Grant No.DMR-1710170 and by a Simons Investigator Grant. The numerical calculations were carried out on the Shared Computing Cluster managed by Boston University’s Research Computing Services.

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