A gapped spin liquid phase with a mean field ansatz: a Bosonic resonating valence-bond description
Gapped spin liquid as the simplest spin liquid has been proved to be the most difficult to realize in realistic models. Here we show that the frustration from a third-neighbor exchange on the spin- model on the square lattice may serve to stabilize such a long-sought state. We argue that a Bosonic RVB description is more appropriate than a Fermonic RVB description for such a gapped spin liquid phase. We show that while the mean field ansatz of the proposed state has a gauge symmetry, the gauge fluctuation spectrum on it is actually gapped and the state should be understood rather as a spin liquid state with topological order. The state is thus locally stable with respect to gauge fluctuation and can emerge continuously from a collinear Neel ordered phase.
Quantum spin liquids are exotic state of matter that support fractionalized excitationsbalents (); palee (); lhuillier (). The resonating valence bond(RVB) picture proposed more than four decades ago by Anderson remains the best way to envisage a quantum spin liquidRVB (). An RVB state is made of the coherent superposition of different singlet pairing patterns of the local spins on the lattice. The short-ranged RVB state is the first proposed and probably the simplest spin liquid state. It features a gapped spectrum in both spin triplet and spin singlet channel and possesses topological order. On a two dimensional torus, it exhibits a typical four fold topological degeneracy. However, in spite of its seemingly innocent appearance, such a gapped spin liquid state has been proved to be the most difficult to realize in realistic models.
Geometric frustration of interaction is the most important way to realize a quantum spin liquid. Numerical studies on frustrated quantum antiferromagnetic models have reported controversial evidences for the existence of gapped spin liquids. For example, DMRG studies on both the spin- Kagome Heisenberg model and the model on the square lattice have reported gapped spin liquid ground state in some parameter regionwhite (); jiang (). However, both claims suffer from uncertainty due to the smallness of the reported gap, especially that in the spin singlet channel, and are both challenged by further studies that report gapless ground state in the claimed parameter regionsran (); liao (); sheng (). This may suggest that these systems are still not sufficiently frustrated.
In this paper, we will focus on the spin- model on the square lattice and ask if a gapped spin liquid can be stabilized when we introduce additional frustration in the model. The model on the square lattice has been investigated by many people in the last two decadesdoucot (); gelfand (); mila (); schulz (); vbs1 (); vbs2 (); capriotti (); sushkov (); mambrini (); darradi (); wangl (); jiang (); boson (); sheng (). At the classical level, the system is Neel ordered for with ordering wave vector . For , the Neel order is replaced by a stripy magnetic order with ordering wave vector or after a first-order transition. Spin wave fluctuations will destroy the magnetic order in the intermediate region of . The nature of this disordered phase is under debate even after twenty years’ intensive study. It is generally believed that some kind of spatial symmetry breaking will happen in this intermediate parameter region.
More recently, a gapped spin liquid phase has been reported by a DMRG study in the intermediate region of , which seems to emerge continuously from the Neel ordered phase for jiang (). This claim, however, is challenged by a further DMRG study, which shows that the Neel ordered phase for is connected to a plaquatte valence bond solid state for , with a small gapless paramagnetic region in betweensheng (). A continuous transition between the Neel ordered phase and a gapped symmetric spin liquid phase is also at odds with a well known effective field theory predictionread (); read1 (); read2 (), which claims that by quantum disordering a collinear antiferromagnetic order one inevitably encounter spatial symmetry breaking as a result of the instanton effect.
The main purpose of this paper is to show that the above prediction of the effective field theory is not necessarily correct. Continuing a previous work on the model on the square latticeboson (), and equipped with a recent development on the gauge field theory of a quantum antiferromagnettri (), we show that by introducing the frustration from a third-neighbor exchange , a fully gapped spin liquid can develop continuously from a Neel ordered phase. We show that although the spin liquid state we proposed has a mean field ansatz with a staggered gauge symmetry, it should be understood as a spin liquid state with intrinsic topological order and gapped gauge fluctuation spectrum.
In this work we will describe the gapped spin liquid phase with the Bosonic RVB theory. As compared to the more commonly used Fermionic RVB theory, the Bosonic RVB theory has the following advantages for our purposeliang (); sandvik (); beach (); xiu (); bilayer (). Firstly, the Bosonic RVB theory can describe the gapped spin liquid phase and the magnetic ordered phase on the same footing and is thus particularly appropriate in the transition region between the two phases. The situation is totally different in the Fermionic RVB theory, in which one need to break the spin rotational symmetry by hand in the magnetic ordered phase. Secondly, the Bosonic RVB theory predicts that only the spin correlation beyond the correlation length would be significantly affected by the opening of the spinon gap. When the correlation length is much larger than the lattice scale, the local correlation would hardly be affected by the gap opening process, just as what we would expect for a spin liquid evolved continuously from the magnetic ordered phaseliang (); brvb (). The situation is again totally different in Fermionic RVB theory, in which gap opening in the spinon spectrum is necessarily accompanied by the appearance of symmetry breaking order parameters at the mean field level, which will affect the local correlation in the standard Landau fashion. This makes the Fermionic RVB state unlikely an appropriate description for a gapped spin liquid phase with incipient magnetic ordering instability.
In this paper, we consider the following model on the square lattice:
in which equals to , and on first-, second- and third-neighbor bonds and is otherwise zero, denotes the spin operator on site .
In the Schwinger Boson representationschwinger (), the spin operator is written as , in which is a Boson operator that is subjected to the no double occupancy constraint of the form , is the Pauli matrix. Such a representation has a built-in gauge redundancy, as the spin operator is unaffected when we perform a gauge transformation of the form , where is an arbitrary phase.
The Heisenberg exchange coupling can be written as , where and schwinger (). In the mean-field treatment, we replace and with their mean-field expectation value and and treat the no double occupancy constraint on average. We then have:
where , , is introduced to enforce the constraint on average. The Bosonic RVB state can be constructed by Gutzwiller projection of the mean field ground state, which has the form of
where is the Gutzwiller projector that enforces the constraint of one Boson per site, is the number of lattice site. The RVB amplitude is determined by the parameters , and .
As a result of the gauge redundancy of the Schwinger Boson representation, the mean-field ansatz for a symmetric spin liquid should be invariant under the gauge projective extension of the physical symmetry groupwen (); wangf (); yang (). The mean field ansatz that meets such a requirement can be classified by the projective symmetry group(PSG) technique, as is done for the square lattice models in  and . As we argued in , when both and are subdominant as compared to , the energetically most favorable Schwinger Boson mean field ansatz belongs to the so called zero-flux class and should satisfy the following rules. First, for sites belonging to different sub-lattices, only a real is allowed. Second, for sites in the same sub-lattice, only a real is allowed. The structure of the mean field ansatz is the most transparent in the sublattice-uniform gauge, in which it takes the form of:
where site belongs to the A sublattice, (with ) denotes the vectors connecting site to its neighbors, up to the third distance. For sites in B sublattice, the sign of should be reversed. An illustration of this mean field ansatz is given in Fig.1.
The mean-field Hamiltonian can be solved most easily in the uniform gaugewangf (); yang (); gauge (), in which the mean field ansatz is manifestly translational invariant. The mean field Hamiltonian in this gauge is given by
in which , , and . Here , , . The mean-field spinon spectrum is given by . The RVB amplitude derived from the mean-field ground state is given by
It can be proved that the RVB amplitude between sites in the same sub-lattice is identically zero.
The mean-field parameters , , and can be determined by solving the mean field self-consistent equations and the particle number equation. These equations have been solved earlier by Mila and collaborators mila () when both and are set to zero. They find that the spin liquid phase is preempted by the stripe magnetic ordered phase when . We hope a gapped spin liquid phase can be stabilized by the additional frustration of .
The evolution of the spinon gap as determined by the solution of the self-consistent equations is shown in Fig.2. Here we restrict the parameters in the region , since the mean field ansatz is constructed on the assumption that both and are sub-dominate. The spinon gap vanishes in the white region of the phase diagram, in which the Neel order with ordering wave vector emerges as a result of the condensation of the gapless spinon at momentum . The Neel ordered phase is separated from the gapped spin liquid phase by a second order phase transition. The spinon gap in the spin liquid phase is found to increase linearly with the exchange coupling near the critical point(see Fig.3). The momentum where the minimal spinon gap is achieved is found to be always at or around in the parameter region considered. It can be seen that the critical value of to open the spinon gap decreases with the increase of . The reduction of the critical makes the gapped spin liquid phase to have better chance to survive the competition with the stripe ordered phase, since the frustration effect caused by is similar in both the Neel ordered and the stripe ordered phase. Here we do not attempt to make a thorough comparison of mean field energies for all possible symmetry breaking phasesmean (), but choose to believe that a finite may stabilize a gapped spin liquid phase somewhere in the phase digram in proximity to the Neel ordered phase, before it is preempted by some unknown symmetry breaking phase.
We now go beyond the saddle point approximation and construct a low energy effective theory for the fluctuation around the saddle point. The fluctuation in the magnitude of the RVB parameters, namely, , and , are all gapped and can be neglected at low energy, since these fluctuations are not related to any symmetry. On the other hand, the saddle point action of the gapped spin liquid phase is manifestly invariant under a staggered gauge transformation of the form: for and for , since is nonzero only for sites in different sub-lattices and is nonzero only for sites in the same sublatticeread (); read1 (); read2 (); wangf (); yang (). This staggered gauge symmetry is a reflection of the collinearity of the Neel ordered phase when the spinon condenses. At the Gaussian level, such a gauge symmetry in the saddle point action would imply a gapless staggered gauge field in the long wave length limit. According to a well known argumentread (); read1 (); read2 (), when the spinon is fully gapped, the singular gauge field configuration called instanton in such a gapless staggered gauge field will proliferate, which will result in spontaneous spatial symmetry breaking and the development of valence bond solid order. A fully symmetric gapped spin liquid phase thus can not emerge continuously from a collinear Neel ordered phase.
Here we show the above argument is incorrect and that a fully symmetric gapped spin liquid phase can emerge continuously from a collinear Neel ordered phase. The key point of our reasoning is the observation that the fluctuation in the Lagrange multiplier, which enforces the no double occupancy constraint on the Schwinger Bosons, can not be treated at the perturbative level. In fact, one should integrate out the Lagrange multiplier exactly to find the correct low energy effective action for the emergent gauge fluctuation. A similar argument has been recently put forward by the present author in the study of the spin liquid phase with a large spinon Fermi surface on the triangular latticetri (). In that work, we find that the effective action of the gauge fluctuation takes a totally different form from that in the Gaussian effective theory when the Lagrange multiplier is exactly integrated out. Here we follow the same reasoning to derive the low energy effective action for the gauge fluctuation in the gapped Bosonic spin liquid.
Here is the Lagrange multiplier introduced to enforce the no double occupancy constraint on the Schwinger Bosons. or is the Hamiltonian written in terms of the Schwinger Bosons, depending on the decoupling channel we will adopt. Introducing the complex Hubbard-Stratonovich field or to decouple the Hamiltonian, the partition function can be written as
here we have omitted the arguments of the Boson fields and the auxiliary fields for clarity, is the Hubbard-Stratonovich field defined on the bond connecting site and . Depending on the bond type, it equals to either or . The action is given by
in which , and denote first-, second- and third-neighbor bonds respectively.
As usual, we neglect the fluctuation in the magnitude of the auxiliary fields and at low energy, which are assumed to be both gapped. The phase of the auxiliary fields and the Lagrange multiplier are then the only degree of freedoms in the low energy effective theory, which are to be interpreted as the spatial and temporal component of a compact gauge field. Such an emergent gauge field has no intrinsic dynamics of its own. An effective dynamics of the gauge fluctuation can be generated by integrating out the Schwinger Bosons. As we have shown in , it is crucial to integrate out the Lagrange multiplier exactly, which enforce the no double occupancy constraint on the slave particles, to derive the correct effective action for the spatial component of the emergent gauge field.
When the Lagrange multiplier is integrated out exactly, the coupling of the gauge field to all gauge non-invariant quantities should vanish identically, since the latter will inevitably violate the no double occupancy constraint. What survives the integration is then the coupling between the gauge invariant fluxes and the corresponding gauge invariant loop operatorstri (). For example, if we only consider RVB parameters on first-neighbor bonds, then to the lowest order of , the coupling between the gauge field and the Schwinger Bosons is given by
in which the sum is over all elementary plaquettes of the square lattice. denotes a plaquette composed of the site , , and (see Fig4(a) for an illustration). is the gauge flux enclosed in this plaquette and is defined as . is a coupling constant that is proportional to . A detailed derivation of a similar result for the spin liquid on the triangular lattice can be found in our recent papertri ().
If we include the second- and the third-neighbor RVB parameters, we can define more gauge invariant fluxes and the corresponding gauge invariant loop operators. For example, we can generate the following coupling by combining first- and second-neighbor RVB parameters
in which is a triangular plaquette as illustrated in Fig.4(b). is the gauge flux enclosed in the plaquette . It is defined as . The coupling constant is proportional to . Obviously, there are infinite number of such gauge invariant fluxes and the corresponding loop operators. However, one can show that all gauge invariant fluxes can be generated by combination of five kinds of elementary gauge fluxes. In Fig.4(b)-4(f), we illustrate the five kinds of loops on which these elementary gauge fluxes are defined.
The dynamics of the gauge fluxes can be determined by studying the spectrum of the loop operators, since the two are linearly coupled with each other. Since the Bosonic spinon is fully gapped in the spin liquid phase, we expect the fluctuation of the loop operators to be also gapped. This implies that the gauge fluctuation in the spin liquid phase is fully gapped, although the mean field ansatz of the spin liquid phase possesses the staggered gauge symmetry. The spin liquid phase should thus be understood as a gapped spin liquid phase with topological order. The gapped nature of the gauge fluctuation also implies that the spin liquid phase is locally stable with respect to gauge fluctuation. A continuous transition between the collinear Neel ordered phase and a gapped spin liquid phase with the full symmetry is thus possible.
In conclusion, we find that the frustration from a third-neighbor exchange may stabilize a fully symmetric gapped spin liquid phase in the spin- model on the square lattice. We argue that such a gapped spin liquid phase is better described by the Bosonic, rather than the Fermionic RVB theory. We find although the mean field ansatz of such a spin liquid phase has a staggered gauge symmetry, it should be understood as a system with gapped gauge fluctuation spectrum and topological order. We argue that such a gapped spin liquid phase is locally stable against gauge fluctuation and can have a continuous transition with the collinear Neel ordered phase. This claim is in strong contradiction with previous field theory predictions based on Gaussian approximation on the Lagrange multiplier, in which the gapped spin liquid phase is argued to be unstable against spatial symmetry breaking as a result of the instanton effect. Our result illustrate again that it is crucial to enforce the no double occupancy constraint on the slave particles exactly to construct the correct low energy effective theory for the emergent gauge field in a quantum spin liquid. Arguments based solely on the gauge symmetry of the mean field ansatz can be misleading.
The author acknowledges the support from NSFC Grant No. 11674391, 973 Project No. 2016YFA0300504, and the research fund from Renmin University of China.
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