A Continuum limit of the mean-field state

Properties of an algebraic spin liquid on the kagome lattice

Abstract

In recent work, we argued that a particular algebraic spin liquid (ASL) may be the ground state of the kagome lattice Heisenberg antiferromagnet. Furthermore, this state, which lacks a spin gap, is appealing in light of recent experiments on herbertsmithite (ZnCu(OH)Cl). Here, we study the properties of this ASL in more detail, using both the low-energy effective field theory and Gutzwiller-projected wavefunctions of fermionic spinons. We identify the competing orders of the ASL, which are observables having slowly-decaying power law correlations – among them we find a set of magnetic orders lying at the M-points of the Brillouin zone, the familiar magnetic ordered state, the “Hastings” valence-bond solid (VBS) state, and a pattern of vector spin chirality ordering corresponding to one of the Dzyaloshinskii-Moriya (DM) interaction terms present in herbertsmithite. Identification of some of these orders requires an understanding of the quantum numbers of magnetic monopole operators in the ASL. We discuss the detection of the magnetic and VBS competing orders in experiments. While we primarily focus on a clean system without DM interaction, we consider the effects of small DM interaction and argue that, surprisingly, it leads to spontaneously broken time reversal symmetry (for DM interaction that preserves XY spin rotation symmetry, there is also XY magnetic order). Our analysis of the projected wavefunction provides an estimate of the “Fermi velocity” that characterizes all low-energy excitations of the ASL – this allows us to estimate the specific heat, which compares favorably with experiments. We also study the spin and bond correlations of the projected wavefunction and compare these results with those of the effective field theory. While the spin correlations in the effective field theory and wavefunction seem to match rather well (although not completely), the bond correlations are more puzzling. We conclude with a discussion of experiments in herbertsmithite and make several predictions.

I Introduction

Recent experiments on herbertsmithite (ZnCu(OH)Cl) have reinvigorated the longstanding interest in the ground state of the Heisenberg antiferromagnet on the kagome lattice.Helton et al. (2007); Mendels et al. (2007); Ofer et al. (); Imai et al. (2008); Lee et al. (2007); de Vries et al. (2008); Bert et al. (2007); Olariu et al. (2008) This material contains kagome layers of antiferromagnetically coupled Cu moments, with an exchange energy . The coupling between adjacent kagome layers is expected to be very small. Remarkably, no sign of ordering – magnetic or otherwise – has been observed down to the lowest temperatures measured ( for some probes). Frozen magnetic moments and spin glass behavior are also not observed. Herbertsmithite is thus a candidate for the experimental realization of a quantum spin liquid in two dimensions.Anderson (1973, 1987)

Various experimental probes point to a vanishing spin gap. Spin liquids with this property are variously (and equivalently) referred to as gapless spin liquids, critical spin liquids, or long-range resonating valence bond (RVB) states. So far, most works studying nonmagnetic ground states of the kagome antiferromagnet have focused on gapped spin liquid states,Marston and Zeng (1991); Sachdev (1992) or valence-bond solid (VBS) states that break lattice symmetries.Marston and Zeng (1991); Hastings (2000); Nikolic and Senthil (2003) A variety of microscopic calculations have provided interesting information, but are unable to establish the nature of the ground state.Leung and Elser (1993); Elstner and Young (1994); Zeng and Elser (1995); Lecheminant et al. (1997); Waldtmann et al. (1998); Sindzingre et al. (2000); Misguich and Sindzingre (2007); Singh and Huse (2007, 2008); Yang et al. () (See also Ref. Misguich and Lhuillier, 2005 for a review.)

The experimental work on herbertsmithite led us, in recent work, to investigate the possibility of a gapless spin liquid ground state in the kagome lattice Heisenberg model.Ran et al. (2007) By considering a class of Gutzwiller-projected fermion wavefunctions, we concluded that the variational ground state of the kagome model is a particular kind of gapless spin liquid known as an algebraic spin liquid (ASL).Rantner and Wen (2001, 2002) Some of us studied the effect of a Zeeman magnetic field, and argued that it leads to spontaneous breaking of parity and XY spin rotation.Ran et al. () Gregor and Motrunich considered the effect of non-magnetic impurities in the ASL, finding results consistent with the NMR experiments on herbertsmithite.Gregor and Motrunich (2008) In this paper, we shall work out the properties of this realization of the ASL in more detail; this leads to a number of predictions that may be relevant for herbertsmithite.

We note that Ma and Marston have considered a different gapless spin liquid (Fermi surface state) using Gutzwiller projected wavefunctions, and have argued it can be stabilized by addition of further-neighbor ferromagnetic exchange.Ma and Marston () Also, a different route to a gapless spin liquid on the kagome lattice has recently been discussed by Ryu et. al.Ryu et al. (2007)

The effective field theory describing the ASL consists of flavors of massless, two-component Dirac fermions coupled to a gauge field. This description is complementary to the projected wavefunction approach – the former correctly captures universal long-distance properties, while the latter can provide information about energetics and other more microscopic properties. Significant progress has been made recently in understanding algebraic spin liquids using effective field theory,Rantner and Wen (2001, 2002); Borokhov et al. (2002); Vafek et al. (2002); Hermele et al. (2004, 2005); Alicea et al. (2005a, b, 2006); Ryu et al. (2007) and it has been found that such states exhibit striking observable properties. The ASL is a quantum critical phase – like a quantum critical point, it supports gapless excitations and nontrivial critical exponents, but can exist as a stable zero-temperature phase and can be accessed with no fine tuning of parameters. The ASL is Lorentz invariant, and all excitations propagate with the same “Fermi velocity” . The symmetry of the ASL is much larger than that of the microscopic spin model, and this emergent symmetry unifies together a variety of superficially unrelated observables. Some of these observables have slowly decaying power law correlations in space and time; these are referred to as “competing orders.”

In this paper, we show how the observable properties of the ASL are manifested in the kagome lattice model. In particular, we identify the competing orders most likely to have slowly decaying correlations – these include magnetic orders and valence-bond solid (VBS) states, as well as patterns of order involving vector and scalar spin chiralities. We discuss the detection of the magnetic and VBS orders in experiments. We also present further results in the projected wavefunction approach – in particular, we give an estimate of the velocity , and study the spin-spin and bond-bond correlations of the wavefunction. Taken together, these results inform a variety of predictions that may be relevant for herbertsmithite. Furthermore, the estimate of allows us to estimate the low-temperature specific heat and magnetic susceptibility, and we find reasonable agreement with experiments.

It bears mentioning that herbertsmithite is almost certainly not described by a Heisenberg model alone. There is now significant evidence that impurities play an important role, especially at low temperature.Imai et al. (2008); de Vries et al. (2008); Bert et al. (2007); Olariu et al. (2008) It has been suggested that antisite defects, where Zn and Cu trade places, constitute the dominant disorder – this leads to both magnetic impurities (Cu occupying Zn sites) and site dilution in the kagome layers (Zn occupying Cu sites).de Vries et al. (2008) Estimates of the concentration of magnetic impurities per kagome lattice site are in the range of to . It has also been suggested that Dzyaloshinskii-Moriya (DM) interaction is an important perturbation to the Heisenberg model.Rigol and Singh (2007a, b) While DM interaction is certainly present, its magnitude is uncertain.

Here, we have not attempted to address the effects of impurities and DM interaction in detail. Instead, our approach is to first understand the spin liquid physics in a clean model with only Heisenberg exchange. One can then include impurities and DM as perturbations to this idealized model. While we do discuss the effects of small DM interaction, the effects of impurities, and of strong DM interaction, are left for future work.

We now give an outline of the paper, which also serves as a more detailed overview of our main results. Readers primarily interested in predictions for experiment can skip Secs. II and III. We begin in Sec. II.1 by discussing how the effective field theory is obtained starting from the Heisenberg model. The field theory can be analyzed in a large- expansion, and our understanding of the physical case () derives from the analysis in this limit. In Sec. II.2, we show how the microscopic symmetries are realized as transformations on the fields of the continuum theory.

Section III describes our main results from the effective field theory approach. Section III.1 reviews the physics of competing orders in the ASL. There are two sets of field theory operators whose correlations are likely to decay slowly for . These are 15 fermion bilinears, and a set of 12 magnetic monopole operators. In Sec. III.2 we relate each of the fermion bilinears to an ordering pattern in the kagome spin model. The ordering patterns arising are the valence-bond solid (VBS) state considered before by Hastings,Hastings (2000) a set of magnetic orders with crystal momenta lying at the M-points in the Brillouin zone, and a pattern of vector spin chirality ordering, which also corresponds to one of the DM terms allowed by symmetry in herbertsmithite. In Sec. III.3, we relate the monopole operators to ordering patterns in the spin model. This is less straightforward than the analysis for the fermion bilinears, as there is an ambiguity in determining the symmetry transformations of the monopoles. This ambiguity is discussed, and partially resolved using algebraic relations among generators of the space group.Alicea et al. (2005a, b, 2006); Ryu et al. (2007) Because the ambiguity in monopole transformations can only be partially resolved, we are left with three free parameters describing the possible transformation laws. We make a conjecture on the value of two of these parameters; based on this, we find among the monopoles a pattern of magnetic order corresponding to the familiar “” state of the kagome lattice. The conjecture is supported by the spin correlations of the projected wavefunction (Sec. VI.2). Depending on the remaining parameter, there are monopoles corresponding to either a pattern of spin-chirality ordering that breaks time reversal symmetry, or a VBS ordering pattern.

In Sec. IV, we discuss how to detect some of the competing orders in experiment. In particular, the M-point and magnetic orders can be detected via NMR relaxation rate, and also by inelastic neutron scattering – both these quantities obey universal scaling forms determined by the critical properties of the ASL. The VBS competing order can be detected via its coupling to an appropriate optical phonon. The lineshape of this phonon can be related to the VBS susceptibility of the ASL, which again obeys a scaling form as a function of frequency and temperature.

In Sec. V, we consider the effect of a small DM interaction. Considering first DM interactions that preserve XY spin rotation symmetry, we argue that, surprisingly, DM interaction induces XY magnetic ordering, which is likely to be in the pattern. More generically, DM interaction completely breaks spin rotation symmetry, and in this case the same argument shows that it leads to spontaneous breaking of time reversal.

In Sec. VI we turn to a further analysis of the projected wavefunction. By construction of variational excited states, in Sec. VI.1 we estimate the velocity . Later on in Sec. VII, this allows us to calculate the specific heat and magnetic susceptibility in mean-field theory; these quantities compare favorably with the experimental data. In Secs. VI.2 and VI.3 we investigate the spin-spin and bond-bond correlators of the projected wavefunction. The spin-spin correlator is dominated by correlations in the pattern of the state – in real space these correlations fall off roughly as . Although the correlations of the M-point order are expected to decay more slowly than this, they are not seen; this may indicate that these correlations have a small coefficient and thus only become important at very long distances. The bond-bond correlation function exhibits power law decay, but the observed correlations are apparently dominated by Fourier components near and thus do not correspond to the Hastings VBS state (or to the VBS state that may arise from the monopoles, which also has crystal momenta at the M-points). We speculate on the possible meaning of this in Sec. VI.3.

We conclude in Sec. VII with a discussion of experiments on herbertsmithite, and open theoretical issues. Various technical details are contained in the appendices.

Ii Description of algebraic spin liquid

ii.1 Effective theory

Figure 1: (a) Unit cell of the kagome lattice, showing the lattice vectors and , and the numbering of the three sites within the unit cell. (b) Brillouin zone. The M, K and points are labeled. The M points will play an important role in our discussion, and have thus been numbered as shown to distinguish among them. The reduced Brillouin zone of the enlarged real-space unit cell (Fig. 2) is denoted by the dashed rectangle. The Dirac nodes lie at , where .

We are interested in the Heisenberg antiferromagnet on the kagome lattice, with Hamiltonian

(1)

where the sum is over nearest-neighbor pairs of sites. The kagome lattice has a three-site unit cell; we label the sites by pairs , where is the lattice vector labeling the unit cell, and labels the three sites (Fig. 1a). We choose and , so the distance between nearest neighbor sites is . The reciprocal lattice primitive vectors can be chosen as and , and the Brillouin zone is as shown in Fig. 1b.

Although our main focus is on the pure Heisenberg model, we shall also consider Dzyaloshinskii-Moriya (DM) interactions, of the type allowed by the crystal symmetries in ZnCu(OH)Cl. We refer the reader to Ref. Rigol and Singh, 2007b for the definition of the two allowed DM terms. In one of these, the -term, the DM vector points along the -axis. The -term, on the other hand, has DM vectors lying in the -plane.

We begin by representing the single-site Hilbert space in terms of fermionic spinons:

(2)

where . We choose fermions (as opposed to bosons) because they allow for the description of stable, gapless spin liquid phases. This representation involves an enlargement of the Hilbert space, and must be accompanied by the local constraint to eliminate unphysical empty and doubly occupied sites.

These variables allow one to construct low-energy effective theories for spin liquid phases, as well as corresponding trial ground state wavefunctions. The starting point for these constructions is a decoupling of the quartic exchange interaction using an auxiliary field residing on the bonds of the lattice. Neglecting the fluctuations of the auxiliary field, one arrives at a quadratic mean-field spinon Hamiltonian. In order to obtain a correct description of any spin liquid ground state, one needs to go beyond the mean-field description and consider the fluctuations of the auxiliary field, which play an important role. One way to do this is to solve the mean-field Hamiltonian, and then perform a Gutzwiller projection onto the subspace with one fermion per site; this results in a legitimate trial wavefunction for the spin model. Alternatively, one can recognize that the fluctuations about the mean-field saddle point take the form of a gauge field coupled to the spinons. One can then write down an effective gauge theory Hamiltonian, which will correctly capture the universal features of a given spin liquid phase.

In a recent paper, we studied the kagome antiferromagnet using Gutzwiller projected wavefunctions.Ran et al. (2007) The main result of this study was that a particular spin liquid state, first discussed by Hastings,Hastings (2000) has a very low energy, even without any tuning of variational parameters. This state has the mean-field Hamiltonian

(3)

The hopping parameter is sometimes written , so that gives the mean-field hopping in units of . Also, encodes the background magnetic flux of the gauge field coupled to the spinons; it is chosen so that -flux pierces the kagome hexagons, and zero flux pierces the triangular plaquettes. The total number of spinons is chosen so that .

Instead of a Fermi surface, has gapless Dirac points at the Fermi energy, near which the fermions obey a massless Dirac dispersion with velocity . One can diagonalize the Hamiltonian using the 6-site unit cell of Fig. 2, with the signs of chosen as shown; the Dirac nodes lie in the reduced Brillouin zone at positions , where (Fig. 1). One can describe the low-energy excitations near the nodes in terms of the Lagrange density for Dirac fermions in -dimensional spacetime:

(4)

Here, is a two-component fermion field, with labeling the spin, and labeling the two nodes at and , respectively. The two components of correspond to the two branches of the Dirac dispersion. Moreover, the index , and , where the are Pauli matrices. Finally, . More details on the band structure of , as well as its continuum limit, and the detailed relationship between and the lattice spinon fields, are given in Appendix A.

Figure 2: The doubled unit cell used to diagonalize the mean-field spinon Hamiltonian Eq. (3). The lattice vectors are and , as shown. It should be stressed that the unit cell doubling is a gauge artifact, and does not represent a breaking of translation symmetry. The thick bonds are those where , and the sites are numbered as shown.

The important fluctuations about the mean-field saddle point specified by are encapsulated in the coupling of the spinons to a compact gauge field, and the spin liquid is thus an algebraic spin liquid. Because of this structure, this state has been referred to as the -Dirac state;Ran et al. (2007, ) here, however, to emphasize the connections with earlier work we shall refer to it as an algebraic spin liquid. The coupling to the gauge field is encoded in the following lattice gauge theory Hamiltonian:

(5)

Here and are the lattice electric field and vector potential, respectively. They reside on nearest-neighbor bonds of the lattice and satisfy the canonical commutation relation . The electric field has integer-valued eigenvalues, and the vector potential’s eigenvalues are -periodic and can be taken in the interval . The notation denotes a sum over all triangular and hexagonal kagome lattice plaquettes, and denotes the discrete (oriented) line integral of around the corresponding plaquette, i.e. the magnetic flux of the gauge field. In general we should allow the energy to differ on triangular and hexagonal plaquettes, but this will not be important for our discussion here. This Hamiltonian must be supplemented by the Gauss’ law constraint

(6)

where is the lattice divergence of the electric field (the sum is over nearest neighbors of ).

In the limit , one recovers a spin model Hamiltonian. However, this effective description is most useful in the limit of large , where fluctuations of the magnetic field are suppressed and the spinons become good variables to describe the physics (at least for short length scales). The short-distance physics of the Heisenberg model is presumably not similar to the short-distance physics of in the large- limit. Instead, the idea is that the two Hamiltonians may have the same long-distance physics, i.e. they are in the same phase.

In the large- limit one obtains the algebraic spin liquid, which is described by the Lagrangian density

(7)

This is the so-called QED3 Lagrangian, which includes the minimal coupling of the gauge field to the spinons, as well as an explicit Maxwell term for the gauge field. In general we may (and must) add other perturbations as allowed by microscopic symmetries – such terms are represented by the ellipsis. While QED3 is a strongly coupled problem, in the sense that the interaction between spinons and gauge field is strongly relevant in the RG sense, it can be understood in a large limit, where the number of two-component fermions fields is increased from 4 to (e.g. ). The theory can be solved for , and one can calculate perturbatively in powers of . This large- expansion underpins the understanding of the algebraic spin liquid fixed point, and has been discussed in great detail elsewhere.Appelquist et al. (1986); Borokhov et al. (2002); Franz et al. (2002); Rantner and Wen (2002); Hermele et al. (2004, 2005)

It shall be convenient to organize the four two-component fermions into the eight-component object

(8)

The Pauli matrices act in the two-component space of each Dirac fermion, so that

(9)

We also define Pauli matrices acting in the spin space, and Pauli matrices acting in the “nodal” space connecting the two nodes at . For example, we have

(10)

and

(11)

Different types of Pauli matrices commute with one another:

(12)

ii.2 Symmetries

Figure 3: Depiction of the generators of the point group symmetry of the kagome lattice, and .

For our analysis later on, we need to work out how the symmetries of the microscopic Hamiltonian Eq. (1) are realized in the continuum theory. The procedure for obtaining these results is described in Appendix B; here, we shall simply enumerate the symmetries and quote the results. The space group of the kagome lattice is generated by translations by the Bravais lattice vector (), rotations by about the center of a hexagonal plaquette (), and a reflection mapping . These point group symmetries are depicted in Fig. 3. For definiteness in defining the rotations and reflections, we take the origin of coordinates at the center of a hexagon. The remaining symmetries are spin rotations and time reversal.

For each symmetry operation, its action on the lattice spinon fields is determined by two requirements. First, it must reproduce the correct action of the symmetry on the spin operator – this ensures that the action of the symmetry on all physical operators is correctly realized. This requirement does not completely specify the action of the symmetry on , because is invariant under local gauge transformations of the spinons – the symmetry may therefore be supplemented by an arbitrary gauge transformation. However, we should also impose a second requirement, which is that the effective Hamiltonian Eq. (5) be invariant. In the present case, this requirement will fix the symmetry operation completely up to multiplication by a global phase factor. This also determines the transformation properties of the electric field and vector potential . Due to the presence of the gauge structure, symmetries have nontrivial action on the spinon fields ; the mathematical structure describing the realization of symmetries in such a situation is referred to as the projective symmetry group (PSG).Wen (2002)

If is a space group symmetry mapping , then the above requirements dictate that the fermions transform as

(13)

where is a gauge transformation depending on ; in the gauge chosen in Eq. (5) we can take . Spin rotations send , where is an matrix. Time reversal acts as follows:

(14)

These transformation laws also imply that the electric field and vector potential transform as scalars under translations, and as vectors under rotations and reflections. Under time reversal, the electric field is even and the vector potential is odd.

Following the procedure in Appendix B, we obtain the following transformations for the (real space) continuum field :

(15)
(16)
(17)
(18)
(19)

where

(20)
(21)

Iii Many competing orders: fermion bilinears and magnetic monopoles

iii.1 General discussion

The focus of this paper is on observable properties of the algebraic spin liquid on the kagome lattice; the most dramatic such property is the unification of many competing orders within the ASL.Hermele et al. (2005) This is manifested in the fact that correlations of a variety of superficially unrelated observables obey power law decay in space and time with the same critical exponent. This unification is accomplished mathematically through the presence of an emergent symmetry that contains the microscopic spin rotation symmetry as a subgroup. This means, for example, that magnetic order parameters can be rotated into nonmagnetic ones. Furthermore, because the ASL is an interacting critical state, the critical exponents governing many physically interesting correlation functions are not equal to their mean field values. In particular, for those observables we refer to as competing orders, the correlation functions decay more slowly than in mean field theory. Such slow decay of a correlation function indicates that the system is closer to being ordered in a particular channel, and is likely to be observable in both experiments and numerical studies.

In the language of critical phenomena, the dominant long-distance correlations within the ASL, and hence the dominant “competing order parameters,” are given by finding the operators in the critical theory with smallest scaling dimension. Suppose we are interested in the correlations of some microscopic observable , which is some function of spin operators for near the lattice point . Its long distance correlations can be understood from the following expression:

(22)

On the right hand side of this expression are a set of operators in the field theory , also located at the spatial point . The meaning of Eq. (22) is that the quantities on the left and right hand sides have the same correlations, at distances much greater than a short-distance cutoff. For any operator in the field theory, generically the coefficient will be nonzero if and only if transforms identically to under microscopic symmetries (in this case, spin rotations, time reversal and space group operations). Apart from this condition, the are nonuniversal. This tells us, then, that the dominant long-distance correlations of are given by the appearing in Eq. (22) with the smallest scaling dimension ; so, for example, .

Which operators in the ASL critical theory have smallest scaling dimension? In the large- limit, these are a set of fermion bilinears transforming as the adjoint of . In the physical case of , these operators are

(23)

where and the are the 15 generators of . One can choose a basis for the generators where they are expressed in terms of the and Pauli matrices:

(24)

In the large- limit, Rantner and Wen calculated the scaling dimension of one of the in the context of the “staggered flux” spin liquid on the square lattice.Rantner and Wen (2002) In Ref. Hermele et al., 2005 the symmetry was emphasized, which implies that all the have the same scaling dimension. The result of Ref. Rantner and Wen, 2002 is

(25)

Because the gauge interaction, which is stronger for smaller , tends to bind together the and fermions in , and make the mode it creates propagate more coherently as opposed to decaying into its two constituent fermions, it is expected that for all values of .

There are other operators in the ASL that may have even smaller scaling dimension than the when . The most likely candidates are magnetic monopole operators – these are topological disorder operators for the gauge field. Because the gauge field is compact, its magnetic flux is quantized in multiples of , and monopole operators are those which insert integer multiples of flux. Therefore they carry nonzero integer charge under the symmetry, which emerges at the ASL fixed point and corresponds to the irrelevance of monopole operators there.Hermele et al. (2004) It is important to emphasize that, for the ASL fixed point to be stable, all monopoles that are allowed as perturbations to the action by microscopic symmetries must be irrelevant. The monopoles we will consider here carry nontrivial quantum numbers and are not allowed perturbations, so they may become relevant without destabilizing the ASL.

As is typical for topological disorder operators, it is difficult to construct monopole operators explicitly in terms of the fermions and gauge field. However, they can be constructed by exploiting the state-operator correspondence from conformal field theory, and it has been foundBorokhov et al. (2002) that monopoles with smallest scaling dimension at large- have unit magnetic charge and transform under the completely antisymmetric self-conjugate representation of the flavor symmetry. In the case this representation is 6 dimensional. Formally we can represent these six operators by , where are indices. There is another set of six monopoles with charge and the same scaling dimension, . In the large- limit the scaling dimension of these operators was found to be

(26)

where .Borokhov et al. (2002) If we naïvely put into the leading order large- result, we obtain , compared to putting into Eq. (25). So it may well be the case that the monopoles give the dominant long distance correlations for the physical case of .

Below, we shall work out the quantum numbers of and under microscopic symmetries, and determine the spin model observables to which they correspond.

iii.2 Fermion bilinears

It is convenient to break the into three different classes of operators. These are

(27)
(28)
(29)

Using the results of Sec. II.2 we can determine how each of these observables transforms under all microscopic symmetries. Clearly form a set of 3 spin triplets, and is also a spin triplet, while the are spin singlets. Under time reversal is odd, while and are even. This implies that is the order parameter for a magnetically ordered state.

For each of these operators, we can identify one or more microscopic operators that transform identically under all the symmetries of the kagome model. Then, by Eq. 22, the will contribute to the long-distance correlations of these microscopic operators. Actually, rather than looking for operators, it is easier to look for ordering patterns (i.e. states as opposed to operators) with the corresponding symmetry transformations. To understand how to do this, let us consider the transformations of . If is a space group operation, then we have

(30)

where is an matrix. These matrices form the irreducible representation of the space group (see Appendix C for more details). If we can find a set of magnetically ordered states for which transforms under the same representation of the space group, then is an order parameter for this state. Furthermore, we can go on to construct microscopic operators that are also order parameters for this state, and identify correlation functions of these operators that obey power law decay with exponent .

The relevant details of the group theory and representation theory of the kagome space group are summarized in Appendix C. As stated above, each component in spin space of transforms in the irreducible representation. The also transform as , that is , and each component in spin space of transforms as the representation.

We have already established that the correspond to magnetically ordered states. We are primarily interested in finding a combination of spin operators corresponding to each , and for this purpose it is enough to consider collinear ordering patterns only. We therefore focus on the -component in spin space .

Our task is then to find three ordering patterns of kagome spins pointing along the -axis in spin space. One approach is simply to guess the right pattern, but it is possible to be more systematic. We are interested in patterns of site ordering on the kagome lattice that are invariant under translations by and , since the are also invariant under such translations. So we can consider all possible site ordering patterns on a cluster of unit cells with 12 sites and periodic boundary conditions. For each site of the cluster we can associate a vector , and the various site ordering patterns can be represented as real linear combinations of these states, where the coefficients of should be associated with . The space group acts on this vector space by , where is a space group operation, so the vector space is a 12 dimensional representation of the space group. This can be decomposed into irreducible representations using the character table of Appendix C. The irreducible representations we find in this manner are a complete catalog of kagome site ordering patterns invariant under and . We find that the representation of interest occurs precisely once in this catalog. The corresponding magnetic ordering patterns can be read off from the basis vectors of this representation, and are shown in Fig. 4. We stress that this is the unique site-ordering pattern corresponding to .

Figure 4: Magnetic ordering patterns corresponding to . These can be thought of as stripes of up- and down-spin hexagons, which are labeled with “+” and “-”, respectively. A spin between two up-spin hexagons points up, and between two down-spin hexagons points down. Spins between an up-spin and down-spin hexagon have no average moment. The three patterns are labeled according to their crystal momenta, which lies at the three M-points of the Brillouin zone , where . The labeling of the M-points is as in Fig. 1.

Next, we wish to find ordering patterns corresponding to . Because these operators are spin singlets and invariant under time reversal, it is natural to look for bond ordering patterns, with order parameters that can be built from , where and are nearest neighbors. This observable measures the strength of singlet formation on a bond. Following the same procedure as for the site ordering patterns, we note that the same unit cell contains 24 bonds; the resulting 24-dimensional vector space is decomposed into irreducible representations in Appendix C. In this case, we find the representation occurs twice, and leads to the patterns shown in Figs. 1718 and 19. Taken together with the “uniform” state, where is the same on every bond, we can superpose the two patterns of Fig. 19 to form the Hastings VBS state, as shown in Fig. 5. It has three inequivalent bonds in its unit cell, and this is precisely because it is built from a superposition of three bond ordering patterns belonging to distinct irreducible representations of the space group. The fourfold degenerate Hastings state is not the only ground state that can be built from the bond ordering patterns obtained in Appendix C, but we restrict our attention to it for simplicity. Furthermore, the ordering patterns obtained in Appendix C contain enough information to work out the contributions of to the bond-bond correlation function within the ASL.

Figure 5: Depiction of the Hastings valence bond solid state. There are three inequivalent sets of bonds, these are distinguished by shading and vertical or horizontal hatching. In previous treatments in the literature,Hastings (2000); Ran et al. (2007) the shaded bonds were taken to have strong dimerization, and the vertically and horizontally hatched bonds to have weak (and equal) dimerization. However, other choices are possible without changing the symmetry of the ground state.

Finally we turn to . This object is a spin triplet, but is even under time reversal; the simplest microscopic operator with these properties is the vector chirality defined on nearest-neighbor bonds. Rather than searching for states as above, in this case it is simpler to find a spin operator transforming identically to . We define an object that naturally resides on the hexagonal plaquettes of the kagome lattice, which we label with the position vectors :

(31)

This can be interpreted as the vector spin chirality of a kagome hexagon. Here, the sum is over the 6 bonds contained in the perimeter of the hexagon, following the orientation convention shown in Fig. 6. More precisely, is always taken to be the the “first” site on the bond according to Fig. 6, and the “second” site. It is straightforward to check that has identical transformation properties as (once a suitable long-wavelength average is taken). The observable is also related to the Dzyaloshinskii-Moriya (DM) interaction on the kagome lattice, and, in particular,

(32)

is precisely the -term allowed by the crystal symmetries in ZnCu(OH)ClRigol and Singh (2007b). The effects of DM interaction on the ASL are discussed further in Sec. V.

Figure 6: Orientation of kagome lattice bonds used to define the vector spin chirality of a hexagon, .

iii.3 Magnetic monopoles

Here, we work out the quantum numbers of the magnetic monopole , which was formally defined in Sec. III.1, based on the results on Ref. Borokhov et al., 2002. Our analysis builds on results of Refs. Alicea et al., 2005a, b, 2006; Ryu et al., 2007, where monopole quantum numbers were worked out in different but closely related settings to the present one. However, we adopt a different perspective; we believe this clarifies some of the issues involved, and we comment on this at the end of this section. Our strategy is first to determine how each microscopic symmetry is embedded in the full symmetry group of the low-energy effective theory. We do not have enough information to determine this completely, so several free parameters have to be introduced. To proceed further, we argue that these free parameters must satisfy certain constraints, determined by relations among the generators of the space group. After taking the constraints into account, we shall be left with three free parameters, and we make a conjecture that determines two of them, based on a physical argument and on calculations using the projected wavefunction.

First, it is useful to recall how transforms under the continuous symmetries of the low-energy ASL fixed point. It is a scalar under Lorentz and continuous translation symmetry. Under rotations we have

(33)

and under a rotation we have

(34)
(35)

We now decompose the symmetry into its subgroup. The first is simply spin rotation symmetry (generated by the ), and the second consists of “nodal” rotations (generated by the ) that mix the two Dirac nodes but commute with spin rotations. This decomposition is useful because it separates spin rotations from space group operations, which are realized in the space as nodal rotations. We replace the index by a pair of indices: . Here, transforms under spin , and under nodal . Under an rotation, we have

(36)

where and are matrices in the spin and nodal spaces, respectively. We can also decompose the monopole operators according to their transformations under the subgroup. We have

(37)
(38)

Here, is a spin triplet and a nodal singlet, and is a spin singlet and nodal triplet.

We now wish to specify the action of microscopic symmetries on and . In doing this, we encounter two problems. First, we determined the rotation corresponding to each microscopic symmetry by working out the transformations of the fermion field . This only determines the rotation up to multiplication by the matrix , which generates the center of . The reason for this ambiguity is that multiplication of by is indistinguishable from a gauge transformation. However, is odd under , but is of course gauge invariant. Second, each microscopic symmetry may come along with a rotation in the space. Because carries no charge, we have no information about this rotation. Both of these ambiguities can be taken into account by multiplying by an undetermined phase factor associated with each symmetry operation. We note that there are no such ambiguities associated with continuous spin rotations – it can be shown that adding an additional phase factor to a spin rotation is not consistent with spin rotation symmetry.

We can now enumerate how the space group symmetries act on the monopoles:

(39)
(40)
(41)
(42)
(43)
(44)

Note that the reflection symmetry sends to , and similarly for . This is because reflections change the sign of the magnetic charge. Here we have introduced matrices describing the rotations in the nodal space, which are obtained from the symmetry operations of Sec. II.2 and are

(45)
(46)
(47)

We can immediately eliminate the phase factor by redefining , and similarly for . This does not affect the other symmetries, and we have

(48)
(49)

We shall now determine the unknown phase factors and to the greatest extent possible, exploiting relations among the space group generators. For example, we shall demand the relation holds when acting on monopole operators. In general, such relations need only hold up to a phase when acting on quantum states. However, in the present case, these relations hold with no additional phase factors when acting on any physical state in either the spin model or the effective lattice gauge theory Hilbert spaces. Since nothing forbids the insertion of a single monopole in the lattice gauge theory, the relations must hold for the monopole operators with no extra phase factors. The relation implies that , where is an integer. Next, the relation

(50)

implies . So we are left only with the free parameter .

We also need to consider time reversal. We have

(51)
(52)

where . Note that time reversal changes the sign of magnetic charge, so it takes monopoles to antimonopoles. Furthermore, time reversal commutes with the subgroup of [see Eq. (19)], so it must take the spin triplet to the spin triplet , and similarly for the nodal triplets and . It is not consistent for and to be arbitrary phases, given that , which holds for all physical states of the spin model and of the effective lattice gauge theory. The relations of Eq. (51) and Eq. (52) are the most general transformations consistent with these considerations.

We proceed by using some physical input to conjecture the likely values of and – the resulting conjecture is supported by the numerical results on the projected wavefunction (see Sec. VI.2). The source of physical input is the following puzzle about the kagome ASL: in the large- limit of the kagome antiferromagnet, there are an infinite number of nearly degenerate magnetically ordered ground states. These consist of all states where the vector sum of classical spins on every triangle is zero – among these states, those with coplanar spin configurations are selected by the zero point energy of spin wave fluctuations about the ground state. Finally, among the coplanar states, the state is selected by anharmonic fluctuations.Chubukov (1992); Chandra et al. (1993); Henley and Chan (1995) Up to this point in our analysis, we have not found signatures of any of these classical low energy states within the ASL. (The ordered states of Fig. 4 have distinct symmetry from the classical ground states lying at the M points, which transform in the representation of the space group.) It is possible that no sign of the large- physics survives down to , but it would perhaps be more natural if at least a hint of it remained, especially given that the ASL has no spin gap.

Figure 7: The magnetically ordered state.

This leads us to the conjecture that will be the order parameter for one of these classical ground states. We know from the analysis above that carries zero crystal momentum, and the unique classical low-energy state with this property is the so-called state, shown in Fig. 7. It is the classical ground state for ferromagnetic third-neighbor exchange .Chubukov (1992) It turns out that if we choose and , then is an order parameter for the state. This can be seen by following the construction of the state order parameter in Appendix D. Based on these considerations, from now on we shall fix and . This choice is supported by the presence of substantial spin correlations in the projected wavefunction.

The parameter still remains to be determined. The Hermitian operators and are spin singlets, with crystal momenta lying at the M-points. If , they are even under time reversal, and then likely correspond to bond ordering patterns. If , they are odd under time reversal, and correspond to ordering patterns in the scalar spin chirality , where the spins can be taken on any three distinct nearby lattice sites. Since the bond-bond correlations in the projected wavefunction are apparently dominated by Fourier components near , we are led to speculate that and the monopoles correspond to spin chirality order.

We now contrast the approach taken here with that of Refs. Alicea et al., 2005a, b, 2006; Ryu et al., 2007, where symmetry transformations of monopole operators were obtained by construction of mean-field ground state wavefunctions with a background flux – this is the quantum state one obtains after insertion of a monopole operator. The correspondence of the resulting states to scaling operators of the low-energy critical theory, in which one is ultimately interested, is not clear, and this issue makes it difficult to interpret the results of Refs. Alicea et al., 2005a, b, 2006; Ryu et al., 2007. However, our approach here, which focuses directly on the scaling operator , is formally equivalent to the analysis of Refs. Alicea et al., 2005a, b, 2006; Ryu et al., 2007, and puts it on firm ground. On the other hand, Refs. Alicea et al., 2005a, b, 2006; Ryu et al., 2007 obtained additional information on symmetries by carrying out an adiabatic insertion of flux numerically. In our opinion, because the ASL has gapless excitations, the correspondence between this procedure and insertion of the scaling operator is not established.

Iv Detection of competing orders

The competing orders arising in the kagome ASL can be detected by a variety of experimental probes. Here, we outline some predictions that we hope will be tested in future experiments on ZnCu(OH)Cl. In this section we focus on the case of an ideal system free of perturbations such as impurities and Dzyaloshinskii-Moriya interaction. We postpone discussion of these issues to the following section, and also Sec. VII.

Two critical exponents enter into this discussion. The first is , which characterizes the correlations of the fermion bilinears, and hence the M-point magnetic order, the Hastings VBS order, and the hexagon vector chirality . The second, , characterizes the monopole operators, and hence the correlations of magnetic order.

The magnetic competing orders can be detected via neutron scattering and NMR. In neutron scattering, the structure factor will exhibit critical scaling behavior near the and points in reciprocal space, with the scaling form

(53)

where is a nonuniversal constant prefactor, and is much smaller than the Brillouin zone size. if lies at one of the M points, and if . There are two different universal scaling functions , again depending on whether lies at the point or one of the M points.

Based on Eq. (53), the NMR relaxation rate is predicted to be

(54)

Here, depends on the nuclear site considered – Cu, O and Cl sites are all sensitive to the M-point magnetic order, but only Cu and O are sensitive to order. Therefore we expect

(55)
(56)

The VBS competing order can be detected via its coupling to phonons. We consider, for simplicity, a single optical phonon that couples to one of the patterns of VBS order shown in Figs. 17 and 18, and hence to some linear combination of the fermion bilinears of the ASL. Using inelastic X-ray or neutron scattering, the lineshape of this phonon can be measured. The lineshape is determined by , the retarded Green’s function for the phonon mode’s normal coordinate. We treat the phonon mode using the RPA of Ref. Cross and Fisher, 1979, which neglects the influence of the gapped optical phonon on the gapless spin system, and has been successfully applied to quasi-one-dimensional magnets, for temperatures above the spin-Peierls transition.Abel et al. (2007) We then have, for the phonon spectral function .

(57)

Here, characterizes the spin-phonon coupling, is the bare phonon frequency, and and are the real and imaginary parts, respectively, of the retarded Green’s function of the VBS order parameter. We have the scaling form

(58)

where is a nonuniversal prefactor. Also, is related to the scaling function appearing in Eq. (53) by , for lying at one of the M points. It may be possible to test this scaling form by measuring the -dependence of .

V Dzyaloshinskii-Moriya interaction

We now consider the effect of a small DM interaction, . First, we must understand which new perturbations to the ASL fixed point are allowed by the now reduced microscopic symmetry. While spin rotation symmetry is completely broken, the space group symmetry remains unchanged. However, lattice reflections and rotations must now be accompanied by appropriate operations in spin space. In the three-dimensional Herbertsmithite structure, the reflection is realized as a -rotation about the -axis (passing through the center of a hexagon), along with a corresponding rotation in spin space. The rotation is realized by first making the same -rotation about the -axis, followed by mirror reflection through the plane with normal (intersecting the center of the same hexagon). In the continuum, the resulting modified symmetry operations are

(59)
(60)

Using these operations, combined with translations and time reversal (which are unchanged), it can be shown that the only fermion bilinear allowed by symmetry is . Furthermore, the monopole operators are still forbidden by symmetry – the spin triplet monopoles are odd under time reversal, and the spin singlet monopoles carry nonzero crystal momentum. Therefore we expect to be the most relevant perturbation to the ASL generated by the DM terms.

We now consider adding the term

(61)

to the ASL fixed point. We wish to understand how depends on and . As noted in Sec. III.2, the -term and are symmetry-equivalent, so will contain a term linear in . We now give an argument that contains no term linear in . Consider a microscopic Hamiltonian with given values of and . Upon some amount of coarse-graining, we obtain a continuum field theory with a given value of . Now we make a -rotation in spin space about the -axis. This sends , but otherwise leaves the microscopic Hamiltonian unchanged. Various operators in the continuum field theory will be affected by this operation, but the value of remains the same. We have shown that does not depend on the sign of . Therefore, up to second order in and ,

(62)

where , and are dimensionless coefficients.

Ignoring coupling between the gauge field and the fermions, the effect of the perturbation Eq. (61) is to open a spin gap. However, coupling to the gauge field plays a surprising and important role. As the fermions are now massive, magnetic monopoles will condense, leading to confinement.Polyakov (1977) Naïvely this leads to a fully gapped spectrum, but we argue below that, in this case it leads instead to XY magnetic order.

Let us first imagine , but , so that we have spin rotation symmetry about the -axis in spin space. The mass term will be induced, and we observe that this term can be written

(63)

That is, this term has a mass for the up-spin fermions, and a mass of opposite sign for the down-spin fermions. Ignoring coupling to the gauge field for the moment, such a mass term leads to a quantum Hall effect (QHE) for the up-spin fermions, and a QHE for spin-down fermions.Haldane (1988)

Now, imagine we adiabatically insert a localized flux of the gauge field. The QHE implies that a single extra spin-up spinon will be induced along with the gauge flux, while one spin-down spinon will be depleted. This is equivalent to a spin flip operation, and we see that insertion of a monopole is accompanied by a spin flip. Furthermore, because the fermions are gapped, the gauge field dynamics is controlled by the Maxwell term , and therefore insertion of a monopole only costs finite action in the imaginary time functional integral. This implies that the monopole propagator is long-ranged – that is, if is a monopole creation operator, then approaches a constant as . Because monopole insertion is accompanied by a spin flip, in the ground state we will have , which corresponds to XY magnetic ordering. Furthermore, within the resulting ordered state, the photon of the gauge field is expected to correspond to the XY Goldstone mode. We note that a similar situation arises, in the absence of DM interaction, when a Zeeman magnetic field is applied.Ran et al. ()

In Sec. III.3, we argued that some of the monopole operators in the ASL correspond to the ordered state. We therefore expect that the XY order induced in the presence of the term is in the pattern, and therefore that the ASL is unstable toward magnetic ordering in the presence of . We remark that, once , the spin rotation symmetry is lost and there is no longer an XY Goldstone mode. On the other hand, time-reversal is still a good symmetry of the Hamiltonian, but is broken in the state. So, for more general DM interaction (), we expect the ASL is unstable to a time-reversal broken ground state.

Vi Analysis of projected wavefunction

We now turn to an analysis of the Gutzwiller projected variational wavefunction for the ASL. To obtain this wavefunction, one begins with , the mean-field ground state of the Hamiltonian Eq. (3). The trial wavefunction is obtained from this by writing

(64)

where the Gutzwiller projection operator,

(65)

with , enforces the single occupancy constraint. Properties of these wavefunctions can be calculated numerically using a standard Monte Carlo technique.Gros (1989)

In Ref. Ran et al., 2007, we found that has the lowest energy among a class of spin liquid wavefunctions. Furthermore, the energy is very low, less than above an extrapolation of the exact ground state energy to the thermodynamic limit;Waldtmann et al. (1998) this is achieved without tuning any continuous variational parameters to minimize the energy. By continuously deforming to include a small VBS ordering of the Hastings type and measuring the resulting variational energy, we argued that ASL is locally stable to this type of ordering. Here, we discuss some of the properties of this wavefunction. First, we construct variational excited states to estimate the velocity characteristic of the ASL. Next, we proceed to discuss the spin-spin and bond-bond correlation functions of the wavefunction, and connect them to our understanding of the low-energy effective theory.

We consider finite systems with periodic boundary conditions for the physical spin operators, so that

(66)

Such a system has sites. This still allows the consideration of twisted boundary conditions for the fermions – we consider both periodic and antiperiodic fermion boundary conditions. A technical point that plays a role in some of our analysis is that, depending on the system size, it is not always possible to construct a projected wavefunction that is fully symmetric under the kagome space group. The details of this subtlety are given in Appendix E. There we show how to construct symmetric wavefunctions for . The technique of Ref. Gros, 1989 is easily extended to calculate with these wavefunctions, which are superpositions of three projected single Slater determinants, and all the correlation function results shown here are for these fully symmetric wavefunctions.

vi.1 Fermi velocity

Figure 8: Plot of the mean-field band structure (solid lines) and projected band structure (solid rectangles), obtained as discussed in the text. The energy is in units of . The band structure is plotted along the line from M to M in the Brillouin zone. The projected band structure can be fit by the mean-field band structure, choosing . The band indices are shown on the right.

An important parameter in the algebraic spin liquid is the “Fermi velocity” . Physically, is defined as the ratio of energy and momentum for any low energy excitation – because the ASL is Lorentz invariant at low energies, all excitations propagate with the same velocity. In principle, can be directly measured via inelastic neutron scattering, by tracking the leading edge of the scattering continuum near the point and the M points, where the spin gap is predicted to vanish. Furthermore, it appears in the coefficients of specific heat and magnetic susceptibility, and in various critical scaling functions that may be measurable.

To estimate , we need to generate low energy excitations of the ASL. A natural way to do this is to make a particle-hole excitation of the mean-field state, then act with the projection operator: