Valence bond solid and possible deconfined quantum criticality in an extended kagome lattice Heisenberg antiferromagnet

TL;DR

This paper provides numerical evidence for a valence bond solid phase in an extended kagome lattice Heisenberg antiferromagnet, suggesting a possible continuous quantum phase transition and revealing complex competing phases.

Contribution

It introduces the existence of an extended VBS phase near the Heisenberg point with detailed singlet patterning, and explores the nature of quantum phase transitions in this system.

Findings

Identification of a diamond-like VBS pattern with a 12-site unit cell.

Evidence for a possible continuous transition from magnetic order to VBS.

Discovery of a spin-nematic-like phase near the ferromagnetic transition.

Abstract

We present numerical evidence for the emergence of an extended valence bond solid (VBS) phase at $T=0$ in the kagome $S=1/2$ Heisenberg antiferromagnet with ferromagnetic further-neighbor interactions. The VBS is located at the boundary between two magnetically ordered regions and extends close to the nearest-neighbor Heisenberg point. It exhibits a diamond-like singlet covering pattern with a $12$-site unit-cell. Our results suggest the possibility of a direct, possibly continuous, quantum phase transition from the neighboring $\mathbf{q}=0$ coplanar magnetically ordered phase into the VBS phase. Moreover, a second phase which breaks lattice symmetries, and is of likely spin-nematic type, is found close to the transition to the ferromagnetic phase. The results have been obtained using numerical Exact Diagonalization. We discuss implications of our results on the nature of nearest-neighbor Heisenberg antiferromagnet.

Equations

$$\begin{array}{rl}H=& J_1\sum _{⟨i,j⟩}{\vec{S}}_i\cdot {\vec{S}}_j+J_2\sum _{⟨⟨i,j⟩⟩}{\vec{S}}_i\cdot {\vec{S}}_j\\ & +J_3\sum _{⟨⟨⟨i,j⟩⟩{⟩}_{\text{h}}}{\vec{S}}_i\cdot {\vec{S}}_j,\end{array}$$

$$S(\mathbf{q})=\frac{1}{N}\sum _{k,l=1}^N{\text{e}}^{-i\mathbf{q}\cdot ({\mathbf{r}}_k-{\mathbf{r}}_l)}⟨{\vec{S}}_k\cdot {\vec{S}}_l⟩.$$

$${\text{M}}^{\prime }=(2\pi ,2\pi /\sqrt{3})$$

$${\text{K}}^{\prime }=(8\pi /3,0)$$

$$D_{kl}\equiv ⟨({\vec{S}}_k\cdot {\vec{S}}_l)({\vec{S}}_1\cdot {\vec{S}}_0)⟩-⟨{\vec{S}}_k\cdot {\vec{S}}_l⟩⟨{\vec{S}}_1\cdot {\vec{S}}_0⟩,$$

$$D_{kl}^z\equiv ⟨(S_k^z\cdot S_l^z)(S_1^z\cdot S_0^z)⟩-⟨S_k^z\cdot S_l^z⟩⟨S_1^z\cdot S_0^z⟩.$$

$$O_{\text{VBS}}\equiv \frac{1}{N}\sum _{⟨k,l⟩}{\theta }_{kl}^{\text{VBS}}{D}_{kl},$$

$$O_{\text{nem}}\equiv \frac{1}{N}\sum _{⟨k,l⟩}{\theta }_{kl}^{\text{nem}}{D}_{kl},$$

$$Q_{kl}\equiv ⟨(S_k^{+}{S}_l^{+})(S_1^{-}{S}_0^{-})⟩,$$

Full text

Introduction —

We expect the unexpected when strong electron interactions meet geometric frustration. The emergence of novel exotic states of matter in frustrated quantum magnets is intensely studied in experiments, theory, and numerical computations. Several materials and theoretical models exhibit a lack of magnetic ordering even at lowest temperatures. Instead, genuine quantum many-body states, like quantum spin liquids Balents (2010); Savary and Balents (2016) or valence bond solids (VBS) can be observed Majumdar and Ghosh (1969); Fouet et al. (2003); Mambrini et al. (2006); Zhitomirsky and Ueda (1996); Iqbal et al. (2011a). Several experiments also have given evidence for emerging VBS phases in a variety of materials Matan et al. (2010); Rüegg (2010); Sheckelton et al. (2012); Smaha et al. (2019).

The nearest-neighbor kagome lattice Heisenberg spin $1/2$ antiferromagnet arguably remains one of the most puzzling conundrum in frustrated magnetism. Various scenarios on the nature of its ground state have been proposed. It has been found early, that a VBS is energetically competitive Marston and Zeng (1991); Leung and Elser (1993); Nikolic and Senthil (2003); Singh and Huse (2007); Capponi et al. (2013). However, more recent numerical studies suggest, that different spin disordered states are a more likely scenario. Several density-matrix renormalization group (DMRG) studies later suggested the possibility of a gapped spin liquid ground state Yan et al. (2011); Depenbrock et al. (2012). More recently, variational Monte Carlo and tensor network studies also suggested a gapless spin liquid state being realized Ran et al. (2007); Iqbal et al. (2015, 2011b, 2013); He et al. (2017); Liao et al. (2017). While conclusion on the nature of its ground state has not unanimously been reached to date Läuchli et al. (2019), several exotic new states of matter have been clearly identified in close proximity to the nearest-neighbor model Bauer et al. (2014); Gong et al. (2014); Wietek et al. (2015); He et al. (2014). Among those, a chiral spin liquid has been found in an extended Heisenberg model with antiferromagnetic second and third nearest-neighbor interactions Gong et al. (2014); Wietek et al. (2015); He et al. (2014). The classical ground state phase diagram of this model has previously been established Messio et al. (2011, 2012). A phase transition between two magnetic orders has been found for antiferromagnetic interactions. In the quantum case, the chiral spin liquid phase is located along the transition line between these two magnetic phases and extends close to the nearest-neighbor point. The classical phase diagram also contains a phase transition line between two types of coplanar magnetic orders for ferromagnetic second and third nearest-neighbor interactions. Given that some frustrated kagome materials involving both ferromagnetic and antiferromagnetic couplings are known to exist Fåk et al. (2012); Kermarrec et al. (2014); Iqbal et al. (2016), there is a strong interest to explore whether novel phases also emerge at or in the vicinity of the classical transition line at $J_{3}=2J_{2}<0$.

Here, we investigate the kagome spin $1/2$ Heisenberg antiferromagnet with additional ferromagnetic second and third nearest-neighbor interactions. We present conclusive numerical evidence for the appearance of a diamond VBS phase in an extended parameter range. The VBS phase is located in the vicinity of the classical transition line between the $\mathbf{q}=0$ and $\sqrt{3}\times\sqrt{3}$ magnetic orders. Interestingly, the phase extends close up to the nearest-neighbor Heisenberg point.

Model and phase diagram —

We consider the Hamiltonian,

[TABLE]

on a kagome lattice geometry, where $\vec{S}_{i}=(S^{x}_{i},S^{y}_{i},S^{z}_{i})$ denotes spin $1/2$ operators, $\langle\ldots\rangle$ and $\langle\langle\ldots\rangle\rangle$ denotes the sum over nearest- and second nearest-neighbor sites, and $\langle\langle\langle\dots\rangle\rangle\rangle_{\textrm{h}}$ denotes sum over third nearest-neighbor interactions only across the hexagons of the kagome lattice, cf. Fig. 1(b). In the following, we set $J_{1}=1$ and focus on the case of ferromagnetic couplings $J_{2}<0$ and $J_{3}<0$.

Most of our results are obtained by Exact Diagonalization (ED) calculations on a $N=36$ site kagome lattice with periodic boundary conditions Wietek and Läuchli (2018); Läuchli et al. (2011). Its Brillouin zone features the K and M points and is hence suited to stabilize both the $\sqrt{3}\times\sqrt{3}$ and $\mathbf{q}=0$ order. Selected results have been obtained on smaller clusters as well as on the larger $N=48$ cluster Läuchli et al. (2019); Wietek and Läuchli (2018). We detect ordering by investigating suitably chosen order parameters and performing tower-of-states analysis, i.e. comparing quantum numbers of finite-size energy eigenstates with theoretical predictions. The order parameters of the ground state and finite-size energy spectra are calculated on a grid for $J_{2}\in[-1,0]$ with spacing $0.05$ and $J_{3}\in[-2,0]$ with spacing $0.1$.

For classical Heisenberg spins the phase diagram of this model has been established in Ref. Messio et al. (2012). The $\sqrt{3}\times\sqrt{3}$ magnetic phase is separated from the $\mathbf{q}=0$ magnetic phase by a transition line located at $J_{3}=2J_{2}$. For $J_{3}<-2$ and $J_{2}<-1$ a ferromagnetic state is stabilized.

In Fig. 1 we present a first exploration of the quantum ($S=1/2$) phase diagram based on a map organized by the quantum numbers of the first excitation above the ground state. The assignment of the phases is performed based on a tower-of-states analysis for different candidate phases. According to this rationale, the blue region indicates the $\sqrt{3}\times\sqrt{3}$ magnetic order, the green region indicates the $\mathbf{q}=0$ magnetic order and the pink region the VBS phase. The nematic phase extends in the yellow and orange region, where two different quantum numbers are the first excitation. The gray lines serve as a guide to the eye and determine approximate phase boundaries. Apart from the expected $\sqrt{3}\times\sqrt{3}$ and $\mathbf{q}=0$ coplanar magnetic order phases, we find an unanticipated diamond VBS and a lattice symmetry breaking spin nematic phase located in the vicinity of the classical transition line. In Fig. 2 we corroborate the spectroscopy picture with an analysis of corresponding order parameters. The spin nematic phase extends close to the classical ferromagnetic phase, while the VBS phase extends close to the nearest-neighbor point. We now proceed to characterize the reported phases in more detail.

Magnetic order —

The $\mathbf{q}=0$ and $\sqrt{3}\times\sqrt{3}$ magnetic phases break spin rotational SU($2$) symmetry and exhibit patterns of magnetic ordering shown in the supplementary material Note (2). We consider the static spin structure factor,

[TABLE]

For the two magnetic orders, the structure factor is peaked at the points

[TABLE]

in the extended Brillouin zone, cf. Messio et al. (2011). Hence, $S(\textrm{{M}}^{\prime})$ and $S(\textrm{{K}}^{\prime})$ shown in Fig. 2(a) and (b) identify both magnetic phases, respectively. The regions where these structure factors are peaked coincide with the blue and green regions in Fig. 1. The blue region in Fig. 1(a) is given by the points, where the first excitation is a triplet, $S=1$, state with K.D3.A1 or $\Gamma$.D6.B1 space group quantum numbers Note (2); Lecheminant et al. (1997). In the green region in Fig. 1(a), the triplet states, $S=1$, with $\Gamma$.D6.A2 and $\Gamma$.D6.E2 space group quantum numbers are the first excitation. Thus, the spin structure factor and energy spectroscopy yield consistent results on the extent of these two phases.

Diamond VBS phase —

To identify the VBS and the lattice symmetry breaking spin-nematic phase we consider the connected dimer correlations,

[TABLE]

where the sites [math] and $1$ are an arbitrary nearest-neighbor bond chosen as reference. These correlations are long-ranged in the VBS phase and exhibit specific patterns of positive and negative correlation that can be predicted for model VBS state. For the $48$ site cluster we computed the diagonal $S^{z}$-dimer correlations

[TABLE]

The sign structure of these correlations serves as a first fingerprint of the particular VBS phase realized. For the diamond VBS state the expected sign structure of the dimer correlations are shown in Note (2). Thereby, we define an order parameter of the VBS phase,

[TABLE]

where $\theta_{kl}^{\text{VBS}}=\pm 1$ denotes the sign as defined in the supplementary material.

This diamond VBS parameter $O_{\text{VBS}}$ is shown in Fig. 2(c), indicating the extent of the VBS phase. It is located between the two magnetic orders and extends basically along the whole classical transition line from $J_{2}=-1$ to $J_{2}=0$. The region of pronounced $O_{\text{VBS}}$ also coincides with the pink region in Fig. 1. There, the first excited state is a singlet $S=0$ state with M.D2.A2 space group quantum number.

The precise nature of the reported VBS itself requires some more care. There are two basic candidate VBS model states with a twelve site unit cell Yan et al. (2011); Poilblanc and Misguich (2011); Huh et al. (2011); Hwang et al. (2015). A pinwheel VBS, where all dimers are static and the pinwheels all share the same orientation. This particular state is eightfold degenerate, a factor four from the translations and a factor two from the pinwheel orientation. On the other hand, like in many other VBS scenarios, there is a resonant version of this VBS, where we consider resonances involving eight-site loops in the shape of a diamond lozenge. A fully packed state of non-overlapping resonances is shown in Fig. 1(c). This state is actually twelve-fold degenerate, a factor four from the translations, and a factor three from the orientations of the diamond lozenges. The dimer-dimer correlations in these two model states are identical, so that dimer correlations alone can not distinguish the two states. However the spectral decomposition Note (2) reveals that beyond some common levels the diamond VBS features a characteristic spin singlet $\Gamma$.D6.E2 level, while the pinwheel VBS comes with a characteristic $\Gamma$.D6.A2 level. A close inspection of the energy spectrum of the VBS phase in Fig. 4(a)&(c) reveals a low-lying spin singlet $\Gamma$.D6.E2 level, and the absence of a low-lying $\Gamma$.D6.A2 level, thus clarifying the presence of a diamond VBS phase in this parameter region.

The spectral features of the VBS phase can be detected across various system sizes from $N=24$ to $N=48$ (only selected sectors) as shown in Fig. 4. The lowest excited state on all clusters we studied belongs to the same $M$ momentum and space group sector, consistent with the diamond VBS order in the thermodynamic limit. Hence, the evidence for a VBS is not only robust in the dimer correlations in Fig. 3 for system sizes $N=36$ and $N=48$ but also in the energy spectra for all system sizes we studied.

Spin nematic-plaquette phase —

The dimer correlations also exhibit a different peculiar sign structure in another parameter region, as shown for $J_{2}=-1$ and $J_{3}=-2$ in Fig. 5(a). We see characteristic positively correlated hexagon patterns suggesting a $2\times 2$ unit cell superstructure. However we are unaware of a singlet VBS model state exhibiting such a correlation pattern. We analogously define an order parameter for this lattice symmetry breaking pattern,

[TABLE]

where the sign $\theta_{kl}^{\text{nem}}=\pm 1$ is defined in the supplementary material. The region in parameter space where its signal is strong is shown in Fig. 1(d).

Since we are unaware of a singlet VBS with this structure, and due to the vicinity of the ferromagnet, we explore the possibility of a phase with additional spin-nematic character, for example of quadrupolar type Penc and Läuchli (2010). Several examples of frustrated ferromagnets giving rise to spin nematic phases have been discussed Shannon et al. (2006); Momoi et al. (2006); Hikihara et al. (2008); Sudan et al. (2009); Iqbal et al. (2016). In Fig. 5(b) we display the quadrupolar bond correlations,

[TABLE]

exhibiting sizeable correlations. We notice a peculiar hexagon-ring sign structure, where the correlations on hexagons surrounding the middle hexagon are negative, while correlations on the other hexagons are positive. In Fig. 5(c) we show an energy spectrum resolved by total $S^{z}$ and we can see a low-lying $S=2$ level, which could be due to the quadrupolar character. The lowest singlet excited state is a M.D2.A1 level, which is in agreement with the reported $2\times 2$ hexagon plaquette superstructure. So we see quite strong evidence for a novel phase, distinct from the other reported phases, but a detailed characterization of the phase (e.g. a corroboration of the spin nematic character) has to be left for future research.

Discussion and Outlook—

We have explored the appearance of two unexpected phases along the classical transition line in the $S=1/2$ kagome Heisenberg antiferromagnet with competing ferromagnetic further neighbor couplings. The first phase is a diamond VBS with a twelve site unit cell. This VBS or variants thereof have been seen in quantum dimer models Huh et al. (2011); Capponi et al. (2013); Hao et al. (2014); Hwang et al. (2015); Ralko et al. (2018) and hinted at by fluctuations or weak correlations in quantum spin models at the nearest-neighbor point ($J_{2}=J_{3}=0$) in Refs. Yan et al. (2011); Läuchli et al. (2019). We have now firmly established this VBS phase in the extended model (1). The location of this VBS phase in the immediate vicinity of the $\mathbf{q}=0$ magnetic order, and the apparent second-order nature of the phase transition between the two phases in exact diagonalization, places this transition into a contender role for an example of a deconfined quantum critical transition, with possibly deconfined spin excitations at the transition Senthil et al. (2004). Recent analytical work on the triangular lattice Jian et al. (2018) and the analysis of the matching VBS and Néel monopoles in the Dirac spin liquid Song et al. (2019) combined with our numerical results renders this scenario at least plausible. It will also be important to work out the connection between the VBS phase and the Dirac spin liquid state, which is currently a prime candidate to describe the kagome antiferromagnet at small antiferromagnetic $J_{2}$ coupling Ran et al. (2007); Iqbal et al. (2011b, 2013); He et al. (2017); Liao et al. (2017), before entering the $\mathbf{q}=0$ magnetic ordered phase.

This part of the phase diagram is then separated by a likely first order phase transition from the $\sqrt{3}\times\sqrt{3}$ magnetically ordered phase phase and the lattice symmetry breaking spin nematic phase close to the ferromagnetic phase. The precise nature of the latter phase is left for future studies.

Acknowledgements.

We acknowledge useful discussions with Y.-C. He, G. Misguich, and C. Wang. The Flatiron Institute is a division of the Simons Foundation. We thank the Austrian Science Fund FWF for support within the project DFG-FOR1807 (I-2868). This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science, and Economic Development, and by the Province of Ontario through the Ministry of Research and Innovation. The computational results presented have been achieved in part using the HPC infrastructure LEO of the University of Innsbruck. The computational results presented have been achieved in part using the Vienna Scientific Cluster (VSC). We acknowledge PRACE for granting access to ”Joliot Curie” HPC resources at TGCC/CEA under grant number 2019204846.