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arXiv:cond-mat/0506457 · Physical Review B

Superfluid-insulator transitions on the triangular lattice

A. A. Burkov, Leon Balents

Published Oct 3, 2005• 38 citations
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Abstract

We report on a phenomenological study of superfluid to Mott insulator transitions of bosons on the triangular lattice, focusing primarily on the interplay between Mott localization and geometrical charge frustration at $1∕2$-filling. A general dual-vortex field theory is developed for arbitrary rational filling factors $f$, based on the appropriate projective symmetry group. At the simple nonfrustrated density $f=1∕3$, we uncover an example of a deconfined quantum critical point very similar to that found on the half-filled square lattice. Turning to $f=1∕2$, the behavior is quite different. Here, we find that the low-energy action describing the Mott transition has an emergent non-Abelian $SU(2)\ifmmode\times\else\texttimes\fi{}U(1)$ symmetry, which is not present at the microscopic level. This large non-Abelian symmetry is directly related to the frustration-induced quasidegeneracy between many charge-ordered states not related by microscopic symmetries. Through this ``pseudospin'' $SU(2)$ symmetry, the charged excitations in the insulator close to the Mott transition develop a Skyrmion-like character. This leads to an understanding of the supersolid phase of the triangular lattice $XXZ$ model as a ``partially melted'' Mott insulator, for more information see Melko et al. [Phys. Rev. Lett. 95, 127207 (2005)], Heidarian and Damle [Phys. Rev. Lett. 95, 127206 (2005)], and Wessel and Troyer [Phys. Rev. Lett. 95, 127205 (2005)]. The latter picture naturally explains a number of puzzling numerical observations of the properties of this supersolid. Moreover, we predict that the nearby quantum phase transition from this supersolid to the Mott insulator is in the noncompact ${\mathrm{CP}}^{1}$ critical universality class as presented in Motrunich and Vishwanath [Phys. Rev. B. 70, 075104 (2004)]. A description of a broad range of other Mott and supersolid states, and a diverse set of quantum critical points between them, is also provided.

DOI
10.1103/physrevb.72.134502

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We report on a phenomenological study of superfluid to Mott insulator transitions of bosons on the triangular lattice, focusing primarily on the interplay between Mott localization and geometrical charge frustration at 1∕21∕21∕2-filling. A general dual-vortex field theory is developed for arbitrary rational filling factors fff, based on the appropriate projective symmetry group. At the simple nonfrustrated density f=1∕3f=1∕3f=1∕3, we uncover an example of a deconfined quantum critical point very similar to that found on the half-filled square lattice. Turning to f=1∕2f=1∕2f=1∕2, the behavior is quite different. Here, we find that the low-energy action describing the Mott transition has an emergent non-Abelian SU(2)\ifmmode×\else\texttimes\fiU(1)SU(2)\ifmmode\times\else\texttimes\fi{}U(1)SU(2)\ifmmode×\else\texttimes\fiU(1) symmetry, which is not present at the microscopic level. This large non-Abelian symmetry is directly related to the frustration-induced quasidegeneracy between many charge-ordered states not related by microscopic symmetries. Through this pseudospin'' $SU(2)$ symmetry, the charged excitations in the insulator close to the Mott transition develop a Skyrmion-like character. This leads to an understanding of the supersolid phase of the triangular lattice $XXZ$ model as a partially melted'' Mott insulator, for more information see Melko et al. [Phys. Rev. Lett. 95, 127207 (2005)], Heidarian and Damle [Phys. Rev. Lett. 95, 127206 (2005)], and Wessel and Troyer [Phys. Rev. Lett. 95, 127205 (2005)]. The latter picture naturally explains a number of puzzling numerical observations of the properties of this supersolid. Moreover, we predict that the nearby quantum phase transition from this supersolid to the Mott insulator is in the noncompact CP1{\mathrm{CP}}^{1}CP1 critical universality class as presented in Motrunich and Vishwanath [Phys. Rev. B. 70, 075104 (2004)]. A description of a broad range of other Mott and supersolid states, and a diverse set of quantum critical points between them, is also provided.
Using large-scale quantum Monte Carlo simulations we study bosons hopping on a triangular lattice with nearest (V)(V)(V) and next-nearest $({V}^{\ensuremath{'}})$ neighbor repulsive interactions. In the limit where V=0V=0V=0 but ${V}^{\ensuremath{'}}$ is large, we find an example of an unusual period-three striped supersolid state that is stable at 1∕21∕21∕2-filling. We discuss the relationship of this state to others on the rich ground-state phase diagram, which include a previously discovered supersolid, a uniform superfluid, as well as several Mott insulating phases. We study several superfluid- and supersolid-to-Mott phase transitions, including one proposed by a recent phenomenological dual vortex field theory as a candidate for an exotic deconfined quantum critical point. We find no examples of unconventional quantum criticality among any of the interesting phase transitions in the model.
We describe the localization transition of superfluids on two-dimensional lattices into commensurate Mott insulators with average particle density p∕qp∕qp∕q (p,qp,qp,q relatively prime integers) per lattice site. For bosons on the square lattice, we argue that the superfluid has at least qqq degenerate species of vortices which transform under a projective representation of the square-lattice space group (a PSG). The formation of a single-vortex condensate produces the Mott insulator, which is required by the PSG to have density wave order at wavelengths of q∕nq∕nq∕n lattice sites (nnn integer) along the principle axes; such a second-order transition is forbidden in the Landau-Ginzburg-Wilson frame-work. We also discuss the superfluid-insulator transition in the direct boson representation and find that an interpretation of the quantum criticality in terms of deconfined fractionalized bosons is only permitted at special values of qqq for which a permutative representation of the PSG exists. We argue [and demonstrate in detail in a companion paper: L. Balents et al., following paper, Phys. Rev. B 71, 144509 (2005)] that our results apply essentially unchanged to electronic systems with short-range pairing, with the PSG determined by the particle density of Cooper pairs. We also describe the effect of static impurities in the superfluid: the impurities locally break the degeneracy between the qqq vortex species, and this induces density-wave order near each vortex. We suggest that such a theory offers an appealing rationale for the local density-of-states modulations observed by Hoffman et al. [Science 295, 466 (2002)], in scanning tunneling microscopy (STM) studies of the vortex lattice of ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ and allows a unified description of the nucleation of density-wave order in zero and finite magnetic fields. We note signatures of our theory that may be tested by future STM experiments.
We study ground state properties and excitation spectra for hard-core Bosons on square and triangular lattices, at half filling, using series expansion methods. Nearest-neighbor repulsion between the Bosons leads to the development of short-range density order at the antiferromagnetic wavevector, and simultaneously a roton minima in the density excitation spectra. On the square-lattice, the model maps on to the well studied XXZ model, and the roton gap collapses to zero precisely at the Heisenberg symmetry point, leading to the well known spectra for the Heisenberg antiferromagnet. On the triangular-lattice, the collapse of the roton gap signals the onset of the supersolid phase. Our results suggest that the transition from the superfluid to the supersolid phase maybe weakly first order. We also find several features in the density of states, including two-peaks and a sharp discontinuity, which maybe observable in experimental realization of such systems.
We study the phases of doped spin S=1∕2S=1∕2S=1∕2 quantum antiferromagnets on the square lattice as they evolve from paramagnetic Mott insulators with valence bond solid (VBS) order at doping $\ensuremath{\delta}=0$ to superconductors at moderate $\ensuremath{\delta}$. The interplay between density-wave and VBS order and superconductivity is efficiently described by the quantum dimer model, which acts as an effective theory for the total spin S=0S=0S=0 sector. We extend the dimer model to include fermionic S=1∕2S=1∕2S=1∕2 excitations and show that its mean-field, static gauge-field saddle points have projective symmetries (PSG's) similar to those of ``slave''-particle U(1) and SU(2) gauge theories. We account for the nonperturbative effects of gauge fluctuations by a duality mapping of the S=0S=0S=0 dimer model. The dual theory of vortices has a PSG identical to that found in a previous paper [L. Balents et al., preceding paper, Phys. Rev. B 71, 144508 (2005)] by a duality analysis of bosons on the square lattice. The previous theory therefore also describes fluctuations across superconducting, supersolid, and Mott insulating phases of the present electronic model. Finally, with the aim of describing neutron scattering experiments, we present a phenomenological model for collective S=1S=1S=1 excitations and their coupling to superflow and density-wave fluctuations.
Kinetically frustrated bosons at half filling in the presence of a competing nearest-neighbor repulsion support a wide supersolid regime on the two-dimensional triangular lattice. We study this model on a two-leg ladder using the finite-size density-matrix renormalization-group method, obtaining a phase diagram which contains three phases: a uniform superfluid (SF), an insulating charge density wave (CDW) crystal, and a bond ordered insulator (BO). We show that the transitions from SF to CDW and SF to BO are continuous in nature, with critical exponents varying continuously along the phase boundaries, while the transition from CDW to BO is found to be first order. The phase diagram is also found to contain an exactly solvable Majumdar Ghosh point, and reentrant SF to CDW phase transitions.