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arXiv:0902.4210 · Physical Review B

Spin Bose-metal phase in a spin- 12 model with ring exchange on a two-leg triangular strip

D. N. Sheng, Olexei I. Motrunich, Matthew P. A. Fisher

Published May 20, 2009• 129 citations
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Abstract

Recent experiments on triangular lattice organic Mott insulators have found evidence for a two-dimensional (2D) spin liquid in close proximity to the metal-insulator transition. A Gutzwiller wave function study of the triangular lattice Heisenberg model with a four-spin ring exchange term appropriate in this regime has found that the projected spinon Fermi sea state has a low variational energy. This wave function, together with a slave particle-gauge theory analysis, suggests that this putative spin liquid possesses spin correlations that are singular along surfaces in momentum space, i.e., ``Bose surfaces.'' Signatures of this state, which we will refer to as a ``spin Bose metal'' (SBM), are expected to manifest in quasi-one-dimensional (quasi-1D) ladder systems: the discrete transverse momenta cut through the 2D Bose surface leading to a distinct pattern of 1D gapless modes. Here, we search for a quasi-1D descendant of the triangular lattice SBM state by exploring the Heisenberg plus ring model on a two-leg triangular strip (zigzag chain). Using density matrix renormalization group (DMRG) supplemented by variational wave functions and a bosonization analysis, we map out the full phase diagram. In the absence of ring exchange the model is equivalent to the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain, and we find the expected Bethe-chain and dimerized phases. Remarkably, moderate ring exchange reveals a new gapless phase over a large swath of the phase diagram. Spin and dimer correlations possess singular wave vectors at particular ``Bose points'' (remnants of the 2D Bose surface) and allow us to identify this phase as the hoped for quasi-1D descendant of the triangular lattice SBM state. We use bosonization to derive a low-energy effective theory for the zigzag spin Bose metal and find three gapless modes and one Luttinger parameter controlling all power law correlations. Potential instabilities out of the zigzag SBM give rise to other interesting phases such as a period-3 valence bond solid or a period-4 chirality order, which we discover in the DMRG. Another interesting instability is into a spin Bose-metal phase with partial ferromagnetism (spin polarization of one spinon band), which we also find numerically using the DMRG.

DOI
10.1103/physrevb.79.205112

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Recent experiments on triangular lattice organic Mott insulators have found evidence for a two-dimensional (2D) spin liquid in close proximity to the metal-insulator transition. A Gutzwiller wave function study of the triangular lattice Heisenberg model with a four-spin ring exchange term appropriate in this regime has found that the projected spinon Fermi sea state has a low variational energy. This wave function, together with a slave particle-gauge theory analysis, suggests that this putative spin liquid possesses spin correlations that are singular along surfaces in momentum space, i.e., Bose surfaces.'' Signatures of this state, which we will refer to as a spin Bose metal'' (SBM), are expected to manifest in quasi-one-dimensional (quasi-1D) ladder systems: the discrete transverse momenta cut through the 2D Bose surface leading to a distinct pattern of 1D gapless modes. Here, we search for a quasi-1D descendant of the triangular lattice SBM state by exploring the Heisenberg plus ring model on a two-leg triangular strip (zigzag chain). Using density matrix renormalization group (DMRG) supplemented by variational wave functions and a bosonization analysis, we map out the full phase diagram. In the absence of ring exchange the model is equivalent to the J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ Heisenberg chain, and we find the expected Bethe-chain and dimerized phases. Remarkably, moderate ring exchange reveals a new gapless phase over a large swath of the phase diagram. Spin and dimer correlations possess singular wave vectors at particular ``Bose points'' (remnants of the 2D Bose surface) and allow us to identify this phase as the hoped for quasi-1D descendant of the triangular lattice SBM state. We use bosonization to derive a low-energy effective theory for the zigzag spin Bose metal and find three gapless modes and one Luttinger parameter controlling all power law correlations. Potential instabilities out of the zigzag SBM give rise to other interesting phases such as a period-3 valence bond solid or a period-4 chirality order, which we discover in the DMRG. Another interesting instability is into a spin Bose-metal phase with partial ferromagnetism (spin polarization of one spinon band), which we also find numerically using the DMRG.
We study a spin-1/2 system with Heisenberg plus ring exchanges on a four-leg triangular ladder using the density matrix renormalization group and Gutzwiller variational wave functions. Near an isotropic lattice regime, for moderate to large ring exchanges we find a spin Bose-metal phase with a spinon Fermi sea consisting of three partially filled bands. Going away from the triangular towards the square lattice regime, we find a staggered dimer phase with dimers in the transverse direction, while for small ring exchanges the system is in a featureless rung phase. We also discuss parent states and a possible phase diagram in two dimensions.
We study the effects of nonmagnetic impurities in a spin Bose-metal (SBM) phase discovered in a two-leg triangular strip spin-1/2 model with ring exchanges (D. N. Sheng et al., arXiv:0902.4210). This phase is a quasi-one-dimensional (quasi-1D) descendant of a two-dimensional (2D) spin liquid with spinon Fermi sea and the present study aims at interpolating between the 1D and 2D cases. Different types of defects can be treated as local-energy perturbations, which we find are always relevant. As a result, a nonmagnetic impurity generically cuts the system into two decoupled parts. We calculate bond energy and local spin susceptibility near the defect, both of which can be measured in experiments. The spin Bose metal has dominant correlations at characteristic incommensurate wave vectors that are revealed near the defect. Thus, the bond energy shows a static texture oscillating as a function of distance from the defect and decaying as a slow power law. The local spin susceptibility also oscillates and actually increases as a function of distance from the defect, similar to the effect found in the 1D chain [S. Eggert and I. Affleck, Phys. Rev. Lett. 75, 934 (1995)]. We calculate the corresponding power-law exponents for the textures as a function of one Luttinger parameter of the SBM theory.
We establish compelling evidence for the existence of new quasi-one-dimensional descendants of the ddd-wave Bose liquid (DBL), an exotic two-dimensional quantum phase of uncondensed itinerant bosons characterized by surfaces of gapless excitations in momentum space [O. I. Motrunich and M. P. A. Fisher, Phys. Rev. B 75, 235116 (2007)]. In particular, motivated by a strong-coupling analysis of the gauge theory for the DBL, we study a model of hard-core bosons moving on the NNN-leg square ladder with frustrating four-site ring exchange. Here, we focus on four- and three-leg systems where we have identified two novel phases: a compressible gapless Bose metal on the four-leg ladder and an incompressible gapless Mott insulator on the three-leg ladder. The former is conducting along the ladder and has five gapless modes, one more than the number of legs. This represents a significant step forward in establishing the potential stability of the DBL in two dimensions. The latter, on the other hand, is a fundamentally quasi-one-dimensional phase that is insulating along the ladder but has two gapless modes and incommensurate power-law transverse density-density correlations. While we have already presented results on this latter phase elsewhere [M. S. Block et al., Phys. Rev. Lett. 106, 046402 (2011)], we will expand upon those results in this work. In both cases, we can understand the nature of the phase using slave-particle-inspired variational wave functions consisting of a product of two distinct Slater determinants, the properties of which compare impressively well to a density matrix renormalization group solution of the model Hamiltonian. Stability arguments are made in favor of both quantum phases by accessing the universal low-energy physics with a bosonization analysis of the appropriate quasi-1D gauge theory. We will briefly discuss the potential relevance of these findings to high-temperature superconductors, cold atomic gases, and frustrated quantum magnets.
Using tensor network states to unravel the physics of quantum spin liquids in minimal, yet generic microscopic spin or electronic models remains notoriously challenging. A prominent open question concerns the nature of the insulating ground state of two-dimensional half-filled Hubbard-type models on the triangular lattice in the vicinity of the Mott metal-insulator transition, a regime which can be approximated microscopically by a spin-1/2 Heisenberg model supplemented with additional "ring-exchange" interactions. Using a novel and efficient state preparation technique whereby we initialize full density matrix renormalization group (DMRG) calculations with highly entangled Gutzwiller-projected Fermi surface trial wave functions, we show -- contrary to previous works -- that the simplest triangular lattice JJJ-KKK spin model with four-site ring exchange likely does not harbor a fully gapless U(1) spinon Fermi surface (spin Bose metal) phase on four- and six-leg wide ladders. Our methodology paves the way to fully resolve with DMRG other controversial problems in the fields of frustrated quantum magnetism and strongly correlated electrons.
Developing a theoretical framework to access the quantum phases of itinerant bosons or fermions in two dimensions that exhibit singular structure along surfaces in momentum space but have no quasiparticle description remains a central challenge in the field of strongly correlated physics. In this paper we propose that distinctive signatures of such two-dimensional (2D) strongly correlated phases will be manifest in quasi-one-dimensional ``NNN-leg ladder'' systems. Characteristic of each parent 2D quantum liquid would be a precise pattern of one-dimensional (1D) gapless modes on the NNN-leg ladder. These signatures could be potentially exploited to approach the 2D phases from controlled numerical and analytical studies in quasi-one-dimension. As a first step we explore itinerant-boson models with a frustrating ring-exchange interaction on the two-leg ladder, searching for signatures of the recently proposed two-dimensional ddd-wave-correlated Bose liquid (DBL) phase. A combination of exact diagonalization, density-matrix renormalization-group, variational Monte Carlo, and bosonization analysis of a quasi-1D gauge theory provide compelling evidence for the existence of an unusual strong-coupling phase of bosons on the two-leg ladder, which can be understood as a descendant of the two-dimensional DBL. We suggest several generalizations to quantum spin and electron Hamiltonians on ladders, which could likewise reveal fingerprints of such 2D non-Fermi-liquid phases.
We consider an electronic model for realizing the spin Bose-metal (SBM) phase on a two-leg triangular strip---a spin liquid phase found by Sheng et al. [Phys. Rev. B 79, 205112 (2009)] in a spin-1/2 model with ring exchanges. The SBM can be viewed as a ``C1S2'' Mott insulator of electrons where the overall charge transporting mode is gapped out. We start from a two-band ``C2S2'' metal and consider extended repulsion motivated by recent ab initio derivation of electronic model for $\ensuremath{\kappa}\text{-ET}$ spin liquid material [K. Nakamura et al., J. Phys. Soc. Jpn. 78, 083710 (2009)]. Using weak coupling renormalization group analysis, we find that the extended interactions allow much wider C2S2 metallic phase than in the Hubbard model with on-site repulsion only. An eight-fermion umklapp term plays a crucial role in producing a Mott insulator but cannot be treated in weak coupling. We use bosonization to extend the analysis to intermediate coupling and study phases obtained out of the C2S2 metal upon increasing overall repulsion strength, finding that the SBM phase is a natural outcome for extended interactions.