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arXiv:0805.0255 · Physical Review B

Strong-coupling phases of frustrated bosons on a two-leg ladder with ring exchange

D. N. Sheng, Olexei I. Motrunich, Simon Trebst, Emanuel Gull, Matthew P. A. Fisher

Published Aug 26, 2008• 52 citations
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Abstract

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 ``$N$-leg ladder'' systems. Characteristic of each parent 2D quantum liquid would be a precise pattern of one-dimensional (1D) gapless modes on the $N$-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 $d$-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.

DOI
10.1103/physrevb.78.054520

Related papers

Bose metals and insulators on multileg ladders with ring exchange

Ryan V. Mishmash, Matthew S. Block, Ribhu K. Kaul +3 more

We establish compelling evidence for the existence of new quasi-one-dimensional descendants of the $d$-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 $N$-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.

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Spin, Bose, and Non-Fermi Liquid Metals in Two Dimensions: Accessing via Multi-Leg Ladders

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Fermions in two dimensions, when subject to anisotropic spin-dependent hopping, can potentially give rise to unusual paired states in unpolarized mixtures that can behave as non-Fermi liquids. One possibility is a fully paired state with a gap for fermion excitations in which the Cooper pairs remain uncondensed. Such a ``Cooper-pair Bose-metal'' phase would be expected to have a singular Bose surface in momentum space. As demonstrated in the context of two-dimensional bosons hopping with a frustrating ring-exchange interaction, an analogous Bose-metal phase has a set of quasi-one-dimensional descendant states when put on a ladder geometry. Here we present a density matrix renormalization group study of the attractive Hubbard model with spin-dependent hopping on a two-leg ladder geometry. In our setup, one spin species moves preferentially along the leg direction, while the other does so along the rung direction. We find compelling evidence for the existence of a novel Cooper-pair Bose-metal phase in a region of the phase diagram at intermediate coupling. We further explore the phase diagram of this model as a function of hopping anisotropy, density, and interaction strength, finding a conventional superfluid phase as well as a phase of paired Cooper pairs with $d$-wave symmetry, similar to the one found in models of hard-core bosons with ring exchange. We argue that simulating this model with cold Fermi gases on spin-dependent optical lattices is a promising direction for realizing exotic quantum states.

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Variational Monte Carlo Methods for Strongly Correlated Quantum Systems on Multileg Ladders

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Quantum mechanical systems of strongly interacting particles in two dimensions comprise a realm of condensed matter physics for which there remain many unanswered theoretical questions. In particular, the most formidable challenges may lie in cases where the ground states show no signs of ordering, break no symmetries, and support many gapless excitations. Such systems are known to exhibit exotic, disordered ground states that are notoriously difficult to study analytically using traditional perturbation techniques or numerically using the most recent methods (e.g., tensor network states) due to the large amount of spatial entanglement. Slave particle descriptions provide a glimmer of hope in the attempt to capture the fundamental, low-energy physics of these highly non-trivial phases of matter. To this end, this dissertation describes the construction and implementation of trial wave functions for use with variational Monte Carlo techniques that can easily model slave particle states. While these methods are extremely computationally tractable in two dimensions, we have applied them here to quasi-one-dimensional systems so that the results of other numerical techniques, such as the density matrix renormalization group, can be directly compared to those determined by the trial wave functions and so that exclusively one-dimensional analytic approaches, namely bosonization, can be employed. While the focus here is on the use of variational Monte Carlo, the sum of these different numerical and analytical tools has yielded a remarkable amount of insight into several exotic quantum ground states. In particular, the results of research on the d-wave Bose liquid phase, an uncondensed state of strongly correlated hard-core bosons living on the square lattice whose wave function exhibits a d-wave sign structure, and the spin Bose-metal phase, a spin-1/2, SU(2) invariant spin liquid of strongly correlated spins living on the triangular lattice, will be presented. Both phases support gapless excitations along surfaces in momentum space in two spatial dimensions and at incommensurate wave vectors in quasi-one dimension, where we have studied them on three- and four-leg ladders. An extension of this work to the study of d-wave correlated itinerant electrons will be discussed.

•Jan 1, 2011•0 citations

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

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.

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Exotic Gapless Mott Insulators of Bosons on Multileg Ladders

Matthew S. Block, Ryan V. Mishmash, Ribhu K. Kaul +3 more

We present evidence for an exotic gapless insulating phase of hard-core bosons on multileg ladders with a density commensurate with the number of legs. In particular, we study in detail a model of bosons moving with direct hopping and frustrating ring exchange on a 3-leg ladder at ν=1/3 filling. For sufficiently large ring exchange, the system is insulating along the ladder but has two gapless modes and power law transverse density correlations at incommensurate wave vectors. We propose a determinantal wave function for this phase and find excellent comparison between variational Monte Carlo and density matrix renormalization group calculations on the model Hamiltonian, thus providing strong evidence for the existence of this exotic phase. Finally, we discuss extensions of our results to other N-leg systems and to N-layer two-dimensional structures.

arXiv:1008.4105•Physical Review Letters•Jan 27, 2011•40 citations
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 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.
Characterizing and accessing quantum phases of itinerant bosons or fermions in two dimensions (2D) that exhibit singular structure along surfaces in momentum space but have no quasi-particle description remains as a central challenge in the field of strongly correlated physics. Fortuitously, signatures of such 2D strongly correlated phases are expected to be manifest in quasi-one-dimensional "NNN-leg ladder" systems. The ladder discretization of the transverse momentum cuts through the 2D surface, leading to a quasi-1D descendant state with a set of low-energy modes whose number grows with the number of legs and whose momenta are inherited from the 2D surfaces. These multi-mode quasi-1D liquids constitute a new and previously unanticipated class of quantum states interesting in their own right. But more importantly they carry a distinctive quasi-1D "fingerprint" of the parent 2D quantum fluid state. This can be exploited to access the 2D phases from controlled numerical and analytical studies in quasi-1D models. The preliminary successes and future prospects in this endeavor will be briefly summarized.
Fermions in two dimensions, when subject to anisotropic spin-dependent hopping, can potentially give rise to unusual paired states in unpolarized mixtures that can behave as non-Fermi liquids. One possibility is a fully paired state with a gap for fermion excitations in which the Cooper pairs remain uncondensed. Such a ``Cooper-pair Bose-metal'' phase would be expected to have a singular Bose surface in momentum space. As demonstrated in the context of two-dimensional bosons hopping with a frustrating ring-exchange interaction, an analogous Bose-metal phase has a set of quasi-one-dimensional descendant states when put on a ladder geometry. Here we present a density matrix renormalization group study of the attractive Hubbard model with spin-dependent hopping on a two-leg ladder geometry. In our setup, one spin species moves preferentially along the leg direction, while the other does so along the rung direction. We find compelling evidence for the existence of a novel Cooper-pair Bose-metal phase in a region of the phase diagram at intermediate coupling. We further explore the phase diagram of this model as a function of hopping anisotropy, density, and interaction strength, finding a conventional superfluid phase as well as a phase of paired Cooper pairs with ddd-wave symmetry, similar to the one found in models of hard-core bosons with ring exchange. We argue that simulating this model with cold Fermi gases on spin-dependent optical lattices is a promising direction for realizing exotic quantum states.
Quantum mechanical systems of strongly interacting particles in two dimensions comprise a realm of condensed matter physics for which there remain many unanswered theoretical questions. In particular, the most formidable challenges may lie in cases where the ground states show no signs of ordering, break no symmetries, and support many gapless excitations. Such systems are known to exhibit exotic, disordered ground states that are notoriously difficult to study analytically using traditional perturbation techniques or numerically using the most recent methods (e.g., tensor network states) due to the large amount of spatial entanglement. Slave particle descriptions provide a glimmer of hope in the attempt to capture the fundamental, low-energy physics of these highly non-trivial phases of matter. To this end, this dissertation describes the construction and implementation of trial wave functions for use with variational Monte Carlo techniques that can easily model slave particle states. While these methods are extremely computationally tractable in two dimensions, we have applied them here to quasi-one-dimensional systems so that the results of other numerical techniques, such as the density matrix renormalization group, can be directly compared to those determined by the trial wave functions and so that exclusively one-dimensional analytic approaches, namely bosonization, can be employed. While the focus here is on the use of variational Monte Carlo, the sum of these different numerical and analytical tools has yielded a remarkable amount of insight into several exotic quantum ground states. In particular, the results of research on the d-wave Bose liquid phase, an uncondensed state of strongly correlated hard-core bosons living on the square lattice whose wave function exhibits a d-wave sign structure, and the spin Bose-metal phase, a spin-1/2, SU(2) invariant spin liquid of strongly correlated spins living on the triangular lattice, will be presented. Both phases support gapless excitations along surfaces in momentum space in two spatial dimensions and at incommensurate wave vectors in quasi-one dimension, where we have studied them on three- and four-leg ladders. An extension of this work to the study of d-wave correlated itinerant electrons will be discussed.
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.
We present evidence for an exotic gapless insulating phase of hard-core bosons on multileg ladders with a density commensurate with the number of legs. In particular, we study in detail a model of bosons moving with direct hopping and frustrating ring exchange on a 3-leg ladder at ν=1/3 filling. For sufficiently large ring exchange, the system is insulating along the ladder but has two gapless modes and power law transverse density correlations at incommensurate wave vectors. We propose a determinantal wave function for this phase and find excellent comparison between variational Monte Carlo and density matrix renormalization group calculations on the model Hamiltonian, thus providing strong evidence for the existence of this exotic phase. Finally, we discuss extensions of our results to other N-leg systems and to N-layer two-dimensional structures.