Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

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

Matthew S. Block

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

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.

Related papers

Exotic Quantum Phases and Phase Transitions of Strongly Interacting Electrons in Low-Dimensional Systems

Ryan V. Mishmash

Experiments on strongly correlated quasi-two-dimensional electronic materials---for example, the high-temperature cuprate superconductors and the putative quantum spin liquids kappa-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2---routinely reveal highly mysterious quantum behavior which cannot be explained in terms of weakly interacting degrees of freedom. Theoretical progress thus requires the introduction of completely new concepts and machinery beyond the traditional framework of the band theory of solids and its interacting counterpart, Landau's Fermi liquid theory. In full two dimensions, controlled and reliable analytical approaches to such problems are severely lacking, as are numerical simulations of even the simplest of model Hamiltonians due to the infamous fermionic sign problem.Here, we attempt to circumvent some of these difficulties by studying analogous problems in quasi-one dimension. In this lower dimensional setting, theoretical and numerical tractability are on much stronger footing due to the methods of bosonization and the density matrix renormalization group, respectively. Using these techniques, we attack two problems: (1) the Mott transition between a Fermi liquid metal and a quantum spin liquid as potentially directly relevant to the organic compounds kappa-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2 and (2) non-Fermi liquid metals as strongly motivated by the strange metal phase observed in the cuprates. In both cases, we are able to realize highly exotic quantum phases as ground states of reasonable microscopic models. This lends strong credence to respective underlying slave-particle descriptions of the low-energy physics, which are inherently strongly interacting and also unconventional in comparison to weakly interacting alternatives.Finally, working in two dimensions directly, we propose a new slave-particle theory which explains in a universal way many of the intriguing experimental results of the triangular lattice organic spin liquid candidates kappa-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2. With use of large-scale variational Monte Carlo calculations, we show that this new state has very competitive trial energy in an effective spin model thought to describe the essential features of the real materials.

•Jan 1, 2014•0 citations

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

D. N. Sheng, Olexei I. Motrunich, Simon Trebst +2 more

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.

arXiv:0805.0255•Physical Review B•Aug 26, 2008•52 citations

Variational Studies of Exotic Bose Liquid, Spin Liquid, and Magnetic Phases

Tiamhock Tay

The strong interest in strongly correlated systems in condensed matter physics has continued unabated for the past few decades. In recent years, the number of novel, exotic quantum phases found in theoretical studies has seen a phenomenal rise. Among those interesting quantum states are bose liquids and spin liquids, where strong quantum fluctuations have prevented the systems from developing a long range order. Our work in this thesis seeks to further the understanding of frustrated systems. In the study of a hard-core boson model with ring-only exchange interactions on a square lattice, we obtain concrete numerical realization of the unconventional Exciton Bose Liquid (EBL) phase, which possesses interesting properties such as a "Bose surface" which resembles the Fermi surface in a metal, as well as unusual thermodynamic properties such as a T Log T dependence for specific heat. An equally important result from this work is the demonstration that the widely used Gutzwiller projection on slave-particle wave functions may generally fail to capture the correct long wavelength physics in the respective systems. For the Heisenberg antiferromagnet on the kagome lattice, which is a promising candidate for realizing a spin-disordered ground state, our variational study shows that the projected Schwinger boson wave function is energetically better than the Dirac spin liquid wave function when a small antiferromagnetic second-neighbor spin coupling is added to the nearest-neighbor model. We also study the anisotropic triangular Heisenberg antiferromagnetic in magnetic field, and find simple, yet accurate wave functions for various regions of the surprisingly rich phase diagram, thus providing insights into the energetics of the competing phases in this interesting model. Finally, our work also highlights permanent-type wave functions as potentially useful constructions in variational studies of systems with short-ranged correlations, e.g., a Mott insulator and a gapped spin liquid.

•May 31, 2011•1 citation

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

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

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 "$N$-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.

arXiv:0812.2955•Dec 16, 2008•1 citation

Computational Studies Of Strongly Correlated Quantum Matter

Hao Shi

The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.

•Jan 1, 2017•0 citations

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.

arXiv:1110.4607•Physical Review B•Dec 27, 2011•41 citations
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.
Experiments on strongly correlated quasi-two-dimensional electronic materials---for example, the high-temperature cuprate superconductors and the putative quantum spin liquids kappa-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2---routinely reveal highly mysterious quantum behavior which cannot be explained in terms of weakly interacting degrees of freedom. Theoretical progress thus requires the introduction of completely new concepts and machinery beyond the traditional framework of the band theory of solids and its interacting counterpart, Landau's Fermi liquid theory. In full two dimensions, controlled and reliable analytical approaches to such problems are severely lacking, as are numerical simulations of even the simplest of model Hamiltonians due to the infamous fermionic sign problem.Here, we attempt to circumvent some of these difficulties by studying analogous problems in quasi-one dimension. In this lower dimensional setting, theoretical and numerical tractability are on much stronger footing due to the methods of bosonization and the density matrix renormalization group, respectively. Using these techniques, we attack two problems: (1) the Mott transition between a Fermi liquid metal and a quantum spin liquid as potentially directly relevant to the organic compounds kappa-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2 and (2) non-Fermi liquid metals as strongly motivated by the strange metal phase observed in the cuprates. In both cases, we are able to realize highly exotic quantum phases as ground states of reasonable microscopic models. This lends strong credence to respective underlying slave-particle descriptions of the low-energy physics, which are inherently strongly interacting and also unconventional in comparison to weakly interacting alternatives.Finally, working in two dimensions directly, we propose a new slave-particle theory which explains in a universal way many of the intriguing experimental results of the triangular lattice organic spin liquid candidates kappa-(BEDT-TTF)2Cu2(CN)3 and EtMe3Sb[Pd(dmit)2]2. With use of large-scale variational Monte Carlo calculations, we show that this new state has very competitive trial energy in an effective spin model thought to describe the essential features of the real materials.
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.
The strong interest in strongly correlated systems in condensed matter physics has continued unabated for the past few decades. In recent years, the number of novel, exotic quantum phases found in theoretical studies has seen a phenomenal rise. Among those interesting quantum states are bose liquids and spin liquids, where strong quantum fluctuations have prevented the systems from developing a long range order. Our work in this thesis seeks to further the understanding of frustrated systems. In the study of a hard-core boson model with ring-only exchange interactions on a square lattice, we obtain concrete numerical realization of the unconventional Exciton Bose Liquid (EBL) phase, which possesses interesting properties such as a "Bose surface" which resembles the Fermi surface in a metal, as well as unusual thermodynamic properties such as a T Log T dependence for specific heat. An equally important result from this work is the demonstration that the widely used Gutzwiller projection on slave-particle wave functions may generally fail to capture the correct long wavelength physics in the respective systems. For the Heisenberg antiferromagnet on the kagome lattice, which is a promising candidate for realizing a spin-disordered ground state, our variational study shows that the projected Schwinger boson wave function is energetically better than the Dirac spin liquid wave function when a small antiferromagnetic second-neighbor spin coupling is added to the nearest-neighbor model. We also study the anisotropic triangular Heisenberg antiferromagnetic in magnetic field, and find simple, yet accurate wave functions for various regions of the surprisingly rich phase diagram, thus providing insights into the energetics of the competing phases in this interesting model. Finally, our work also highlights permanent-type wave functions as potentially useful constructions in variational studies of systems with short-ranged correlations, e.g., a Mott insulator and a gapped spin liquid.
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.
The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.
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.