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CRC Press eBooks

Low-dimensional Quantum Spin Systems

Indrani Bose

Published Apr 24, 2019• 0 citations
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Abstract

Low-dimensional quantum spin systems is interacting many body systems for which several rigorous results are known. A large number of compounds exist which can effectively be described as low-dimensional spin systems. Frustration occurs when the interaction energies associated with all the spin pairs cannot be simultaneously minimized. The parallel spin pair may be located on any one of the bonds of the triangle. The entropy/spin has a non-zero value in the ground state and there is no ordering of the spins. The sum over permutations is in accordance with the bosonic nature of the waves, the spin waves, propagating along the chain. The energy-momentum relations suggest that the low-lying spectrum is actually a combination of two elementary excitations known as spinons. In the case of integer spin chains, the spinons axe bound and the excitation spectrum consists of spin-wave-like modes exhibiting the Haldane gap.

DOI
10.1201/9780429187520-8

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Low-dimensional quantum spin systems are interacting many body systems for which several rigorous results are known. Powerful techniques like the Bethe Ansatz provide exact knowledge of the ground state energy and low-lying excitation spectrum. A large number of compounds exist which can effectively be described as low-dimensional spin systems. Thus many of the rigorous results are also of experimental relevance. In these lectures, an introduction is given to quantum spin systems and the rigorous results known for such systems. Some general theorems and exactly-solvable models are discussed. An elementary introduction to the Bethe Ansatz technique is given.

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Theory of Quantum Spin Liquids in Quasi 1-D Systems

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Ever since the introduction of resonating valence bond states by P.\\ W.\\ Anderson in 1970s, the search for exotic many--body quantum behavior in peculiar states of matter known as quantum spin liquids has been a central challenge in the field of condensed matter physics. An important searching ground for such \\emph{quantum--ness} is the low energy physics of frustrated magnets. In a quantum system, frustration---the inability of a physical system to achieve a global state that consistently minimizes its energy locally---enables the unavoidable quantum fluctuations to hybridize an extensive number of classical states into highly entangled quantum states at low temperatures. It can be argued that this is the most practical approach to achieve genuine many--body quantum behavior at large scales. The fundamental source of frustration in many physical systems, is the existence of local non--commuting terms. 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I illustrate that the standard Heisenberg antiferromagnet Hamiltonian over kagome strip is an extended gapless quantum phase, that is well characterized by two fermionic/bosonic gapless modes and power--law decaying spin and bond--energy correlations. I also demonstrate that the correlation functions oscillate at tunably incommensurate wave vectors. It turns out that this phase can be identified by a particular marginal instability of a two-band spinon Fermi surface coupled to an emergent U(1) gauge field. This interpretation is supported by analytic Abelian bosonization and with extensive numerical large--scale density matrix renormalization group study as well as variational Monte Carlo calculations on Gutzwiller ans\\"atz wave functions. This intriguing result is the first numerical demonstration of emergent fermionic spinons in a simple SU(2) invariant nearest-neighbor Heisenberg model beyond the strictly 1D (Bethe chain) limit. The unexpected success of the fermionic spinons in describing the low energy physics of kagome antiferromagnet, reintroduces the questions about the validity of spinon physics as an effective theory of quantum spin liquids. However, numerical methods are still far behind the theoretical advances. And still even today, an accurate description of the quantum spin liquid states using tensor network methods is notoriously challenging. It is known that for large quasi-1D systems, the density matrix renormalization group and related methods usually require significant computational resources and sometimes fail to converge to a satisfactory state. On the other hand, variational wavefunctions acquired from the Gutzwiller projection of gaussian fermionic theories have long served as both a theoretical starting point for the construction of such spin liquid states and as an inspiration for numerical variational Monte Carlo (VMC) to calculate observables of interest. Noting this observation I examine a different method by exploring the possibility of constructing a matrix product state (MPS) representation for a Gutzwiller--projected state from two given MPS representations of gaussian fermionic theories in the \\ref{chap:gutz} chapter. I investigate the complexity of different approaches to achieve Gutzwiller projection for MPSs and introduce the novel algorithm which we call the Guzwiller zipper method. The performance of the algorithm is tested against two copies of a single half-filled band of spin-1/2 fermionic spinons. In a successful attempt to describe the nature of spin liquid states on quasi-1D strips of triangular and kagome-like lattices, I apply this method to two MPS of multi-band fermionic spinon theories and compare with the complexity of the traditional VMC approach. In particular, we methodically disprove the conjecture of a spinon fermi surface spin liquid for the triangular lattice. Finally, I conclude this thesis by laying out a bird's eye view of the current and future of spin liquid physics in kagome and triangular lattices and pointing out numerous possible applications of the novel Gutzwiller zipper method. In addition, I would also discuss its possible extensions to more complicated tensor networks as well as versions for higher spatial dimensions.

•Jan 1, 2020•0 citations

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Inelastic light scattering in low-dimensional spin systems

Kwang‐Yong Choi

In this work diverse results obtained from a number of low-dimensional quantum spin systems using Raman spectroscopy have been presented. The main motivation of this thesis is to hunt for new types of elementary excitations which may be present in low-dimensional spin systems with exotic spin topology. The studied systems can be on the one hand characterized by the formation of spin dimers which originate from a special spin topology like ACuCl3 (A=Tl, K), LiCu2O2, and SrCu(BO3)2. On the other hand, the system such as K2V3O8 and Cu2Te2O5X2 (X=Br,Cl) is dominated by quantum fluctuations due to reduced dimensionalilty. A powerful tool for studying low-dimensional spin systems is Raman spectroscopy which is sensitive to nearly all possible types of collective excitations as well as the interplay among them. In particular, magnetic light scattering can provide information about exchange paths and coupling constants, spinon excitations, spin gaps, and triplet-triplet interactions. The main results are summarized as follows; (i) the observation of a coherent coexistence of long-rang order and dimerized state in the frustrated spin chain system LiCu2O2, (ii) the presence of three-magnon scattering and dynamic spin-phonon coupling in the three-dimensional coupled spin dimer systems ACuCl3 (A=Tl, K), (iii) anomalous low-energy excitations including two-magnon scattering and electronic/orbital excitations in the two-dimensional Heisenberg antiferromagnetic system K2V3O8, (iv) the observation of inter-plane soft mode and a coupling of spin to soft mode in the quasi two-dimensional orthogonal dimer system SrCu(BO3)2, and (v) the existence of longitudinal magnons in the spin tetrahedra system Cu2Te2O5X2 (X=Br,Cl).

•Jan 1, 2003•0 citations
Low-dimensional quantum spin systems is interacting many body systems for which several rigorous results are known. A large number of compounds exist which can effectively be described as low-dimensional spin systems. Frustration occurs when the interaction energies associated with all the spin pairs cannot be simultaneously minimized. The parallel spin pair may be located on any one of the bonds of the triangle. The entropy/spin has a non-zero value in the ground state and there is no ordering of the spins. The sum over permutations is in accordance with the bosonic nature of the waves, the spin waves, propagating along the chain. The energy-momentum relations suggest that the low-lying spectrum is actually a combination of two elementary excitations known as spinons. In the case of integer spin chains, the spinons axe bound and the excitation spectrum consists of spin-wave-like modes exhibiting the Haldane gap.
Low-dimensional quantum spin systems are interacting many body systems for which several rigorous results are known. Powerful techniques like the Bethe Ansatz provide exact knowledge of the ground state energy and low-lying excitation spectrum. A large number of compounds exist which can effectively be described as low-dimensional spin systems. Thus many of the rigorous results are also of experimental relevance. In these lectures, an introduction is given to quantum spin systems and the rigorous results known for such systems. Some general theorems and exactly-solvable models are discussed. An elementary introduction to the Bethe Ansatz technique is given.
Low-dimensional quantum spin systems are interacting many body systems for which several rigorous results are known. Powerful techniques like the Bethe Ansatz provide exact knowledge of the ground state energy and low-lying excitation spectrum. A large number of compounds exists which can effectively be described as low-dimensional spin systems. Thus many of the rigorous results are also of experimental relevance. In this Lecture Note, an introduction is given to quantum spin systems and the rigorous results known for such systems. Some general theorems and exactly-solvable models are discussed. An elementary introduction to the Bethe Ansatz technique is given.
Observing constituent particles with fractional quantum numbers in confined and deconfined states is an interesting and challenging problem in quantum many-body physics. Here, we further explore a computational scheme [Y. Tang and A. W. Sandvik, Phys. Rev. Lett. 107, 157201 (2011)] based on valence-bond quantum Monte Carlo simulations of quantum spin systems. Using several different one-dimensional models, we characterize S=12S=\frac{1}{2}S=21​ spinon excitations using the intrinsic spinon size $\ensuremath{\lambda}$ and confinement length $\mathrm{\ensuremath{\Lambda}}$ (the size of a bound state). The spinons have finite size in valence-bond-solid states, infinite size in the critical region (with overlaps characterized by power laws), and become ill defined (completely unlocalizable) in the N\'eel state (which we stabilize in one dimension by introducing long-range interactions). We also verify that pairs of spinons are deconfined in uniform spin chains but become confined upon introducing a pattern of alternating coupling strengths (dimerization) or coupling two chains (forming a ladder). In the dimerized system, an individual spinon can be small when the confinement length is large; this is the case when the imposed dimerization is weak but the ground state of the corresponding uniform chain is a spontaneously formed valence-bond solid (where the spinons are deconfined). Based on our numerical results, we argue that a system with $\ensuremath{\lambda}\ensuremath{\ll}\mathrm{\ensuremath{\Lambda}}$ is associated with weak repulsive short-range spinon-spinon interactions. In principle, both the length scales $\ensuremath{\lambda}$ and $\mathrm{\ensuremath{\Lambda}}$ can still be individually tuned from small to infinite (with $\ensuremath{\lambda}\ensuremath{\le}\mathrm{\ensuremath{\Lambda}}$) by varying model parameters. In contrast, in the ladder system the two lengths are always similar, and this is the case also in the weakly dimerized systems when the corresponding uniform chain is in the critical phase. In these systems, the effective spinon-spinon interactions are purely attractive and there is only a single large length scale close to criticality, which is reflected in the standard spin correlations as well as in the spinon characteristics.
Ever since the introduction of resonating valence bond states by P.\\ W.\\ Anderson in 1970s, the search for exotic many--body quantum behavior in peculiar states of matter known as quantum spin liquids has been a central challenge in the field of condensed matter physics. An important searching ground for such \\emph{quantum--ness} is the low energy physics of frustrated magnets. In a quantum system, frustration---the inability of a physical system to achieve a global state that consistently minimizes its energy locally---enables the unavoidable quantum fluctuations to hybridize an extensive number of classical states into highly entangled quantum states at low temperatures. It can be argued that this is the most practical approach to achieve genuine many--body quantum behavior at large scales. The fundamental source of frustration in many physical systems, is the existence of local non--commuting terms. In the case of antiferromagnet spin theories the simplest examples are triangles of nearest--neighboring sites. Consequently, lattices with triangles as building blocks have long been studied in hopes of discovering spin liquids physics. In particular the kagome lattice, which is composed of corner--sharing triangles, has been the center of at least four decades of spin liquid research. This is due to the underconstrained nature of system made out of corner--sharing triangles. After the first three chapters, introduction in chapter~\\ref{chap:intro}, preliminary physics in chapter~\\ref{chap:prelim}, and a brief introduction to spin--liquids in chapter~\\ref{chap:QSLs}, in chapter~\\ref{chap:kagome} I tackle a simpler version of the long-standing problem of spin--1/21/21/2 Heisenberg antiferromagnets by considering a quasi--1D lattice consisting entirely of corner--sharing triangles, \\emph{kagome strip}. I illustrate that the standard Heisenberg antiferromagnet Hamiltonian over kagome strip is an extended gapless quantum phase, that is well characterized by two fermionic/bosonic gapless modes and power--law decaying spin and bond--energy correlations. I also demonstrate that the correlation functions oscillate at tunably incommensurate wave vectors. It turns out that this phase can be identified by a particular marginal instability of a two-band spinon Fermi surface coupled to an emergent U(1) gauge field. This interpretation is supported by analytic Abelian bosonization and with extensive numerical large--scale density matrix renormalization group study as well as variational Monte Carlo calculations on Gutzwiller ans\\"atz wave functions. This intriguing result is the first numerical demonstration of emergent fermionic spinons in a simple SU(2) invariant nearest-neighbor Heisenberg model beyond the strictly 1D (Bethe chain) limit. The unexpected success of the fermionic spinons in describing the low energy physics of kagome antiferromagnet, reintroduces the questions about the validity of spinon physics as an effective theory of quantum spin liquids. However, numerical methods are still far behind the theoretical advances. And still even today, an accurate description of the quantum spin liquid states using tensor network methods is notoriously challenging. It is known that for large quasi-1D systems, the density matrix renormalization group and related methods usually require significant computational resources and sometimes fail to converge to a satisfactory state. On the other hand, variational wavefunctions acquired from the Gutzwiller projection of gaussian fermionic theories have long served as both a theoretical starting point for the construction of such spin liquid states and as an inspiration for numerical variational Monte Carlo (VMC) to calculate observables of interest. Noting this observation I examine a different method by exploring the possibility of constructing a matrix product state (MPS) representation for a Gutzwiller--projected state from two given MPS representations of gaussian fermionic theories in the \\ref{chap:gutz} chapter. I investigate the complexity of different approaches to achieve Gutzwiller projection for MPSs and introduce the novel algorithm which we call the Guzwiller zipper method. The performance of the algorithm is tested against two copies of a single half-filled band of spin-1/2 fermionic spinons. In a successful attempt to describe the nature of spin liquid states on quasi-1D strips of triangular and kagome-like lattices, I apply this method to two MPS of multi-band fermionic spinon theories and compare with the complexity of the traditional VMC approach. In particular, we methodically disprove the conjecture of a spinon fermi surface spin liquid for the triangular lattice. Finally, I conclude this thesis by laying out a bird's eye view of the current and future of spin liquid physics in kagome and triangular lattices and pointing out numerous possible applications of the novel Gutzwiller zipper method. In addition, I would also discuss its possible extensions to more complicated tensor networks as well as versions for higher spatial dimensions.
Using large-scale determinant quantum Monte Carlo simulations in combination with the stochastic analytical continuation, we study two-particle dynamical correlation functions in the anisotropic square lattice of weakly coupled one-dimensional (1D) Hubbard chains at half filling and in the presence of weak frustration. The evolution of the static spin structure factor upon increasing the interchain coupling is suggestive of the transition from the power-law decay of spin-spin correlations in the 1D limit to long-range antiferromagnetic order in the quasi-1D regime and at T=0T=0T=0. In the numerically accessible regime of interchain couplings, the charge sector remains gapped. The low-energy momentum dependence of the spin excitations is well described by the linear spin-wave theory with the largest intensity located around the antiferromagnetic wave vector. This magnon mode corresponds to a bound state of two spinons. At higher energies the spinons deconfine and we observe signatures of the two-spinon continuum which progressively fade away as a function of interchain hopping.
In this work diverse results obtained from a number of low-dimensional quantum spin systems using Raman spectroscopy have been presented. The main motivation of this thesis is to hunt for new types of elementary excitations which may be present in low-dimensional spin systems with exotic spin topology. The studied systems can be on the one hand characterized by the formation of spin dimers which originate from a special spin topology like ACuCl3 (A=Tl, K), LiCu2O2, and SrCu(BO3)2. On the other hand, the system such as K2V3O8 and Cu2Te2O5X2 (X=Br,Cl) is dominated by quantum fluctuations due to reduced dimensionalilty. A powerful tool for studying low-dimensional spin systems is Raman spectroscopy which is sensitive to nearly all possible types of collective excitations as well as the interplay among them. In particular, magnetic light scattering can provide information about exchange paths and coupling constants, spinon excitations, spin gaps, and triplet-triplet interactions. The main results are summarized as follows; (i) the observation of a coherent coexistence of long-rang order and dimerized state in the frustrated spin chain system LiCu2O2, (ii) the presence of three-magnon scattering and dynamic spin-phonon coupling in the three-dimensional coupled spin dimer systems ACuCl3 (A=Tl, K), (iii) anomalous low-energy excitations including two-magnon scattering and electronic/orbital excitations in the two-dimensional Heisenberg antiferromagnetic system K2V3O8, (iv) the observation of inter-plane soft mode and a coupling of spin to soft mode in the quasi two-dimensional orthogonal dimer system SrCu(BO3)2, and (v) the existence of longitudinal magnons in the spin tetrahedra system Cu2Te2O5X2 (X=Br,Cl).