In quantum spin chains, it has been observed that the incommensurability occurs near valence-bond-solid (VBS)-type solvable points, and the correlation length becomes shortest at VBS-type points. Besides, the correlation function decays purely exponentially at VBStype points, in contrast with the two-dimensional (2D) Ornstein-Zernicke type behavior in other regions with an excitation gap. We propose a mechanism to explain the onset of the incommensurability and the shortest correlation length at VBS-like points. This theory can be applicable for more general cases. Frustration is one of the important problem in many-body physics. Recently, many things in frustrated systems become clearer with field theoretical approaches and numerical calculations (density matrix renormalization group (DMRG) etc.). However, relatively little is known about commensurateincommensurate (C-IC) transitions induced by frustration in quantum spin systems. Among them, it is observed the C-IC change in the region with an excitation gap (therefore, it is not a phase transition) in one-dimensional (1D) quantum spin models. For example, the C-IC change begins from the Majumdar-Ghosh point [1] in the S=1/2 next-nearest-neighbor (NNN) chainIn quantum spin chains, it has been observed that the incommensurability occurs near valence-bond-solid (VBS)-type solvable points, and the correlation length becomes shortest at VBS-type points. Besides, the correlation function decays purely exponentially at VBStype points, in contrast with the two-dimensional (2D) Ornstein-Zernicke type behavior in other regions with an excitation gap. We propose a mechanism to explain the onset of the incommensurability and the shortest correlation length at VBS-like points. This theory can be applicable for more general cases. Frustration is one of the important problem in many-body physics. Recently, many things in frustrated systems become clearer with field theoretical approaches and numerical calculations (density matrix renormalization group (DMRG) etc.). However, relatively little is known about commensurateincommensurate (C-IC) transitions induced by frustration in quantum spin systems. Among them, it is observed the C-IC change in the region with an excitation gap (therefore, it is not a phase transition) in one-dimensional (1D) quantum spin models. For example, the C-IC change begins from the Majumdar-Ghosh point [1] in the S=1/2 next-nearest-neighbor (NNN) chain

Jan 1, 20087 citations

Incommensurability and Edge State in Quantum Spin Chain

K. Nomura, Takahiro Murashima

In quantum spin chains, it has been observed that the incommensurability occurs near valence-bond-solid (VBS) type points. It was difficult to study the commensurate–incommensurate (C–IC) change. On the one hand field theoretical approaches are not justified because of the short correlation length. On the other hand numerical calculations are not suitable to study the incommensurability since it is needed to treat the large size data. We discuss the relation between the edge state and the incommensurability, partially using the previous our study on the C–IC change.In quantum spin chains, it has been observed that the incommensurability occurs near valence-bond-solid (VBS) type points. It was difficult to study the commensurate–incommensurate (C–IC) change. On the one hand field theoretical approaches are not justified because of the short correlation length. On the other hand numerical calculations are not suitable to study the incommensurability since it is needed to treat the large size data. We discuss the relation between the edge state and the incommensurability, partially using the previous our study on the C–IC change.

Journal of the Physical Society of JapanJan 1, 20059 citations

Onset of incommensurability at the valence-bond-solid point in theS=1 quantum spin chain

Ulrich Schollwöck, Th. Jolicœur, Thomas Garel

Most of the present understanding of the S=1 quantum spin chains displaying the Haldane gap is coming from the so-called valence-bond-solid (VBS) Hamiltonian which has an exactly known ground state. We show that this point is characterized by the onset of short-range incommensurate spin correlations in the one-parameter family of Hamiltonians ${\mathit{H}}_{\mathrm{\ensuremath{\theta}}}$= cos\ensuremath{\theta}${\ensuremath{\sum}}_{\mathit{i}}$ ${\mathit{S}}_{\mathit{i}}$ \ensuremath{\cdot} ${\mathit{S}}_{\mathit{i}+1}$ +sin\ensuremath{\theta}${\ensuremath{\sum}}_{\mathit{i}}$ ( ${\mathit{S}}_{\mathit{i}}$ \ensuremath{\cdot} ${\mathit{S}}_{\mathit{i}+1}$ ${)}^{2}$ . This gives a physical meaning to this special point. We establish precise values for the gaps, correlations, the string order parameter, and identify the VBS point as a disorder point in the sense of classical statistical mechanics. It is a quantum remnant of the classical transition between a ground state with long-range N\'eel order and a ground state with incommensurate long-range order. \textcopyright{} 1996 The American Physical Society.Most of the present understanding of the S=1 quantum spin chains displaying the Haldane gap is coming from the so-called valence-bond-solid (VBS) Hamiltonian which has an exactly known ground state. We show that this point is characterized by the onset of short-range incommensurate spin correlations in the one-parameter family of Hamiltonians ${\mathit{H}}_{\mathrm{\ensuremath{\theta}}}$= cos\ensuremath{\theta}${\ensuremath{\sum}}_{\mathit{i}}$${\mathit{S}}_{\mathit{i}}$\ensuremath{\cdot}${\mathit{S}}_{\mathit{i}+1}$+sin\ensuremath{\theta}${\ensuremath{\sum}}_{\mathit{i}}$ (${\mathit{S}}_{\mathit{i}}$\ensuremath{\cdot}${\mathit{S}}_{\mathit{i}+1}$${)}^{2}$. This gives a physical meaning to this special point. We establish precise values for the gaps, correlations, the string order parameter, and identify the VBS point as a disorder point in the sense of classical statistical mechanics. It is a quantum remnant of the classical transition between a ground state with long-range N\'eel order and a ground state with incommensurate long-range order. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9505086Physical review. B, Condensed matterFeb 1, 1996161 citations

The asymmetric valence-bond-solid states in quantum spin chains: The difference between odd and even spins

Daisuke Maekawa, Hal Tasaki

The qualitative difference in low-energy properties of spin S quantum antiferromagnetic chains with integer S and half-odd-integer S discovered by Haldane [F. D. M. Haldane, arXiv:1612.00076 (1981); Phys. Lett. A 93, 464–468 (1983); Phys. Rev. Lett. 50, 1153–1156 (1983)] and Tasaki [Tasaki, Graduate Texts in Physics (Springer, 2020)] can be intuitively understood in terms of the valence-bond picture proposed by Affleck et al. [I. Affleck, Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477–528 (1988)]. Here, we develop a similarly intuitive diagrammatic explanation of the qualitative difference between chains with odd S and even S, which is at the heart of the theory of symmetry-protected topological (SPT) phases. (There is a 24 min video in which the essence of the present work is discussed: https://youtu.be/URsf9e_PLlc.) More precisely, we define one-parameter families of states, which we call the asymmetric valence-bond solid (VBS) states, that continuously interpolate between the Affleck–Kennedy–Lieb–Tasaki (AKLT) state and the trivial zero state in quantum spin chains with S = 1 and 2. The asymmetric VBS state is obtained by systematically modifying the AKLT state. It always has exponentially decaying truncated correlation functions and is a unique gapped ground state of a short-ranged Hamiltonian. We also observe that the asymmetric VBS state possesses the time-reversal, the Z2×Z2, and the bond-centered inversion symmetries for S = 2 but not for S = 1. This is consistent with the known fact that the AKLT model belongs to the trivial SPT phase if S = 2 and to a nontrivial SPT phase if S = 1. Although such interpolating families of disordered states were already known, our construction is unified and is based on a simple physical picture. It also extends to spin chains with general integer S and provides us with an intuitive explanation of the essential difference between models with odd and even spins.The qualitative difference in low-energy properties of spin S quantum antiferromagnetic chains with integer S and half-odd-integer S discovered by Haldane [F. D. M. Haldane, arXiv:1612.00076 (1981); Phys. Lett. A 93, 464–468 (1983); Phys. Rev. Lett. 50, 1153–1156 (1983)] and Tasaki [Tasaki, Graduate Texts in Physics (Springer, 2020)] can be intuitively understood in terms of the valence-bond picture proposed by Affleck et al. [I. Affleck, Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477–528 (1988)]. Here, we develop a similarly intuitive diagrammatic explanation of the qualitative difference between chains with odd S and even S, which is at the heart of the theory of symmetry-protected topological (SPT) phases. (There is a 24 min video in which the essence of the present work is discussed: https://youtu.be/URsf9e_PLlc.) More precisely, we define one-parameter families of states, which we call the asymmetric valence-bond solid (VBS) states, that continuously interpolate between the Affleck–Kennedy–Lieb–Tasaki (AKLT) state and the trivial zero state in quantum spin chains with S = 1 and 2. The asymmetric VBS state is obtained by systematically modifying the AKLT state. It always has exponentially decaying truncated correlation functions and is a unique gapped ground state of a short-ranged Hamiltonian. We also observe that the asymmetric VBS state possesses the time-reversal, the Z2×Z2, and the bond-centered inversion symmetries for S = 2 but not for S = 1. This is consistent with the known fact that the AKLT model belongs to the trivial SPT phase if S = 2 and to a nontrivial SPT phase if S = 1. Although such interpolating families of disordered states were already known, our construction is unified and is based on a simple physical picture. It also extends to spin chains with general integer S and provides us with an intuitive explanation of the essential difference between models with odd and even spins.

arXiv:2205.00653Journal of Mathematical PhysicsMar 1, 20237 citations

Order Parameter to Characterize Valence-Bond-Solid States in Quantum Spin Chains

Masaaki Nakamura, Synge Todo

We propose an order parameter to characterize valence-bond-solid (VBS) states in quantum spin chains, given by the ground-state expectation value of a unitary operator appearing in the Lieb-Schultz-Mattis argument. We show that the order parameter changes the sign according to the number of valence bonds (broken valence bonds) at the boundary for periodic (open) systems. This allows us to determine the phase transition point in between different VBS states. We demonstrate this theory in the successive dimerization transitions of the bond-alternating Heisenberg chains, using the quantum Monte Carlo method.We propose an order parameter to characterize valence-bond-solid (VBS) states in quantum spin chains, given by the ground-state expectation value of a unitary operator appearing in the Lieb-Schultz-Mattis argument. We show that the order parameter changes the sign according to the number of valence bonds (broken valence bonds) at the boundary for periodic (open) systems. This allows us to determine the phase transition point in between different VBS states. We demonstrate this theory in the successive dimerization transitions of the bond-alternating Heisenberg chains, using the quantum Monte Carlo method.

arXiv:cond-mat/0112377Physical Review LettersJul 30, 2002107 citations

Thermal valence-bond-solid transition of quantum spins in two dimensions

Songbo Jin, Anders W. Sandvik

We study the $S=1/2$ Heisenberg ( $J$ ) model on the two-dimensional (2D) square lattice in the presence of additional higher-order spin interactions ( $Q$ ) which lead to a valence-bond-solid (VBS) ground state. Using quantum Monte Carlo simulations, we analyze the thermal VBS transition. We find continuously varying exponents, with the correlation-length exponent $\ensuremath{\nu}$ close to the Ising value for large $Q/J$ and diverging when $Q/J$ approaches the quantum-critical point (the critical temperature ${T}_{c}\ensuremath{\rightarrow}0$). We identify the transition with a class of conformal field theories with charge $c=1$ and critical exponents varying between those of the 2D Ising model and the Kosterlitz-Thouless (KT) fixed point. We find explicit evidence for KT physics by studying the emergence of $U(1)$ symmetry of the order parameter at $T={T}_{c}$ when ${T}_{c}\ensuremath{\rightarrow}0$.We study the $S=1/2$ Heisenberg ($J$) model on the two-dimensional (2D) square lattice in the presence of additional higher-order spin interactions ($Q$) which lead to a valence-bond-solid (VBS) ground state. Using quantum Monte Carlo simulations, we analyze the thermal VBS transition. We find continuously varying exponents, with the correlation-length exponent $\ensuremath{\nu}$ close to the Ising value for large $Q/J$ and diverging when $Q/J$ approaches the quantum-critical point (the critical temperature ${T}_{c}\ensuremath{\rightarrow}0$). We identify the transition with a class of conformal field theories with charge $c=1$ and critical exponents varying between those of the 2D Ising model and the Kosterlitz-Thouless (KT) fixed point. We find explicit evidence for KT physics by studying the emergence of $U(1)$ symmetry of the order parameter at $T={T}_{c}$ when ${T}_{c}\ensuremath{\rightarrow}0$.

arXiv:1303.0703Physical Review BMay 22, 201320 citations

{\mathit{S}}_{\mathit{i}}

\ensuremath{\cdot}

{\mathit{S}}_{\mathit{i}+1}

+sin\ensuremath{\theta}${\ensuremath{\sum}}_{\mathit{i}}$ (

{\mathit{S}}_{\mathit{i}}

\ensuremath{\cdot}

{\mathit{S}}_{\mathit{i}+1}

{)}^{2}

. This gives a physical meaning to this special point. We establish precise values for the gaps, correlations, the string order parameter, and identify the VBS point as a disorder point in the sense of classical statistical mechanics. It is a quantum remnant of the classical transition between a ground state with long-range N\'eel order and a ground state with incommensurate long-range order. \textcopyright{} 1996 The American Physical Society.

We study the

S=1/2

Heisenberg (

J

) model on the two-dimensional (2D) square lattice in the presence of additional higher-order spin interactions (

Q

) which lead to a valence-bond-solid (VBS) ground state. Using quantum Monte Carlo simulations, we analyze the thermal VBS transition. We find continuously varying exponents, with the correlation-length exponent $\ensuremath{\nu}$ close to the Ising value for large

Q/J

and diverging when

Q/J

approaches the quantum-critical point (the critical temperature ${T}_{c}\ensuremath{\rightarrow}0$). We identify the transition with a class of conformal field theories with charge

c=1

and critical exponents varying between those of the 2D Ising model and the Kosterlitz-Thouless (KT) fixed point. We find explicit evidence for KT physics by studying the emergence of

U(1)

symmetry of the order parameter at

T={T}_{c}

when ${T}_{c}\ensuremath{\rightarrow}0$.