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

arXiv:cond-mat/9609064 · Physical review. B, Condensed matter

Variational and density-matrix renormalization-group studies of the frustrated antiferromagnetic Heisenberg S=1 quantum spin chain

Alexei K. Kolezhuk, R. Roth, Ulrich Schollwöck

Published Apr 1, 1997• 51 citations
arXivHTMLPDF

Abstract

In this paper we study a frustrated antiferromagnetic isotropic Heisenberg S=1 quantum spin chain H=${\ensuremath{\sum}}_{\mathrm{i}}$${\mathbf{S}}_{\mathrm{i}}$${\mathbf{S}}_{\mathrm{i}+1}$+\ensuremath{\alpha}${\ensuremath{\sum}}_{\mathrm{i}}$${\mathbf{S}}_{\mathrm{i}}$${\mathbf{S}}_{\mathrm{i}+2}$, using a variational ansatz starting from valence bond states and the density-matrix renormalization group. We find both methods to give results in very good qualitative and good quantitative agreement, which clarify the phase diagram as follows: At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{D}}$=0.284(1), there is a disorder point of the first kind, marking the onset of incommensurate spin-spin correlations in the chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{L}}$=0.3725(25) there is a Lifshitz point, at which the excitation spectrum is found to develop a particular doubly degenerate structure. These points are the quantum remnants of the transition from antiferromagnetic to spiral order in the classical frustrated chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{T}}$=0.7444(6) there is a first-order phase transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase characterized by nonvanishing string order to a phase which can be understood as a next-nearest-neighbor generalization of the AKLT model. At the transition, the string order parameter shows (to numerical precision) a discontinuous jump of 0.085 to zero; the correlation length and the gap are both finite at the transition. The problem of edge states in open frustrated chains is discussed at length.

DOI
10.1103/physrevb.55.8928

Related papers

First Order Transition in the Frustrated Antiferromagnetic Heisenberg S=1 Quantum Spin Chain

Alexei K. Kolezhuk, Roland Roth, Ulrich Schollwöck

We study a frustrated Heisenberg $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ quantum spin chain with next-nearest neighbor (NNN) coupling $\ensuremath{\alpha}$ using a variational ansatz and the density matrix renormalization group. We find as quantum remnants of the phase transition in the classical chain a disorder point at ${\ensuremath{\alpha}}_{D}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.284(1)$ and a Lifshitz point at ${\ensuremath{\alpha}}_{L}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.3725(25)$. Our main finding is a first-order transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase to a NNN generalization of the AKLT phase at ${\ensuremath{\alpha}}_{T}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.7444(6)$. At the transition, string order jumps discontinuously by $\ensuremath{\approx}0.085$ to 0; correlation length and gap are finite.

Physical Review Letters•Dec 16, 1996•89 citations

Absence of string order in the anisotropic S=2 Heisenberg antiferromagnet

Harald N. Aschauer, Ulrich Schollwöck

In this paper we study an antiferromagnetic Heisenberg $S=2$ quantum spin chain at $T=0$ with both interaction and on-site anisotropy, $H={\ensuremath{\sum}}_{i}\frac{1}{2}{(S}_{i}^{+}{S}_{i+1}^{\ensuremath{-}}{+S}_{i}^{\ensuremath{-}}{S}_{i+1}^{+}{)+J}^{z}{S}_{i}^{z}{S}_{i+1}^{z}{+D(S}_{i}^{z}{)}^{2}$. While the phase diagram of this chain for $S=1$ and the isotropic point for $S=2$ are by now excellently understood, contradictory scenarios exist for the $S=2$ anisotropic phase diagram, which imply completely different mechanisms of the emergence of the classical $\stackrel{\ensuremath{\rightarrow}}{S}\ensuremath{\infty}$ limit, hence the importance of the question. One main scenario predicts the emergence of a cascade of phase transitions not seen in the $S=1$ case, based on an analysis using the Affleck-Kennedy-Lieb-Tasaki (AKLT) model. Another scenario is in favor of an almost classical phase diagram for $S=2$, from which the $S=\ensuremath{\infty}$ phase diagram evolves quite trivially; the $S=1$ case then emerges as very special with its dominant quantum effects. Numerical studies have so far not been conclusive. Using the density matrix renormalization group, we study gaps and correlation functions in the anisotropic $S=2$ chain. The question of the existence of hidden topological order in the anisotropic $S=2$ chain is emphasized, as it distinguishes between the proposed scenarios. Careful extrapolation allows us to show that the topological order is zero in the thermodynamical limit in all disordered phases, in particular in the new phase interposed between the Haldane and large-$D$ phases. The decay of the hidden order is substantially faster than that of ${S}^{+}{S}^{\ensuremath{-}}$ correlations and excellently fitted by power laws typical of massless phases. This excludes the AKLT-model-based scenario in favor of an almost classical phase diagram for the $S=2$ and thus higher spin chains.

arXiv:cond-mat/9802189•Physical review. B, Condensed matter•Jul 1, 1998•26 citations

Connectivity transition in the frustrated S=1 chain revisited

Alexei K. Kolezhuk, U. Schollwöck

The phase transition in the antiferromagnetic isotropic Heisenberg $S=1$ chain with frustrating next-nearest-neighbor coupling $\ensuremath{\alpha}$ is reconsidered. We identify the order parameter of the large-\ensuremath{\alpha} phase as describing two intertwined strings, each possessing a usual string order. The transition has a topological nature determined by the change in the string connectivity. Numerical evidence from the density matrix renormalization group results is supported by the effective theory based on soliton states.

arXiv:cond-mat/0110095•Physical review. B, Condensed matter•Feb 11, 2002•38 citations

Frustrated quantum Heisenberg ferrimagnetic chains

Nedko B. Ivanov, Johannes Richter, Ulrich Schollwöck

We study the ground-state properties of weakly frustrated Heisenberg ferrimagnetic chains with nearest- and next-nearest-neighbor antiferromagnetic exchange interactions and two types of alternating sublattice spins ${S}_{1}>{S}_{2},$ using $1/S$ spin-wave expansions, density-matrix renormalization group, and exact diagonalization techniques. It is argued that the zero-point spin fluctuations completely destroy the classical commensurate-incommensurate continuous transition. Instead, the long-range ferrimagnetic state disappears through a discontinuous transition to a singlet state at a larger value of the frustration parameter. In the ferrimagnetic phase we find a disorder point marking the onset of incommensurate real-space short-range spin-spin correlations.

arXiv:cond-mat/9803150•Physical review. B, Condensed matter•Dec 1, 1998•30 citations

Frustrated ferromagnetic spin- 12 chain in a magnetic field: The phase diagram and thermodynamic properties

Fabian Heidrich‐Meisner, A. Honecker, T. Vekua

The frustrated ferromagnetic spin-$1∕2$ Heisenberg chain is studied by means of a low-energy field theory as well as the density-matrix renormalization group and exact diagonalization methods. First, we study the ground-state phase diagram in a magnetic field and find an ``even-odd'' (EO) phase characterized by bound pairs of magnons in the region of two weakly coupled antiferromagnetic chains. A jump in the magnetization curves signals a first-order transition at the boundary of the EO phase, but otherwise the curves are smooth. Second, we discuss thermodynamic properties at zero field, where we confirm a double-peak structure in the specific heat for moderate frustrating next-nearest-neighbor interactions.

arXiv:cond-mat/0605204•Physical Review B•Jul 27, 2006•139 citations

A comparative study of the phase diagrams of spin- and spin-1 antiferromagnetic chains with dimerization and frustration

Swapan Kumar Pati, R. Chitra, Diptiman Sen +2 more

The ground-state `phase' diagrams and some low-energy properties of isotropic antiferromagnetic spin- and spin-1 chains with a next-nearest-neighbour exchange and an alternation of the nearest-neighbour exchanges have been compared for the first time using the density matrix renormalization group method. In the spin- chain, the system is gapless for and , and is gapped everywhere else in the plane. At , for small , the gap increases as , where is a disorder line. To the left of this line, the structure factor S(q) peaks at (the Néel `phase'), while to the right, decreases from to (the spiral `phase') as increases. We also discover a ` phase' for large values of both and . In the spin-1 case, we find a line running from a gapless point at up to a `gapless' point at such that the open-chain ground state is fourfold degenerate below the line and is unique above it. There is a disorder line in this case also and it has the same equation as in the spin- case, but the line ends at about . Similarly to the spin- case, to the left of this line, the peak in the structure factor is at (the Néel `phase'), while to the right of the line, it is at less than (the spiral `phase'). For , the system corresponds to a spin ladder and the system is gapped for all values of the interchain coupling for both spin- and spin-1 ladders.

arXiv:cond-mat/9603107•Journal of Physics Condensed Matter•Jan 6, 1997•23 citations
In this paper we study a frustrated antiferromagnetic isotropic Heisenberg S=1 quantum spin chain H={\ensuremath{\sum}}_{\mathrm{i}}$${\mathbf{S}}_{\mathrm{i}}$${\mathbf{S}}_{\mathrm{i}+1}+\ensuremath{\alpha}{\ensuremath{\sum}}_{\mathrm{i}}$${\mathbf{S}}_{\mathrm{i}}$${\mathbf{S}}_{\mathrm{i}+2}, using a variational ansatz starting from valence bond states and the density-matrix renormalization group. We find both methods to give results in very good qualitative and good quantitative agreement, which clarify the phase diagram as follows: At \ensuremathαD{\mathrm{\ensuremath{\alpha}}}_{\mathrm{D}}\ensuremathαD​=0.284(1), there is a disorder point of the first kind, marking the onset of incommensurate spin-spin correlations in the chain. At \ensuremathαL{\mathrm{\ensuremath{\alpha}}}_{\mathrm{L}}\ensuremathαL​=0.3725(25) there is a Lifshitz point, at which the excitation spectrum is found to develop a particular doubly degenerate structure. These points are the quantum remnants of the transition from antiferromagnetic to spiral order in the classical frustrated chain. At \ensuremathαT{\mathrm{\ensuremath{\alpha}}}_{\mathrm{T}}\ensuremathαT​=0.7444(6) there is a first-order phase transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase characterized by nonvanishing string order to a phase which can be understood as a next-nearest-neighbor generalization of the AKLT model. At the transition, the string order parameter shows (to numerical precision) a discontinuous jump of 0.085 to zero; the correlation length and the gap are both finite at the transition. The problem of edge states in open frustrated chains is discussed at length.
We study a frustrated Heisenberg S=1S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1S=1 quantum spin chain with next-nearest neighbor (NNN) coupling $\ensuremath{\alpha}$ using a variational ansatz and the density matrix renormalization group. We find as quantum remnants of the phase transition in the classical chain a disorder point at ${\ensuremath{\alpha}}_{D}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.284(1)$ and a Lifshitz point at ${\ensuremath{\alpha}}_{L}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.3725(25)$. Our main finding is a first-order transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase to a NNN generalization of the AKLT phase at ${\ensuremath{\alpha}}_{T}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.7444(6)$. At the transition, string order jumps discontinuously by $\ensuremath{\approx}0.085$ to 0; correlation length and gap are finite.
In this paper we study an antiferromagnetic Heisenberg S=2S=2S=2 quantum spin chain at T=0T=0T=0 with both interaction and on-site anisotropy, $H={\ensuremath{\sum}}_{i}\frac{1}{2}{(S}_{i}^{+}{S}_{i+1}^{\ensuremath{-}}{+S}_{i}^{\ensuremath{-}}{S}_{i+1}^{+}{)+J}^{z}{S}_{i}^{z}{S}_{i+1}^{z}{+D(S}_{i}^{z}{)}^{2}$. While the phase diagram of this chain for S=1S=1S=1 and the isotropic point for S=2S=2S=2 are by now excellently understood, contradictory scenarios exist for the S=2S=2S=2 anisotropic phase diagram, which imply completely different mechanisms of the emergence of the classical $\stackrel{\ensuremath{\rightarrow}}{S}\ensuremath{\infty}$ limit, hence the importance of the question. One main scenario predicts the emergence of a cascade of phase transitions not seen in the S=1S=1S=1 case, based on an analysis using the Affleck-Kennedy-Lieb-Tasaki (AKLT) model. Another scenario is in favor of an almost classical phase diagram for S=2S=2S=2, from which the $S=\ensuremath{\infty}$ phase diagram evolves quite trivially; the S=1S=1S=1 case then emerges as very special with its dominant quantum effects. Numerical studies have so far not been conclusive. Using the density matrix renormalization group, we study gaps and correlation functions in the anisotropic S=2S=2S=2 chain. The question of the existence of hidden topological order in the anisotropic S=2S=2S=2 chain is emphasized, as it distinguishes between the proposed scenarios. Careful extrapolation allows us to show that the topological order is zero in the thermodynamical limit in all disordered phases, in particular in the new phase interposed between the Haldane and large-DDD phases. The decay of the hidden order is substantially faster than that of ${S}^{+}{S}^{\ensuremath{-}}$ correlations and excellently fitted by power laws typical of massless phases. This excludes the AKLT-model-based scenario in favor of an almost classical phase diagram for the S=2S=2S=2 and thus higher spin chains.
The phase transition in the antiferromagnetic isotropic Heisenberg S=1S=1S=1 chain with frustrating next-nearest-neighbor coupling $\ensuremath{\alpha}$ is reconsidered. We identify the order parameter of the large-\ensuremath{\alpha} phase as describing two intertwined strings, each possessing a usual string order. The transition has a topological nature determined by the change in the string connectivity. Numerical evidence from the density matrix renormalization group results is supported by the effective theory based on soliton states.
We study the ground-state properties of weakly frustrated Heisenberg ferrimagnetic chains with nearest- and next-nearest-neighbor antiferromagnetic exchange interactions and two types of alternating sublattice spins S1>S2,{S}_{1}>{S}_{2},S1​>S2​, using 1/S1/S1/S spin-wave expansions, density-matrix renormalization group, and exact diagonalization techniques. It is argued that the zero-point spin fluctuations completely destroy the classical commensurate-incommensurate continuous transition. Instead, the long-range ferrimagnetic state disappears through a discontinuous transition to a singlet state at a larger value of the frustration parameter. In the ferrimagnetic phase we find a disorder point marking the onset of incommensurate real-space short-range spin-spin correlations.
The frustrated ferromagnetic spin-1∕21∕21∕2 Heisenberg chain is studied by means of a low-energy field theory as well as the density-matrix renormalization group and exact diagonalization methods. First, we study the ground-state phase diagram in a magnetic field and find an ``even-odd'' (EO) phase characterized by bound pairs of magnons in the region of two weakly coupled antiferromagnetic chains. A jump in the magnetization curves signals a first-order transition at the boundary of the EO phase, but otherwise the curves are smooth. Second, we discuss thermodynamic properties at zero field, where we confirm a double-peak structure in the specific heat for moderate frustrating next-nearest-neighbor interactions.
The ground-state `phase' diagrams and some low-energy properties of isotropic antiferromagnetic spin- and spin-1 chains with a next-nearest-neighbour exchange and an alternation of the nearest-neighbour exchanges have been compared for the first time using the density matrix renormalization group method. In the spin- chain, the system is gapless for and , and is gapped everywhere else in the plane. At , for small , the gap increases as , where is a disorder line. To the left of this line, the structure factor S(q) peaks at (the Néel `phase'), while to the right, decreases from to (the spiral `phase') as increases. We also discover a ` phase' for large values of both and . In the spin-1 case, we find a line running from a gapless point at up to a `gapless' point at such that the open-chain ground state is fourfold degenerate below the line and is unique above it. There is a disorder line in this case also and it has the same equation as in the spin- case, but the line ends at about . Similarly to the spin- case, to the left of this line, the peak in the structure factor is at (the Néel `phase'), while to the right of the line, it is at less than (the spiral `phase'). For , the system corresponds to a spin ladder and the system is gapped for all values of the interchain coupling for both spin- and spin-1 ladders.