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

Physical review. B, Condensed matter

Phase transitions in antiferromagnetic quantum spin chains with bond alternation

Shoji Yamamoto

Published Feb 1, 1997• 52 citations
Publisher

Abstract

We study the phase transitions in antiferromagnetic quantum spin chains with bond alternation, H=J${\ensuremath{\sum}}_{\mathrm{i}}$[1-(-1${)}^{\mathrm{i}}$\ensuremath{\delta}]${\mathbf{S}}_{\mathrm{i}}$\ensuremath{\cdot}${\mathbf{S}}_{\mathrm{i}+1}$, of spin S=1, 3/2, and 2. On the one hand, using a transfer matrix technique, we make a simple variational approach, which results in the better estimates of the transition points than by the O(3) nonlinear-\ensuremath{\sigma}-model quantum field theory. On the other hand, employing a quantum Monte Carlo method, we calculate the generalized string order parameter, ${\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}$(\ensuremath{\theta})=${\mathrm{lim}}_{\mathrm{L}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}$(\ensuremath{\theta};L) with ${\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}$(\ensuremath{\theta};L)=〈${\mathrm{S}}_{\mathrm{L}\mathrm{/}4}^{\mathrm{z}}$${\ensuremath{\prod}}_{\mathrm{j}\mathrm{=}\mathrm{L}\mathrm{/}4}^{3\mathrm{L}\mathrm{/}4\mathrm{\ensuremath{-}}1}$exp[i\ensuremath{\theta}${\mathrm{S}}_{\mathrm{j}}^{\mathrm{z}}$]${\mathrm{S}}_{3\mathrm{L}\mathrm{/}4}^{\mathrm{z}}$〉. It turns out that the transition points are successfully detected by observing the overall behavior of ${\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}$(\ensuremath{\theta}) at various values of \ensuremath{\delta}. Investigating the dependences of ${\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}$(\ensuremath{\theta};L) on \ensuremath{\theta}, L, and S, we discuss the applicability of the valence-bond-solid picture to the ground states of the present Hamiltonian.

DOI
10.1103/physrevb.55.3603

Related papers

S=3/2Antiferromagnetic Spin Chain with Bond Alternation

Makoto Yajima, Minoru Takahashi

The successive phase transitions of the Heisenberg spin chain with bond alternation, H = ∑ i =1 L -1 [1-(-1) i δ] S i · S i +1 , for the spin S =3/2 case under the open boundary condition are investigated by the density-matrix renormalization-group method proposed by White. Besides the Heisenberg point, we find that another gapless point exists at δ c = 0.42 ±0.02. This result qualitatively supports the field-theoretical prediction by Affleck and Haldane. However, our estimation of δ c deviates from the field-theoretical prediction, δ c = 2/3. We also find a fourfold degeneracy of the ground state in the region 0 <δ< δ c in the thermodynamic limit. Our results show that the degeneracy is described by the edge excitations, which is essentially similar to the one in the S =1 case. We calculate the string correlation function and find that its behavior is consistent with the valence-bond-solid (VBS) picture.

Journal of the Physical Society of Japan•Jan 15, 1996•23 citations

Elementary excitations of S =1 antiferromagnetic Heisenberg chains with bond alternation

Shoji Yamamoto

A world-line quantum Monte Carlo method is employed to study the elementary excitations of S=1 antiferromagnetic Heisenberg chains with bond alternation described by the Hamiltonian, scrH=J ${\mathit{tsum}}_{\mathit{i}}$ [1-(-1${)}^{\mathit{i}}$\ensuremath{\delta}]${\mathrm{S}}_{\mathit{i}}$\ensuremath{\cdot}${\mathrm{S}}_{\mathit{i}+1}$. Imaginary-time correlation functions of this system at low-enough temperatures exhibit almost single-exponential decay and thus the lowest excitation spectrum can be successfully extracted from them. As \ensuremath{\delta} increases from 0 to 1, the system encounters a continuous phase transition at \ensuremath{\delta}\ensuremath{\simeq}0.25, where the energy(E)-momentum(q) relation is fitted well to the form E(q)=(${\mathit{v}}^{2}$ ${\mathrm{sin}}^{2}$q+${\mathrm{\ensuremath{\Delta}}}^{2}$${)}^{1/2}$ with the effective light velocity v=2.46\ifmmode\pm\else\textpm\fi{}0.08 and the energy gap \ensuremath{\Delta} which vanishes in the long chain limit. The central charge c at the massless critical point is estimated to be 1.02\ifmmode\pm\else\textpm\fi{}0.09, which suggests the universality class of the Gaussian model.

Physical review. B, Condensed matter•Jun 1, 1995•26 citations

Valence bond mapping of antiferromagnetic spin chains

J. Dukelsky, Stuart Pittel

Boson mapping techniques are developed to describe valence bond correlations in quantum spin chains. Applying the method to the alternating bond Hamiltonian for a generic spin chain, we derive an analytic expression for the transition points which gives perfect agreement with existing density-matrix renormalization-group and quantum Monte Carlo calculations.

arXiv:cond-mat/9703168•Physical review. B, Condensed matter•Nov 1, 1997•2 citations

Hidden Order and Dimerization Transition inS=2Chains

Masanori Yamanaka, Masaki Oshikawa, Seiji Miyashita

We study ground state properties of the $S=2$ quantum antiferromagnetic chain with a bond alternation \[ H = \sum_{j} [ 1 + δ(-1)^j ] \mbox{\boldmath $S$}_{j} \cdot \mbox{\boldmath $S$}_{j+1} \] by a Quantum Monte Carlo calculation. We find that the hidden $Z_2 \times Z_2$ symmetry is broken for $0.3 < |δ| < 0.5$ while it is unbroken in the other regions. This confirms the successive dimerization transitions first predicted by Affleck and Haldane. Our result shows that these transitions can be understood in terms of the hidden $Z_2 \times Z_2$ symmetry breaking, as was discussed using the Valence-Bond-Solid states. Furthermore, we find that the behavior of the generalized string correlation is qualitatively very similar to that in the Valence-Bond-Solid states, including the location of zeroes as a function of the angle parameter.

arXiv:cond-mat/9604107•Journal of the Physical Society of Japan•Jun 15, 1996•18 citations

Ground-State Properties ofS=1 Antiferromagnetic Heisenberg Chains with Bond Alternation

Shoji Yamamoto

Ground-state properties of S =1 antiferromagnetic Heisenberg chains with bond alternation, H = J ∑ i {1-(-1) i δ} S i · S i +1 , are studied by a quantum Monte Carlo method. Spin correlation between chain ends on the open chain appears for δ<0.25, while it vanishes for δ≥0.25. At δ≃0.25, the spin correlation length has a peak and the S =1/2 edge states for the open chain disappear. The fourfold degeneracy of the ground state of the open chain holds for δ<0.25, while it is removed for δ≥0.25. All the results support the continuous phase transition at δ≃0.25 demonstrated in the previous works.

Journal of the Physical Society of Japan•Dec 15, 1994•54 citations

Successive valence-bond-state transitions in quantum mixed spin chains

Zhaoxin Xu, Jianhui Dai, He-Ping Ying +1 more

We study quantum mixed-spin chains consisting of two integer spins ${S}^{1}$ and two half integer spins ${S}^{2}$ arrayed as ${S}^{1}\ensuremath{-}{S}^{1}\ensuremath{-}{S}^{2}\ensuremath{-}{S}^{2}$ in a unit cell with antiferromagnetic nearest-neighbor couplings ${J}_{1}$ ${(J}_{2})$ between the spins of equal (different) magnitudes. By varying the ratio of two competing couplings, $\ensuremath{\alpha}{=J}_{2}{/J}_{1},$ the systems undergo successive quantum phase transitions between different valence-bond-solid (VBS) states accompanied by the vanishing of the energy gap. By a quantum Monte Carlo simulation with an improved loop cluster algorithm, we find one critical point ${\ensuremath{\alpha}}_{c}=0.762(1)$ in a $1\ensuremath{-}1\ensuremath{-}\frac{1}{2}\ensuremath{-}\frac{1}{2}$ chain and two critical points ${\ensuremath{\alpha}}_{c1}=0.479(1)$ and ${\ensuremath{\alpha}}_{c2}=1.318(1)$ in a $1\ensuremath{-}1\ensuremath{-}\frac{3}{2}\ensuremath{-}\frac{3}{2}$ chain, respectively. The calculated expectation values of the Lieb-Shultz-Mattis twist operator show the characteristic features of the single VBS transition in the $1\ensuremath{-}1\ensuremath{-}\frac{1}{2}\ensuremath{-}\frac{1}{2}$ chain and two successive VBS transitions in the $1\ensuremath{-}1\ensuremath{-}\frac{3}{2}\ensuremath{-}\frac{3}{2}$ chain at these critical points.

Physical review. B, Condensed matter•Jun 19, 2003•8 citations
We study the phase transitions in antiferromagnetic quantum spin chains with bond alternation, H=J\ensuremath∑i{\ensuremath{\sum}}_{\mathrm{i}}\ensuremath∑i​[1-(-1)i{)}^{\mathrm{i}})i\ensuremath{\delta}]Si{\mathbf{S}}_{\mathrm{i}}Si​\ensuremath{\cdot}Si+1{\mathbf{S}}_{\mathrm{i}+1}Si+1​, of spin S=1, 3/2, and 2. On the one hand, using a transfer matrix technique, we make a simple variational approach, which results in the better estimates of the transition points than by the O(3) nonlinear-\ensuremath{\sigma}-model quantum field theory. On the other hand, employing a quantum Monte Carlo method, we calculate the generalized string order parameter, Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta})={\mathrm{lim}}_{\mathrm{L}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$${\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}(\ensuremath{\theta};L) with Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta};L)=〈{\mathrm{S}}_{\mathrm{L}\mathrm{/}4}^{\mathrm{z}}$${\ensuremath{\prod}}_{\mathrm{j}\mathrm{=}\mathrm{L}\mathrm{/}4}^{3\mathrm{L}\mathrm{/}4\mathrm{\ensuremath{-}}1}exp[i\ensuremath{\theta}Sjz{\mathrm{S}}_{\mathrm{j}}^{\mathrm{z}}Sjz​]S3L/4z{\mathrm{S}}_{3\mathrm{L}\mathrm{/}4}^{\mathrm{z}}S3L/4z​〉. It turns out that the transition points are successfully detected by observing the overall behavior of Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta}) at various values of \ensuremath{\delta}. Investigating the dependences of Ostringz{\mathrm{O}}_{\mathrm{string}}^{\mathrm{z}}Ostringz​(\ensuremath{\theta};L) on \ensuremath{\theta}, L, and S, we discuss the applicability of the valence-bond-solid picture to the ground states of the present Hamiltonian.
The successive phase transitions of the Heisenberg spin chain with bond alternation, H = ∑ i =1 L -1 [1-(-1) i δ] S i · S i +1 , for the spin S =3/2 case under the open boundary condition are investigated by the density-matrix renormalization-group method proposed by White. Besides the Heisenberg point, we find that another gapless point exists at δ c = 0.42 ±0.02. This result qualitatively supports the field-theoretical prediction by Affleck and Haldane. However, our estimation of δ c deviates from the field-theoretical prediction, δ c = 2/3. We also find a fourfold degeneracy of the ground state in the region 0 <δ< δ c in the thermodynamic limit. Our results show that the degeneracy is described by the edge excitations, which is essentially similar to the one in the S =1 case. We calculate the string correlation function and find that its behavior is consistent with the valence-bond-solid (VBS) picture.
A world-line quantum Monte Carlo method is employed to study the elementary excitations of S=1 antiferromagnetic Heisenberg chains with bond alternation described by the Hamiltonian, scrH=J tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​ [1-(-1)i{)}^{\mathit{i}})i\ensuremath{\delta}]Si{\mathrm{S}}_{\mathit{i}}Si​\ensuremath{\cdot}Si+1{\mathrm{S}}_{\mathit{i}+1}Si+1​. Imaginary-time correlation functions of this system at low-enough temperatures exhibit almost single-exponential decay and thus the lowest excitation spectrum can be successfully extracted from them. As \ensuremath{\delta} increases from 0 to 1, the system encounters a continuous phase transition at \ensuremath{\delta}\ensuremath{\simeq}0.25, where the energy(E)-momentum(q) relation is fitted well to the form E(q)=(v2{\mathit{v}}^{2}v2 sin2{\mathrm{sin}}^{2}sin2q+${\mathrm{\ensuremath{\Delta}}}^{2}$)1/2{)}^{1/2})1/2 with the effective light velocity v=2.46\ifmmode\pm\else\textpm\fi{}0.08 and the energy gap \ensuremath{\Delta} which vanishes in the long chain limit. The central charge c at the massless critical point is estimated to be 1.02\ifmmode\pm\else\textpm\fi{}0.09, which suggests the universality class of the Gaussian model.
We study ground state properties of the S=2S=2S=2 quantum antiferromagnetic chain with a bond alternation \[ H = \sum_{j} [ 1 + δ(-1)^j ] \mbox{\boldmath $S$}_{j} \cdot \mbox{\boldmath $S$}_{j+1} \] by a Quantum Monte Carlo calculation. We find that the hidden Z2×Z2Z_2 \times Z_2Z2​×Z2​ symmetry is broken for 0.3<∣δ∣<0.50.3 < |δ| < 0.50.3<∣δ∣<0.5 while it is unbroken in the other regions. This confirms the successive dimerization transitions first predicted by Affleck and Haldane. Our result shows that these transitions can be understood in terms of the hidden Z2×Z2Z_2 \times Z_2Z2​×Z2​ symmetry breaking, as was discussed using the Valence-Bond-Solid states. Furthermore, we find that the behavior of the generalized string correlation is qualitatively very similar to that in the Valence-Bond-Solid states, including the location of zeroes as a function of the angle parameter.
Ground-state properties of S =1 antiferromagnetic Heisenberg chains with bond alternation, H = J ∑ i {1-(-1) i δ} S i · S i +1 , are studied by a quantum Monte Carlo method. Spin correlation between chain ends on the open chain appears for δ<0.25, while it vanishes for δ≥0.25. At δ≃0.25, the spin correlation length has a peak and the S =1/2 edge states for the open chain disappear. The fourfold degeneracy of the ground state of the open chain holds for δ<0.25, while it is removed for δ≥0.25. All the results support the continuous phase transition at δ≃0.25 demonstrated in the previous works.
We study quantum mixed-spin chains consisting of two integer spins S1{S}^{1}S1 and two half integer spins S2{S}^{2}S2 arrayed as ${S}^{1}\ensuremath{-}{S}^{1}\ensuremath{-}{S}^{2}\ensuremath{-}{S}^{2}$ in a unit cell with antiferromagnetic nearest-neighbor couplings J1{J}_{1}J1​ (J2){(J}_{2})(J2​) between the spins of equal (different) magnitudes. By varying the ratio of two competing couplings, $\ensuremath{\alpha}{=J}_{2}{/J}_{1},$ the systems undergo successive quantum phase transitions between different valence-bond-solid (VBS) states accompanied by the vanishing of the energy gap. By a quantum Monte Carlo simulation with an improved loop cluster algorithm, we find one critical point ${\ensuremath{\alpha}}_{c}=0.762(1)$ in a $1\ensuremath{-}1\ensuremath{-}\frac{1}{2}\ensuremath{-}\frac{1}{2}$ chain and two critical points ${\ensuremath{\alpha}}_{c1}=0.479(1)$ and ${\ensuremath{\alpha}}_{c2}=1.318(1)$ in a $1\ensuremath{-}1\ensuremath{-}\frac{3}{2}\ensuremath{-}\frac{3}{2}$ chain, respectively. The calculated expectation values of the Lieb-Shultz-Mattis twist operator show the characteristic features of the single VBS transition in the $1\ensuremath{-}1\ensuremath{-}\frac{1}{2}\ensuremath{-}\frac{1}{2}$ chain and two successive VBS transitions in the $1\ensuremath{-}1\ensuremath{-}\frac{3}{2}\ensuremath{-}\frac{3}{2}$ chain at these critical points.