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/0003271 · Physical review. B, Condensed matter

Thermodynamics of spin S = 1 / 2 antiferromagnetic uniform and alternating-exchange Heisenberg chains

D. C. Johnston, Reinhard Karl Kremer, Matthias Troyer, X. Wang, Andreas Klümper, Sergey L. Bud’ko, Alex F. Panchula, Paul C. Canfield

Published Apr 1, 2000• 534 citations
arXivHTMLPDF

Abstract

The magnetic susceptibility ${\ensuremath{\chi}}^{*}(t)$ and specific heat $C(t)$ versus temperature t of the spin $S=1/2$ antiferromagnetic (AF) alternating-exchange ${(J}_{1}$ and ${J}_{2})$ Heisenberg chain are studied for the entire range $0<~\ensuremath{\alpha}<~1$ of the alternation parameter $\ensuremath{\alpha}\ensuremath{\equiv}{J}_{2}{/J}_{1}{(J}_{1},$ ${J}_{2}>~0,$ ${J}_{2}<~{J}_{1},$ ${t=k}_{\mathrm{B}}{T/J}_{1},$ ${\ensuremath{\chi}}^{*}=\ensuremath{\chi}{J}_{1}{/Ng}^{2}{\ensuremath{\mu}}_{B}^{2}).$ For the uniform chain $(\ensuremath{\alpha}=1),$ the high-accuracy ${\ensuremath{\chi}}^{*}(t)$ and $C(t)$ Bethe ansatz data of Kl\"umper and Johnston (unpublished) are shown to agree very well at low t with the respective exact theoretical low-$t$ logarithmic correction predictions of Lukyanov [Nucl. Phys. B $522,$ 533 (1998)]. Accurate $(\ensuremath{\sim}{10}^{\ensuremath{-}7})$ independent empirical fits to the respective data are obtained over t ranges spanning 25 orders of magnitude, $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}25}<~t<~5,$ which contain extrapolations to the respective exact $t=0$ limits. The infinite temperature entropy calculated using our $C(t)$ fit function is within 8 parts in ${10}^{8}$ of the exact value $\mathrm{ln}2.$ Quantum Monte Carlo (QMC) simulations and transfer-matrix density-matrix renormalization group (TMRG) calculations of ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ are presented for $0.002<~t<~10$ and $0.05<~\ensuremath{\alpha}<~1,$ and an accurate $(2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4})$ two-dimensional $(\ensuremath{\alpha},t)$ fit to the combined data is obtained for $0.01<~t<~10$ and $0<~\ensuremath{\alpha}<~1.$ From the low-$t$ TMRG data, the spin gap $\ensuremath{\Delta}(\ensuremath{\alpha})$ is extracted for $0.8<~\ensuremath{\alpha}<~0.995$ and compared with previous results, and a fit function is formulated for $0<~\ensuremath{\alpha}<~1$ by combining these data with literature data. We infer from our data that the asymptotic critical regime near the uniform chain limit is only entered for $\ensuremath{\alpha}\ensuremath{\gtrsim}0.99.$ We examine in detail the theoretical predictions of Bulaevskii [Sov. Phys. Solid State $11,$ 921 (1969)], for ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ and compare them with our results. To illustrate the application and utility of our theoretical results, we model our experimental $\ensuremath{\chi}(T)$ and specific heat ${C}_{\mathrm{p}}(T)$ data for ${\mathrm{NaV}}_{2}{\mathrm{O}}_{5}$ single crystals in detail. The $\ensuremath{\chi}(T)$ data above the spin dimerization temperature ${T}_{\mathrm{c}}\ensuremath{\approx}34\mathrm{K}$ are not in quantitative agreement with the prediction for the $S=1/2$ uniform Heisenberg chain, but can be explained if there is a moderate ferromagnetic interchain coupling and/or if J changes with T. Fitting the $\ensuremath{\chi}(T)$ data using our ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ fit function, we obtain the sample-dependent spin gap and range $\ensuremath{\Delta}{(T=0)/k}_{\mathrm{B}}=103(2)\mathrm{K},$ alternation parameter $\ensuremath{\delta}(0)\ensuremath{\equiv}(1\ensuremath{-}\ensuremath{\alpha})/(1+\ensuremath{\alpha})=0.034(6)$ and average exchange constant ${J(0)/k}_{\mathrm{B}}=640(80)\mathrm{K}.$ The $\ensuremath{\delta}(T)$ and $\ensuremath{\Delta}(T)$ are derived from the data. A spin pseudogap with magnitude $\ensuremath{\approx}0.4\ensuremath{\Delta}(0)$ is consistently found just above ${T}_{\mathrm{c}},$ which decreases with increasing temperature. From our ${C}_{\mathrm{p}}(T)$ measurements on two crystals, we infer that the magnetic specific heat at low temperatures $T\ensuremath{\lesssim}15\mathrm{K}$ is too small to be resolved experimentally, and that the spin entropy at ${T}_{\mathrm{c}}$ is too small to account for the entropy of the transition. A quantitative analysis indicates that at ${T}_{\mathrm{c}},$ at least 77% of the entropy change due to the transition at ${T}_{\mathrm{c}}$ and associated order parameter fluctuations arise from the lattice and/or charge degrees of freedom and less than 23% from the spin degrees of freedom.

DOI
10.1103/physrevb.61.9558

Related papers

Thermodynamics of the Spin- 1/2 Antiferromagnetic Uniform Heisenberg Chain

Andreas Klümper, D. C. Johnston

We present a new application of the traditional thermodynamic Bethe ansatz to the spin-1/2 antiferromagnetic uniform Heisenberg chain and derive exact nonlinear integral equations for just two functions describing the elementary excitations. By using this approach, the magnetic susceptibility chi and specific heat C versus temperature T are calculated to high accuracy for 5x10(-25)</=T/J</=5. The chi(T) data agree very well at low T with the asymptotically exact theoretical low- T prediction of Lukyanov [Nucl. Phys. B522, 533 (1998)]. The unknown coefficients of the second and third lowest-order logarithmic correction terms in Lukyanov's theory for C(T) are estimated from the C(T) data.

arXiv:cond-mat/0002140•Physical Review Letters•May 15, 2000•125 citations

Low-temperature thermodynamics of the antiferromagnetic J1−J2 model: Entropy, critical points, and spin gap

Sudip Kumar Saha, Manodip Routh, Manoranjan Kumar +1 more

The antiferromagnetic ${J}_{1}\ensuremath{-}{J}_{2}$ model is a spin-1/2 chain with isotropic exchange ${J}_{1}>0$ between first neighbors and ${J}_{2}=\ensuremath{\alpha}{J}_{1}$ between second neighbors. The model supports both gapless quantum phases with nondegenerate ground states and gapped phases with $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})>0$ and doubly degenerate ground states. Exact thermodynamics is limited to $\ensuremath{\alpha}=0$, the linear Heisenberg antiferromagnet (HAF). Exact diagonalization of small systems at frustration $\ensuremath{\alpha}$ followed by density matrix renormalization group calculations returns the entropy density $S(T,\ensuremath{\alpha},N)$ and magnetic susceptibility $\ensuremath{\chi}(T,\ensuremath{\alpha},N)$ of progressively larger systems up to $N=96$ or 152 spins. Convergence to the thermodynamic limit, $S(T,\ensuremath{\alpha})$ or $\ensuremath{\chi}(T,\ensuremath{\alpha})$, is demonstrated down to $T/J\ensuremath{\sim}0.01$ in the sectors $\ensuremath{\alpha}<1$ and $\ensuremath{\alpha}>1$. $S(T,\ensuremath{\alpha})$ yields the critical points between gapless phases with ${S}^{\ensuremath{'}}(0,\ensuremath{\alpha})>0$ and gapped phases with ${S}^{\ensuremath{'}}(0,\ensuremath{\alpha})=0$. The ${S}^{\ensuremath{'}}(T,\ensuremath{\alpha})$ maximum at ${T}^{*}(\ensuremath{\alpha})$ is obtained directly in chains with large $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$ and by extrapolation for small gaps. A phenomenological approximation for $S(T,\ensuremath{\alpha})$ down to $T=0$ indicates power-law deviations ${T}^{\ensuremath{-}\ensuremath{\gamma}(\ensuremath{\alpha})}$ from $exp[\ensuremath{-}\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})/T]$ with exponent $\ensuremath{\gamma}(\ensuremath{\alpha})$ that increases with $\ensuremath{\alpha}$. The $\ensuremath{\chi}(T,\ensuremath{\alpha})$ analysis also yields power-law deviations, but with exponent $\ensuremath{\eta}(\ensuremath{\alpha})$ that decreases with $\ensuremath{\alpha}$. Spin correlation functions account for $S(T,\ensuremath{\alpha})$ differences between frustration $\ensuremath{\alpha}<1$ within a chain and $\ensuremath{\alpha}>1$ between HAFs on sublattices. $S(T,\ensuremath{\alpha})$ and the spin density $\ensuremath{\rho}(T,\ensuremath{\alpha})=4T\ensuremath{\chi}(T,\ensuremath{\alpha})$ probe the thermal and magnetic fluctuations, respectively, of strongly correlated spin states. Gapless chains have constant $S(T,\ensuremath{\alpha})/\ensuremath{\rho}(T,\ensuremath{\alpha})$ for $T<0.10$. Remarkably, the ratio decreases (increases) with $T$ in chains with large (small) $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$.

arXiv:2106.14775•Physical review. B./Physical review. B•Jun 24, 2021•3 citations

Thermodynamic properties of tetrameric bond-alternating spin chains

Hantao Lu, Y. H. Su, L. Q. Sun +4 more

Thermodynamic properties of a tetrameric bond-alternating Heisenberg spin chain with ferromagnetic-ferromagnetic-antiferromagnetic-antiferromagnetic exchange interactions are studied using the transfer-matrix renormalization group and compared to experimental measurements. The temperature dependence of the uniform susceptibility exhibits typical ferrimagnetic features. Both the uniform and staggered magnetic susceptibilities diverge in the limit $T\ensuremath{\rightarrow}0$, indicating that the ground state has both ferromagnetic and antiferromagnetic long-range orders. A double-peak structure appears in the temperature dependence of the specific heat. Our numerical calculation gives a good account for the temperature and field dependence of the susceptibility, the magnetization, and the specific heat for $\mathrm{Cu}{(3\text{\ensuremath{-}}\mathrm{Clpy})}_{2}{({\mathrm{N}}_{3})}_{2}$ $(3\text{\ensuremath{-}}\mathrm{Clpy}=3\text{\ensuremath{-}}\text{Chloroyridine})$.

arXiv:cond-mat/0412275•Physical Review B•Apr 29, 2005•14 citations

Theory of Alternating Antiferromagnetic Heisenberg Linear Chains

William Duffy, Kevin P. Barr

The exact eigenvalue spectrum and thermodynamic properties of the spin Hamiltonian $\mathcal{H}=2|J|\ensuremath{\Sigma}\stackrel{\frac{N}{2}}{i=1}({\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i\ensuremath{-}1}+a{\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i+1})$ are calculated for short chains of $N=4,6,8, \mathrm{and} 10$ spins, each of spin \textonehalf{}, for alternation parameter $a=0.2,0.4,0.6, \mathrm{and} 0.8$, with ${\mathrm{S}}_{N+1}\ensuremath{\equiv}{\mathrm{S}}_{1}$. The behavior for $N=\ensuremath{\infty}$ is estimated by extrapolation. Comparison is made with the known results for $a=0$ and 1. The ground-state energy, ground-state short-range order, energy gap, and triplet excitation spectrum are compared with various approximate theories. Zero-temperature infinite-chain magnetization curves are inferred from the finite-chain results. The energy, entropy, specific heat, and magnetic susceptibility for $N=10$ are shown to approximate well the behavior for $N=\ensuremath{\infty}$ when $\frac{\mathrm{kT}}{|J|}>a$. The magnetic-susceptibility data on the free radical 2,2-bis ($p$-nitrophenyl)-1-picrylhydrazyl are shown to agree well with the theoretical results for an alternating chain with $a\ensuremath{\approx}0.6$.

Physical Review•Jan 10, 1968•227 citations

Magnetism and thermodynamics of spin- 1∕2 Heisenberg diamond chains in a magnetic field

Bo Gu, Gang Su

The magnetic and thermodynamic properties of spin-$1∕2$ Heisenberg diamond chains are investigated in three different cases: (a) ${J}_{1}$, ${J}_{2}$, ${J}_{3}>0$ (frustrated), (b) ${J}_{1}$, ${J}_{3}<0$, ${J}_{2}>0$ (frustrated), and (c) ${J}_{1}$, ${J}_{2}>0$, ${J}_{3}<0$ (nonfrustrated), where the bond coupling ${J}_{i}$ $(i=1,2,3)>0$ stands for an antiferromagnetic (AF) interaction, and $<0$ for a ferromagnetic (F) interaction. The density-matrix renormalization-group (DMRG) technique is invoked to study the properties of the system in the ground state, while the transfer-matrix renormalization-group (TMRG) technique is applied to explore the thermodynamic properties. The local magnetic moments, spin-correlation functions, and static structure factors are discussed in the ground state for the three cases. It is shown that the static structure factor $S(q)$ shows peaks at wave vectors $q=a\ensuremath{\pi}∕3$ $(a=0,1,2,3,4,5)$ for different couplings in a zero magnetic field, which, however, in the magnetic fields where the magnetization plateau with $m=1∕6$ pertains, exhibits the peaks only at $q=0$, $2\ensuremath{\pi}∕3$, and $4\ensuremath{\pi}∕3$, which are found to be couplings independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is observed that the six modes are closely related to the low-lying excitations of the system. At finite temperatures, the magnetization, susceptibility, and specific heat show various behaviors for different couplings. The double-peak structures of the susceptibility, and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the competition of the couplings. It is also uncovered that the $XXZ$ anisotropy of F and AF couplings leads the system of case (c) to display quite different behaviors. In addition, the experimental data of the susceptibility, specific heat, and magnetization for the compound ${\mathrm{Cu}}_{3}{(\mathrm{C}{\mathrm{O}}_{3})}_{2}{(\mathrm{O}\mathrm{H})}_{2}$ are fairly compared with our TMRG results.

arXiv:0704.0150•Physical Review B•May 24, 2007•74 citations

Thermodynamics of spin- 1 ∕ 2 antiferromagnet-antiferromagnet-ferromagnet and ferromagnet-ferromagnet-antiferromagnet trimerized quantum Heisenberg chains

Bo Gu, Gang Su, Song Wei Gao

The magnetization process, the susceptibility, and the specific heat of the spin-$1∕2$ antiferromagnet (AF)-AF-ferromagnet (F) and F-F-AF trimerized quantum Heisenberg chains have been investigated by means of the transfer matrix renormalization group (TMRG) technique as well as the modified spin-wave (MSW) theory. A magnetization plateau at $m=1∕6$ for both trimerized chains is observed at low temperature. The susceptibility and the specific heat show various behaviors for different ferromagnetic and antiferromagnetic interactions and in different magnetic fields. The TMRG results of susceptibility and the specific heat can be nicely fitted by a linear superposition of double two-level systems, where two fitting equations are proposed. Three branch excitations, one gapless excitation and two gapful excitations, for both systems are found within the MSW theory. It is observed that the MSW theory captures the main characteristics of the thermodynamic behaviors at low temperatures. The TMRG results are also compared with the possible experimental data.

arXiv:cond-mat/0512652•Physical Review B•Apr 27, 2006•41 citations
The magnetic susceptibility \ensuremathχ∗(t){\ensuremath{\chi}}^{*}(t)\ensuremathχ∗(t) and specific heat C(t)C(t)C(t) versus temperature t of the spin S=1/2S=1/2S=1/2 antiferromagnetic (AF) alternating-exchange (J1{(J}_{1}(J1​ and J2){J}_{2})J2​) Heisenberg chain are studied for the entire range 0< \ensuremathα< 10<~\ensuremath{\alpha}<~10< \ensuremathα< 1 of the alternation parameter \ensuremathα\ensuremath≡J2/J1(J1,\ensuremath{\alpha}\ensuremath{\equiv}{J}_{2}{/J}_{1}{(J}_{1},\ensuremathα\ensuremath≡J2​/J1​(J1​, J2> 0,{J}_{2}>~0,J2​> 0, J2< J1,{J}_{2}<~{J}_{1},J2​< J1​, t=kBT/J1,{t=k}_{\mathrm{B}}{T/J}_{1},t=kB​T/J1​, \ensuremathχ∗=\ensuremathχJ1/Ng2\ensuremathμB2).{\ensuremath{\chi}}^{*}=\ensuremath{\chi}{J}_{1}{/Ng}^{2}{\ensuremath{\mu}}_{B}^{2}).\ensuremathχ∗=\ensuremathχJ1​/Ng2\ensuremathμB2​). For the uniform chain (\ensuremathα=1),(\ensuremath{\alpha}=1),(\ensuremathα=1), the high-accuracy \ensuremathχ∗(t){\ensuremath{\chi}}^{*}(t)\ensuremathχ∗(t) and C(t)C(t)C(t) Bethe ansatz data of Kl"umper and Johnston (unpublished) are shown to agree very well at low t with the respective exact theoretical low-ttt logarithmic correction predictions of Lukyanov [Nucl. Phys. B 522,522,522, 533 (1998)]. Accurate (\ensuremath∼10\ensuremath−7)(\ensuremath{\sim}{10}^{\ensuremath{-}7})(\ensuremath∼10\ensuremath−7) independent empirical fits to the respective data are obtained over t ranges spanning 25 orders of magnitude, 5\ifmmode×\else\texttimes\fi10\ensuremath−25< t< 5,5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}25}<~t<~5,5\ifmmode×\else\texttimes\fi10\ensuremath−25< t< 5, which contain extrapolations to the respective exact t=0t=0t=0 limits. The infinite temperature entropy calculated using our C(t)C(t)C(t) fit function is within 8 parts in 108{10}^{8}108 of the exact value ln2.\mathrm{ln}2.ln2. Quantum Monte Carlo (QMC) simulations and transfer-matrix density-matrix renormalization group (TMRG) calculations of \ensuremathχ∗(\ensuremathα,t){\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)\ensuremathχ∗(\ensuremathα,t) are presented for 0.002< t< 100.002<~t<~100.002< t< 10 and 0.05< \ensuremathα< 1,0.05<~\ensuremath{\alpha}<~1,0.05< \ensuremathα< 1, and an accurate (2\ifmmode×\else\texttimes\fi10\ensuremath−4)(2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4})(2\ifmmode×\else\texttimes\fi10\ensuremath−4) two-dimensional (\ensuremathα,t)(\ensuremath{\alpha},t)(\ensuremathα,t) fit to the combined data is obtained for 0.01< t< 100.01<~t<~100.01< t< 10 and 0< \ensuremathα< 1.0<~\ensuremath{\alpha}<~1.0< \ensuremathα< 1. From the low-ttt TMRG data, the spin gap \ensuremathΔ(\ensuremathα)\ensuremath{\Delta}(\ensuremath{\alpha})\ensuremathΔ(\ensuremathα) is extracted for 0.8< \ensuremathα< 0.9950.8<~\ensuremath{\alpha}<~0.9950.8< \ensuremathα< 0.995 and compared with previous results, and a fit function is formulated for 0< \ensuremathα< 10<~\ensuremath{\alpha}<~10< \ensuremathα< 1 by combining these data with literature data. We infer from our data that the asymptotic critical regime near the uniform chain limit is only entered for \ensuremathα\ensuremath≳0.99.\ensuremath{\alpha}\ensuremath{\gtrsim}0.99.\ensuremathα\ensuremath≳0.99. We examine in detail the theoretical predictions of Bulaevskii [Sov. Phys. Solid State 11,11,11, 921 (1969)], for \ensuremathχ∗(\ensuremathα,t){\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)\ensuremathχ∗(\ensuremathα,t) and compare them with our results. To illustrate the application and utility of our theoretical results, we model our experimental \ensuremathχ(T)\ensuremath{\chi}(T)\ensuremathχ(T) and specific heat Cp(T){C}_{\mathrm{p}}(T)Cp​(T) data for NaV2O5{\mathrm{NaV}}_{2}{\mathrm{O}}_{5}NaV2​O5​ single crystals in detail. The \ensuremathχ(T)\ensuremath{\chi}(T)\ensuremathχ(T) data above the spin dimerization temperature Tc\ensuremath≈34K{T}_{\mathrm{c}}\ensuremath{\approx}34\mathrm{K}Tc​\ensuremath≈34K are not in quantitative agreement with the prediction for the S=1/2S=1/2S=1/2 uniform Heisenberg chain, but can be explained if there is a moderate ferromagnetic interchain coupling and/or if J changes with T. Fitting the \ensuremathχ(T)\ensuremath{\chi}(T)\ensuremathχ(T) data using our \ensuremathχ∗(\ensuremathα,t){\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)\ensuremathχ∗(\ensuremathα,t) fit function, we obtain the sample-dependent spin gap and range \ensuremathΔ(T=0)/kB=103(2)K,\ensuremath{\Delta}{(T=0)/k}_{\mathrm{B}}=103(2)\mathrm{K},\ensuremathΔ(T=0)/kB​=103(2)K, alternation parameter \ensuremathδ(0)\ensuremath≡(1\ensuremath−\ensuremathα)/(1+\ensuremathα)=0.034(6)\ensuremath{\delta}(0)\ensuremath{\equiv}(1\ensuremath{-}\ensuremath{\alpha})/(1+\ensuremath{\alpha})=0.034(6)\ensuremathδ(0)\ensuremath≡(1\ensuremath−\ensuremathα)/(1+\ensuremathα)=0.034(6) and average exchange constant J(0)/kB=640(80)K.{J(0)/k}_{\mathrm{B}}=640(80)\mathrm{K}.J(0)/kB​=640(80)K. The \ensuremathδ(T)\ensuremath{\delta}(T)\ensuremathδ(T) and \ensuremathΔ(T)\ensuremath{\Delta}(T)\ensuremathΔ(T) are derived from the data. A spin pseudogap with magnitude \ensuremath≈0.4\ensuremathΔ(0)\ensuremath{\approx}0.4\ensuremath{\Delta}(0)\ensuremath≈0.4\ensuremathΔ(0) is consistently found just above Tc,{T}_{\mathrm{c}},Tc​, which decreases with increasing temperature. From our Cp(T){C}_{\mathrm{p}}(T)Cp​(T) measurements on two crystals, we infer that the magnetic specific heat at low temperatures T\ensuremath≲15KT\ensuremath{\lesssim}15\mathrm{K}T\ensuremath≲15K is too small to be resolved experimentally, and that the spin entropy at Tc{T}_{\mathrm{c}}Tc​ is too small to account for the entropy of the transition. A quantitative analysis indicates that at Tc,{T}_{\mathrm{c}},Tc​, at least 77% of the entropy change due to the transition at Tc{T}_{\mathrm{c}}Tc​ and associated order parameter fluctuations arise from the lattice and/or charge degrees of freedom and less than 23% from the spin degrees of freedom.
We present a new application of the traditional thermodynamic Bethe ansatz to the spin-1/2 antiferromagnetic uniform Heisenberg chain and derive exact nonlinear integral equations for just two functions describing the elementary excitations. By using this approach, the magnetic susceptibility chi and specific heat C versus temperature T are calculated to high accuracy for 5x10(-25)</=T/J</=5. The chi(T) data agree very well at low T with the asymptotically exact theoretical low- T prediction of Lukyanov [Nucl. Phys. B522, 533 (1998)]. The unknown coefficients of the second and third lowest-order logarithmic correction terms in Lukyanov's theory for C(T) are estimated from the C(T) data.
The antiferromagnetic ${J}_{1}\ensuremath{-}{J}_{2}$ model is a spin-1/2 chain with isotropic exchange J1>0{J}_{1}>0J1​>0 between first neighbors and ${J}_{2}=\ensuremath{\alpha}{J}_{1}$ between second neighbors. The model supports both gapless quantum phases with nondegenerate ground states and gapped phases with $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})>0$ and doubly degenerate ground states. Exact thermodynamics is limited to $\ensuremath{\alpha}=0$, the linear Heisenberg antiferromagnet (HAF). Exact diagonalization of small systems at frustration $\ensuremath{\alpha}$ followed by density matrix renormalization group calculations returns the entropy density $S(T,\ensuremath{\alpha},N)$ and magnetic susceptibility $\ensuremath{\chi}(T,\ensuremath{\alpha},N)$ of progressively larger systems up to N=96N=96N=96 or 152 spins. Convergence to the thermodynamic limit, $S(T,\ensuremath{\alpha})$ or $\ensuremath{\chi}(T,\ensuremath{\alpha})$, is demonstrated down to $T/J\ensuremath{\sim}0.01$ in the sectors $\ensuremath{\alpha}<1$ and $\ensuremath{\alpha}>1$. $S(T,\ensuremath{\alpha})$ yields the critical points between gapless phases with ${S}^{\ensuremath{'}}(0,\ensuremath{\alpha})>0$ and gapped phases with ${S}^{\ensuremath{'}}(0,\ensuremath{\alpha})=0$. The ${S}^{\ensuremath{'}}(T,\ensuremath{\alpha})$ maximum at ${T}^{*}(\ensuremath{\alpha})$ is obtained directly in chains with large $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$ and by extrapolation for small gaps. A phenomenological approximation for $S(T,\ensuremath{\alpha})$ down to T=0T=0T=0 indicates power-law deviations ${T}^{\ensuremath{-}\ensuremath{\gamma}(\ensuremath{\alpha})}$ from $exp[\ensuremath{-}\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})/T]$ with exponent $\ensuremath{\gamma}(\ensuremath{\alpha})$ that increases with $\ensuremath{\alpha}$. The $\ensuremath{\chi}(T,\ensuremath{\alpha})$ analysis also yields power-law deviations, but with exponent $\ensuremath{\eta}(\ensuremath{\alpha})$ that decreases with $\ensuremath{\alpha}$. Spin correlation functions account for $S(T,\ensuremath{\alpha})$ differences between frustration $\ensuremath{\alpha}<1$ within a chain and $\ensuremath{\alpha}>1$ between HAFs on sublattices. $S(T,\ensuremath{\alpha})$ and the spin density $\ensuremath{\rho}(T,\ensuremath{\alpha})=4T\ensuremath{\chi}(T,\ensuremath{\alpha})$ probe the thermal and magnetic fluctuations, respectively, of strongly correlated spin states. Gapless chains have constant $S(T,\ensuremath{\alpha})/\ensuremath{\rho}(T,\ensuremath{\alpha})$ for T<0.10T<0.10T<0.10. Remarkably, the ratio decreases (increases) with TTT in chains with large (small) $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$.
Thermodynamic properties of a tetrameric bond-alternating Heisenberg spin chain with ferromagnetic-ferromagnetic-antiferromagnetic-antiferromagnetic exchange interactions are studied using the transfer-matrix renormalization group and compared to experimental measurements. The temperature dependence of the uniform susceptibility exhibits typical ferrimagnetic features. Both the uniform and staggered magnetic susceptibilities diverge in the limit $T\ensuremath{\rightarrow}0$, indicating that the ground state has both ferromagnetic and antiferromagnetic long-range orders. A double-peak structure appears in the temperature dependence of the specific heat. Our numerical calculation gives a good account for the temperature and field dependence of the susceptibility, the magnetization, and the specific heat for $\mathrm{Cu}{(3\text{\ensuremath{-}}\mathrm{Clpy})}_{2}{({\mathrm{N}}_{3})}_{2}$ $(3\text{\ensuremath{-}}\mathrm{Clpy}=3\text{\ensuremath{-}}\text{Chloroyridine})$.
The exact eigenvalue spectrum and thermodynamic properties of the spin Hamiltonian $\mathcal{H}=2|J|\ensuremath{\Sigma}\stackrel{\frac{N}{2}}{i=1}({\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i\ensuremath{-}1}+a{\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i+1})$ are calculated for short chains of N=4,6,8,and10N=4,6,8, \mathrm{and} 10N=4,6,8,and10 spins, each of spin \textonehalf{}, for alternation parameter a=0.2,0.4,0.6,and0.8a=0.2,0.4,0.6, \mathrm{and} 0.8a=0.2,0.4,0.6,and0.8, with ${\mathrm{S}}_{N+1}\ensuremath{\equiv}{\mathrm{S}}_{1}$. The behavior for $N=\ensuremath{\infty}$ is estimated by extrapolation. Comparison is made with the known results for a=0a=0a=0 and 1. The ground-state energy, ground-state short-range order, energy gap, and triplet excitation spectrum are compared with various approximate theories. Zero-temperature infinite-chain magnetization curves are inferred from the finite-chain results. The energy, entropy, specific heat, and magnetic susceptibility for N=10N=10N=10 are shown to approximate well the behavior for $N=\ensuremath{\infty}$ when kT∣J∣>a\frac{\mathrm{kT}}{|J|}>a∣J∣kT​>a. The magnetic-susceptibility data on the free radical 2,2-bis (ppp-nitrophenyl)-1-picrylhydrazyl are shown to agree well with the theoretical results for an alternating chain with $a\ensuremath{\approx}0.6$.
The magnetic and thermodynamic properties of spin-1∕21∕21∕2 Heisenberg diamond chains are investigated in three different cases: (a) J1{J}_{1}J1​, J2{J}_{2}J2​, J3>0{J}_{3}>0J3​>0 (frustrated), (b) J1{J}_{1}J1​, J3<0{J}_{3}<0J3​<0, J2>0{J}_{2}>0J2​>0 (frustrated), and (c) J1{J}_{1}J1​, J2>0{J}_{2}>0J2​>0, J3<0{J}_{3}<0J3​<0 (nonfrustrated), where the bond coupling Ji{J}_{i}Ji​ (i=1,2,3)>0(i=1,2,3)>0(i=1,2,3)>0 stands for an antiferromagnetic (AF) interaction, and <0<0<0 for a ferromagnetic (F) interaction. The density-matrix renormalization-group (DMRG) technique is invoked to study the properties of the system in the ground state, while the transfer-matrix renormalization-group (TMRG) technique is applied to explore the thermodynamic properties. The local magnetic moments, spin-correlation functions, and static structure factors are discussed in the ground state for the three cases. It is shown that the static structure factor S(q)S(q)S(q) shows peaks at wave vectors $q=a\ensuremath{\pi}∕3$ (a=0,1,2,3,4,5)(a=0,1,2,3,4,5)(a=0,1,2,3,4,5) for different couplings in a zero magnetic field, which, however, in the magnetic fields where the magnetization plateau with m=1∕6m=1∕6m=1∕6 pertains, exhibits the peaks only at q=0q=0q=0, $2\ensuremath{\pi}∕3$, and $4\ensuremath{\pi}∕3$, which are found to be couplings independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is observed that the six modes are closely related to the low-lying excitations of the system. At finite temperatures, the magnetization, susceptibility, and specific heat show various behaviors for different couplings. The double-peak structures of the susceptibility, and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the competition of the couplings. It is also uncovered that the XXZXXZXXZ anisotropy of F and AF couplings leads the system of case (c) to display quite different behaviors. In addition, the experimental data of the susceptibility, specific heat, and magnetization for the compound Cu3(CO3)2(OH)2{\mathrm{Cu}}_{3}{(\mathrm{C}{\mathrm{O}}_{3})}_{2}{(\mathrm{O}\mathrm{H})}_{2}Cu3​(CO3​)2​(OH)2​ are fairly compared with our TMRG results.
The magnetization process, the susceptibility, and the specific heat of the spin-1∕21∕21∕2 antiferromagnet (AF)-AF-ferromagnet (F) and F-F-AF trimerized quantum Heisenberg chains have been investigated by means of the transfer matrix renormalization group (TMRG) technique as well as the modified spin-wave (MSW) theory. A magnetization plateau at m=1∕6m=1∕6m=1∕6 for both trimerized chains is observed at low temperature. The susceptibility and the specific heat show various behaviors for different ferromagnetic and antiferromagnetic interactions and in different magnetic fields. The TMRG results of susceptibility and the specific heat can be nicely fitted by a linear superposition of double two-level systems, where two fitting equations are proposed. Three branch excitations, one gapless excitation and two gapful excitations, for both systems are found within the MSW theory. It is observed that the MSW theory captures the main characteristics of the thermodynamic behaviors at low temperatures. The TMRG results are also compared with the possible experimental data.