Physical review. B, Condensed matter

Critical behavior of the SU(3) ferromagnetic Heisenberg chain

Young-Ju Jee, Kong-Ju-Bock Lee, P. Schlottmann

Published Feb 1, 1989• 16 citations

Abstract

We have extracted the leading low-temperature contributions to the specific heat and the magnetic susceptibility from the thermodynamic Bethe-Ansatz equations of the SU(3) invariant ferromagnetic Heisenberg chain of spin 1. The critical exponents for the specific heat, the susceptibility, and the correlation length are $\ensuremath{\alpha}=\ensuremath{-}\frac{1}{2}$, $\ensuremath{\gamma}=2$, and $\ensuremath{\nu}=1$, respectively. The susceptibility and the correlation length exhibit logarithmic corrections, which are quenched by relatively small magnetic fields. In large fields, the energy required to flip a spin gives rise to an exponential activation.

DOI: 10.1103/physrevb.39.2815

Related papers

Critical behavior of the isotropic ferromagnetic Heisenberg chain with arbitrary spinS

Kong-Ju-Bock Lee, P. Schlottmann

Physical review. B, Condensed matterJul 1, 198720 citations

Critical Behavior of the Isotropic Ferromagnetic Quantum Heisenberg Chain

P. Schlottmann

Physical Review LettersMay 13, 198570 citations

Critical Behavior of Spin-1/2 One-Dimensional Heisenberg Ferromagnet at Low Temperatures

Miki Yamada, Minoru Takahashi

Journal of the Physical Society of JapanJun 15, 198686 citations

Thermodynamics of the one-dimensional frustrated Heisenberg ferromagnet with arbitrary spin

M. Härtel, Johannes Richter, Dieter Ihle +2 more

arXiv:1106.1013Physical Review BSep 6, 201125 citations

Introduction to the Thermodynamics of Spin Chains

Luca Mezincescu, Rafael I. Nepomechie

We review how to obtain the thermodynamic Bethe Ansatz (TBA) equations for the antiferromagnetic Heisenberg ring in an external magnetic field. We review how to solve these equations for low temperature and small field, and calculate the specific heat and magnetic susceptibility.

arXiv:hep-th/9212124Submitted toDec 21, 199228 citations

Thermodynamics of quantum Heisenberg spin chains

Tao Xiang

arXiv:cond-mat/9808179Physical review. B, Condensed matterOct 1, 199892 citations

Physical review. B, Condensed matter

Critical behavior of the SU(3) ferromagnetic Heisenberg chain

Young-Ju Jee, Kong-Ju-Bock Lee, P. Schlottmann

Published Feb 1, 1989• 16 citations

Publisher

Abstract

\ensuremath{\alpha}=\ensuremath{-}\frac{1}{2}

\ensuremath{\gamma}=2

, and

\ensuremath{\nu}=1

, respectively. The susceptibility and the correlation length exhibit logarithmic corrections, which are quenched by relatively small magnetic fields. In large fields, the energy required to flip a spin gives rise to an exponential activation.

DOI: 10.1103/physrevb.39.2815

Related papers

Critical behavior of the isotropic ferromagnetic Heisenberg chain with arbitrary spinS

Kong-Ju-Bock Lee, P. Schlottmann

We have extracted the leading low-temperature contributions to the specific heat and the magnetic susceptibility from the thermodynamic Bethe-ansatz equations of the SU(2) invariant ferromagnetic Heisenberg chain of arbitrary spin S. This extends previous analytic results for S=1/2 to other spin values. The critical exponent for the specific heat is \ensuremath{\alpha}=-1/2, that for the susceptibility is \ensuremath{\gamma}=2, and that for the correlation length is \ensuremath{\nu}=1. The susceptibility and the correlation length exhibit logarithmic corrections, reflecting the analogy to the Kondo problem. These logarithmic corrections are quenched by relatively small magnetic fields. In large fields the energy required to flip a spin gives rise to an exponential activation.We have extracted the leading low-temperature contributions to the specific heat and the magnetic susceptibility from the thermodynamic Bethe-ansatz equations of the SU(2) invariant ferromagnetic Heisenberg chain of arbitrary spin S. This extends previous analytic results for S=1/2 to other spin values. The critical exponent for the specific heat is \ensuremath{\alpha}=-1/2, that for the susceptibility is \ensuremath{\gamma}=2, and that for the correlation length is \ensuremath{\nu}=1. The susceptibility and the correlation length exhibit logarithmic corrections, reflecting the analogy to the Kondo problem. These logarithmic corrections are quenched by relatively small magnetic fields. In large fields the energy required to flip a spin gives rise to an exponential activation.

Physical review. B, Condensed matterJul 1, 198720 citations

Critical Behavior of the Isotropic Ferromagnetic Quantum Heisenberg Chain

P. Schlottmann

The thermodynamic Bethe-Ansatz equations for the isotropic ferromagnetic $S=\frac{1}{2}$ Heisenberg chain have been solved numerically. At low temperatures, I find a power-law dependence on $T$ for the specific heat and the susceptibility with critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.49\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\gamma}=2.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\Delta}\ensuremath{\simeq}\ensuremath{\gamma}$. The exponent $\ensuremath{\alpha}$ is compatible with effective spin waves and $\ensuremath{\gamma}$ and $\ensuremath{\Delta}$ are the exponents of the classical chain. Amplitudes and corrections to scaling are obtained and differences with previous results are discussed.The thermodynamic Bethe-Ansatz equations for the isotropic ferromagnetic $S=\frac{1}{2}$ Heisenberg chain have been solved numerically. At low temperatures, I find a power-law dependence on $T$ for the specific heat and the susceptibility with critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.49\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\gamma}=2.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\Delta}\ensuremath{\simeq}\ensuremath{\gamma}$. The exponent $\ensuremath{\alpha}$ is compatible with effective spin waves and $\ensuremath{\gamma}$ and $\ensuremath{\Delta}$ are the exponents of the classical chain. Amplitudes and corrections to scaling are obtained and differences with previous results are discussed.

Physical Review LettersMay 13, 198570 citations

Critical Behavior of Spin-1/2 One-Dimensional Heisenberg Ferromagnet at Low Temperatures

Miki Yamada, Minoru Takahashi

A linear chain of S =1/2 spins with the isotropic Hamiltonian H = J Σ( S i x S i +1 x + S i y S i +1 y + S i z S i +1 z -1/4), ( J <0) , at low temperatures is studied numerically by solving a set of nonlinear integral equations which are derived from the Bethe Ansatz. We find that the free energy and susceptibility are both expanded in $\sqrt{T/|J|}$ about T =0. The critical exponents are a =-1/2 and y =2. The coefficient of T 3/2 term in the free energy agrees with the spin wave theory with the accuracy of 0.1%. The coefficient of T -2 term in the zero-field susceptibility agrees with Fisher's solution of the classical Heisenberg model with the accuracy of 0.3%. The contribution from the second and higher order terms of the expansions in these thermodynamic quantities is not negligible in the region T /| J |>0.002.A linear chain of S =1/2 spins with the isotropic Hamiltonian H = J Σ( S i x S i +1 x + S i y S i +1 y + S i z S i +1 z -1/4), ( J <0) , at low temperatures is studied numerically by solving a set of nonlinear integral equations which are derived from the Bethe Ansatz. We find that the free energy and susceptibility are both expanded in $\sqrt{T/|J|}$ about T =0. The critical exponents are a =-1/2 and y =2. The coefficient of T 3/2 term in the free energy agrees with the spin wave theory with the accuracy of 0.1%. The coefficient of T -2 term in the zero-field susceptibility agrees with Fisher's solution of the classical Heisenberg model with the accuracy of 0.3%. The contribution from the second and higher order terms of the expansions in these thermodynamic quantities is not negligible in the region T /| J |>0.002.

Journal of the Physical Society of JapanJun 15, 198686 citations

Thermodynamics of the one-dimensional frustrated Heisenberg ferromagnet with arbitrary spin

M. Härtel, Johannes Richter, Dieter Ihle +2 more

The thermodynamic quantities (spin-spin correlation functions $\ensuremath{\langle}{\mathbf{S}}_{0}{\mathbf{S}}_{n}\ensuremath{\rangle}$, correlation length $\ensuremath{\xi}$, spin susceptibility $\ensuremath{\chi}$, and specific heat ${C}_{V}$ ) of the frustrated one-dimensional ${J}_{1}$ - ${J}_{2}$ Heisenberg ferromagnet with arbitrary spin quantum number $S$ below the quantum critical point, i.e., for ${J}_{2}<|{J}_{1}|/4$ , are calculated using a rotation-invariant Green-function formalism and full diagonalization as well as a finite-temperature Lanczos technique for finite chains of up to $N=20$ sites. The low-temperature behavior of the susceptibility $\ensuremath{\chi}$ and the correlation length $\ensuremath{\xi}$ are well described by $\ensuremath{\chi}=\frac{2}{3}{S}^{4}\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right){T}^{\ensuremath{-}2}+A{S}^{5/2}{\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right)}^{1/2}{T}^{\ensuremath{-}3/2}$ and $\ensuremath{\xi}={S}^{2}\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right){T}^{\ensuremath{-}1}+B{S}^{1/2}{\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right)}^{1/2}{T}^{\ensuremath{-}1/2}$ with $A\ensuremath{\approx}1.1\ensuremath{\cdots}1.2$ and $B\ensuremath{\approx}0.84\ensuremath{\cdots}0.89$. The vanishing of the factors in front of the temperature at ${J}_{2}=|{J}_{1}|/4$ indicates a change of the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}$ at $T\ensuremath{\rightarrow}0$. The specific heat may exhibit an additional frustration-induced low-temperature maximum when approaching the quantum critical point. Such a maximum appears for $S=1/2$ and $S=1$ , but was not found for $S>1$ .The thermodynamic quantities (spin-spin correlation functions $\ensuremath{\langle}{\mathbf{S}}_{0}{\mathbf{S}}_{n}\ensuremath{\rangle}$, correlation length $\ensuremath{\xi}$, spin susceptibility $\ensuremath{\chi}$, and specific heat ${C}_{V}$) of the frustrated one-dimensional ${J}_{1}$-${J}_{2}$ Heisenberg ferromagnet with arbitrary spin quantum number $S$ below the quantum critical point, i.e., for ${J}_{2}<|{J}_{1}|/4$, are calculated using a rotation-invariant Green-function formalism and full diagonalization as well as a finite-temperature Lanczos technique for finite chains of up to $N=20$ sites. The low-temperature behavior of the susceptibility $\ensuremath{\chi}$ and the correlation length $\ensuremath{\xi}$ are well described by $\ensuremath{\chi}=\frac{2}{3}{S}^{4}\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right){T}^{\ensuremath{-}2}+A{S}^{5/2}{\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right)}^{1/2}{T}^{\ensuremath{-}3/2}$ and $\ensuremath{\xi}={S}^{2}\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right){T}^{\ensuremath{-}1}+B{S}^{1/2}{\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right)}^{1/2}{T}^{\ensuremath{-}1/2}$ with $A\ensuremath{\approx}1.1\ensuremath{\cdots}1.2$ and $B\ensuremath{\approx}0.84\ensuremath{\cdots}0.89$. The vanishing of the factors in front of the temperature at ${J}_{2}=|{J}_{1}|/4$ indicates a change of the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}$ at $T\ensuremath{\rightarrow}0$. The specific heat may exhibit an additional frustration-induced low-temperature maximum when approaching the quantum critical point. Such a maximum appears for $S=1/2$ and $S=1$, but was not found for $S>1$.

arXiv:1106.1013Physical Review BSep 6, 201125 citations

Introduction to the Thermodynamics of Spin Chains

Luca Mezincescu, Rafael I. Nepomechie

arXiv:hep-th/9212124Submitted toDec 21, 199228 citations

Thermodynamics of quantum Heisenberg spin chains

Tao Xiang

Thermodynamic properties of the quantum Heisenberg spin chains with $S=1/2,$ 1, and $3/2$ are investigated using the transfer-matrix renormalization-group method. The temperature dependence of the magnetization, susceptibility, specific heat, spin-spin correlation length, and several other physical quantities in a zero or finite applied field are calculated and compared. Our data agree well with the Bethe ansatz, exact diagonalization, and quantum Monte Carlo results and provide further insight into the quantum effects in the antiferromagnetic Heisenberg spin chains.Thermodynamic properties of the quantum Heisenberg spin chains with $S=1/2,$ 1, and $3/2$ are investigated using the transfer-matrix renormalization-group method. The temperature dependence of the magnetization, susceptibility, specific heat, spin-spin correlation length, and several other physical quantities in a zero or finite applied field are calculated and compared. Our data agree well with the Bethe ansatz, exact diagonalization, and quantum Monte Carlo results and provide further insight into the quantum effects in the antiferromagnetic Heisenberg spin chains.

arXiv:cond-mat/9808179Physical review. B, Condensed matterOct 1, 199892 citations

\ensuremath{\alpha}=\ensuremath{-}\frac{1}{2}

\ensuremath{\gamma}=2

, and

\ensuremath{\nu}=1

The thermodynamic Bethe-Ansatz equations for the isotropic ferromagnetic

S=\frac{1}{2}

Heisenberg chain have been solved numerically. At low temperatures, I find a power-law dependence on

T

for the specific heat and the susceptibility with critical exponents $\ensuremath{\alpha}=\ensuremath{-}0.49\ifmmode\pm\else\textpm\fi{}0.02$, $\ensuremath{\gamma}=2.00\ifmmode\pm\else\textpm\fi{}0.02$, and $\ensuremath{\Delta}\ensuremath{\simeq}\ensuremath{\gamma}$. The exponent $\ensuremath{\alpha}$ is compatible with effective spin waves and $\ensuremath{\gamma}$ and $\ensuremath{\Delta}$ are the exponents of the classical chain. Amplitudes and corrections to scaling are obtained and differences with previous results are discussed.

\sqrt{T/|J|}

about T =0. The critical exponents are a =-1/2 and y =2. The coefficient of T 3/2 term in the free energy agrees with the spin wave theory with the accuracy of 0.1%. The coefficient of T -2 term in the zero-field susceptibility agrees with Fisher's solution of the classical Heisenberg model with the accuracy of 0.3%. The contribution from the second and higher order terms of the expansions in these thermodynamic quantities is not negligible in the region T /| J |>0.002.

{C}_{V}

) of the frustrated one-dimensional

{J}_{1}

{J}_{2}

Heisenberg ferromagnet with arbitrary spin quantum number

S

below the quantum critical point, i.e., for

{J}_{2}<|{J}_{1}|/4

, are calculated using a rotation-invariant Green-function formalism and full diagonalization as well as a finite-temperature Lanczos technique for finite chains of up to

N=20

sites. The low-temperature behavior of the susceptibility $\ensuremath{\chi}$ and the correlation length $\ensuremath{\xi}$ are well described by $\ensuremath{\chi}=\frac{2}{3}{S}^{4}\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right){T}^{\ensuremath{-}2}+A{S}^{5/2}{\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right)}^{1/2}{T}^{\ensuremath{-}3/2}$ and $\ensuremath{\xi}={S}^{2}\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right){T}^{\ensuremath{-}1}+B{S}^{1/2}{\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right)}^{1/2}{T}^{\ensuremath{-}1/2}$ with $A\ensuremath{\approx}1.1\ensuremath{\cdots}1.2$ and $B\ensuremath{\approx}0.84\ensuremath{\cdots}0.89$. The vanishing of the factors in front of the temperature at

{J}_{2}=|{J}_{1}|/4

indicates a change of the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}$ at $T\ensuremath{\rightarrow}0$. The specific heat may exhibit an additional frustration-induced low-temperature maximum when approaching the quantum critical point. Such a maximum appears for

S=1/2

and

S=1

, but was not found for

S>1

Thermodynamic properties of the quantum Heisenberg spin chains with

S=1/2,

1, and

3/2

are investigated using the transfer-matrix renormalization-group method. The temperature dependence of the magnetization, susceptibility, specific heat, spin-spin correlation length, and several other physical quantities in a zero or finite applied field are calculated and compared. Our data agree well with the Bethe ansatz, exact diagonalization, and quantum Monte Carlo results and provide further insight into the quantum effects in the antiferromagnetic Heisenberg spin chains.