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

Thermodynamics of the frustrated ferromagnetic spin-1/2 Heisenberg chain

Johannes Richter, M. Härtel, Dieter Ihle, S. -l. Drechsler

Published Nov 21, 2012• 4 citations
OpenAlex

Abstract

Abstract. We studied the thermodynamics of the one-dimensional J1-J2 spin-1/2 Heisenberg chain for ferromagnetic nearest-neighbor bonds J1 < 0 and frustrating antiferromagnetic nextnearest-neighbor bonds J2> 0 using full diagonalization of finite rings and a second-order Green-function formalism. Thereby we focus on J2 < |J1|/4 where the ground state is still ferromagnetic, but the frustration influences the thermodynamic properties. We found that their critical indices are not changed by J2. The analysis of the low-temperature behavior of the susceptibility χ leads to the conclusion that this behavior changes from χ ∝ T −2 at J2 < |J1|/4 to χ ∝ T −3/2 at the quantum-critical point J2 = |J1|/4. Another effect of the frustration is the appearance of an extra low-T maximum in the specific heat Cv(T) for J2 � |J1|/8, indicating its strong influence on the low-energy spectrum. Introduction: In low-dimensional frustrated quantum magnets thermal and quantum fluctuations strongly influence the low-temperature physics [1,2]. Special attention has been paid to one-dimensional (1D) J1-J2 quantum Heisenberg magnets, see Ref. [3] and references therein. Recent experimental studies have shown that edge-shared chain cuprates, such as LiVCuO4, Li(Na)Cu2O2, Li2ZrCuO4, and Li2CuO2 [4–13], represent a family of quantum magnets for

Related papers

Thermodynamics of the frustrated ferromagnetic spin-1/2 Heisenberg chain

Johannes Richter, M Härtel, D Ihle +1 more

We studied the thermodynamics of the one-dimensional J 1 - J 2 spin-1/2 Heisenberg chain for ferromagnetic nearest-neighbor bonds J 1 < 0 and frustrating antiferromagnetic next- nearest-neighbor bonds J 2 > 0 using full diagonalization of finite rings and a second-order Green-function formalism. Thereby we focus on J 2 < | J 1 |/4 where the ground state is still ferromagnetic, but the frustration influences the thermodynamic properties. We found that their critical indices are not changed by J 2 . The analysis of the low-temperature behavior of the susceptibility χ leads to the conclusion that this behavior changes from χ ∝ T -2 at J 2 < | J 1 |/4 to χ ∝ T -3/2 at the quantum-critical point J 2 = | J 1 |/4. Another effect of the frustration is the appearance of an extra low- T maximum in the specific heat C v ( T ) for J 2 & | J 1 |/8, indicating its strong influence on the low-energy spectrum.

arXiv:0811.3549•Journal of Physics Conference Series•Jan 1, 2009•6 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 $S>1$.

arXiv:1106.1013•Physical Review B•Sep 6, 2011•25 citations

Thermodynamics of frustrated ferromagnetic spin- 12 Heisenberg chains: Role of interchain coupling

P. Müller, Johannes Richter, Dieter Ihle

The thermodynamics of coupled frustrated ferromagnetic chains is studied within a spin-rotation-invariant Green's-function approach. We consider an isotropic Heisenberg spin-half system with a ferromagnetic in-chain coupling ${J}_{1}<0$ between nearest neighbors and a frustrating antiferromagnetic next-nearest-neighbor in-chain coupling ${J}_{2}>0$. We focus on the moderate strength of frustration ${J}_{2}<|{J}_{1}|/4$ such that the in-chain spin-spin correlations are predominantly ferromagnetic. We consider two interchain couplings (ICs) ${J}_{\ensuremath{\perp},y}$ and ${J}_{\ensuremath{\perp},z}$, corresponding to the two axes perpendicular to the chain, where ferromagnetic as well as antiferromagnetic ICs are taken into account. We discuss the influence of frustration on the ground-state properties for antiferromagnetic ICs, where the ground state is of a quantum nature. The major part of our study is devoted to the finite-temperature properties. We calculate the critical temperature ${T}_{c}$ as a function of the competing exchange couplings ${J}_{2},{J}_{\ensuremath{\perp},y},{J}_{\ensuremath{\perp},z}$. We find that for fixed ICs, ${T}_{c}$ decreases monotonically with increasing frustration ${J}_{2}$, where as ${J}_{2}\ensuremath{\rightarrow}|{J}_{1}|/4$ the ${T}_{c}({J}_{2})$ curve drops down rapidly. To characterize the magnetic ordering below and above ${T}_{c}$, we calculate the spin-spin correlation functions $\ensuremath{\langle}{\mathbf{S}}_{0}{\mathbf{S}}_{\mathbf{R}}\ensuremath{\rangle}$, the magnetic order parameter $M$, the uniform static susceptibility ${\ensuremath{\chi}}_{0}$, as well as the correlation length $\ensuremath{\xi}$. Moreover, we discuss the specific heat ${C}_{V}$ and the temperature dependence of the excitation spectrum ${\ensuremath{\omega}}_{\mathbf{q}}$. As ${J}_{2}\ensuremath{\rightarrow}|{J}_{1}|/4$, some unusual frustration-induced features were found, such as an increase of the in-chain spin stiffness (in the case of ferromagnetic ICs) or of the in-chain spin-wave velocity (in the case of antiferromagnetic ICs) with growing temperature.

arXiv:1703.04573•Physical review. B./Physical review. B•Apr 5, 2017•15 citations

Thermodynamics of a one-dimensional frustrated spin- 12 Heisenberg ferromagnet

M. Härtel, Johannes Richter, Dieter Ihle +1 more

We calculate the thermodynamic quantities (correlation functions $⟨{\mathbf{S}}_{0}{\mathbf{S}}_{n}⟩$, correlation length $\ensuremath{\xi}$, spin susceptibility $\ensuremath{\chi}$, and specific heat ${C}_{V}$) of the frustrated one-dimensional spin-half ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg ferromagnet, i.e., for ${J}_{2}<0.25|{J}_{1}|$, using a rotation-invariant Green's-function formalism and full diagonalization of finite lattices. We find that the critical indices are not changed by ${J}_{2}$, i.e., $\ensuremath{\chi}={y}_{0}{T}^{\ensuremath{-}2}$ and $\ensuremath{\xi}={x}_{0}{T}^{\ensuremath{-}1}$ at $T\ensuremath{\rightarrow}0$. However, the coefficients ${y}_{0}$ and ${x}_{0}$ linearly decrease with increasing ${J}_{2}$ according to the relations ${y}_{0}=(1\ensuremath{-}4{J}_{2}/|{J}_{1}|)/24$ and ${x}_{0}=(1\ensuremath{-}4{J}_{2}/|{J}_{1}|)/4$, i.e., both coefficients vanish at ${J}_{2}=0.25|{J}_{1}|$ indicating the zero-temperature phase transition that is accompanied by a change in the low-temperature behavior of $\ensuremath{\chi}$ $(\ensuremath{\xi})$ from $\ensuremath{\chi}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ $(\ensuremath{\xi}\ensuremath{\propto}{T}^{\ensuremath{-}1})$ at ${J}_{2}<0.25|{J}_{1}|$ to $\ensuremath{\chi}\ensuremath{\propto}{T}^{\ensuremath{-}3/2}$ $(\ensuremath{\xi}\ensuremath{\propto}{T}^{\ensuremath{-}1/2})$ at ${J}_{2}=0.25|{J}_{1}|$. In addition, we detect the existence of an additional low-temperature maximum in the specific heat when approaching the critical point at ${J}_{2}=0.25|{J}_{1}|$.

arXiv:0807.1391•Physical Review B•Nov 12, 2008•50 citations

Thermodynamics of the frustrated one-dimensional spin- 12 Heisenberg ferromagnet in a magnetic field

M. Härtel, Johannes Richter, Dieter Ihle

We calculate the low-temperature thermodynamic quantities (magnetization, correlation functions, transverse and longitudinal correlation lengths, spin susceptibility, and specific heat) of the frustrated one-dimensional spin-half ${J}_{1}$-${J}_{2}$ Heisenberg ferromagnet, i.e., for ${J}_{2}<0.25|{J}_{1}|$, in an external magnetic field using a second-order Green-function formalism and full diagonalization of finite systems. We determine power-law relations for the field dependence of the position and the height of the maximum of the uniform susceptibility. Considering the specific heat at low magnetic fields, two maxima in its temperature dependence are found.

arXiv:1102.3581•Physical Review B•Jun 13, 2011•17 citations

J1−J2 Heisenberg model at and close to its z=4 quantum critical point

Jesko Sirker, V. Ya. Krivnov, Д. В. Дмитриев +5 more

We study the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model with ferromagnetic nearest-neighbor coupling ${J}_{1}<0$ and antiferromagnetic next-nearest-neighbor coupling ${J}_{2}>0$ at and close to the $z=4$ quantum critical point (QCP) at ${J}_{1}/{J}_{2}=\ensuremath{-}4$. The ${J}_{1}\ensuremath{-}{J}_{2}$ model plays an important role for recently synthesized chain cuprates as well as in supersymmetric Yang-Mills theories. We study the thermodynamic properties using field theory, a modified spin-wave theory, as well as numerical density-matrix renormalization group calculations. Furthermore, we compare with results for the classical model obtained by analytical methods and Monte Carlo simulations. As one of our main results, we present numerical evidence that the susceptibility at the QCP seems to diverge with temperature $T$ as $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}1.2}$ in the quantum case, in contrast to the classical model where $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}4/3}$.

arXiv:1106.1008•Physical Review B•Oct 3, 2011•41 citations
Abstract. We studied the thermodynamics of the one-dimensional J1-J2 spin-1/2 Heisenberg chain for ferromagnetic nearest-neighbor bonds J1 < 0 and frustrating antiferromagnetic nextnearest-neighbor bonds J2> 0 using full diagonalization of finite rings and a second-order Green-function formalism. Thereby we focus on J2 < |J1|/4 where the ground state is still ferromagnetic, but the frustration influences the thermodynamic properties. We found that their critical indices are not changed by J2. The analysis of the low-temperature behavior of the susceptibility χ leads to the conclusion that this behavior changes from χ ∝ T −2 at J2 < |J1|/4 to χ ∝ T −3/2 at the quantum-critical point J2 = |J1|/4. Another effect of the frustration is the appearance of an extra low-T maximum in the specific heat Cv(T) for J2 � |J1|/8, indicating its strong influence on the low-energy spectrum. Introduction: In low-dimensional frustrated quantum magnets thermal and quantum fluctuations strongly influence the low-temperature physics [1,2]. Special attention has been paid to one-dimensional (1D) J1-J2 quantum Heisenberg magnets, see Ref. [3] and references therein. Recent experimental studies have shown that edge-shared chain cuprates, such as LiVCuO4, Li(Na)Cu2O2, Li2ZrCuO4, and Li2CuO2 [4–13], represent a family of quantum magnets for
We studied the thermodynamics of the one-dimensional J 1 - J 2 spin-1/2 Heisenberg chain for ferromagnetic nearest-neighbor bonds J 1 < 0 and frustrating antiferromagnetic next- nearest-neighbor bonds J 2 > 0 using full diagonalization of finite rings and a second-order Green-function formalism. Thereby we focus on J 2 < | J 1 |/4 where the ground state is still ferromagnetic, but the frustration influences the thermodynamic properties. We found that their critical indices are not changed by J 2 . The analysis of the low-temperature behavior of the susceptibility χ leads to the conclusion that this behavior changes from χ ∝ T -2 at J 2 < | J 1 |/4 to χ ∝ T -3/2 at the quantum-critical point J 2 = | J 1 |/4. Another effect of the frustration is the appearance of an extra low- T maximum in the specific heat C v ( T ) for J 2 & | J 1 |/8, indicating its strong influence on the low-energy spectrum.
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 CV{C}_{V}CV​) of the frustrated one-dimensional J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg ferromagnet with arbitrary spin quantum number SSS below the quantum critical point, i.e., for J2<∣J1∣/4{J}_{2}<|{J}_{1}|/4J2​<∣J1​∣/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=20N=20N=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 J2=∣J1∣/4{J}_{2}=|{J}_{1}|/4J2​=∣J1​∣/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=1/2S=1/2S=1/2 and S=1S=1S=1, but was not found for S>1S>1S>1.
The thermodynamics of coupled frustrated ferromagnetic chains is studied within a spin-rotation-invariant Green's-function approach. We consider an isotropic Heisenberg spin-half system with a ferromagnetic in-chain coupling J1<0{J}_{1}<0J1​<0 between nearest neighbors and a frustrating antiferromagnetic next-nearest-neighbor in-chain coupling J2>0{J}_{2}>0J2​>0. We focus on the moderate strength of frustration J2<∣J1∣/4{J}_{2}<|{J}_{1}|/4J2​<∣J1​∣/4 such that the in-chain spin-spin correlations are predominantly ferromagnetic. We consider two interchain couplings (ICs) ${J}_{\ensuremath{\perp},y}$ and ${J}_{\ensuremath{\perp},z}$, corresponding to the two axes perpendicular to the chain, where ferromagnetic as well as antiferromagnetic ICs are taken into account. We discuss the influence of frustration on the ground-state properties for antiferromagnetic ICs, where the ground state is of a quantum nature. The major part of our study is devoted to the finite-temperature properties. We calculate the critical temperature Tc{T}_{c}Tc​ as a function of the competing exchange couplings ${J}_{2},{J}_{\ensuremath{\perp},y},{J}_{\ensuremath{\perp},z}$. We find that for fixed ICs, Tc{T}_{c}Tc​ decreases monotonically with increasing frustration J2{J}_{2}J2​, where as ${J}_{2}\ensuremath{\rightarrow}|{J}_{1}|/4$ the Tc(J2){T}_{c}({J}_{2})Tc​(J2​) curve drops down rapidly. To characterize the magnetic ordering below and above Tc{T}_{c}Tc​, we calculate the spin-spin correlation functions $\ensuremath{\langle}{\mathbf{S}}_{0}{\mathbf{S}}_{\mathbf{R}}\ensuremath{\rangle}$, the magnetic order parameter MMM, the uniform static susceptibility ${\ensuremath{\chi}}_{0}$, as well as the correlation length $\ensuremath{\xi}$. Moreover, we discuss the specific heat CV{C}_{V}CV​ and the temperature dependence of the excitation spectrum ${\ensuremath{\omega}}_{\mathbf{q}}$. As ${J}_{2}\ensuremath{\rightarrow}|{J}_{1}|/4$, some unusual frustration-induced features were found, such as an increase of the in-chain spin stiffness (in the case of ferromagnetic ICs) or of the in-chain spin-wave velocity (in the case of antiferromagnetic ICs) with growing temperature.
We calculate the thermodynamic quantities (correlation functions ⟨S0Sn⟩⟨{\mathbf{S}}_{0}{\mathbf{S}}_{n}⟩⟨S0​Sn​⟩, correlation length $\ensuremath{\xi}$, spin susceptibility $\ensuremath{\chi}$, and specific heat CV{C}_{V}CV​) of the frustrated one-dimensional spin-half ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg ferromagnet, i.e., for J2<0.25∣J1∣{J}_{2}<0.25|{J}_{1}|J2​<0.25∣J1​∣, using a rotation-invariant Green's-function formalism and full diagonalization of finite lattices. We find that the critical indices are not changed by J2{J}_{2}J2​, i.e., $\ensuremath{\chi}={y}_{0}{T}^{\ensuremath{-}2}$ and $\ensuremath{\xi}={x}_{0}{T}^{\ensuremath{-}1}$ at $T\ensuremath{\rightarrow}0$. However, the coefficients y0{y}_{0}y0​ and x0{x}_{0}x0​ linearly decrease with increasing J2{J}_{2}J2​ according to the relations ${y}_{0}=(1\ensuremath{-}4{J}_{2}/|{J}_{1}|)/24$ and ${x}_{0}=(1\ensuremath{-}4{J}_{2}/|{J}_{1}|)/4$, i.e., both coefficients vanish at J2=0.25∣J1∣{J}_{2}=0.25|{J}_{1}|J2​=0.25∣J1​∣ indicating the zero-temperature phase transition that is accompanied by a change in the low-temperature behavior of $\ensuremath{\chi}$ $(\ensuremath{\xi})$ from $\ensuremath{\chi}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ $(\ensuremath{\xi}\ensuremath{\propto}{T}^{\ensuremath{-}1})$ at J2<0.25∣J1∣{J}_{2}<0.25|{J}_{1}|J2​<0.25∣J1​∣ to $\ensuremath{\chi}\ensuremath{\propto}{T}^{\ensuremath{-}3/2}$ $(\ensuremath{\xi}\ensuremath{\propto}{T}^{\ensuremath{-}1/2})$ at J2=0.25∣J1∣{J}_{2}=0.25|{J}_{1}|J2​=0.25∣J1​∣. In addition, we detect the existence of an additional low-temperature maximum in the specific heat when approaching the critical point at J2=0.25∣J1∣{J}_{2}=0.25|{J}_{1}|J2​=0.25∣J1​∣.
We calculate the low-temperature thermodynamic quantities (magnetization, correlation functions, transverse and longitudinal correlation lengths, spin susceptibility, and specific heat) of the frustrated one-dimensional spin-half J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg ferromagnet, i.e., for J2<0.25∣J1∣{J}_{2}<0.25|{J}_{1}|J2​<0.25∣J1​∣, in an external magnetic field using a second-order Green-function formalism and full diagonalization of finite systems. We determine power-law relations for the field dependence of the position and the height of the maximum of the uniform susceptibility. Considering the specific heat at low magnetic fields, two maxima in its temperature dependence are found.
We study the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model with ferromagnetic nearest-neighbor coupling J1<0{J}_{1}<0J1​<0 and antiferromagnetic next-nearest-neighbor coupling J2>0{J}_{2}>0J2​>0 at and close to the z=4z=4z=4 quantum critical point (QCP) at ${J}_{1}/{J}_{2}=\ensuremath{-}4$. The ${J}_{1}\ensuremath{-}{J}_{2}$ model plays an important role for recently synthesized chain cuprates as well as in supersymmetric Yang-Mills theories. We study the thermodynamic properties using field theory, a modified spin-wave theory, as well as numerical density-matrix renormalization group calculations. Furthermore, we compare with results for the classical model obtained by analytical methods and Monte Carlo simulations. As one of our main results, we present numerical evidence that the susceptibility at the QCP seems to diverge with temperature TTT as $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}1.2}$ in the quantum case, in contrast to the classical model where $\ensuremath{\chi}\ensuremath{\sim}{T}^{\ensuremath{-}4/3}$.