arXiv:cond-mat/9803122 · Physical Review Letters

Thermodynamics of Frustrated Quantum Spin Chains

K. Maisinger, Ulrich Schollwöck

Published Jul 13, 1998• 44 citations

Abstract

We apply the transfer-matrix density matrix renormalization group (DMRG) approach to frustrated quantum spin chains, down to $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.025\mathrm{K}$ in the thermodynamic limit. The incommensurability problem of exact diagonalization and the negative sign problem of quantum Monte Carlo vanish completely. For illustration, we give results for chains with next-nearest-neighbor frustration and for the delta chain, which has been a testbed for many thermodynamic methods. The DMRG approach proves to be an extremely powerful method for the old problem of the thermodynamics of frustrated systems.

DOI: 10.1103/physrevlett.81.445

Related papers

Hybrid ED/DMRG approach to the thermodynamics of 1D quantum models

Sudip Kumar Saha, Dayasindhu Dey, Manoranjan Kumar +1 more

arXiv:1905.05956May 15, 20190 citations

Hybrid exact diagonalization and density matrix renormalization group approach to the thermodynamics of one-dimensional quantum models

Sudip Kumar Saha, Dayasindhu Dey, Manoranjan Kumar +1 more

Physical review. B./Physical review. BMay 24, 201912 citations

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

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

arXiv:cond-mat/0605204Physical Review BJul 27, 2006139 citations

Accurate computation of low-temperature thermodynamics for quantum spin chains

Yukun Huang, Pochung Chen, Ying-Jer Kao

arXiv:1203.4848Physical Review BDec 3, 201225 citations

Finite-temperature density matrix renormalization using an enlarged Hilbert space

Adrian E. Feiguin, Steven R. White

We apply a generalization of the time-dependent density matrix renormalization group (DMRG) to study finite-temperature properties of several quantum spin chains, including the frustrated ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model. We discuss several practical issues with the method, including use of quantum numbers and finite-size effects. We compare with transfer-matrix DMRG, finding that both methods produce excellent results.

arXiv:cond-mat/0510124Physical Review BDec 2, 2005327 citations

Thermodynamics of quantum Heisenberg spin chains

Tao Xiang

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

arXiv:cond-mat/9803122 · Physical Review Letters

Thermodynamics of Frustrated Quantum Spin Chains

K. Maisinger, Ulrich Schollwöck

Published Jul 13, 1998• 44 citations

arXiv HTML PDF

Abstract

We apply the transfer-matrix density matrix renormalization group (DMRG) approach to frustrated quantum spin chains, down to

T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.025\mathrm{K}

in the thermodynamic limit. The incommensurability problem of exact diagonalization and the negative sign problem of quantum Monte Carlo vanish completely. For illustration, we give results for chains with next-nearest-neighbor frustration and for the delta chain, which has been a testbed for many thermodynamic methods. The DMRG approach proves to be an extremely powerful method for the old problem of the thermodynamics of frustrated systems.

DOI: 10.1103/physrevlett.81.445

Related papers

Hybrid ED/DMRG approach to the thermodynamics of 1D quantum models

Sudip Kumar Saha, Dayasindhu Dey, Manoranjan Kumar +1 more

Exact diagonalization (ED) of small model systems gives the thermodynamics of spin chains or quantum cell models at high temperature $T$ . Density matrix renormalization group (DMRG) calculations of progressively larger systems are used to obtain excitations up to a cutoff $W_C$ and the low- $T$ thermodynamics. The hybrid approach is applied to the magnetic susceptibility $χ(T)$ and specific heat $C(T)$ of spin- $1/2$ chains with isotropic exchange such as the linear Heisenberg antiferromagnet (HAF) and the frustrated $J_1-J_2$ model with ferromagnetic (F) $J_1 < 0$ and antiferromagnetic (AF) $J_2 > 0$ . The hybrid approach is fully validated by comparison with HAF results. It extends $J_1-J_2$ thermodynamics down to $T \sim 0.01|J_1|$ for $J_2/|J_1| \geq α_c = 1/4$ and is consistent with other methods. The criterion for the cutoff $W_C(N)$ in systems of $N$ spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific $T(N)$ at system size $N$ .Exact diagonalization (ED) of small model systems gives the thermodynamics of spin chains or quantum cell models at high temperature $T$. Density matrix renormalization group (DMRG) calculations of progressively larger systems are used to obtain excitations up to a cutoff $W_C$ and the low-$T$ thermodynamics. The hybrid approach is applied to the magnetic susceptibility $χ(T)$ and specific heat $C(T)$ of spin-$1/2$ chains with isotropic exchange such as the linear Heisenberg antiferromagnet (HAF) and the frustrated $J_1-J_2$ model with ferromagnetic (F) $J_1 < 0$ and antiferromagnetic (AF) $J_2 > 0$. The hybrid approach is fully validated by comparison with HAF results. It extends $J_1-J_2$ thermodynamics down to $T \sim 0.01|J_1|$ for $J_2/|J_1| \geq α_c = 1/4$ and is consistent with other methods. The criterion for the cutoff $W_C(N)$ in systems of $N$ spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific $T(N)$ at system size $N$.

arXiv:1905.05956May 15, 20190 citations

Hybrid exact diagonalization and density matrix renormalization group approach to the thermodynamics of one-dimensional quantum models

Sudip Kumar Saha, Dayasindhu Dey, Manoranjan Kumar +1 more

Exact diagonalization of small model systems gives the thermodynamics of spin chains or quantum cell models at high temperatures $T$ . Density matrix renormalization group calculations of progressively larger systems are used to obtain excitations up to a cutoff ${W}_{C}$ and the low- $T$ thermodynamics. The hybrid approach is applied to the magnetic susceptibility $\ensuremath{\chi}(T)$ and specific heat $C(T)$ of spin- $1/2$ chains with isotropic exchange such as the linear Heisenberg antiferromagnet and the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ model with ferromagnetic ${J}_{1}<0$ and antiferromagnetic ${J}_{2}>0$ . The hybrid approach is fully validated by comparison with Heisenberg antiferromagnet results. It extends ${J}_{1}\ensuremath{-}{J}_{2}$ thermodynamics down to $T\ensuremath{\sim}0.01|{J}_{1}|$ for ${J}_{2}/|{J}_{1}|\ensuremath{\ge}{\ensuremath{\alpha}}_{c}=1/4$ and is consistent with other methods. The criterion for the cutoff ${W}_{C}(N)$ in systems of $N$ spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific $T(N)$ at system size $N$ .Exact diagonalization of small model systems gives the thermodynamics of spin chains or quantum cell models at high temperatures $T$. Density matrix renormalization group calculations of progressively larger systems are used to obtain excitations up to a cutoff ${W}_{C}$ and the low-$T$ thermodynamics. The hybrid approach is applied to the magnetic susceptibility $\ensuremath{\chi}(T)$ and specific heat $C(T)$ of spin-$1/2$ chains with isotropic exchange such as the linear Heisenberg antiferromagnet and the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ model with ferromagnetic ${J}_{1}<0$ and antiferromagnetic ${J}_{2}>0$. The hybrid approach is fully validated by comparison with Heisenberg antiferromagnet results. It extends ${J}_{1}\ensuremath{-}{J}_{2}$ thermodynamics down to $T\ensuremath{\sim}0.01|{J}_{1}|$ for ${J}_{2}/|{J}_{1}|\ensuremath{\ge}{\ensuremath{\alpha}}_{c}=1/4$ and is consistent with other methods. The criterion for the cutoff ${W}_{C}(N)$ in systems of $N$ spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific $T(N)$ at system size $N$.

Physical review. B./Physical review. BMay 24, 201912 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.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/0605204Physical Review BJul 27, 2006139 citations

Accurate computation of low-temperature thermodynamics for quantum spin chains

Yukun Huang, Pochung Chen, Ying-Jer Kao

We apply the biorthonormal transfer-matrix renormalization group (BTMRG) [Huang, Phys. Rev. E 83, 036702 (2011)] to study low-temperature properties of quantum spin chains. Simulations on anisotropic Heisenberg spin- $1/2$ chains demonstrate that the BTMRG outperforms the conventional transfer-matrix renormalization group by successfully accessing far lower temperature than previously reported, while retaining the same level of accuracy. The power of the method is further illustrated by the calculation of the low-temperature specific heat for a frustrated spin chain.We apply the biorthonormal transfer-matrix renormalization group (BTMRG) [Huang, Phys. Rev. E 83, 036702 (2011)] to study low-temperature properties of quantum spin chains. Simulations on anisotropic Heisenberg spin-$1/2$ chains demonstrate that the BTMRG outperforms the conventional transfer-matrix renormalization group by successfully accessing far lower temperature than previously reported, while retaining the same level of accuracy. The power of the method is further illustrated by the calculation of the low-temperature specific heat for a frustrated spin chain.

arXiv:1203.4848Physical Review BDec 3, 201225 citations

Finite-temperature density matrix renormalization using an enlarged Hilbert space

Adrian E. Feiguin, Steven R. White

arXiv:cond-mat/0510124Physical Review BDec 2, 2005327 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

We apply the transfer-matrix density matrix renormalization group (DMRG) approach to frustrated quantum spin chains, down to

T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.025\mathrm{K}

Exact diagonalization (ED) of small model systems gives the thermodynamics of spin chains or quantum cell models at high temperature

T

. Density matrix renormalization group (DMRG) calculations of progressively larger systems are used to obtain excitations up to a cutoff

W_C

and the low-

T

thermodynamics. The hybrid approach is applied to the magnetic susceptibility

χ(T)

and specific heat

C(T)

of spin-

1/2

chains with isotropic exchange such as the linear Heisenberg antiferromagnet (HAF) and the frustrated

J_1-J_2

model with ferromagnetic (F)

J_1 < 0

and antiferromagnetic (AF)

J_2 > 0

. The hybrid approach is fully validated by comparison with HAF results. It extends

J_1-J_2

thermodynamics down to

T \sim 0.01|J_1|

for

J_2/|J_1| \geq α_c = 1/4

and is consistent with other methods. The criterion for the cutoff

W_C(N)

in systems of

N

spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific

T(N)

at system size

N

Exact diagonalization of small model systems gives the thermodynamics of spin chains or quantum cell models at high temperatures

T

. Density matrix renormalization group calculations of progressively larger systems are used to obtain excitations up to a cutoff

{W}_{C}

and the low-

T

thermodynamics. The hybrid approach is applied to the magnetic susceptibility $\ensuremath{\chi}(T)$ and specific heat

C(T)

of spin-

1/2

chains with isotropic exchange such as the linear Heisenberg antiferromagnet and the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ model with ferromagnetic

{J}_{1}<0

and antiferromagnetic

{J}_{2}>0

. The hybrid approach is fully validated by comparison with Heisenberg antiferromagnet results. It extends ${J}_{1}\ensuremath{-}{J}_{2}$ thermodynamics down to $T\ensuremath{\sim}0.01|{J}_{1}|$ for ${J}_{2}/|{J}_{1}|\ensuremath{\ge}{\ensuremath{\alpha}}_{c}=1/4$ and is consistent with other methods. The criterion for the cutoff

{W}_{C}(N)

in systems of

N

spins is discussed. The cutoff leads to bounds for the thermodynamic limit that are best satisfied at a specific

T(N)

at system size

N

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.

1/2

chains demonstrate that the BTMRG outperforms the conventional transfer-matrix renormalization group by successfully accessing far lower temperature than previously reported, while retaining the same level of accuracy. The power of the method is further illustrated by the calculation of the low-temperature specific heat for a frustrated spin chain.

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.