arXiv:cond-mat/0306365

Field-Induced Ising Criticality and Incommensurability in Anisotropic Spin-1 Chains

Y. J. Wang

Published Jun 13, 2003• 2 citations

Abstract

Quantum phase transitions induced by an external magnetic field in the Haldane-gapped spin-1 chains are studied in a fermionic field-theoretic description of the model. In the case with broken axial symmetry, two transitions occurs consecutively as increasing the field: $h_c=(m_1 m_2)^{1/2}$ (with $m_1$ and $m_2$ the transverse masses) is the critical Ising point and $\tilde{h}_c^* (> h_c)$ is the commensurate-incommensurate transition point. Although the latter is thermodynamically indiscernible, its property is revealed in the spin correlation functions (both uniform and staggered) which are calculated upon expressing the correlations of coupled Ising pair fields in terms of block-Toeplitz determinants.

DOI: 10.48550/arxiv.cond-mat/0306365

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arXiv:cond-mat/0306365

Field-Induced Ising Criticality and Incommensurability in Anisotropic Spin-1 Chains

Y. J. Wang

Published Jun 13, 2003• 2 citations

arXiv HTML PDF

Abstract

h_c=(m_1 m_2)^{1/2}

(with

m_1

and

m_2

the transverse masses) is the critical Ising point and

\tilde{h}_c^* (> h_c)

is the commensurate-incommensurate transition point. Although the latter is thermodynamically indiscernible, its property is revealed in the spin correlation functions (both uniform and staggered) which are calculated upon expressing the correlations of coupled Ising pair fields in terms of block-Toeplitz determinants.

DOI: 10.48550/arxiv.cond-mat/0306365

Related papers

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Anup Kumar Bera, B. Lake, A. T. M. N. Islam +1 more

We use inelastic neutron scattering to measure magnetic fluctuations as a function of transverse magnetic field and temperature in a single crystal of the coupled Haldane chain compound $\mathrm{SrNi}{}_{2}{\text{V}}_{2}{\text{O}}_{8}$ with uniaxial anisotropy. At the base temperature (2 K), spin excitations confirm a field-induced quantum phase transition from the disordered Haldane (gapped singlet) state to a gapped long-range ordered state through a quantum critical point (QCP) (${\ensuremath{\mu}}_{0}{H}_{c}\ensuremath{\approx}11.5$ T). At elevated temperatures in the vicinity of the QCP, both the energy and linewidth of the lowest triplet mode scale linearly with temperature, in accordance with quantum criticality. We also demonstrate that the experimental field dependence of the triplet excitation modes at the base temperature is in agreement with the bosonic and macroscopic models, but is in contrast to the fermionic and perturbative models.We use inelastic neutron scattering to measure magnetic fluctuations as a function of transverse magnetic field and temperature in a single crystal of the coupled Haldane chain compound $\mathrm{SrNi}{}_{2}{\text{V}}_{2}{\text{O}}_{8}$ with uniaxial anisotropy. At the base temperature (2 K), spin excitations confirm a field-induced quantum phase transition from the disordered Haldane (gapped singlet) state to a gapped long-range ordered state through a quantum critical point (QCP) (${\ensuremath{\mu}}_{0}{H}_{c}\ensuremath{\approx}11.5$ T). At elevated temperatures in the vicinity of the QCP, both the energy and linewidth of the lowest triplet mode scale linearly with temperature, in accordance with quantum criticality. We also demonstrate that the experimental field dependence of the triplet excitation modes at the base temperature is in agreement with the bosonic and macroscopic models, but is in contrast to the fermionic and perturbative models.

Physical Review BAug 25, 20159 citations

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Mixed-order transition in the antiferromagnetic quantum Ising chain in a field

Péter Lajkó, Ferenc Iglói

The antiferromagnetic quantum Ising chain has a quantum critical point which belongs to the universality class of the transverse Ising model (TIM). When a longitudinal field ( $h$ ) is switched on, the phase transition is preserved, which turns to first order for $h/\mathrm{\ensuremath{\Gamma}}\ensuremath{\rightarrow}\ensuremath{\infty}, \mathrm{\ensuremath{\Gamma}}$ being the strength of the transverse field. Here we will re-examine the critical properties along the phase transition line. During a quantum block renormalization group calculation, the TIM fixed point for $h/\mathrm{\ensuremath{\Gamma}}>0$ is found to be unstable. Using DMRG techniques, we calculated the entanglement entropy and the spin-spin correlation function, both of which signaled a divergent correlation length at the transition point with the TIM exponents. At the same time, the bulk correlation function has a jump and the end-to-end correlation function has a discontinuous derivative at the transition point. Consequently for finite $h/\mathrm{\ensuremath{\Gamma}}$ the transition is of mixed order.The antiferromagnetic quantum Ising chain has a quantum critical point which belongs to the universality class of the transverse Ising model (TIM). When a longitudinal field ($h$) is switched on, the phase transition is preserved, which turns to first order for $h/\mathrm{\ensuremath{\Gamma}}\ensuremath{\rightarrow}\ensuremath{\infty}, \mathrm{\ensuremath{\Gamma}}$ being the strength of the transverse field. Here we will re-examine the critical properties along the phase transition line. During a quantum block renormalization group calculation, the TIM fixed point for $h/\mathrm{\ensuremath{\Gamma}}>0$ is found to be unstable. Using DMRG techniques, we calculated the entanglement entropy and the spin-spin correlation function, both of which signaled a divergent correlation length at the transition point with the TIM exponents. At the same time, the bulk correlation function has a jump and the end-to-end correlation function has a discontinuous derivative at the transition point. Consequently for finite $h/\mathrm{\ensuremath{\Gamma}}$ the transition is of mixed order.

arXiv:2012.07470Physical review. B./Physical review. BMay 4, 20219 citations

h_c=(m_1 m_2)^{1/2}

(with

m_1

and

m_2

the transverse masses) is the critical Ising point and

\tilde{h}_c^* (> h_c)

We use inelastic neutron scattering to measure magnetic fluctuations as a function of transverse magnetic field and temperature in a single crystal of the coupled Haldane chain compound

\mathrm{SrNi}{}_{2}{\text{V}}_{2}{\text{O}}_{8}

with uniaxial anisotropy. At the base temperature (2 K), spin excitations confirm a field-induced quantum phase transition from the disordered Haldane (gapped singlet) state to a gapped long-range ordered state through a quantum critical point (QCP) (${\ensuremath{\mu}}_{0}{H}_{c}\ensuremath{\approx}11.5$ T). At elevated temperatures in the vicinity of the QCP, both the energy and linewidth of the lowest triplet mode scale linearly with temperature, in accordance with quantum criticality. We also demonstrate that the experimental field dependence of the triplet excitation modes at the base temperature is in agreement with the bosonic and macroscopic models, but is in contrast to the fermionic and perturbative models.

{\mathrm{H}}_{\mathrm{c}1}

and an upper critical field

{\mathrm{H}}_{\mathrm{c}2}

, where it is gapless and has power-law correlations. We calculate the critical exponents for such a generic gapped system in the incommensurate regime at the critical field

{\mathrm{H}}_{\mathrm{c}1}

and in its vicinity. Our analysis also applies to the spin- ladder. We compute the full dynamical susceptibilities at finite temperature. We use the same to discuss the thermal broadening of various modes and obtain the low-temperature behavior of the nuclear spin relaxation rate ${\mathrm{T}}_{1}^{\mathrm{\ensuremath{-}}1}$. We discuss the results obtained here for the special cases of the dimerized chain, frustrated chain, and the spin- ladder.

We study the anisotropic Heisenberg

(\mathrm{XYZ})

spin-1/2 chain placed in a magnetic field pointing along the x axis. We use bosonization and a renormalization group analysis to show that the model has a nontrivial fixed point at a certain value of the

\mathrm{XY}

anisotropy a and the magnetic field h. Hence there is a line of critical points in the

(a,h)

plane on which the system is gapless, even though the Hamiltonian has no continuous symmetry. The quantum critical line corresponds to a spin-flip transition; it separates two gapped phases in one of which the

{Z}_{2}

symmetry of the Hamiltonian is broken. Our study has a bearing on one of the transitions of the axial next-nearest neighbor Ising chain in a transverse magnetic field. We also discuss the properties of the model when the magnetic field is increased further, in particular, the disorder line on which the ground state is a direct product of single spin states.

c=1/2

c = 1 / 2 , which terminates at a critical end-point where

c=7/10

c = 7 / 10 . Calculating the critical exponents of the order parameter, neutral gap and spin–spin-correlation function, we find

\beta =1/8

β = 1 / 8 (1/24),

\nu =1

ν = 1 (5/9), and

\eta =1/4

η = 1 / 4 (3/20), respectively, which proves the Ising (tricritical Ising) universality class in accordance with field-theoretical predictions.

The antiferromagnetic quantum Ising chain has a quantum critical point which belongs to the universality class of the transverse Ising model (TIM). When a longitudinal field (

h

) is switched on, the phase transition is preserved, which turns to first order for $h/\mathrm{\ensuremath{\Gamma}}\ensuremath{\rightarrow}\ensuremath{\infty}, \mathrm{\ensuremath{\Gamma}}$ being the strength of the transverse field. Here we will re-examine the critical properties along the phase transition line. During a quantum block renormalization group calculation, the TIM fixed point for $h/\mathrm{\ensuremath{\Gamma}}>0$ is found to be unstable. Using DMRG techniques, we calculated the entanglement entropy and the spin-spin correlation function, both of which signaled a divergent correlation length at the transition point with the TIM exponents. At the same time, the bulk correlation function has a jump and the end-to-end correlation function has a discontinuous derivative at the transition point. Consequently for finite $h/\mathrm{\ensuremath{\Gamma}}$ the transition is of mixed order.