Physical review. B, Condensed matter

Quantum critical dynamics with mode coupling. II. Antiferromagnets

W. L. Ridgway

Published Jul 1, 1980• 0 citations

Abstract

We extend a previ\'ous study of quantum critical dynamics in ferromagnets to the isotropic antiferromagnet. The model we consider here also includes so-called "mode-mode coupling" terms. The quantum limit is found to have the characteristic dimensionality ${d}_{Q}=6$ versus the classical value ${d}_{C}=4$. A renormalization-group expansion to order $\ensuremath{\epsilon}=6\ensuremath{-}d$ is used to evaluate spin and staggered spin transport quantities ${\ensuremath{\Gamma}}_{M}$ and ${\ensuremath{\Gamma}}_{N}$ in the limit of low critical temperature ${T}_{N}$. Dynamic scaling is found to break down in the quantum case and scaling regions of quantum, classical, and "mixed" statistics are discussed in light of this breakdown.

DOI: 10.1103/physrevb.22.245

Related papers

Quantum critical dynamics with mode coupling. I. Ferromagnets

W. L. Ridgway, John Hertz

Physical review. B, Condensed matterJul 1, 19802 citations

Critical spin dynamics of Heisenberg ferromagnets revisited

Dmytro Tarasevych, Peter Kopietz

arXiv:2111.07621Physical review. B./Physical review. BJan 4, 20229 citations

Hydrodynamic modes and reversible mode couplings

Uwe Claus Täuber

Cambridge University Press eBooksMar 6, 20140 citations

Critical dynamics of isotropic antiferromagnets using renormalization-group methods: T≥TN

R. Feeedman, Gene F. Mazenko

Physical review. B, Solid stateJun 1, 197648 citations

Dynamical kinetic energy quenching in antiferromagnetic quantum critical metals

Anton Borissov, Vladimir Calvera, Sung-Sik Lee

arXiv:2408.07158Physical review. B./Physical review. BApr 10, 20250 citations

Quantum Ferrimagnets

Ramin Abolfath, H. Hamidian, A. Langari

arXiv:cond-mat/9901063Jan 8, 19991 citation

Physical review. B, Condensed matter

Quantum critical dynamics with mode coupling. II. Antiferromagnets

W. L. Ridgway

Published Jul 1, 1980• 0 citations

Publisher

Abstract

We extend a previ'ous study of quantum critical dynamics in ferromagnets to the isotropic antiferromagnet. The model we consider here also includes so-called "mode-mode coupling" terms. The quantum limit is found to have the characteristic dimensionality

{d}_{Q}=6

versus the classical value

{d}_{C}=4

. A renormalization-group expansion to order

\ensuremath{\epsilon}=6\ensuremath{-}d

is used to evaluate spin and staggered spin transport quantities

{\ensuremath{\Gamma}}_{M}

and

{\ensuremath{\Gamma}}_{N}

in the limit of low critical temperature

{T}_{N}

. Dynamic scaling is found to break down in the quantum case and scaling regions of quantum, classical, and "mixed" statistics are discussed in light of this breakdown.

DOI: 10.1103/physrevb.22.245

Related papers

Quantum critical dynamics with mode coupling. I. Ferromagnets

W. L. Ridgway, John Hertz

We examine the low-temperature critical dynamics of two ferromagnetic models of the disordered state having "mode-mode coupling." A method is developed to incorporate mode-mode coupling contributions in an $O(\ensuremath{\epsilon})$ momentum-space renormalization-group analysis of a quantum system. For each model, both quantum and classical critical regions may exist. The characteristic dimensionality changes from $d=6$ classically to $d=10$ in the quantum limit. The dynamic critical index associated with the quantum fixed point is $z=4\ensuremath{-}\frac{1}{3}(10\ensuremath{-}d)$. The scaling behavior of the spin transport coefficient $\ensuremath{\Gamma}$ is examined in the $T=0$ limit. We discuss the properties of the quantum-to-classical crossover, including the "metastable" nature of the quantum fixed points.We examine the low-temperature critical dynamics of two ferromagnetic models of the disordered state having "mode-mode coupling." A method is developed to incorporate mode-mode coupling contributions in an $O(\ensuremath{\epsilon})$ momentum-space renormalization-group analysis of a quantum system. For each model, both quantum and classical critical regions may exist. The characteristic dimensionality changes from $d=6$ classically to $d=10$ in the quantum limit. The dynamic critical index associated with the quantum fixed point is $z=4\ensuremath{-}\frac{1}{3}(10\ensuremath{-}d)$. The scaling behavior of the spin transport coefficient $\ensuremath{\Gamma}$ is examined in the $T=0$ limit. We discuss the properties of the quantum-to-classical crossover, including the "metastable" nature of the quantum fixed points.

Physical review. B, Condensed matterJul 1, 19802 citations

Critical spin dynamics of Heisenberg ferromagnets revisited

Dmytro Tarasevych, Peter Kopietz

We calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ in the paramagnetic regime of quantum Heisenberg ferromagnets for temperatures $T$ close to the critical temperature ${T}_{c}$ using our recently developed functional renormalization group approach to quantum spin systems. In $d=3$ dimensions, we find that for small momenta $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ the dynamic structure factor assumes the scaling form $S(\mathbit{k},\ensuremath{\omega})=(\ensuremath{\tau}TG(\mathbit{k})/\ensuremath{\pi})\mathrm{\ensuremath{\Phi}}(k\ensuremath{\xi},\ensuremath{\omega}\ensuremath{\tau})$, where $G(\mathbit{k})$ is the static spin-spin correlation function, $\ensuremath{\xi}$ is the correlation length, and the characteristic timescale $\ensuremath{\tau}$ is proportional to ${\ensuremath{\xi}}^{5/2}$. We explicitly calculate the dynamic scaling function $\mathrm{\ensuremath{\Phi}}(x,y)$ and find satisfactory agreement with neutron scattering experiments probing the critical spin dynamics in EuO and EuS. Precisely at the critical point where $\ensuremath{\xi}=\ensuremath{\infty}$, our result for the dynamic structure factor can be written as $S(\mathbit{k},\ensuremath{\omega})={(\ensuremath{\pi}{\ensuremath{\omega}}_{k})}^{\ensuremath{-}1}{T}_{c}G(\mathbit{k}){\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\omega}/{\ensuremath{\omega}}_{k})$, where ${\ensuremath{\omega}}_{k}\ensuremath{\propto}{k}^{5/2}$. We find that ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ vanishes as ${\ensuremath{\nu}}^{\ensuremath{-}13/5}$ for large $\ensuremath{\nu}$, and as ${\ensuremath{\nu}}^{3/5}$ for small $\ensuremath{\nu}$. While the large-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ is consistent with calculations based on mode-coupling theory and with perturbative renormalization group calculations to second order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, our result for small frequencies disagrees with previous calculations. We argue that until now, neither experiments nor numerical simulations have been sufficiently accurate to determine the low-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$. We also calculate the low-temperature behavior of $S(\mathbit{k},\ensuremath{\omega})$ in one- and two-dimensional ferromagnets and find that it satisfies dynamic scaling with exponent $z=2$ and exhibits a pseudogap for small frequencies.We calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ in the paramagnetic regime of quantum Heisenberg ferromagnets for temperatures $T$ close to the critical temperature ${T}_{c}$ using our recently developed functional renormalization group approach to quantum spin systems. In $d=3$ dimensions, we find that for small momenta $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ the dynamic structure factor assumes the scaling form $S(\mathbit{k},\ensuremath{\omega})=(\ensuremath{\tau}TG(\mathbit{k})/\ensuremath{\pi})\mathrm{\ensuremath{\Phi}}(k\ensuremath{\xi},\ensuremath{\omega}\ensuremath{\tau})$, where $G(\mathbit{k})$ is the static spin-spin correlation function, $\ensuremath{\xi}$ is the correlation length, and the characteristic timescale $\ensuremath{\tau}$ is proportional to ${\ensuremath{\xi}}^{5/2}$. We explicitly calculate the dynamic scaling function $\mathrm{\ensuremath{\Phi}}(x,y)$ and find satisfactory agreement with neutron scattering experiments probing the critical spin dynamics in EuO and EuS. Precisely at the critical point where $\ensuremath{\xi}=\ensuremath{\infty}$, our result for the dynamic structure factor can be written as $S(\mathbit{k},\ensuremath{\omega})={(\ensuremath{\pi}{\ensuremath{\omega}}_{k})}^{\ensuremath{-}1}{T}_{c}G(\mathbit{k}){\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\omega}/{\ensuremath{\omega}}_{k})$, where ${\ensuremath{\omega}}_{k}\ensuremath{\propto}{k}^{5/2}$. We find that ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ vanishes as ${\ensuremath{\nu}}^{\ensuremath{-}13/5}$ for large $\ensuremath{\nu}$, and as ${\ensuremath{\nu}}^{3/5}$ for small $\ensuremath{\nu}$. While the large-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ is consistent with calculations based on mode-coupling theory and with perturbative renormalization group calculations to second order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, our result for small frequencies disagrees with previous calculations. We argue that until now, neither experiments nor numerical simulations have been sufficiently accurate to determine the low-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$. We also calculate the low-temperature behavior of $S(\mathbit{k},\ensuremath{\omega})$ in one- and two-dimensional ferromagnets and find that it satisfies dynamic scaling with exponent $z=2$ and exhibits a pseudogap for small frequencies.

arXiv:2111.07621Physical review. B./Physical review. BJan 4, 20229 citations

Hydrodynamic modes and reversible mode couplings

Uwe Claus Täuber

Equipped with the field theory representation of non-linear Langevin equations, the tools of dynamic perturbation theory, and the dynamic renormalization group introduced in Chapters 4 and 5, we are now in the position to revisit models for dynamic critical behavior that entail reversible mode couplings and other conserved hydrodynamic modes. We have already encountered some of these in Section 3.3. In models C and D, respectively, a non-conserved or conserved n -component order parameter is coupled to a conserved scalar field, the energy density. Through a systematic renormalization group analysis, we may critically assess the earlier predictions from scaling theory, and discuss the stability of fixed points characterized by strong dynamic scaling, wherein the order parameter and conserved non-critical mode fluctuate with equal rates, and weak dynamic scaling regimes, where these characteristic time scales differ. Next we investigate isotropic ferromagnets (model J), with the conserved spin density subject to reversible precession in addition to diffusive relaxation. Exploiting rotational invariance, we can now firmly establish the scaling relation z = ( d + 2 − η )/2. Similar symmetry arguments yield a scaling relation for the dynamic exponents associated with the order parameter and the non-critical fields in the O ( n )-symmetric SSS model that encompasses model E for planar ferromagnets and superfluid helium 4 (for n = 2), and model G for isotropic antiferromagnets ( n = 3). There exist competing strong- and weak-scaling fixed points, with the former stable to one-loop order, and characterized by z = d /2 for all slow modes.Equipped with the field theory representation of non-linear Langevin equations, the tools of dynamic perturbation theory, and the dynamic renormalization group introduced in Chapters 4 and 5, we are now in the position to revisit models for dynamic critical behavior that entail reversible mode couplings and other conserved hydrodynamic modes. We have already encountered some of these in Section 3.3. In models C and D, respectively, a non-conserved or conserved n -component order parameter is coupled to a conserved scalar field, the energy density. Through a systematic renormalization group analysis, we may critically assess the earlier predictions from scaling theory, and discuss the stability of fixed points characterized by strong dynamic scaling, wherein the order parameter and conserved non-critical mode fluctuate with equal rates, and weak dynamic scaling regimes, where these characteristic time scales differ. Next we investigate isotropic ferromagnets (model J), with the conserved spin density subject to reversible precession in addition to diffusive relaxation. Exploiting rotational invariance, we can now firmly establish the scaling relation z = ( d + 2 − η )/2. Similar symmetry arguments yield a scaling relation for the dynamic exponents associated with the order parameter and the non-critical fields in the O ( n )-symmetric SSS model that encompasses model E for planar ferromagnets and superfluid helium 4 (for n = 2), and model G for isotropic antiferromagnets ( n = 3). There exist competing strong- and weak-scaling fixed points, with the former stable to one-loop order, and characterized by z = d /2 for all slow modes.

Cambridge University Press eBooksMar 6, 20140 citations

Critical dynamics of isotropic antiferromagnets using renormalization-group methods: T≥TN

R. Feeedman, Gene F. Mazenko

We study a model equation of motion describing the dynamics of isotropic antiferromagnets near the critical point. We perform a renormalization-group analysis of this equation of motion correct to $O(\ensuremath{\epsilon})$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, finding recursion relations and analyzing their fixed points and their stability. For the one stable fixed point we obtain the dynamical index $z=\frac{d}{2}=2\ensuremath{-}\frac{\ensuremath{\epsilon}}{2}$. We also calculate the response and correlation functions in the scaling region using perturbation theory correct to $O(\ensuremath{\epsilon})$. Our calculation gives analytical forms for the correlation functions for the staggered magnetization and the magnetization. These correlation functions can be written in the dynamical scaling form. The shape function that characterizes the frequency spectrum for the staggered magnetization is Lorentzian in the hydrodynamical regime, but shows fluctuation induced peaks at and near the N\ifmmode\dagger\else\textdagger\fi{}eel temperature. The shape function for the magnetization is essentially Lorentzian for all $x=q\ensuremath{\xi}$ (where $q$ is the wave number and $\ensuremath{\xi}$ is the correlation length) and shows a narrow lydrodynamical width as $x\ensuremath{\rightarrow}0$.We study a model equation of motion describing the dynamics of isotropic antiferromagnets near the critical point. We perform a renormalization-group analysis of this equation of motion correct to $O(\ensuremath{\epsilon})$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, finding recursion relations and analyzing their fixed points and their stability. For the one stable fixed point we obtain the dynamical index $z=\frac{d}{2}=2\ensuremath{-}\frac{\ensuremath{\epsilon}}{2}$. We also calculate the response and correlation functions in the scaling region using perturbation theory correct to $O(\ensuremath{\epsilon})$. Our calculation gives analytical forms for the correlation functions for the staggered magnetization and the magnetization. These correlation functions can be written in the dynamical scaling form. The shape function that characterizes the frequency spectrum for the staggered magnetization is Lorentzian in the hydrodynamical regime, but shows fluctuation induced peaks at and near the N\ifmmode\dagger\else\textdagger\fi{}eel temperature. The shape function for the magnetization is essentially Lorentzian for all $x=q\ensuremath{\xi}$ (where $q$ is the wave number and $\ensuremath{\xi}$ is the correlation length) and shows a narrow lydrodynamical width as $x\ensuremath{\rightarrow}0$.

Physical review. B, Solid stateJun 1, 197648 citations

Dynamical kinetic energy quenching in antiferromagnetic quantum critical metals

Anton Borissov, Vladimir Calvera, Sung-Sik Lee

We study the dynamics of critical spin fluctuations and hot electrons at the metallic antiferromagnetic quantum critical points with ${Z}_{2}$ and O(2) spin symmetries, building on earlier works on the O(3) symmetric theory. The interacting theories in $2+1$ dimensions are approached from $3+1$ -dimensional theories in the $\ensuremath{\epsilon}$ expansion that tunes the codimension of Fermi surface as a control parameter. The low-energy physics of the ${Z}_{2}$ and O(2) theories qualitatively differ from each other and also from that of the O(3) theory. The difference is caused by higher-order quantum corrections beyond the one-loop order that are important even to the leading order in $\ensuremath{\epsilon}$. The naive loop expansion breaks down due to dynamical quenching of kinetic energy: the speed of the collective mode ( $c$ ) and the Fermi velocity perpendicular to the magnetic ordering vector ( $v$ ) become vanishingly small at low energies. What sets the three theories apart is the hierarchy that emerges between the quenched kinetic terms. At the infrared fixed point, $c/v$ becomes 0, 1, and $\ensuremath{\infty}$ in the ${Z}_{2}$ , O(2), and O(3) theories, respectively. At intermediate energy scales, the slow renormalization group (RG) flows of $c$ and $v$ toward their fixed point values create approximate scale invariance controlled by approximate marginal parameters. The manifold of those quasifixed points and the RG flow therein determines crossovers from scaling behaviors with transient critical exponents at intermediate energy scales to the universal scaling in the low-energy limit. If the symmetry group is viewed as a tuning parameter, then the O(2) theory corresponds to a multicritical point which has one additional quasimarginal parameter than the other two theories.We study the dynamics of critical spin fluctuations and hot electrons at the metallic antiferromagnetic quantum critical points with ${Z}_{2}$ and O(2) spin symmetries, building on earlier works on the O(3) symmetric theory. The interacting theories in $2+1$ dimensions are approached from $3+1$-dimensional theories in the $\ensuremath{\epsilon}$ expansion that tunes the codimension of Fermi surface as a control parameter. The low-energy physics of the ${Z}_{2}$ and O(2) theories qualitatively differ from each other and also from that of the O(3) theory. The difference is caused by higher-order quantum corrections beyond the one-loop order that are important even to the leading order in $\ensuremath{\epsilon}$. The naive loop expansion breaks down due to dynamical quenching of kinetic energy: the speed of the collective mode ($c$) and the Fermi velocity perpendicular to the magnetic ordering vector ($v$) become vanishingly small at low energies. What sets the three theories apart is the hierarchy that emerges between the quenched kinetic terms. At the infrared fixed point, $c/v$ becomes 0, 1, and $\ensuremath{\infty}$ in the ${Z}_{2}$, O(2), and O(3) theories, respectively. At intermediate energy scales, the slow renormalization group (RG) flows of $c$ and $v$ toward their fixed point values create approximate scale invariance controlled by approximate marginal parameters. The manifold of those quasifixed points and the RG flow therein determines crossovers from scaling behaviors with transient critical exponents at intermediate energy scales to the universal scaling in the low-energy limit. If the symmetry group is viewed as a tuning parameter, then the O(2) theory corresponds to a multicritical point which has one additional quasimarginal parameter than the other two theories.

arXiv:2408.07158Physical review. B./Physical review. BApr 10, 20250 citations

Quantum Ferrimagnets

Ramin Abolfath, H. Hamidian, A. Langari

We study quantum ferrimagnets in one, two, and three dimensions by using a variety of methods and approximations. These include: (i) a treatment based on the spin coherent state path-integral formulation of quantum ferrimagnets by taking into account the leading order quantum and thermal fluctuations (ii) a field-theoretical (non-linear $σ$ -model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h ( $h_{c1} < h < h_{c2}$ ) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.We study quantum ferrimagnets in one, two, and three dimensions by using a variety of methods and approximations. These include: (i) a treatment based on the spin coherent state path-integral formulation of quantum ferrimagnets by taking into account the leading order quantum and thermal fluctuations (ii) a field-theoretical (non-linear $σ$-model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h ($h_{c1} < h < h_{c2}$) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.

arXiv:cond-mat/9901063Jan 8, 19991 citation

{d}_{Q}=6

versus the classical value

{d}_{C}=4

. A renormalization-group expansion to order

\ensuremath{\epsilon}=6\ensuremath{-}d

is used to evaluate spin and staggered spin transport quantities

{\ensuremath{\Gamma}}_{M}

and

{\ensuremath{\Gamma}}_{N}

in the limit of low critical temperature

{T}_{N}

. Dynamic scaling is found to break down in the quantum case and scaling regions of quantum, classical, and "mixed" statistics are discussed in light of this breakdown.

d=6

classically to

d=10

in the quantum limit. The dynamic critical index associated with the quantum fixed point is $z=4\ensuremath{-}\frac{1}{3}(10\ensuremath{-}d)$. The scaling behavior of the spin transport coefficient $\ensuremath{\Gamma}$ is examined in the

T=0

limit. We discuss the properties of the quantum-to-classical crossover, including the "metastable" nature of the quantum fixed points.

We calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ in the paramagnetic regime of quantum Heisenberg ferromagnets for temperatures

T

close to the critical temperature

{T}_{c}

using our recently developed functional renormalization group approach to quantum spin systems. In

d=3

dimensions, we find that for small momenta $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ the dynamic structure factor assumes the scaling form $S(\mathbit{k},\ensuremath{\omega})=(\ensuremath{\tau}TG(\mathbit{k})/\ensuremath{\pi})\mathrm{\ensuremath{\Phi}}(k\ensuremath{\xi},\ensuremath{\omega}\ensuremath{\tau})$, where $G(\mathbit{k})$ is the static spin-spin correlation function, $\ensuremath{\xi}$ is the correlation length, and the characteristic timescale $\ensuremath{\tau}$ is proportional to ${\ensuremath{\xi}}^{5/2}$. We explicitly calculate the dynamic scaling function $\mathrm{\ensuremath{\Phi}}(x,y)$ and find satisfactory agreement with neutron scattering experiments probing the critical spin dynamics in EuO and EuS. Precisely at the critical point where $\ensuremath{\xi}=\ensuremath{\infty}$, our result for the dynamic structure factor can be written as $S(\mathbit{k},\ensuremath{\omega})={(\ensuremath{\pi}{\ensuremath{\omega}}_{k})}^{\ensuremath{-}1}{T}_{c}G(\mathbit{k}){\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\omega}/{\ensuremath{\omega}}_{k})$, where ${\ensuremath{\omega}}_{k}\ensuremath{\propto}{k}^{5/2}$. We find that ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ vanishes as ${\ensuremath{\nu}}^{\ensuremath{-}13/5}$ for large $\ensuremath{\nu}$, and as ${\ensuremath{\nu}}^{3/5}$ for small $\ensuremath{\nu}$. While the large-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ is consistent with calculations based on mode-coupling theory and with perturbative renormalization group calculations to second order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, our result for small frequencies disagrees with previous calculations. We argue that until now, neither experiments nor numerical simulations have been sufficiently accurate to determine the low-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$. We also calculate the low-temperature behavior of $S(\mathbit{k},\ensuremath{\omega})$ in one- and two-dimensional ferromagnets and find that it satisfies dynamic scaling with exponent

z=2

and exhibits a pseudogap for small frequencies.

q

is the wave number and $\ensuremath{\xi}$ is the correlation length) and shows a narrow lydrodynamical width as $x\ensuremath{\rightarrow}0$.

We study the dynamics of critical spin fluctuations and hot electrons at the metallic antiferromagnetic quantum critical points with

{Z}_{2}

and O(2) spin symmetries, building on earlier works on the O(3) symmetric theory. The interacting theories in

2+1

dimensions are approached from

3+1

-dimensional theories in the $\ensuremath{\epsilon}$ expansion that tunes the codimension of Fermi surface as a control parameter. The low-energy physics of the

{Z}_{2}

and O(2) theories qualitatively differ from each other and also from that of the O(3) theory. The difference is caused by higher-order quantum corrections beyond the one-loop order that are important even to the leading order in $\ensuremath{\epsilon}$. The naive loop expansion breaks down due to dynamical quenching of kinetic energy: the speed of the collective mode (

c

) and the Fermi velocity perpendicular to the magnetic ordering vector (

v

) become vanishingly small at low energies. What sets the three theories apart is the hierarchy that emerges between the quenched kinetic terms. At the infrared fixed point,

c/v

becomes 0, 1, and $\ensuremath{\infty}$ in the

{Z}_{2}

, O(2), and O(3) theories, respectively. At intermediate energy scales, the slow renormalization group (RG) flows of

c

and

v

toward their fixed point values create approximate scale invariance controlled by approximate marginal parameters. The manifold of those quasifixed points and the RG flow therein determines crossovers from scaling behaviors with transient critical exponents at intermediate energy scales to the universal scaling in the low-energy limit. If the symmetry group is viewed as a tuning parameter, then the O(2) theory corresponds to a multicritical point which has one additional quasimarginal parameter than the other two theories.

σ

-model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h (

h_{c1} < h < h_{c2}

) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.