Quantum critical systems with multiple dynamics possess not only one but several time scales, ${\ensuremath{\tau}}_{i}\ensuremath{\sim}{\ensuremath{\xi}}^{{z}_{i}}$, which diverge with the correlation length $\ensuremath{\xi}$. We investigate how scaling predictions are modified for the simplest case of multiple dynamics characterized by two dynamical critical exponents, ${z}_{>}$ and ${z}_{<}$ . We argue that one should distinguish the case of coupled and decoupled multiple dynamic scaling depending on whether there exists a scaling exponent which depends on both ${z}_{i}$ . As an example, we study generalized ${\ensuremath{\Phi}}^{4}$ theories with multiple dynamics below their upper critical dimension, $d+{z}_{<}<4$ . We identify under which condition coupled scaling is generated. In this case the interaction of quantum and classical fluctuations leads to an emergent dynamical exponent, ${z}_{e}=\frac{{z}_{>}}{\ensuremath{\nu}({z}_{>}\ensuremath{-}{z}_{<})+1}$.Quantum critical systems with multiple dynamics possess not only one but several time scales, ${\ensuremath{\tau}}_{i}\ensuremath{\sim}{\ensuremath{\xi}}^{{z}_{i}}$, which diverge with the correlation length $\ensuremath{\xi}$. We investigate how scaling predictions are modified for the simplest case of multiple dynamics characterized by two dynamical critical exponents, ${z}_{>}$ and ${z}_{<}$. We argue that one should distinguish the case of coupled and decoupled multiple dynamic scaling depending on whether there exists a scaling exponent which depends on both ${z}_{i}$. As an example, we study generalized ${\ensuremath{\Phi}}^{4}$ theories with multiple dynamics below their upper critical dimension, $d+{z}_{<}<4$. We identify under which condition coupled scaling is generated. In this case the interaction of quantum and classical fluctuations leads to an emergent dynamical exponent, ${z}_{e}=\frac{{z}_{>}}{\ensuremath{\nu}({z}_{>}\ensuremath{-}{z}_{<})+1}$.

arXiv:1205.3400Physical Review BSep 6, 20128 citations

Three studies in quantum criticality

R. R. Ramazashvili

Jan 1, 19980 citations

Scaling Theory of Quantum Critical Phenomena

World Scientific lecture notes in physicsJan 1, 20010 citations

Multiscale quantum criticality: Pomeranchuk instability in isotropic metals

Margit Zacharias, P. Wölfle, Markus Garst

As a paradigmatic example of multiscale quantum criticality, we consider the Pomeranchuk instability of an isotropic Fermi liquid in two spatial dimensions, $d=2$ . The corresponding Ginzburg-Landau theory for the quadrupolar fluctuations of the Fermi surface consists of two coupled modes, critical at the same point, and characterized by different dynamical exponents: one being ballistic with dynamical exponent $z=2$ and the other one is Landau damped with $z=3$ , thus giving rise to multiple dynamical scales. We find that at temperature $T=0$ , the ballistic mode governs the low-energy structure of the theory as it possesses the smaller effective dimension $d+z$ . Its self-interaction leads to logarithmic singularities, which we treat with the help of the renormalization group. At finite temperature, the coexistence of two different dynamical scales gives rise to a modified quantum-to-classical crossover. It extends over a parametrically large regime with intricate interactions of quantum and classical fluctuations leading to a universal $T$ dependence of the correlation length independent of the interaction amplitude. The multiple scales are also reflected in the phase diagram and in the critical thermodynamics. In particular, we find that the latter cannot be interpreted in terms of only a single dynamical exponent whereas, e.g., the critical specific heat is determined by the $z=3$ mode, the critical compressibility is found to be dominated by the $z=2$ fluctuations.As a paradigmatic example of multiscale quantum criticality, we consider the Pomeranchuk instability of an isotropic Fermi liquid in two spatial dimensions, $d=2$. The corresponding Ginzburg-Landau theory for the quadrupolar fluctuations of the Fermi surface consists of two coupled modes, critical at the same point, and characterized by different dynamical exponents: one being ballistic with dynamical exponent $z=2$ and the other one is Landau damped with $z=3$, thus giving rise to multiple dynamical scales. We find that at temperature $T=0$, the ballistic mode governs the low-energy structure of the theory as it possesses the smaller effective dimension $d+z$. Its self-interaction leads to logarithmic singularities, which we treat with the help of the renormalization group. At finite temperature, the coexistence of two different dynamical scales gives rise to a modified quantum-to-classical crossover. It extends over a parametrically large regime with intricate interactions of quantum and classical fluctuations leading to a universal $T$ dependence of the correlation length independent of the interaction amplitude. The multiple scales are also reflected in the phase diagram and in the critical thermodynamics. In particular, we find that the latter cannot be interpreted in terms of only a single dynamical exponent whereas, e.g., the critical specific heat is determined by the $z=3$ mode, the critical compressibility is found to be dominated by the $z=2$ fluctuations.

arXiv:0907.0489Physical Review BOct 12, 200957 citations

Quantum Criticality from Within

ScienceMar 4, 20040 citations

Multiplicative logarithmic corrections to quantum criticality in three-dimensional dimerized antiferromagnets

Yan Qin, Bruce Normand, Anders W. Sandvik +1 more

We investigate the quantum phase transition in an $S=1/2$ dimerized Heisenberg antiferromagnet in three spatial dimensions. By performing large-scale quantum Monte Carlo simulations and detailed finite-size scaling analyses, we obtain high-precision results for the quantum critical properties at the transition from the magnetically disordered dimer-singlet phase to the antiferromagnetically ordered N\'eel phase. This transition breaks $\text{O}(N)$ symmetry with $N=3$ in $D=3+1$ dimensions. This is the upper critical dimension, where multiplicative logarithmic corrections to the leading mean-field critical properties are expected; we extract these corrections, establishing their precise forms for both the zero-temperature staggered magnetization ${m}_{s}$ and the N\'eel temperature ${T}_{N}$ . We present a scaling ansatz for ${T}_{N}$ , including logarithmic corrections, which agrees with our data and indicates exact linearity with ${m}_{s}$ , implying a complete decoupling of quantum and thermal fluctuation effects even arbitrarily close to the quantum critical point. We also demonstrate the predicted $N$ -independent leading and subleading logarithmic corrections in the size dependence of the staggered magnetic susceptibility. These logarithmic scaling forms have not previously been identified or verified by unbiased numerical methods, and we discuss their relevance to experimental studies of dimerized quantum antiferromagnets such as ${\mathrm{TlCuCl}}_{3}$ .We investigate the quantum phase transition in an $S=1/2$ dimerized Heisenberg antiferromagnet in three spatial dimensions. By performing large-scale quantum Monte Carlo simulations and detailed finite-size scaling analyses, we obtain high-precision results for the quantum critical properties at the transition from the magnetically disordered dimer-singlet phase to the antiferromagnetically ordered N\'eel phase. This transition breaks $\text{O}(N)$ symmetry with $N=3$ in $D=3+1$ dimensions. This is the upper critical dimension, where multiplicative logarithmic corrections to the leading mean-field critical properties are expected; we extract these corrections, establishing their precise forms for both the zero-temperature staggered magnetization ${m}_{s}$ and the N\'eel temperature ${T}_{N}$. We present a scaling ansatz for ${T}_{N}$, including logarithmic corrections, which agrees with our data and indicates exact linearity with ${m}_{s}$, implying a complete decoupling of quantum and thermal fluctuation effects even arbitrarily close to the quantum critical point. We also demonstrate the predicted $N$-independent leading and subleading logarithmic corrections in the size dependence of the staggered magnetic susceptibility. These logarithmic scaling forms have not previously been identified or verified by unbiased numerical methods, and we discuss their relevance to experimental studies of dimerized quantum antiferromagnets such as ${\mathrm{TlCuCl}}_{3}$.

arXiv:1506.06073Physical Review BDec 2, 201529 citations

{z}_{>}

and

{z}_{<}

. We argue that one should distinguish the case of coupled and decoupled multiple dynamic scaling depending on whether there exists a scaling exponent which depends on both

{z}_{i}

. As an example, we study generalized ${\ensuremath{\Phi}}^{4}$ theories with multiple dynamics below their upper critical dimension,

d+{z}_{<}<4

. We identify under which condition coupled scaling is generated. In this case the interaction of quantum and classical fluctuations leads to an emergent dynamical exponent, ${z}_{e}=\frac{{z}_{>}}{\ensuremath{\nu}({z}_{>}\ensuremath{-}{z}_{<})+1}$.

As a paradigmatic example of multiscale quantum criticality, we consider the Pomeranchuk instability of an isotropic Fermi liquid in two spatial dimensions,

d=2

. The corresponding Ginzburg-Landau theory for the quadrupolar fluctuations of the Fermi surface consists of two coupled modes, critical at the same point, and characterized by different dynamical exponents: one being ballistic with dynamical exponent

z=2

and the other one is Landau damped with

z=3

, thus giving rise to multiple dynamical scales. We find that at temperature

T=0

, the ballistic mode governs the low-energy structure of the theory as it possesses the smaller effective dimension

d+z

. Its self-interaction leads to logarithmic singularities, which we treat with the help of the renormalization group. At finite temperature, the coexistence of two different dynamical scales gives rise to a modified quantum-to-classical crossover. It extends over a parametrically large regime with intricate interactions of quantum and classical fluctuations leading to a universal

T

dependence of the correlation length independent of the interaction amplitude. The multiple scales are also reflected in the phase diagram and in the critical thermodynamics. In particular, we find that the latter cannot be interpreted in terms of only a single dynamical exponent whereas, e.g., the critical specific heat is determined by the

z=3

mode, the critical compressibility is found to be dominated by the

z=2

fluctuations.

We investigate the quantum phase transition in an

S=1/2

dimerized Heisenberg antiferromagnet in three spatial dimensions. By performing large-scale quantum Monte Carlo simulations and detailed finite-size scaling analyses, we obtain high-precision results for the quantum critical properties at the transition from the magnetically disordered dimer-singlet phase to the antiferromagnetically ordered N\'eel phase. This transition breaks

\text{O}(N)

symmetry with

N=3

D=3+1

dimensions. This is the upper critical dimension, where multiplicative logarithmic corrections to the leading mean-field critical properties are expected; we extract these corrections, establishing their precise forms for both the zero-temperature staggered magnetization

{m}_{s}

and the N\'eel temperature

{T}_{N}

. We present a scaling ansatz for

{T}_{N}

, including logarithmic corrections, which agrees with our data and indicates exact linearity with

{m}_{s}

, implying a complete decoupling of quantum and thermal fluctuation effects even arbitrarily close to the quantum critical point. We also demonstrate the predicted

N

-independent leading and subleading logarithmic corrections in the size dependence of the staggered magnetic susceptibility. These logarithmic scaling forms have not previously been identified or verified by unbiased numerical methods, and we discuss their relevance to experimental studies of dimerized quantum antiferromagnets such as

{\mathrm{TlCuCl}}_{3}