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arXiv:1205.3400 · Physical Review B

Quantum criticality with multiple dynamics

Tobias Meng, Achim Rosch, Markus Garst

Published Sep 6, 2012• 8 citations
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Abstract

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, ${z}_{e}=\frac{{z}_{>}}{\ensuremath{\nu}({z}_{>}\ensuremath{-}{z}_{<})+1}$.

DOI
10.1103/physrevb.86.125107

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Quantum critical systems with multiple dynamics possess not only one but several time scales, \ensuremathτi\ensuremath∼\ensuremathξzi{\ensuremath{\tau}}_{i}\ensuremath{\sim}{\ensuremath{\xi}}^{{z}_{i}}\ensuremathτi​\ensuremath∼\ensuremathξzi​, which diverge with the correlation length \ensuremathξ\ensuremath{\xi}\ensuremathξ. We investigate how scaling predictions are modified for the simplest case of multiple dynamics characterized by two dynamical critical exponents, z>{z}_{>}z>​ and z<{z}_{<}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 zi{z}_{i}zi​. As an example, we study generalized \ensuremathΦ4{\ensuremath{\Phi}}^{4}\ensuremathΦ4 theories with multiple dynamics below their upper critical dimension, d+z<<4d+{z}_{<}<4d+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, ze=z>\ensuremathν(z>\ensuremath−z<)+1{z}_{e}=\frac{{z}_{>}}{\ensuremath{\nu}({z}_{>}\ensuremath{-}{z}_{<})+1}ze​=\ensuremathν(z>​\ensuremath−z<​)+1z>​​.
As a paradigmatic example of multiscale quantum criticality, we consider the Pomeranchuk instability of an isotropic Fermi liquid in two spatial dimensions, d=2d=2d=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=2z=2z=2 and the other one is Landau damped with z=3z=3z=3, thus giving rise to multiple dynamical scales. We find that at temperature T=0T=0T=0, the ballistic mode governs the low-energy structure of the theory as it possesses the smaller effective dimension d+zd+zd+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 TTT 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=3z=3z=3 mode, the critical compressibility is found to be dominated by the z=2z=2z=2 fluctuations.
We investigate the short-time quantum critical dynamics in the imaginary-time relaxation processes of finite-size systems. Universal scaling behaviors exist in the imaginary-time evolution. In particular, the system undergoes a critical initial slip stage characterized by an exponent $\ensuremath{\theta}$, in which an initial power-law increase emerges in the imaginary-time correlation function when the initial state has a zero order parameter and a vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo (QMC) simulations. In the one-dimensional case, we locate the quantum critical point at (h/J)c=1.00003(8){(h/J)}_{c}=1.000\phantom{\rule{0.16em}{0ex}}03(8)(h/J)c​=1.00003(8) in the thermodynamic limit, and we estimate the critical initial slip exponent $\ensuremath{\theta}=0.3734(2)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)$ by analyzing data on chains of length L=32−−256L=32--256L=32−−256 and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by 3.04451(7)3.044\phantom{\rule{0.16em}{0ex}}51(7)3.04451(7) in the thermodynamic limit, while the critical exponents are $\ensuremath{\theta}=0.209(4)$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)$ estimated by data on systems of size L=24L=24L=24--64 and 32--64, respectively. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts due to the essential differences between classical and quantum dynamics. The short-time critical dynamics and the imaginary-time relaxation QMC approach can be readily adapted to various models.
Many-body systems with multiple emergent time scales arise in various contexts, including classical critical systems, correlated quantum materials, and ultracold atoms. We investigate such nontrivial quantum dynamics in a different setting: a spin-1 bilinear-biquadratic chain. It has a solvable entangled ground state, but a gapless excitation spectrum that is poorly understood. By using large-scale density matrix renormalization group simulations, we find that the lowest excitations have a dynamical exponent zzz that varies from 2 to 3.2 as we vary a coupling in the Hamiltonian. We find an additional gapless mode with a continuously varying exponent $2\ensuremath{\le}z<2.7$, which establishes the presence of multiple dynamics. In order to explain these striking properties, we construct a continuum wave function for the ground state, which correctly describes the correlations and entanglement properties. We also give a continuum parent Hamiltonian, but show that additional ingredients are needed to capture the excitations of the chain. By using an exact mapping to the nonequilibrium dynamics of a classical spin chain, we find that the large dynamical exponent is due to subdiffusive spin motion. Finally, we discuss the connections to other spin chains and to a family of quantum critical models in two dimensions.
We study a novel type of quantum criticality of the Lifshitz ${\ensuremath{\varphi}}^{4}$ theory below the upper critical dimension du=z+dc=8{d}_{u}=z+{d}_{c}=8du​=z+dc​=8, where the dynamic critical exponent z=4z=4z=4 and the spatial upper critical dimension dc=4{d}_{c}=4dc​=4. Two fixed points, one Gaussian and the other non-Gaussian, are identified with zero and finite interaction strengths, respectively. At zero temperature the particle density exhibits different power-law dependences on the chemical potential in the weak- and strong-interaction regions. At finite temperatures, critical behaviors in the quantum disordered region are mainly controlled by the chemical potential. In contrast, in the quantum critical region critical scalings are determined by temperature. The scaling ansatz remains valid in the strong-interaction limit for the chemical potential, correlation length, and particle density, while it breaks down in the weak-interaction one. Approaching the upper critical dimension, physical quantities develop logarithmic dependence on dimensionality in the strong-interaction region. These results are applied to spin-orbit coupled bosonic systems, leading to predictions testable by future experiments.
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Conditional mutual information (CMI) characterizes dependence between two regions conditioned on an intervening buffer. Its characteristic decay length, the Markov length, has recently been identified as a diverging length scale at mixed-state phase transitions. Here, we investigate CMI dynamics in one-dimensional classical stochastic models and propose a universal scaling form for systems whose Markov length diverges with time. We derive this form in the inactive phase of parity-conserving directed percolation, where domain-wall coarsening generates a growing length scale and a self-similar probability distribution. We introduce a Hankel-based truncation scheme that enables matrix-product-state simulations at long times and over large spatial regions. Using this method, we verify the proposed scaling both in the inactive phase and at the nonequilibrium critical point, where the dynamical critical exponent governs Markov-length growth. More broadly, the same scaling form captures cases in which collective inference sets the diverging Markov length. Despite these distinct mechanisms, the CMI between fixed microscopic regions exhibits a common scaling dimension of two. Finally, we extend this framework to temporal CMI, establishing an analogous relation between the strength of conditional dependence in time domain and its characteristic memory time.