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arXiv:1909.11540 · Journal of Physics A Mathematical and Theoretical

Comment on ‘fluctuation-dominated phase ordering at a mixed order transition’

Claude Godrèche

Published Dec 24, 2020• 7 citations
arXivHTMLPDF

Abstract

Abstract Renewal processes generated by a power-law distribution of intervals with tail index less than unity are genuinely non-stationary. This issue is illustrated by a critical review of the recent paper by Barma, Majumdar and Mukamel (2019 J. Phys. A 52 254001), devoted to the investigation of the properties of a specific one-dimensional equilibrium spin system with long-range interactions. We explain why discarding the non-stationarity of the process underlying the model leads to an incorrect expression of the critical spin–spin correlation function, even when the system, subjected to periodic boundary conditions, is translation invariant.

DOI
10.1088/1751-8121/abd34a

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Fluctuation-dominated phase ordering at a mixed order transition

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Abstract Mixed order transitions are those which show a discontinuity of the order parameter as well as a divergent correlation length at the transition (critical) point. We show that the behavior of the order parameter correlation function along the transition line of mixed order transitions can change from normal critical behavior with power law decay, to fluctuation-dominated phase ordering as a parameter is varied. The defining features of fluctuation-dominated order are anomalous fluctuations which remain large in the thermodynamic limit, and correlation functions which approach a finite value through a cusp singularity as the separation scaled by the system size approaches zero. We demonstrate that fluctuation-dominated order sets in along a portion of the transition line of an Ising model with truncated long-range interactions which was earlier shown to exhibit mixed order transitions. We also argue that this connection may hold more generally.

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This Comment is dedicated to the investigation of many-body localization in a quantum Ising model with long-range power-law interactions ${r}^{\ensuremath{-}\ensuremath{\alpha}}$, relevant for a variety of systems ranging from electrons in Anderson insulators to spin excitations in chains of cold atoms. It has earlier been argued [arXiv:cond-mat/0611387 (2005); Phys. Rev. B 91, 094202 (2015)] that this model obeys the dimensional constraint suggesting the delocalization of all finite-temperature states in the thermodynamic limit for $\ensuremath{\alpha}\ensuremath{\le}2d$ in a $d$-dimensional system. This expectation conflicts with the recent numerical studies of the specific interacting spin model of Li et al. [Phys. Rev. A 94, 063625 (2016)]. To resolve this controversy we reexamine the model of Li et al. [Phys. Rev. A 94, 063625 (2016)] and demonstrate that the infinite-temperature states there obey the dimensional constraint. The earlier developed scaling theory for the critical system size required for delocalization is extended to small exponents $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}d$. The disagreements between the two works are explained by the nonstandard selection of investigated states in the ordered phase in the work of Li et al. [Phys. Rev. A 94, 063625 (2016)].

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Reply to “Comment on ‘Fluctuation-dissipation relations in the nonequilibrium critical dynamics of Ising models’ ”

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We have recently shown that in nonequilibrium spin systems at criticality the limit ${X}^{\ensuremath{\infty}}$ of the fluctuation-dissipation ratio $X(t,{t}_{w})$ for $t⪢{t}_{w}⪢1$ can be measured using observables such as magnetization or energy [Phys. Rev. E 68, 016116 (2003)]. Pleimling argues in a Comment [preceding paper, Phys. Rev. 70, 018101 (2004)] on our paper that for such observables correlation and response functions are dominated by one-time quantities dependent only on $t$, and are therefore not suitable for a determination of ${X}^{\ensuremath{\infty}}$. Using standard scaling forms of correlation and response functions, as used by Pleimling, we show that our data do have a genuine two-time dependence and allow $X(t,{t}_{w})$ and ${X}^{\ensuremath{\infty}}$ to be measured, so that Pleimling's criticisms are easily refuted. We also compare with predictions from renormalization-group calculations, which are consistent with our numerical observation of a fluctuation-dissipation plot for the magnetization that is very close to a straight line. A key point remains that coherent observables make measurements of ${X}^{\ensuremath{\infty}}$ easier than the traditionally used incoherent ones, producing fluctuation-dissipation plots whose slope is close to ${X}^{\ensuremath{\infty}}$ over a much larger range.

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The breaking of the continuous time-translation symmetry manifests, in Markovian open quantum systems, through the emergence of nonstationary dynamical phases. Systems that display nonequilibrium transitions into these phases are referred to as time crystals, and they can be realized, for example, in many-body systems governed by collective dissipation and long-ranged interactions. Here, we provide a complete analytical characterization of a boundary time-crystal phase transition. This involves exact expressions for the order parameter and for the dynamics of quantum fluctuations, which, in the time-crystalline phase, remains asymptotically non-Markovian as a consequence of the time-translation symmetry breaking. We demonstrate that boundary time crystals are intrinsically critical phases, where fluctuations exhibit a power-law divergence with time. Our results show that a dissipative time-crystal phase is far more than merely a classical nonlinear and nonstationary (limit cycle) dynamics of a macroscopic order parameter. It is rather a genuine many-body phase where the properties of correlations distinctly differ from those of stationary ones.

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Available mathematical results entail the absence of a proper ordering transition taking place at finite temperature in the case $d=1,$ $\ensuremath{\sigma}=1.1,$ where the authors of the article commented on find evidence of such a transition. Moreover, in contrast to the findings of the authors of the article, the case $d=2,$ $\ensuremath{\sigma}=0$ entails a ground-state energy per particle diverging logarithmically with increasing sample size, hence an infinite transition temperature.

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On the spectral properties of inhomogeneous Luttinger liquids

Manuel Simonato

The physics of one-dimensional systems of interacting fermions strongly differs from the picture offered by Fermi liquid theory. The low-energy properties of metallic, paramagnetic 1D Fermi systems are well described by the Luttinger liquid (LL) theory. Among other properties, the homogeneous LL is known to exhibit power-law behaviour for various correlation functions. For spin rotational invariant models, the exponents can be expressed in terms of a single parameter Kρ, which is known to be non-universal. The LL concept emerged in the study of homogeneous systems with periodic boundary condition (PBC). In these models the local spectral function shows a power-law suppression at energies asymptotically close to the chemical potential µ. Theoretically, this is very well understood and the exact LL parameter Kρ is known for a set of integrable models. In the last decades however, the interest shifted towards the investigation of LLs with PBC including impurities. The fact that these systems have been shown to scale to chains with open ends led to the study of several models where open boundary condition (OBC) are introduced. As the translational invariance is broken, the exponent of the power-law suppression close to the boundary differs from the one in the bulk. The present work is devoted to the study of the local spectral function of two lattice models of LLs with open boundaries, namely the spinless fermions model with nearest neighbour interaction and the 1D Hubbard model, with the methods of perturbation theory. It is demonstrated that many aspects of the spectral properties can be understood within the HF theory. Despite what one usually reads about one-dimensional interacting Fermi systems, in the presence of the boundary perturbation theory is already capable of providing meaningful results. Concurrently, this method also allows us to develop an understanding of the dip appearing in the local spectral function at an energy different from the chemical potential.

•Apr 1, 2019•0 citations
Abstract Renewal processes generated by a power-law distribution of intervals with tail index less than unity are genuinely non-stationary. This issue is illustrated by a critical review of the recent paper by Barma, Majumdar and Mukamel (2019 J. Phys. A 52 254001), devoted to the investigation of the properties of a specific one-dimensional equilibrium spin system with long-range interactions. We explain why discarding the non-stationarity of the process underlying the model leads to an incorrect expression of the critical spin–spin correlation function, even when the system, subjected to periodic boundary conditions, is translation invariant.
Abstract Mixed order transitions are those which show a discontinuity of the order parameter as well as a divergent correlation length at the transition (critical) point. We show that the behavior of the order parameter correlation function along the transition line of mixed order transitions can change from normal critical behavior with power law decay, to fluctuation-dominated phase ordering as a parameter is varied. The defining features of fluctuation-dominated order are anomalous fluctuations which remain large in the thermodynamic limit, and correlation functions which approach a finite value through a cusp singularity as the separation scaled by the system size approaches zero. We demonstrate that fluctuation-dominated order sets in along a portion of the transition line of an Ising model with truncated long-range interactions which was earlier shown to exhibit mixed order transitions. We also argue that this connection may hold more generally.
This Comment is dedicated to the investigation of many-body localization in a quantum Ising model with long-range power-law interactions ${r}^{\ensuremath{-}\ensuremath{\alpha}}$, relevant for a variety of systems ranging from electrons in Anderson insulators to spin excitations in chains of cold atoms. It has earlier been argued [arXiv:cond-mat/0611387 (2005); Phys. Rev. B 91, 094202 (2015)] that this model obeys the dimensional constraint suggesting the delocalization of all finite-temperature states in the thermodynamic limit for $\ensuremath{\alpha}\ensuremath{\le}2d$ in a ddd-dimensional system. This expectation conflicts with the recent numerical studies of the specific interacting spin model of Li et al. [Phys. Rev. A 94, 063625 (2016)]. To resolve this controversy we reexamine the model of Li et al. [Phys. Rev. A 94, 063625 (2016)] and demonstrate that the infinite-temperature states there obey the dimensional constraint. The earlier developed scaling theory for the critical system size required for delocalization is extended to small exponents $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}d$. The disagreements between the two works are explained by the nonstandard selection of investigated states in the ordered phase in the work of Li et al. [Phys. Rev. A 94, 063625 (2016)].
We have recently shown that in nonequilibrium spin systems at criticality the limit ${X}^{\ensuremath{\infty}}$ of the fluctuation-dissipation ratio X(t,tw)X(t,{t}_{w})X(t,tw​) for t⪢tw⪢1t⪢{t}_{w}⪢1t⪢tw​⪢1 can be measured using observables such as magnetization or energy [Phys. Rev. E 68, 016116 (2003)]. Pleimling argues in a Comment [preceding paper, Phys. Rev. 70, 018101 (2004)] on our paper that for such observables correlation and response functions are dominated by one-time quantities dependent only on ttt, and are therefore not suitable for a determination of ${X}^{\ensuremath{\infty}}$. Using standard scaling forms of correlation and response functions, as used by Pleimling, we show that our data do have a genuine two-time dependence and allow X(t,tw)X(t,{t}_{w})X(t,tw​) and ${X}^{\ensuremath{\infty}}$ to be measured, so that Pleimling's criticisms are easily refuted. We also compare with predictions from renormalization-group calculations, which are consistent with our numerical observation of a fluctuation-dissipation plot for the magnetization that is very close to a straight line. A key point remains that coherent observables make measurements of ${X}^{\ensuremath{\infty}}$ easier than the traditionally used incoherent ones, producing fluctuation-dissipation plots whose slope is close to ${X}^{\ensuremath{\infty}}$ over a much larger range.
The breaking of the continuous time-translation symmetry manifests, in Markovian open quantum systems, through the emergence of nonstationary dynamical phases. Systems that display nonequilibrium transitions into these phases are referred to as time crystals, and they can be realized, for example, in many-body systems governed by collective dissipation and long-ranged interactions. Here, we provide a complete analytical characterization of a boundary time-crystal phase transition. This involves exact expressions for the order parameter and for the dynamics of quantum fluctuations, which, in the time-crystalline phase, remains asymptotically non-Markovian as a consequence of the time-translation symmetry breaking. We demonstrate that boundary time crystals are intrinsically critical phases, where fluctuations exhibit a power-law divergence with time. Our results show that a dissipative time-crystal phase is far more than merely a classical nonlinear and nonstationary (limit cycle) dynamics of a macroscopic order parameter. It is rather a genuine many-body phase where the properties of correlations distinctly differ from those of stationary ones.
Available mathematical results entail the absence of a proper ordering transition taking place at finite temperature in the case d=1,d=1,d=1, $\ensuremath{\sigma}=1.1,$ where the authors of the article commented on find evidence of such a transition. Moreover, in contrast to the findings of the authors of the article, the case d=2,d=2,d=2, $\ensuremath{\sigma}=0$ entails a ground-state energy per particle diverging logarithmically with increasing sample size, hence an infinite transition temperature.
The physics of one-dimensional systems of interacting fermions strongly differs from the picture offered by Fermi liquid theory. The low-energy properties of metallic, paramagnetic 1D Fermi systems are well described by the Luttinger liquid (LL) theory. Among other properties, the homogeneous LL is known to exhibit power-law behaviour for various correlation functions. For spin rotational invariant models, the exponents can be expressed in terms of a single parameter Kρ, which is known to be non-universal. The LL concept emerged in the study of homogeneous systems with periodic boundary condition (PBC). In these models the local spectral function shows a power-law suppression at energies asymptotically close to the chemical potential µ. Theoretically, this is very well understood and the exact LL parameter Kρ is known for a set of integrable models. In the last decades however, the interest shifted towards the investigation of LLs with PBC including impurities. The fact that these systems have been shown to scale to chains with open ends led to the study of several models where open boundary condition (OBC) are introduced. As the translational invariance is broken, the exponent of the power-law suppression close to the boundary differs from the one in the bulk. The present work is devoted to the study of the local spectral function of two lattice models of LLs with open boundaries, namely the spinless fermions model with nearest neighbour interaction and the 1D Hubbard model, with the methods of perturbation theory. It is demonstrated that many aspects of the spectral properties can be understood within the HF theory. Despite what one usually reads about one-dimensional interacting Fermi systems, in the presence of the boundary perturbation theory is already capable of providing meaningful results. Concurrently, this method also allows us to develop an understanding of the dip appearing in the local spectral function at an energy different from the chemical potential.