We present a mean-field description of the zigzag phase transition of a quasi-one-dimensional system of strongly interacting particles, with interaction potential ${r}^{\ensuremath{-}n}{e}^{\ensuremath{-}r/\ensuremath{\lambda}}$, that are confined by a power-law potential (${y}^{\ensuremath{\alpha}}$). The parameters of the resulting one-dimensional Ginzburg-Landau theory are determined analytically for different values of $\ensuremath{\alpha}$ and $n$ . Close to the transition point for the zigzag phase transition, the scaling behavior of the order parameter is determined. For $\ensuremath{\alpha}=2$, the zigzag transition from a single to a double chain is of second order, while for $\ensuremath{\alpha}>2$, the one-chain configuration is always unstable and, for $\ensuremath{\alpha}<2$, the one-chain ordered state becomes unstable at a certain critical density, resulting in jumps of single particles out of the chain.We present a mean-field description of the zigzag phase transition of a quasi-one-dimensional system of strongly interacting particles, with interaction potential ${r}^{\ensuremath{-}n}{e}^{\ensuremath{-}r/\ensuremath{\lambda}}$, that are confined by a power-law potential (${y}^{\ensuremath{\alpha}}$). The parameters of the resulting one-dimensional Ginzburg-Landau theory are determined analytically for different values of $\ensuremath{\alpha}$ and $n$. Close to the transition point for the zigzag phase transition, the scaling behavior of the order parameter is determined. For $\ensuremath{\alpha}=2$, the zigzag transition from a single to a double chain is of second order, while for $\ensuremath{\alpha}>2$, the one-chain configuration is always unstable and, for $\ensuremath{\alpha}<2$, the one-chain ordered state becomes unstable at a certain critical density, resulting in jumps of single particles out of the chain.

arXiv:1207.4320Physical Review BOct 18, 201118 citations

Scaling at First-Order Phase Transitions

WORLD SCIENTIFIC eBooksMay 1, 20080 citations

Relating Theories via Renormalization

Leo P. Kadanoff

The renormalization method is specifically aimed at connecting theories describing physical processes at different length scales and thereby connecting different theories in the physical sciences. The renormalization method used today is the outgrowth of one hundred and fifty years of scientific study of thermal physics and phase transitions. Different phases of matter show qualitatively different behavior separated by abrupt phase transitions. These qualitative differences seem to be present in experimentally observed condensed-matter systems. However, the "extended singularity theorem" in statistical mechanics shows that sharp changes can only occur in infinitely large systems. Abrupt changes from one phase to another are signaled by fluctuations that show correlation over infinitely long distances, and are measured by correlation functions that show algebraic decay as well as various kinds of singularities and infinities in thermodynamic derivatives and in measured system parameters. Renormalization methods were first developed in field theory to get around difficulties caused by apparent divergences at both small and large scales. The renormalization (semi-)group theory of phase transitions was put together by Kenneth G. Wilson in 1971 based upon ideas of scaling and universality developed earlier in the context of phase transitions and of couplings dependent upon spatial scale coming from field theory. Correlations among regions with fluctuations in their order underlie renormalization ideas. Wilson's theory is the first approach to phase transitions to agree with the extended singularity theorem. Some of the history of the study of these correlations and singularities is recounted, along with the history of renormalization and related concepts of scaling and universality. Applications are summarized.The renormalization method is specifically aimed at connecting theories describing physical processes at different length scales and thereby connecting different theories in the physical sciences. The renormalization method used today is the outgrowth of one hundred and fifty years of scientific study of thermal physics and phase transitions. Different phases of matter show qualitatively different behavior separated by abrupt phase transitions. These qualitative differences seem to be present in experimentally observed condensed-matter systems. However, the "extended singularity theorem" in statistical mechanics shows that sharp changes can only occur in infinitely large systems. Abrupt changes from one phase to another are signaled by fluctuations that show correlation over infinitely long distances, and are measured by correlation functions that show algebraic decay as well as various kinds of singularities and infinities in thermodynamic derivatives and in measured system parameters. Renormalization methods were first developed in field theory to get around difficulties caused by apparent divergences at both small and large scales. The renormalization (semi-)group theory of phase transitions was put together by Kenneth G. Wilson in 1971 based upon ideas of scaling and universality developed earlier in the context of phase transitions and of couplings dependent upon spatial scale coming from field theory. Correlations among regions with fluctuations in their order underlie renormalization ideas. Wilson's theory is the first approach to phase transitions to agree with the extended singularity theorem. Some of the history of the study of these correlations and singularities is recounted, along with the history of renormalization and related concepts of scaling and universality. Applications are summarized.

arXiv:1102.3705Feb 17, 20115 citations

Scaling Theory of First-Order Quantum Phase Transitions

M. A. Contínentino

Cambridge University Press eBooksJan 1, 20171 citation

Perspectives from the theory of phase transitions

G. Toulouse

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento BSep 1, 1974111 citations

Critical exponents for higher order phase transitions: Landau theory and RG flow

Joydeep Chakravarty, Diksha Jain

Abstract In this work, we define and calculate critical exponents associated with higher order thermodynamic phase transitions. Such phase transitions can be classified into two classes: with or without a local order parameter. For phase transitions involving a local order parameter, we write down the Landau theory and calculate critical exponents using the saddle point approximation. Further, we investigate fluctuations about the saddle point and demarcate when such fluctuations dominate over saddle point calculations by introducing the generalized Ginzburg criteria. We use Wilsonian renormalization group (RG) to derive scaling forms for observables near criticality and obtain scaling relations between the critical exponents. Afterwards, we find out fixed points of the RG flow using the one-loop beta function and calculate critical exponents about the fixed points for third and fourth order phase transitions.Abstract In this work, we define and calculate critical exponents associated with higher order thermodynamic phase transitions. Such phase transitions can be classified into two classes: with or without a local order parameter. For phase transitions involving a local order parameter, we write down the Landau theory and calculate critical exponents using the saddle point approximation. Further, we investigate fluctuations about the saddle point and demarcate when such fluctuations dominate over saddle point calculations by introducing the generalized Ginzburg criteria. We use Wilsonian renormalization group (RG) to derive scaling forms for observables near criticality and obtain scaling relations between the critical exponents. Afterwards, we find out fixed points of the RG flow using the one-loop beta function and calculate critical exponents about the fixed points for third and fourth order phase transitions.

arXiv:2102.08398Journal of Statistical Mechanics Theory and ExperimentSep 1, 20218 citations

n

. Close to the transition point for the zigzag phase transition, the scaling behavior of the order parameter is determined. For $\ensuremath{\alpha}=2$, the zigzag transition from a single to a double chain is of second order, while for $\ensuremath{\alpha}>2$, the one-chain configuration is always unstable and, for $\ensuremath{\alpha}<2$, the one-chain ordered state becomes unstable at a certain critical density, resulting in jumps of single particles out of the chain.