We derive a renormalization group to calculate the nontrivial critical exponent of the divergent correlation length, thereby providing a universality classification of essential singularities in infinite-order phase transitions. This method thus resolves the vanishing scaling matrix problem. The exponent is obtained from the maximal eigenvalue of a scaling matrix in this renormalization group, as in the case of ordinary second-order phase transitions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well known Berezinskiĭ-Kosterlitz-Thouless transition.
We derive a renormalization group to calculate the nontrivial critical exponent of the divergent correlation length, thereby providing a universality classification of essential singularities in infinite-order phase transitions. This method thus resolves the vanishing scaling matrix problem. The exponent is obtained from the maximal eigenvalue of a scaling matrix in this renormalization group, as in the case of ordinary second-order phase transitions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well known Berezinskiĭ-Kosterlitz-Thouless transition.
We consider quantum states under the renormalization-group (RG) transformations introduced by Verstraete et al. [Phys. Rev. Lett. 94, 140601 (2005)] and propose a quantification of entanglement under such RGs (via the geometric measure of entanglement). We examine the resulting entanglement under RG transformations for the ground states of ``matrix-product-state'' Hamiltonians constructed by Wolf et al. [Phys. Rev. Lett. 97, 110403 (2006)] that possess quantum phase transitions. We find that near critical points, the ground-state entanglement exhibits singular behavior. The singular behavior within finite steps of the RG obeys a scaling hypothesis and reveals the correlation length exponent. However, under the infinite steps of RG transformation, the singular behavior is rendered different and is universal only when there is an underlying conformal-field-theory description of the critical point.
Finite-size scaling and the renormalization group form the central toolkit
for analyzing quantum phase transitions (QPTs). In this Letter, we introduce
a novel Hilbert-space tailoring scheme to probe quantum critical phenomena.
Applied to the second-order QPT of the one-dimensional (1D) XY model, our method
yields precise critical points and exponents on lattices containing merely 50
unit cells. We further establish the universal applicability of this
framework via investigations of the Berezinskii-Kosterlitz-Thouless
transition in the 1D XXZ chain: critical parameters are recovered
with as few as 12 lattice sites. This technique may open an alternative,
efficient route to universally characterize QPT across many-body lattice systems.
The renormalization-group method is applied to the analysis of phase transitions in systems where the order parameter s is coupled to a nonordering additional variable y. A variety of critical and tricritical behaviors or first-order transitions is found as a function of the physical variables and possible macroscopic constraints imposed on the system. For s2y coupling, the correlation function of y was found to be governed by a correlation length which is proportional to that of the order parameter, and by a critical index ${\ensuremath{\eta}}_{y}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\eta}}_{y}=2\ensuremath{-}2\ensuremath{\alpha}$; $\ensuremath{\alpha}$ here is the specific-heat exponent of the appropriate unconstrained system. The singular part of the susceptibility, ${\ensuremath{\chi}}_{y}$, has a critical exponent equal to $\ensuremath{\alpha}$, the true specific-heat exponent. When the coupling is s2y2, a weaker singularity of ${\ensuremath{\chi}}_{y}$ appears. The crossover between this behavior and the one typical to s2y coupling is calculated. 〈y〉 has a singular part with an exponent $1\ensuremath{-}\ensuremath{\alpha}$, in the unconstrained case. The breakdown of the scaling law related to the correlation function of y in the constrained case is discussed.
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.
Non-equilibrium critical phenomena have attracted a lot of research interest in the recent decades. Similar to equilibrium critical phenomena, the concept of universality remains the major tool to order the great variety of non-equilibrium phase transitions systematically. All systems belonging to a given universality class share the same set of critical exponents, and certain scaling functions become identical near the critical point. It is known that the scaling functions vary more widely between different universality classes than the exponents. Thus, universal scaling functions offer a sensitive and accurate test for a system's universality class. On the other hand, universal scaling functions demonstrate the robustness of a given universality class impressively. Unfortunately, most studies focus on the determination of the critical exponents, neglecting the universal scaling functions. In this work a particular class of non-equilibrium critical phenomena is considered, the so-called absorbing phase transitions. Absorbing phase transitions are expected to occur in physical, chemical as well as biological systems, and a detailed introduction is presented. The universal scaling behavior of two different universality classes is analyzed in detail, namely the directed percolation and the Manna universality class. Especially, directed percolation is the most common universality class of absorbing phase transitions. The presented picture gallery of universal scaling functions includes steady state, dynamical as well as finite size scaling functions. In particular, the effect of an external field conjugated to the order parameter is investigated. Incorporating the conjugated field, it is possible to determine the equation of state, the susceptibility, and to perform a modified finite-size scaling analysis appropriate for absorbing phase transitions. Focusing on these equations, the obtained results can be applied to other non-equilibrium continuous phase transitions observed in numerical simulations or experiments. Thus, we think that the presented picture gallery of universal scaling functions is valuable for future work. Additionally to the manifestation of universality classes, universal scaling functions are useful in order to check renormalization group results quantitatively. Since the renormalization group theory is the basis of our understanding of critical phenomena, it is of fundamental interest to examine the accuracy of the obtained results. Due to the continuing improvement of computer hardware, accurate numerical data have become available, resulting in a fruitful interplay between numerical investigations and renormalization group analyzes.
Non-equilibrium critical phenomena have attracted a lot of research interest in the recent decades. Similar to equilibrium critical phenomena, the concept of universality remains the major tool to order the great variety of non-equilibrium phase transitions systematically. All systems belonging to a given universality class share the same set of critical exponents, and certain scaling functions become identical near the critical point. It is known that the scaling functions vary more widely between different universality classes than the exponents. Thus, universal scaling functions offer a sensitive and accurate test for a system's universality class. On the other hand, universal scaling functions demonstrate the robustness of a given universality class impressively. Unfortunately, most studies focus on the determination of the critical exponents, neglecting the universal scaling functions. In this work a particular class of non-equilibrium critical phenomena is considered, the so-called absorbing phase transitions. Absorbing phase transitions are expected to occur in physical, chemical as well as biological systems, and a detailed introduction is presented. The universal scaling behavior of two different universality classes is analyzed in detail, namely the directed percolation and the Manna universality class. Especially, directed percolation is the most common universality class of absorbing phase transitions. The presented picture gallery of universal scaling functions includes steady state, dynamical as well as finite size scaling functions. In particular, the effect of an external field conjugated to the order parameter is investigated. Incorporating the conjugated field, it is possible to determine the equation of state, the susceptibility, and to perform a modified finite-size scaling analysis appropriate for absorbing phase transitions. Focusing on these equations, the obtained results can be applied to other non-equilibrium continuous phase transitions observed in numerical simulations or experiments. Thus, we think that the presented picture gallery of universal scaling functions is valuable for future work. Additionally to the manifestation of universality classes, universal scaling functions are useful in order to check renormalization group results quantitatively. Since the renormalization group theory is the basis of our understanding of critical phenomena, it is of fundamental interest to examine the accuracy of the obtained results. Due to the continuing improvement of computer hardware, accurate numerical data have become available, resulting in a fruitful interplay between numerical investigations and renormalization group analyzes.