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Register a new userScaling Theories of Kosterlitz-Thouless Phase Transitions
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We propose scaling theories for Kosterlitz-Thouless (KT) phase transitions on the basis of the hallmark exponential growth of their correlation length. Finite-size scaling, finite-entanglement scaling, short-time critical dynamics, and finite-time scaling, as well as some of their combinations are studied. Relaxation times of both a usual power-law and an unusual power-law with a logarithmic factor are considered. Finite-size and finite-entanglement scaling forms somehow similar to a frequently employed ansatz are presented. The Kibble-Zurek scaling of topological defect density for a linear driving across the KT transition point is investigated in detail. An implicit equation for a rate exponent in the theory is derived and the exponent varies with the distance from the critical point and the driving rate consistent with relevant experiments. To verify the theories, we utilize the KT phase transition of a one-dimensional Bose-Hubbard model. The infinite time-evolving-block-decimation algorithm is employed to solve numerically the model for finite bond dimensions. Both a correlation length and an entanglement entropy in imaginary time and only the entanglement entropy in real-time driving are computed. Both the short-time critical dynamics in imaginary time and the finite-time scaling in real-time driving, both including the finite bond dimension, for the measured quantities are found to describe the numerical results quite well via surface collapses. The critical point is also estimated and confirmed to be $0.302(1)$ at the infinite bond dimension on the basis of the scaling form.
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We propose a scaling theory for the many-body localization (MBL) phase transition in one dimension, building on the idea that it proceeds via a quantum avalanche. We argue that the critical properties can be captured at a coarse-grained level by a Kosterlitz-Thouless (KT) renormalization group (RG) flow. On phenomenological grounds, we identify the scaling variables as the density of thermal regions and the lengthscale that controls the decay of typical matrix elements. Within this KT picture, the MBL phase is a line of fixed points that terminates at the delocalization transition. We discuss two possible scenarios distinguished by the distribution of rare, fractal thermal inclusions within the MBL phase. In the first scenario, these regions have a stretched exponential distribution in the MBL phase. In the second scenario, the near-critical MBL phase hosts rare thermal regions that are power-law distributed in size. This points to the existence of a second transition within the MBL phase, at which these power-laws change to the stretched exponential form expected at strong disorder. We numerically simulate two different phenomenological RGs previously proposed to describe the MBL transition. Both RGs display a universal power-law length distribution of thermal regions at the transition with a critical exponent $alpha_c=2$, and continuously varying exponents in the MBL phase consistent with the KT picture.
We test an improved finite-size scaling method for reliably extracting the critical temperature $T_{rm BKT}$ of a Berezinskii-Kosterlitz-Thouless (BKT) transition. Using known single-parameter logarithmic corrections to the spin stiffness $rho_s$ at $T_{rm BKT}$ in combination with the Kosterlitz-Nelson relation between the transition temperature and the stiffness, $rho_s(T_{rm BKT})=2T_{rm BKT}/pi$, we define a size dependent transition temperature $T_{rm BKT}(L_1,L_2)$ based on a pair of system sizes $L_1,L_2$, e.g., $L_2=2L_1$. We use Monte Carlo data for the standard two-dimensional classical XY model to demonstrate that this quantity is well behaved and can be reliably extrapolated to the thermodynamic limit using the next expected logarithmic correction beyond the ones included in defining $T_{rm BKT}(L_1,L_2)$. For the Monte Carlo calculations we use GPU (graphical processing unit) computing to obtain high-precision data for $L$ up to 512. We find that the sub-leading logarithmic corrections have significant effects on the extrapolation. Our result $T_{rm BKT}=0.8935(1)$ is several error bars above the previously best estimates of the transition temperature; $T_{rm BKT} approx 0.8929$. If only the leading log-correction is used, the result is, however, consistent with the lower value, suggesting that previous works have underestimated $T_{rm BKT}$ because of neglect of sub-leading logarithms. Our method is easy to implement in practice and should be applicable to generic BKT transitions.
We demonstrate that a machine learning technique with a simple feedforward neural network can sensitively detect two successive phase transitions associated with the Berezinskii-Kosterlitz-Thouless (BKT) phase in q-state clock models simultaneously by analyzing the weight matrix components connecting the hidden and output layers. We find that the method requires only a data set of the raw spatial spin configurations for the learning procedure. This data set is generated by Monte-Carlo thermalizations at selected temperatures. Neither prior knowledge of, for example, the transition temperatures, number of phases, and order parameters nor processed data sets of, for example, the vortex configurations, histograms of spin orientations, and correlation functions produced from the original spin-configuration data are needed, in contrast with most of previously proposed machine learning methods based on supervised learning. Our neural network evaluates the transition temperatures as T_2/J=0.921 and T_1/J=0.410 for the paramagnetic-to-BKT transition and BKT-to-ferromagnetic transition in the eight-state clock model on a square lattice. Both critical temperatures agree well with those evaluated in the previous numerical studies.
We have considered two classical lattice-gas models, consisting of particles that carry multicomponent magnetic momenta, and associated with a two-dimensional square lattices; each site can host one particle at most, thus implicitly allowing for hard-core repulsion; the pair interaction, restricted to nearest neighbors, is ferromagnetic and involves only two components. The case of zero chemical potential has been investigated by Grand--Canonical Monte Carlo simulations; the fluctuating occupation numbers now give rise to additional fluid-like observables in comparison with the usual saturated--lattice situation; these were investigated and their possible influence on the critical behaviour was discussed. Our results show that the present model supports a Berezinskii-Kosterlitz-Thouless phase transition with a transition temperature lower than that of the saturated lattice counterpart due to the presence of ``vacancies; comparisons were also made with similar models studied in the literature.
We reexamine the two-dimensional linear O(2) model ($varphi^4$ theory) in the framework of the nonperturbative renormalization-group. From the flow equations obtained in the derivative expansion to second order and with optimization of the infrared regulator, we find a transition between a high-temperature (disordered) phase and a low-temperature phase displaying a line of fixed points and algebraic order. We obtain a picture in agreement with the standard theory of the Kosterlitz-Thouless (KT) transition and reproduce the universal features of the transition. In particular, we find the anomalous dimension $eta(Tkt)simeq 0.24$ and the stiffness jump $rho_s(Tkt^-)simeq 0.64$ at the transition temperature $Tkt$, in very good agreement with the exact results $eta(Tkt)=1/4$ and $rho_s(Tkt^-)=2/pi$, as well as an essential singularity of the correlation length in the high-temperature phase as $Tto Tkt$.
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