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Duality in a Model of Layered Superfluids and Sliding Phases

Steven Vayl

Published Jan 1, 2020• 0 citations

Abstract

The intent of my project is to determine if the proposal of sliding phases in XY layered systems has physical ground. It will be done by comparing numerical and analytical results for a layered XY models. Sliding phases were first proposed in the context of DNA complexes and then extended to XY models, 1D coupled wires and superfluid films. The existence of the sliding phase would mean that there is a phase transition from 3D to 2D behavior. Such systems have been studied both in the clean case and with disorder. The idea of the sliding phases is based on applying the criterion of relevancy of Josephson coupling between layers along the line of the Sine-Gordon model. Such a criterion is, strictly speaking, valid for a non-compact variable while the XY model is defined for compact one. Thus, it is not clear if this approximation has any validity. Our analysis of the numerical data obtained within Monte Carlo simulations of a model showed that the scaling behavior of the Josephson coupling disagrees qualitatively (and quantitatively) with the flow equation derived from the renormalization group (RG) prediction. Instead of the 3D to 2D phase transition predicted by the RG we see the behavior consistent with the system retaining its 3D character. Our approach is based on the language of duality. It allows reformulating the model in terms of the objects amenable to effective large scale simulations. Furthermore, using this language it becomes possible to formulate a general criterion for the sliding phases to exist – which conspicuously demonstrates that the RG approach is incorrect when applied to the situations of the dimensional reduction. The dual approach allows finding the asymptotically exact analytical solution of the model for clean and disordered cases. It is also suggested how to achieve sliding phases – by introducing effective gauge type interaction between Josephson currents mediated by some soft mode which does not affect currents along the planes.

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The intent of my project is to determine if the proposal of sliding phases in XY layered systems has physical ground. It will be done by comparing numerical and analytical results for a layered XY models. Sliding phases were first proposed in the context of DNA complexes and then extended to XY models, 1D coupled wires and superfluid films. The existence of the sliding phase would mean that there is a phase transition from 3D to 2D behavior. Such systems have been studied both in the clean case and with disorder. The idea of the sliding phases is based on applying the criterion of relevancy of Josephson coupling between layers along the line of the Sine-Gordon model. Such a criterion is, strictly speaking, valid for a non-compact variable while the XY model is defined for compact one. Thus, it is not clear if this approximation has any validity. Our analysis of the numerical data obtained within Monte Carlo simulations of a model showed that the scaling behavior of the Josephson coupling disagrees qualitatively (and quantitatively) with the flow equation derived from the renormalization group (RG) prediction. Instead of the 3D to 2D phase transition predicted by the RG we see the behavior consistent with the system retaining its 3D character. Our approach is based on the language of duality. It allows reformulating the model in terms of the objects amenable to effective large scale simulations. Furthermore, using this language it becomes possible to formulate a general criterion for the sliding phases to exist – which conspicuously demonstrates that the RG approach is incorrect when applied to the situations of the dimensional reduction. The dual approach allows finding the asymptotically exact analytical solution of the model for clean and disordered cases. It is also suggested how to achieve sliding phases – by introducing effective gauge type interaction between Josephson currents mediated by some soft mode which does not affect currents along the planes.

We analyse the proposal of sliding phases (SP) in layers hosting global U(1) symmetric variables with finite inter-layer Josephson coupling. Based on the Kosterlitz-Thouless renormalization group (RG) approach, such phases were predicted to exist in various layered (or 1D quantum coupled) systems. The key in the RG argument is treating the coupling as though the variables are non-compact. Large scale Monte Carlo simulations of a layered model, where the SP is supposed to exist, finds no indication of such a phase. Instead, 3D behavior is observed. This result is consistent with the asymptotically exact analytical solution. A generic argument against SP in translationally invariant systems with short range interactions is provided. We have also suggested an alternative model for the SP -- adding long-range interactions to the inter-layer Josephson term.

Using large-scale quantum Monte Carlo simulations of lattice bosonic models, we precisely investigate the effect of weak Josephson tunneling between 2D superfluid or superconducting layers. In the clean case, the Kosterlitz-Thouless transition immediately turns into 3DXY, with phase coherence and superflow in all spatial directions, and a strong enhancement of the critical temperature. However, when disorder is present, rare regions fluctuations can lead to an intermediate finite-temperature phase —the so-called sliding regime— where only 2D superflow occurs within the layers without any transverse superfluid coherence, while a true 3D Bose-Einstein condensate exists. Critical properties of such an unconventional regime are carefully investigated.

This is a set of notes recalling some of the most important results on the XY model from the ground up. They are meant for a junior researcher wanting to get accustomed to the Kosterlitz-Thouless phase transition in the context of the 2D classical XY model. The connection to the 2D Coulomb gas is presented in detail, as well as the renormalization group flow obtained from this dual representation. A numerical Monte-Carlo approach to the classical XY model is presented. Finally, two physical setting that realize the celebrated Kosterlitz-Thouless phase transition are presented: superfluid and liquid crystal thin films.

New qualitative picture of vortex length-scale dependence has been found in recent electrical transport measurements performed on strongly anisotropic BSCCO single crystals in zero magnetic field. This indicates the need for a better description of the 3D/2D crossover in vortex dimensionality. The vortex-dominated properties of high transition temperature superconductors with extremely high anisotropy (layered systems) are reasonably well described in the framework of the layered XY model which can be mapped onto the layered sine-Gordon model. For the latter we derive an exact renormalization group (RG) equation using Wegner's and Houghton's approach in the local potential approximation. The agreement of the UV scaling laws find by us by linearizing the RG equations with those obtained previously in the literature in the dilute gas approximation makes the improvement appearant which can be achieved by solving our RG equations numerically.

A new qualitative view of vortex length–scale dependence has been found in recent electrical transport measurements performed on strongly anisotropic Bi2Sr2CaCu2O8 single crystals in zero magnetic field. This indicates the need for a better description of the 3D/2D crossover in vortex dimensionality. The vortex-dominated properties of high transition temperature superconductors with extremely high anisotropy (layered systems) are reasonably well described in the framework of the layered XY model which can be mapped onto the layered sine-Gordon model. For the latter we derive an exact renormalization group (RG) equation using Wegner's and Houghton's approach in the local potential approximation. The agreement of the UV scaling laws found by us by linearizing the RG equations with those obtained previously in the literature in the dilute gas approximation makes apparent the improvement which can be achieved by solving our RG equations numerically.

Superfluid and superconducting phase transitions of interacting bosons are observed in diverse physical systems such as helium, superconducting materials, ultra-cold atomic gases, etc. For all these systems one-, two-, and three-dimensional (1D, 2D, and 3D) realizations are experimentally accessible. However, the current experimental focus is shifting to the low dimensional systems (e.g., films and wires) due to facilitated measurement techniques and a constant demand of technological applications. From a theoretical standpoint large fluctuations in 1D and 2D systems make them a challenging problem, especially when disorder is present. Here we study peculiarities associated with disorder in low-dimensional cases. Specifically, we are interested in the class of the systems where superfluid or superconducting transitions can be mapped to the finite temperature 2D superfluid-normal Kosterlitz-Thouless (KT) transition: (a) 2D bosonic system with a strong diagonal disorder, (b) 1D system of lattice bosons with off-diagonal disorder in the limit of large integer filling factor at zero temperature, and (c) 3D layered superconductors with no tunnelling between layers in zero external magnetic field.