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arXiv:1603.09644

Sliding phases in U(1) symmetric systems -- mirage of the renormalization group

Steven Vayl, Anatoly B. Kuklov, Vadim Oganesyan

Published Mar 31, 2016• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1603.09644

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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.
We show that three-dimensional ``sliding'' analogs of the Kosterlitz-Thouless phase, in stacked classical two-dimensional XY\mathrm{XY}XY models and quantum systems of coupled Luttinger liquids, can be enlarged by the application of a parallel magnetic field, which has the effect of increasing the scaling dimensions of the most relevant operators that can perturb the critical sliding phases. Within our renormalization-group analysis, we also find that for the case of coupled Luttinger liquids, this effect is interleaved with the onset of the integer quantum Hall effect for weak interactions and fields. We comment on experimental implications for a conjectured smectic metal phase in the cuprates.
Abstract. The phase structure of the layered sine-Gordon (LSG) model is investigated in terms of symmetry considerations by means of differential renormalization group (RG) method using the local potential approximation. The RG analysis of the general N-layer model provides us with the possibility to consider the dependence of the vortex dynamics on the number of layers. The Lagrangians are distinguished according to the number of zero eigenvalues of their mass matrices where the number is found to be decisive with respect to the phase structure of the N-layer models, with neighbouring layers being coupled by terms quadratic in the field variables. It is shown that the LSG model with N layers undergoes a Kosterlitz–Thouless type phase transition at the critical value of the parameter β2 c = 8Nπ. In the limit of infinitely many layers the LSG model can be considered as the discretized version of the three-dimensional sine-Gordon model which has been shown to have a single phase within the local potential approximation. The infinite critical value of the parameter β2 c for the LSG model in the continuum limit (N → ∞) is consistent with the latter observation. PACS numbers: 11.10.Hi,11.10.Gh,11.10.Kk
In a recent paper Nahum argued on the basis of a renormalization-group analysis that, contrary to standard lore, ground states of 1D spin chains with short-range interactions can spontaneously break U(1) ``easy-plane'' spin rotation symmetry. True long-range order of (Sx,Sy)(S^x,S^y)(Sx,Sy) arises at the phase transition between two quasi-long-range-ordered phases. Here we present detailed numerical results for an anisotropic spin-1 chain model. We find a magnetization exponent β≈0.29β\approx0.29β≈0.29, consistent with the εεε-expansion prediction β≈0.28β\approx0.28β≈0.28, a divergence of the Luttinger parameter on approaching the transition from both sides with an exponent that we relate to Nahum's transition, and finite-size spectra consistent with a dynamical critical exponent z≈2z\approx2z≈2. Our results support the critical behavior predicted by Nahum's field theory.
We study the phase-locking transition of two coupled low-dimensional superfluids, either two-dimensional superfluids at finite temperature, or one-dimensional superfluids at zero temperature. We find that the superfluids have a strong tendency to phase-lock. The phase-locking is accompanied by a sizeable increase of the transition temperature T c (in 2D systems) of the resulting double-layer superfluid to thermal Bose gas transition, compared to the Kosterlitz-Thouless temperature T KT of the uncoupled 2D systems, which suggests a plausible way of observing the Kibble-Zurek mechanism in two-dimensional cold atom systems by rapidly varying the tunneling rate between the superfluids. If there is also interaction between atoms in different layers present we find additional phases, while no sliding phase, characterized by order or quasi–long-range order (QLRO) either in the symmetric or the antisymmetric sector of the system.
Using a new approximate strong-randomness renormalization group (RG), we study the many-body localized (MBL) phase and phase transition in one-dimensional quantum systems with short-range interactions and quenched disorder. Our RG is built on those of Zhang et al. [Phys. Rev. B 93, 224201 (2016)] and Goremykina et al. [Phys. Rev. Lett. 122, 040601 (2019)], which are based on thermal and insulating blocks. Our main addition is to characterize each insulating block with two lengths: a physical length and an internal decay length $\ensuremath{\zeta}$ for its effective interactions. In this approach, the MBL phase is governed by a RG fixed line that is parametrized by a global decay length $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}$, and the rare large thermal inclusions within the MBL phase have a fractal geometry. As the phase transition is approached from within the MBL phase, $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}$ approaches the finite critical value corresponding to the avalanche instability, and the fractal dimension of large thermal inclusions approaches zero. Our analysis is consistent with a Kosterlitz-Thouless-like RG flow, with no intermediate critical MBL phase.
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.