BKT-paired phase in coupled XY models

TL;DR

This paper investigates how linear tunneling coupling between two 2D XY systems affects their topological phases, revealing a novel BKT-paired phase with unique correlation properties through numerical and theoretical analysis.

Contribution

It introduces and characterizes a new BKT-paired phase in coupled XY models, supported by numerical simulations and renormalization group analysis.

Findings

The BKT-paired phase exists at any finite interlayer coupling.

In this phase, 1-body correlations decay exponentially, while 2-body correlations follow a power-law.

Numerical simulations confirm the phase's presence across a range of couplings.

Abstract

We study the effect of a linear tunneling coupling between 2D systems, each separately exhibiting the topological Berezinskii-Kosterlitz-Thouless (BKT) transition. In the uncoupled limit, there are two phases: one where the 1-body correlation functions are algebraically decaying and the other with exponential decay. When the linear coupling is turned on, a third BKT-paired phase emerges, in which 1-body correlations are exponentially decaying, while 2-body correlation functions exhibit power-law decay. We perform numerical simulations in the paradigmatic case of two coupled XY models at finite temperature, finding that for any finite value of the interlayer coupling, the BKT-paired phase is present. We provide a picture of the phase diagram using a renormalization group approach.

Equations

$$\begin{gathered} \mathcal{H}=-J\sum _{⟨ij⟩}\cos \left({\varphi }_i-{\varphi }_j\right)-J\sum _{⟨ij⟩}\cos \left({\psi }_i-{\psi }_j\right)+ \\ -K\sum _i\cos \left({\varphi }_i-{\psi }_i\right), \end{gathered}$$

$$\mathcal{H}=-2J\sum _{⟨ij⟩}\cos \left({\varphi }_i-{\varphi }_j\right),$$

$$c_{\uparrow }(k)$$

$$c_{\downarrow }(k)$$

$$z(k)$$

$$S[\phi ]=S_{\mathrm{kin}}[\phi ]+\sum _l\int U(|{\phi }_l|)d^{2}x$$

$$S_{\text{kin}}[\varphi ]=\frac{1}{2}\sum _{\sigma ,q}\frac{{\phi }_{\sigma }(q){\phi }_{\sigma }(q)}{K^{\sigma }(q)},$$

$${\phi }_{\pm }(q)=\frac{{\phi }_1(q)\pm {\phi }_2(q)}{\sqrt{2}}.$$

$${\mu }_{\pm }=\left(\frac{1}{K^{\pm }(0)}-\frac{1}{2}\right).$$

$$\displaystyle 0=\frac{\partial S[\varphi]}{\partial\varphi_{\pm}}\bigg{|}_{\varphi=\text{const}}$$

$$S_{\text{c}}[\phi ]=\sum _{\sigma }\int d^{2}x\frac{|\nabla {\phi }_{\sigma }(x){|}^2}{2m_{\sigma }}+\text{local terms}$$

$$S[\theta ]=\sum _{\sigma }\int d^{2}x\frac{{\rho }_{\sigma }}{2m_{\sigma }}{\partial }_{\mu }{\theta }_{\sigma }{\partial }_{\mu }{\theta }_{\sigma }.$$

$${\partial }_t{K}_k$$

$${\partial }_t{g}_k$$

$$K_{\Lambda }$$

$$g_{\Lambda }$$

$$J_{\mathrm{eff}}^{\sigma }=\frac{{\rho }_{\text{MF},\sigma }}{2m_{\sigma }}.$$

Full text

An ample variety of physical properties and phenomena emerge when two many-body systems are coupled. There are, of course, different ways of coupling interacting systems, depending on the geometry of the uncoupled systems, on the relevant degrees of freedom and on the way in which the coupling is established. One of them is the so-called weak coupling, when the Hamiltonian term modeling the coupling is a perturbation with respect to the uncoupled Hamiltonians. In this case, if the two systems are separately described by a macroscopic wavefunction – such as superfluids or superconductors – a relative phase between them emerges, as one can see in the Josephson effect between superconductors Barone82 ; Tafuri05 , Helium containers Pereverzev97 ; Hoskinson05 , Bose-Einstein condensates Cataliotti01 ; Albiez05 and ultracold fermions Valtolina15 . In these cases, the microscopic details of the coupling enter only in the Josephson energy ruling the maximum current that can flow through the junction.

Coupling two superfluids gives rise to a dissipationless drag current that is often referred to as the Andreev-Bashkin effect Andreev76 . Previous investigations Kaurov05 ; Kuklov03 ; Svistunov15 have mainly focused on the three-dimensional case, where strong enough drag densities have been found to modify the order of the symmetry breaking transition Dahl08 . Indeed, when the coupling is no longer a perturbation, one can expect that the bulk properties of the uncoupled systems are significantly altered and that new phases may emerge. A typical phenomenon induced by a strong coupling is that the order in the systems is substituted by a phase between them, therefore giving rise to an order parameter expressed in terms of operators of both systems. Countless different examples of couplings exist. Among the simplest instances, one can imagine taking two one-dimensional systems and couple them lengthwise, thus creating a ladder geometry. Examples of this kind of coupling include magnetic spin ladders, where two quantum spin chains are put in a ladder geometry, see for instance Refs. Gogolin98 ; Dagotto99 . The bilayer structure, in which two two-dimensional systems are coupled, is also a paradigmatic configuration, which has been studied in numerous physical systems, ranging from graphene Novoselov04 to quantum Hall systems Jain07 and dipolar models Baranov12 . The bilayer configuration is of particular relevance because it may refer to two quantum or classical systems at finite temperature, or – via the quantum-classical correspondence – to the ladder geometry in which two quantum one-dimensional models at zero temperature are coupled.

In this Letter, we address the emergence of a paired phase when two models exhibiting a topological phase transition are coupled. We will refer to the Berezinskii-Kosterlitz-Thouless (BKT) universality class Kosterlitz16 and study the effect of the coupling of two BKT systems. We have two main motivations for our study. First, the BKT transition is not characterized by a local order parameter and by the conventional spontaneous symmetry breaking. Rather, one can locate the critical point in which the system becomes superfluid by looking at the correlation functions, exhibiting a power law (exponential) decay in (outside) the superfluid phase. BKT transitions has been observed in several different 2D physical systems, including superfluid Helium Bishop78 , superconducting films and arrays Simanek1994 ; Fazio94 , as well as bosonic Hadzibabic2006 ; Schweikhard07 and fermionic Murthy15 ultracold systems.

The second motivation is related to the presence and relevance of the coupling between 2D systems in a variety of experimental systems, ranging from layered superconductors Tinkham96 ; Klemm12 2D two-component mixtures Svistunov15 and 2D ultracold gases Hadzibabic11 . In the latter case, a single 2D ultracold system is obtained by means of a suitably large transverse confinement: for instance, one can realize a 2D geometry by confining the atoms in a well of an optical lattice along the transverse direction, with the transverse confinement frequency increasing with the power laser and being $\gtrsim 1-10\text{ kHz}$ Bloch08 . When two adjacent minima of the vertical lattice are left populated, one would have two 2D ultracold gases with tunable interlayer coupling. In this way, one would study the effect of the bilayer coupling on the 2D physics already observed in single layers Hadzibabic2006 ; Schweikhard07 ; Murthy15 . Notice that in this case, the coupling is (for bosons) of the form $b^{\dagger}_{1}b_{2}$ (where $b_{\alpha}$ creates a particle in the $\alpha$ layer), thereby corresponding to what we refer to as linear coupling.

Despite the rising interest of the theoretical community in low dimensional binary mixtures Karle19 ; Parisi18 ; Konietin18 ; Sellin18 , the influence of interspecies coupling on universal behavior remains largely unknown. In this context, a compelling question is i) “What is the fate of the quasi-long-range order in the presence of a strong coupling between two BKT systems?”. One may as well ask ii) “Does quasi order occur in the mixed correlators of the coupled system? If so, does it appear already at small couplings?”. Previous attempts to answer these questions mainly focused on the study of effective models for coupled topological excitations, such as the coupled sine-Gordon model Mathey08 and the coupled Coulomb gases Podolsky09 , evidencing the emergence of a composite vortex gas phase as well as a second-order phase transition at strong coupling. However, the results of these investigations cannot be directly applied to our case, since the low energy equivalence between two dimensional superfluids and topological defect gases appears to be spoiled by the presence of the bilayer coupling term.

To address the questions above, we consider the paradigmatic case of two coupled $XY$ models at finite temperature, based on the expectation that similar features emerge for other classical and quantum models in the $XY$ universality class when linearly coupled. The paradigmatic nature of the XY model results in longly certified numerical tools, which allow us to reliably investigate the bilayer case.

The model.

The bilayer $XY$ Hamiltonian can be written as

[TABLE]

where $\vec{S}_{i}=(\cos\phi_{i},\sin\phi_{i})$ and $\vec{T}_{i}=(\cos\psi_{i},\sin\psi_{i})$ are the $XY$ spins with $n=2$ components defined on the first and the second layer, respectively, each being a 2D system. We will also use the notation $S_{i}=e^{\mathrm{i}\phi_{i}}$ and $T_{i}=e^{\mathrm{i}\psi_{i}}$. Notice that we are considering the symmetric case where the intralayer coupling takes the same value $J$ on both layers. The critical line separating the region with power law correlations from the one with exponential correlations in model (1) has been studied in literature parga . Moreover, several properties of coupled and layered XY models and two-components systems have also been investigated Granato88 ; Zhang05 ; Vayl17 ; Rancon17 ; Kobayashi18 ; Karle19 , as well as coupled $3D$ $XY$ models Smiseth05 ; Sellin16 . Here our focus is on the remarkable quasi-order for pairs of spins on different layers that arises in a 2D bilayer system.

When $K=0$, the two layers are decoupled and the behavior of the model is that of the standard 2D XY model Kosterlitz16 ; Svistunov15 : a superfluid phase and the normal state are separated by the BKT transition occurring at $T_{\text{BKT}}\approx 0.893J$ Gupta1988 ; Gupta92 ; tomita ; Hasenbusch05 ; Komura12 . The normal state is characterized by the exponential decay of the phase-phase correlator; in the superfluid phase, on the other hand, one finds the peculiar algebraic decay of correlations functions.

In the other limiting case, $K\to\infty$, the spins on the upper and lower layer are constrained to be parallel, i.e., $\phi_{i}=\psi_{i}\ \forall i$, so that the effective Hamiltonian reduces, up to additive constants, to

[TABLE]

i.e. that of a 2D $XY$ model with coupling constant $J^{\prime}=2J$, therefore with critical temperature $T^{\prime}_{BKT}\approx 1.786\ J$.

Results for the phase diagram.

We want to study the full phase diagram of the model, away from the two limiting cases just analyzed. From the theory of bilayer systems and two-component mixtures Svistunov15 ; Ueda10 one may expect that, apart from the normal and superfluid phases of the BKT transition, a phase involving variables (operators in the quantum case) of both layers can emerge. The question is whether it emerges only for large values of the coupling $K$ or not. To make a comparison with a similar, yet different, analytically solvable case, let us consider the square lattice Askhin-Teller model baxter : the Hamiltonian has a form similar to Eq. (1), $H_{AT}=-J\sum_{\langle ij\rangle}s_{i}s_{j}-J\sum_{\langle ij\rangle}t_{i}t_{j}-K\sum_{\langle ij\rangle}s_{i}s_{j}t_{i}t_{j}$, with $s_{i},t_{i}=\pm 1$. The model has indeed a phase whose order parameter is $\langle st\rangle$ baxter . However, apart from the fact that spin variables are discrete in the Askhin-Teller model and continuous in the bilayer $XY$ model, there are two remarkable differences: first, the coupling is quartic; second, the bilayer $XY$ model exhibits quasi-order and no spontaneous symmetry braking at finite temperature.

To determine the full phase diagram of the bilayer $XY$ model, we perform Monte Carlo (MC) simulations of the Hamiltonian of Eq. (1) on a $2\times L\times L$ lattice, $L$ being the linear dimension of each layer. We vary $\beta J$ and $\beta K$, where as usual $\beta=(k_{B}T)^{-1}$. Monte Carlo updates use the Swendsen-Wang algorithm tomita ; swendsen after embedded cluster decomposition wolff . We sample the following correlation functions:

[TABLE]

where the summations extend over all $i$, $j$ pairs separated by $k$ lattice sites along the $x$ or $y$ direction. Due to the symmetry of the system upon exchange between the upper and lower layer, $c_{\uparrow}(k)$ and $c_{\downarrow}(k)$ coincide across the whole phase diagram, let us call them $c(k)$. All correlation functions are well fitted by the function $Q(r)=(r/r_{\text{alg}})^{\alpha}\exp(-r/r_{\text{exp}})$, where $r_{\text{alg}}$ is the characteristic length associated to the algebraic decay and $r_{\text{exp}}$ is the characteristic length associated to the exponential decay.

To tell apart the exponential and algebraic behaviour we compare $r_{\text{exp}}$ with the linear system size used in the MC simulation: if $r_{\text{exp}}$ is smaller than twice the lattice dimensions, then we mark the decay as exponential, otherwise it is marked as algebraic, since the exponential characteristic length $r_{\text{exp}}$ is neglectable when compared to the dimensions of the system being studied. This allows us to obtain the phase diagram shown in Fig. 1(a), identifying the three phases. We denote by $A$ the usual superfluid BKT phase, characterized by the algebraic decay of $c(k)$. The correlation function $z(k)$ is also found to be algebraically decaying in the $A$ phase. The normal phase, denoted by $B$, is characterized, as usual, by the exponential decay of $c(k)$. We checked that in the $B$ phase $z(k)$ is exponentially decaying, as well. Finally, we observe a third $C$ phase, that is instead characterized by the exponential decay of $c(k)$ and the algebraic decay of $z(k)$. Notice that, within the precision we can achieve, for small values of $K$ one always has a range of values of $J$ for which there is the $C$ phase. In other words, the $C$ phase extends up to vanishing values of $K$.

We dub the $C$ phase ‘BKT-paired phase’, since each layer has the $1$-body correlation function $c(k)$ decaying exponentially, while the $2$-body correlation function $z(k)$ has an algebraic behaviour. Since $z(k)$ is the correlation function between $S_{i}T_{i}$ and $S_{j}T_{j}$ for $i$ and $j$ far apart from each other, this is related to a quasi-long-range order for the pair variables $S_{i}T_{i}$ in the different sites of the bilayer system. The phase transition between $A$ and $C$ appears to be a conventional BKT line, and in agreement with Ref. parga . It is worth noting that despite some similarities between the phase diagram presented in Fig. 1 and the one discussed in Ref. Mathey08 , we find no trace of any actual symmetry breaking while our paired BKT phase appears also at infinitely small coupling strengths. Moreover, the presence of power law four point correlations was not noticed in previous investigations.

To further numerically investigate these two transition lines we resort to probability-changing cluster (PCC) algorithm, introduced in Ref. tomita for the single-layer BKT transition. Following the approach therein, at first the XY variables are mapped onto two Ising models, using Wolff’s embedded cluster formalism wolff . Subsequently the temperature of the system is tuned in progressively smaller increments, looking for the $\nicefrac{{1}}{{2}}$ percolation threshold of the Kasteleyn-Fortuin clusters defined from each Ising model tomita , signaling the BKT transition. It can in fact be shown coniglio ; kf that the spin-spin correlation function $\langle S_{i}S_{j}\rangle$ equals $p_{ij}$, the probability that the site $i$ and $j$ belong to the Kasteleyn-Fortuin same cluster, effectively linking the presence of a percolating cluster spanning the whole system to the onset of quasi-long-range order. In the present case one can identify two different critical lines through the $\nicefrac{{1}}{{2}}$ percolation threshold criterion, depending on whether the clusters by which percolation is identified are allowed to extend on a single layer or on both layers, respectively. The results we obtain, shown in Fig. 1(b), are compatible with the identification of the $A$, $B$ and $C$ phases obtained by the study of correlation functions. We now supplement this analysis by a RG calculation.

RG approach.

Mermin-Wagner theorem Mermin1966 forbids spontaneous breaking of continuous symmetries in two dimensions and, thus, only quasi-order, characterised by power law correlations, can arise in our model. Such effect, which cannot be reproduced within traditional perturbative approaches, can only be described by RG theory via the explicit introduction of the topological defects Jose1977 ; Kosterlitz16 . Within the functional RG framework Delamotte2012 ; Wetterich1993 , it has been possible to derive non-perturbative flow equations which correctly reproduce the Mermin-Wagner theorem Codello2013 ; Codello2015 ; Defenu2015 for $O(N)$ symmetric model, but the presence of quasi-order in the thermodynamic limit for $N=2$ and $d=2$ could not be found Grater1995 . Recently, a new method has been proposed Defenu2017 which allows to include in the traditional BKT formalism the non-universal corrections arising at small and intermediate scales.

We shall now employ a simplified version of this approach to the bilayer case, showing how it is possible to reproduce the MC phase diagram. First of all, we are going to derive the mean field solution of our problem; then we are going to use it to compute the effective spin stiffness in the low temperature phase, which will serve as the initial condition of the traditional BKT flow equations Kosterlitz1973 ; Kosterlitz1974 . Finally, solving the BKT flow equations with the MF initial condition, we will be able to locate the vortex unbinding transition.

Within our framework it is convenient to employ a Hubbard-Stratonovich transformation Simanek1994 ; Machado2010 ; Nishimori2011 , in order to introduce the continuous field representation of the bilayer model under study. Following this procedure, the bilayer XY model can be exactly mapped into two coupled complex $|\varphi|^{4}$ theories with the action

[TABLE]

where $U(|\varphi|)=\log(I_{0}(|\varphi|))$ is the local potential of the model, written in terms of the zeroth order Bessel function $I_{0}$, and $S_{\mathrm{kin}}[\varphi]$ is the kinetic action. The index $l\in\{1,2\}$ label two different complex fields, which represent the spin variables in the two different layers.

The kinetic part of the action is readily written in Fourier space

[TABLE]

where $K^{\sigma}(q)=2J\varepsilon_{0}(q)+2\mu+\sigma 2K$, $\sigma\in\{+,-\}$ and a multiplicative $\beta$ factor has been absorbed into the definition of the couplings. The coefficient $\mu$ has to be introduced in order to have a positively defined exchange matrix, as needed in the Hubbard-Stratonovich transformation  Machado2010 . The relation between the $\varphi_{\pm}$ fields and the original $\varphi_{1,2}$ counterparts reads

[TABLE]

Rewriting the local potential in Eq. (6) in terms of the $\varphi_{\pm}$ fields just introduced, one can identify the masses

[TABLE]

In order to proceed further with the analysis, one has to choose a suitable value of $\mu$. In principle any value of $\mu>2J+K$ is a valid choice and the choice of $\mu$ shall not alter the behavior of the model Machado2010 . However, when approximations are employed, a $\mu$-dependence of the result can be found Jakubczyk2016 ; Defenu2017 . In the following, we are going to only consider the continuum limit of the kinetic action in Eq. (7), discarding lattice effects, thus we may extend our choices also to $\mu<2J+K$ and, in particular, we will fix $\mu=K$, consistently with the optimal choice discussed in Defenu2017 for a single XY layer.

It is now possible to compute the two MF phase boundaries imposing the vanishing of one of the two masses $\mu_{\pm}$ at each phase boundary. According to this argument the lower phase boundary in Fig. 1(c) is obtained by the vanishing of $\mu_{+}$, while $\mu_{-}$ becomes zero at the upper boundary. Above each boundary one of the fields $\varphi_{\pm}$ acquires a finite expectation value thus violating the Mermin-Wagner theorem. The field expectation values can be obtained solving the saddle point equation

[TABLE]

where $S[\varphi]$ is defined in Eq. (6). The field density is defined according to $\rho_{\text{MF},\sigma}=|\varphi_{\text{MF},\sigma}|^{2}$, with $\varphi_{\text{MF},\sigma}$ solution of the saddle point equation, see Eq. (10).

As anticipated, the low energy behaviour of the model cannot be described simply considering the static MF solution, which provides an unphysical symmetry breaking scenario. In order to correct the MF picture and provide a reliable phase diagram, the effect of vortex fluctuations at low energy shall be included. In this perspective, it is convenient to consider the action (6) in the continuum limit

[TABLE]

where $m_{\pm}^{-1}=\frac{J}{K^{\pm}(0)^{2}}$. The explicit introduction of vortex fluctuations shall be done in the amplitude and phase representation $\varphi_{\sigma}=\sqrt{\rho_{\sigma}}e^{\mathrm{i}\theta_{\sigma}(x)}$. Assuming that the field amplitude remains frozen at the MF expectation $\rho_{\sigma}=\rho_{\text{MF},\sigma}$, the model action only contains phase terms

[TABLE]

In the latter action the two phases are uncoupled and two distinguished BKT transitions are found. Latter statement can be proven by mapping each independent quadratic phase term into a sine-Gordon model Defenu2017 .

The physics of the transitions is fully described, at this approximation level, by the effective phase stiffness $\rho_{\sigma}/2m_{\sigma}$ which is non vanishing only for finite density expectation $\rho_{\sigma}>0$. Once again for fixed $K$ and $J<1-K+\mu$ both the effective phase stiffnesses vanish and all the system correlation functions are exponentially decaying. Then for $1-K+\mu<J<1+K+\mu$ the effective stiffness of the $\theta_{+}$ phase becomes finite and the phase dynamics becomes relevant. The low energy phase dynamics is conveniently described by the BKT flow equations

[TABLE]

At the bare level, $K$ and $g$ assume the values

[TABLE]

where the bare effective spin system is simply given by

[TABLE]

In order to derive Eqs. (13)–(14) one has to assume a UV regularization, which traditionally relies in considering the Coulomb gas charges as hard disks of finite radius Kosterlitz1973 .

Using the MF expectations obtained solving the saddle point Eq. (10) we can now derive the theoretical prediction for the two BKT transition line reported in panel (b) of Fig. 1. The upper transition line, which corresponds to the conventional BKT transition, comes from the unbinding of the $\theta_{-}$ vortexes. Such phase boundary appears to be flat at our approximation level since we solve Eq. (10) separately for $\varphi_{\text{MF},-}$ posing $\varphi_{\text{MF},+}=0$, due to the difficulties encountered in solving the two coupled equations for the expectation values. On the other hand the $K$ dependence of the lower transition line is only due to the explicit $K$ dependence in the $m_{+}$ factor and it is in good agreement with the MC prediction.

Finally, we should compare our results with the corresponding results for two linearly-coupled Ising models. Denoting by $S_{i}$ and $T_{i}$ the spins in the two layers, a phase with order parameter $\langle ST\rangle\neq 0$ cannot exist (in the paramagnetic phase with $\langle S\rangle=\langle T\rangle=0$). The same is occuring for the two linear coupled $XY$ models treated in this Letter, i.e.  there is no phase with order parameter $\langle\cos(\phi+\psi)\rangle$, in agreement with the Mermin-Wagner theorem. What is present is indeed a BKT of the pairs, specific of coupled two-dimensional systems with $O(2)$ symmetry, not contradicting the previous result.

Conclusions.

We investigated the occurrence of a BKT-paired phase in coupled $XY$ models, corresponding to an exponential (algebraic) decay for the $1$-body ($2$-body) correlation functions. We found that the third phase is present even for very small linear couplings between the layers, within the precision of our simulations. Furthermore, we presented an RG treatment which was found to be in agreement with the aforementioned results. Our findings call for a more systematic study of finite size-scaling effects in the $J-K$ phase diagram: experiments with coupled 2D ultracold gases are performed on lattices with a linear size of tens up to a few hundreds of lattices sites, therefore motivating the study of the presence of the BKT-paired phase in realistic setups.

Furthermore, an exciting perspective is represented by a comparison of the present results with the ones obtained for other types of couplings, such as quartic ones, and to consider the case of different $J$’s in the layer. A systematic analysis for small $K$ done at larger sizes would also be important in clarifying whether the critical value for the presence of the BKT transition is zero as the RG approach seems to suggest. Moreover, we think as well that it would be a deserving subject of future investigations to further analyse the order to quasi-order transition between phases $B$ and $C$ of Fig. 1. Finally, we mention that an interesting perspective is represented by the comparison of our results with analytical findings from suitably derived theories where two sine-Gordon models are coupled in a way fixed by the microscopic coupling.

Acknowledgements.

We thank S. Chiacchiera, G. Delfino, N. Dupuis, T. Enss, M. Fabrizio and G. Gori for many stimulating discussions. G.B. acknowledges support from the Austrian Science Fund (FWF), under project No. M2461-N27. N.D. acknowledges support from Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster) and from the DFG Collaborative Research Centre “SFB 1225 ISOQUANT”. Support from the CNR/MTA Italy-Hungary 2019-2021 Joint Project “Strongly interacting systems in confined geometries” is gratefully acknowledged.