Crystalline droplets with emergent topological color-charge in many-body systems with sign-changing interactions

TL;DR

This paper introduces a new type of self-bound droplet with an emergent topological color charge in many-body systems, formed via sign-changing interactions, with potential realization in cold atom experiments.

Contribution

It presents a novel self-bound droplet model with topological color charge arising from sign-changing interactions, and proposes an experimental realization with cold atoms in optical cavities.

Findings

Droplets form through spontaneous crystallization into topologically distinct domains.

Each droplet's color charge influences mutual interactions, being attractive or repulsive.

The model predicts nontrivial viscous dynamics in droplet plasmas.

Abstract

We introduce a novel type of self-bound droplet which carries an emergent color charge. We consider a system of particles hopping on a lattice and interacting via a commensurately sign-changing potential which is attractive at a short range. The droplet formation is heralded by spontaneous crystallization into topologically distinct domains. This endows each droplet with an emergent color charge governing their mutual interactions: attractive for equal colors and repulsive otherwise. The number of allowed colors is fixed only by the discrete spatial symmetries of the sign-changing part of the interaction potential. With increasing interaction range, the droplets become progressively more mobile, with their color charge still being energetically protected, allowing for nontrivial viscous dynamics of the interacting droplet plasmas formed during cooling. Sign-changing potentials with a short-range attraction appear quite naturally for light-mediated interactions and we concretely propose a realization in state-of-the-art experiments with cold atoms in a multimode optical cavity.

Figures

Equations

$$T_{\mathrm{DW}}=\frac{N(U_A+U_B)}{L^D\cdot 2\ln 2}\sim \frac{N{U}_0{\xi }^D}{L^D}$$

$$T_{\mathrm{dr}}=\frac{N{U}_0}{2\ln (L/\xi {)}^D}$$

Full text

Introduction. Classical and quantum many-body systems often feature competitions between attractive and repulsive forces, including kinetic terms such as thermodynamic or quantum pressure. Attractive forces favor the formation of clusters of matter, while repulsive and kinetic terms prevent the collapse of the system. Through competition between attraction and repulsion various self-bound objects may emerge: from atomic nuclei Bender:2003 , van der Waals nanoclusters Chalasinski:1994 , droplets in conventional liquids Frohn:2000-book and in liquid helium Barranco:2006 , to stars, galaxies Padmanabhan:1993-book , and black holes Dvali:2014 . (Ultra)cold-atomic systems represent a convenient playground for studying such self-bound objects. Quantum droplets were recently observed in bosonic dipolar gases Pfau:2016-Nature ; Chomaz:2016 and in bosonic mixtures Tarruell:2018-Science ; Semeghini:2018 .

Here, we introduce another type of droplet with the distinguishing property of carrying an emergent color charge of topological origin. We show that such droplets can form and be stably self-bound if the interparticle interactions are (i) sign changing in space commensurately with the lattice, (ii) attractive on the same lattice site, and (iii) finite-ranged. Features (ii) and (iii) allow for the formation of droplets which are self-bound below a certain critical temperature. Crucially, it is feature (i) which endows the droplets with the charge, as they can then only be stable if their constituent particles occupy a specific sublattice, i.e., take a specific “color”. The latter indeed determines the sign of the interaction between droplets: attractive for the same color and repulsive otherwise. The integer number of allowed colors thus depends solely on the discrete spatial symmetries of the lattice and the interaction potential. We find that the color charge of a droplet is conserved due to an energetic protection, as it would take an extensive number of local single-particle processes to remove it. Moreover, each of those processes is hindered by an energy barrier which also scales extensively in the thermodynamic limit.

The emergence of a droplet color charge is intimately connected with the phenomenon of the spontaneous breaking of discrete spatial translational invariance by choosing a specific sublattice. Indeed, we observe that droplet formation is necessarily heralded by crystallization: upon cooling down the system, crystalline domains are formed, and each one then subsequently condenses into one or several droplets with the same color charge. The following cooling dynamics is one of a viscous plasma of droplets with different color charges governing their mutual interactions, whose range is set by the microscopic interaction range. For sufficiently long interaction ranges, we also observe stable, translationally invariant crystalline phases at intermediate temperatures.

While the microscopic features (i) and (ii) might seem exotic in the context of condensed-matter physics, they are quite generically found in systems of particles with light-mediated interactions, provided that specific electromagnetic modes are selected via laser driving and/or light confinement. This is, for instance, the case for experiments involving laser driven-atoms in optical resonators Ritsch:2013 ; Black:2003 ; Baumann:2010 ; Arnold:2012-PRL ; Hemmerich:2014 ; Leonard:2017 ; Vaidya:2018 ; Lev:2019-short ; Lev:2019-long or photonic crystals Tudela:2015 . In particular, the formation of noncharged droplets due to light-mediated interactions in a noncommensurate case has been predicted recently Pohl:2018 .

Model. We consider particles under the influence of the following potential, $H=\sum_{k}V(\mathbf{x}_{k})+\sum_{k<l}U(\mathbf{x}_{k},\mathbf{x}_{l})$. For a deep external potential $|V|\gg|U|$ we can consider a simplified lattice version of the model. Provided that the interaction potential $U$ satisfies the properties (i)-(iii) listed in the Introduction, the phenomenology we are going to describe further depends solely on the spatial symmetry of the interaction potential relative to the external lattice. Being in a commensurate case, the above symmetry would be the one being spontaneously broken by entering a crystalline, specifically a density-wave (DW), phase. This discrete symmetry will also determine the set of possible colors of the droplet charge, as we shall see below. For concreteness, we will restrict ourselves to a $\mathbb{Z}_{2}$ symmetry emerging in the two-dimensional configuration illustrated in Fig. 1. We parametrize the interaction potential between two particles, one of which is situated at the origin and another at site $\mathbf{r}_{i,j}=i\mathbf{a}_{1}+j\mathbf{a}_{2}$, as $U_{i,j}\equiv U(0,\mathbf{r}_{i,j})=(-1)^{i+j}f(r_{i,j})$ (where $\mathbf{a}_{1},\mathbf{a}_{2}$ are the basis vectors of the lattice). The envelope function $f(r)<0$ is monotonously increasing up to zero with some characteristic range $\xi$. In the following, we consider a thermal regime where the dynamics is classical (the regime of applicability is discussed in SM ). We performed Metropolis Monte Carlo (MC) simulations of the discrete two-dimensional model with a particular choice of the interaction potential $U_{i,j}=-U_{0}(-1)^{i+j}\exp(-(i^{2}+j^{2})/\xi^{2})$, which possesses the generic properties described above (we use the lattice constant $a$ as the unit of length). It is crucial to note that it is the sign-changing part of the interaction potential $(-1)^{i+j}$ that possesses the aforementioned $\mathbb{Z}_{2}$ symmetry defining the two possible types of color charges.

Thermodynamics. The qualitative form of the equilibrium phase diagram of our model, shown in Fig. 1, can be simply understood by considering the free energy, which also provides analytical estimates for the transition temperatures. At high temperatures we expect all the particles to be uniformly spread over all sites of the lattice. By decreasing the temperature, the sign-alternating nature of the interaction potential favors a checkerboard DW state, while the $\xi$-range attraction favors a droplet state with all the particles concentrated within one cell of size set by the range $\xi\times\xi$ of the interaction potential (note that while at $T\rightarrow 0$ we expect a droplet to be localized to just one site, but for nonzero temperatures, the droplet can be smeared up to size $\xi$ by the entropic force). Let us compare the free energies of these three states: uniform, DW, and droplet. For generality we consider a $D$-dimensional case, when the particles sit in a box with volume $L^{D}$. In the uniform state, all the sites are populated by an approximately equal number of particles $n\equiv N/L^{D}$. Each particle gives a contribution $\frac{1}{2}(-U_{A}+U_{B})n$ to the energy, where $-U_{A}=-\sum_{i+j=\mathrm{even}}|U_{i,j}|$ is the negative (attractive) part coming from the interaction with all particles sharing the same sublattice, while $U_{B}=\sum_{i+j=\mathrm{odd}}U_{i,j}$ is the positive (repulsive) part from the other sublattice. The entropy of the state is $S_{\mathrm{unif}}\approx N\ln L^{D}$, resulting in a free energy of the uniform state given by $F_{\mathrm{unif}}=-\frac{1}{2}(U_{A}-U_{B})n^{2}L^{D}-TN\ln L^{D}.$ In the checkerboard DW state, all the particles occupy one sublattice with $2n$ particles per site on average. The free energy of the DW state is thus $F_{\mathrm{DW}}=-\frac{1}{2}U_{A}(2n)^{2}\frac{L^{D}}{2}-TN\ln(L^{D}/2)\ .$ In the droplet state, all particles occupy a given sublattice within a box of size $\xi^{D}$ and each of them interacts with all the others with a corresponding energy of the order of $-U_{0}\equiv U(0)$, so that the free energy can be estimated as $F_{\mathrm{dr}}\approx-\frac{1}{2}U_{0}N^{2}-TN\ln\xi^{D}.$ In the particular case of a short-range potential, $\xi\lesssim 1$, all the atoms would collapse to one lattice site.

The droplet state is the lowest energy one and thus the most favorable at low $T$. Hence there are two scenarios for phase transitions with lowering the temperature: (a) transition from the high-temperature uniform state to the DW state (and then to the droplet state at some lower temperature); (b) transition from the uniform state directly to the droplet state. For case (a), from $F_{\mathrm{unif}}=F_{\mathrm{DW}}$ we get the critical temperature

[TABLE]

where we have used the identity $C_{D}\xi^{D}U_{0}\equiv U_{A}+U_{B}=\sum_{i,j}|U_{i,j}|$, with $C_{D}$ being the dimensionless factor for the volume of a $D$-dimensional sphere with radius $r$: $V=C_{D}r^{D}$. Using mean-field theory it is possible to quantitatively determine $T_{\mathrm{DW}}$ [SM, ]. For case (b), from $F_{\mathrm{unif}}=F_{\mathrm{dr}}$ we get

[TABLE]

Comparing expressions (1) and (2) we see that if $\xi=\text{const}$ in the thermodynamic limit $N,L\rightarrow\infty$ (while $n=N/L^{D}=\text{const.}$) we always have $T_{\mathrm{dr}}>T_{\mathrm{DW}}$, so that the droplet phase is always more favorable than the DW phase. However, if the range of the potential also scales with the size of the system as $\xi\sim L$, then both transition temperatures scale identically as $L\rightarrow\infty$, which allows for the existence of the DW phase at intermediate temperatures even in the thermodynamic limit. In SM we present an exactly solvable toy model for the droplet-formation phase transition. In general, the qualitative dependence of the critical temperatures on the interaction range is shown by the black and orange lines in the phase diagram of Fig. 1.

Our MC simulations for the case $T_{\mathrm{dr}}<T_{\mathrm{DW}}$ confirm the existence of a stable DW phase. In order to trace the DW transition we use the even-odd sublattice imbalance order parameter $m=(N_{\mathrm{even}}-N_{\mathrm{odd}})/N$; for the droplet-formation phase transition we use the order parameter related to breaking of translational symmetry of the system $O_{\mathrm{dr}}=\sqrt{\frac{L^{2}}{L^{2}-1}\sum_{i}\left(\frac{n_{i}}{N}-\frac{1}{L^{2}}\right)^{2}},$ so that $O_{\mathrm{dr}}=0$ corresponds to the uniform phase and $O_{\mathrm{dr}}=1$ to the single-site droplet. For $\xi=6$ (corresponding to the dashed arrow labeled by $a$ in Fig. 1), we indeed observe both phase transitions with lowering the temperature: a continuous transition from the uniform to the DW phase at $T\approx 1130U_{0}$ and a first-order phase transition to the droplet phase at lower $T\approx 790U_{0}$ on cooling and $T\approx 1010U_{0}$ on heating (Fig. 2). Figure 3 directly visualizes the sequence of the phase transitions showing the snapshots of the system at different temperatures. The initial droplet of size $\xi\times\xi$ (Fig. 3c) with lowering the temperature undergoes a crossover to the state where it becomes localized on one site (Fig. 3d). Note that even such a localized droplet has a “memory” about the sublattice of the intermediate DW state it originates from.

The situation is more complex in the case $T_{\mathrm{dr}}>T_{\mathrm{DW}}$, where the free-energy arguments above however do not provide the full picture. There we have assumed one crucial feature of the droplet, namely that it occupies a single sublattice, as confirmed already by the numerical simulations shown in Fig. 3. This is due to the fact that only such a droplet is energetically favorable against the uniform phase because of the sign-changing nature of the interactions. Droplets in our system have to choose a sublattice, which demonstrates a deep interconnection between the formation of a DW and droplets. This has important implications both for thermodynamics and dynamics. Deferring the discussion of the cooling dynamics to the next section, we report here the result of our MC simulations for the critical temperature of droplet formation out of the homogeneous phase for the case $T_{\mathrm{dr}}>T_{\mathrm{DW}}$. We observe that on cooling the critical temperature is not given by our estimated $T_{\mathrm{dr}}$, but rather by the DW-critical temperature $T_{\mathrm{DW}}$. In the temperature region $T_{\mathrm{DW}}<T<T_{\mathrm{dr}}$ the uniform phase is still metastable and we observe hysteresis (see Fig. 2c,d and dashed orange line in Fig. 1). This is because during cooling the long-wavelength density fluctuations do not gain energy (while losing the entropy) due to the sign-changing nature of the interaction potential. On the contrary, below $T_{\mathrm{DW}}$ one sublattice is spontaneously chosen so that long-wavelength density fluctuations gain energy and allow for the formation of the droplets. During cooling the second-order DW-transition thus triggers the first-order droplet transition. This picture will be confirmed by our simulations of the cooling dynamics that we discuss next.

Cooling dynamics. The fact that droplet formation is heralded by crystallization into a DW where particles spontaneously choose a sublattice effectively endows droplets with an emergent charge (set by the choice of sublattice) governing their mutual interaction. The latter has a range $\xi$ and, as a consequence of the commensurate sign-changing nature, is attractive between the similar-color charges and repulsive otherwise (see SM ). The discrete number of possible charge colors is determined by the number of distinct sublattices, i.e., by the discrete spatial symmetry which is spontaneously broken by the DW. Despite our specific example involving a $\mathbb{Z}_{2}$ symmetry, i.e., two charge types, other numbers of such “color” charges can be implemented using appropriate geometries (see SM for the discussion of the four-color case). The set of droplet colors has a topological nature in the same sense domain-wall defects in a DW state do. For the latter, however, only the domain wall separating two infinite domains is topologically protected as it cannot be removed by local perturbations. For finite domains the protection is reduced to a weaker energetic one – there is an extensive energetic barrier for shifting a domain to a different sublattice. In the same fashion, our color-charged droplets are local objects that can not be topologically protected but are rather energetically protected by an extensive energetic barrier, as we discuss below.

We consider the case $T_{\mathrm{DW}}<T_{\mathrm{dr}}$ (for $\xi=5$, $L=100$) and use a rapid quench from a random configuration to a low temperature $T\ll T_{\mathrm{DW}}$, tracing then the MC evolution of the state at this fixed temperature (Fig. 4). Within several MC sweeps particles form domains corresponding to even and odd sublattices (Fig. 4a). Because of the repulsion between particles occupying different sublattices, the domains separate from each other with the formation of intermediate low-density regions of width $\sim\xi$ (Fig. 4b). After the formation of domains, the attractive interaction between particles within a domain becomes efficient and induces self-collapse. Each domain, depending on its size, decomposes into one or several droplets, preserving the $\mathbb{Z}_{2}$-color charge of the initial domain (Fig. 4c).

The droplets can however perform a diffusive motion, with the diffusion constant being determined by $T$ and the interaction range $\xi$. For droplets collapsed to a single site (i.e., at $T\ll T_{\mathrm{dr}}$), the particle-by-particle flow of the droplet becomes suppressed by an energy barrier $\sim(U_{1,1}-U_{0})N_{0}^{2}$ (where $N_{0}$ is the number of atoms in the droplet). However, for droplets localized in a $\xi\times\xi$ region (i.e., at $T\lesssim T_{\mathrm{dr}}$) the barrier is estimated to be not higher than $\sim(U_{1,1}-U_{0})(N_{0}/\xi^{2})^{2}\sim U_{0}N_{0}^{2}/\xi^{6}$, so that with increasing $\xi$ the movement of the droplets becomes possible, while the barrier $\sim U_{0}N_{0}^{2}$ still protects the total color charge of the droplets of each species. This protection is a required feature for an emergent charge: it needs indeed to be conserved during the dynamics. As anticipated, the protection in our case is of an energetic nature since, as long as $N_{0}$ is extensive, there are required $N_{0}$ local single-particle steps, each of which is suppressed by an energy barrier scaling itself with $N_{0}$. By tuning $\xi$ we can find a temperature interval where the droplets are mobile (i.e., can perform a diffusive motion), but with the total color charge still being conserved. Figure 1 qualitatively shows the mobility of the droplets in the region of their stability in the phase diagram.

During the cooling and after droplet formation we thus deal with an interesting emergent physics: an overall neutral (apart from statistical fluctuations) plasma of color-charged droplets performing viscous movement and interacting over a range $\xi$. Despite the fact that the true thermodynamically stable state is the single droplet discussed in the previous section, the droplet plasma eventually reaches a metastable state with an extensive lifetime. As droplets with equal color charges merge with one another the plasma will be eventually composed of two droplets (one for each color). Since the remaining droplets repel each other, an extensive energy barrier prevents the system from reaching the single-droplet state.

Implementation with light-mediated interactions. In the regime of strong coupling where the back-action between light and matter cannot be neglected, photons effectively mediate interactions between particles. Sign-changing interaction potentials appear naturally if specific electromagnetic modes are selected via driving and/or confinement. For example, consider an ensemble of neutral atoms inside an optical cavity Ritsch:2013 . Let us assume the atomic motion being confined to the $x-y$ plane transverse to the cavity axis and driving a given atomic transition by two off-resonant laser beams forming standing waves, so that the laser field can be written as $\Omega(\mathbf{x})=\Omega_{0}(\cos(2\pi\tilde{x}/\lambda)+\cos(2\pi\tilde{y}/\lambda))$, where $\tilde{x}=(x+y)/\sqrt{2}$, $\tilde{y}=(x-y)/\sqrt{2}$, and $\lambda$ is the pump wavelength which settles the lattice constant to be $a=\lambda/\sqrt{2}$. For red detuning from the atomic transition this creates a two-dimensional optical square lattice potential $V(\mathbf{x})=-V_{0}\Omega^{2}(\mathbf{x})$ (Fig. 1; see also SM for the details of the microscopic model). In addition, two-photon transitions involving one laser- and one cavity-photon generate an interaction potential between the atoms which takes the form $U(\mathbf{x},\mathbf{x}^{\prime})=\Omega(\mathbf{x})\Omega(\mathbf{x}^{\prime})f(|\mathbf{x}-\mathbf{x^{\prime}}|)$ [Ritsch:2013, ], where the envelope function $f(r)$ is determined by the geometry of the cavity. For a near-planar cavity with only one mode close to resonance, we have $f(r)=const$, i.e., a global interaction with $\xi=\infty$, while for an infinite number of resonant modes (for example, a concentric or confocal cavity) we have $f(r)=-\delta(r)$ (completely localized interaction). For a realistic case of large but finite number of nearly degenerate modes realized recently in Stanford Vaidya:2018 , $f(r)<0$ is typically a monotonously increasing up to zero function with some characteristic range $\xi$, exactly as we have considered in the present work. A modification of the described scheme using pump lasers with orthogonal polarizations also allows us to realize a more complex case of four color-charge species (see SM for the details).

DW phases of atoms in optical cavities so far have been experimentally studied in the regime where the interaction range $\xi$ is larger than the cloud’s size Black:2003 ; Baumann:2010 ; Arnold:2012-PRL ; Hemmerich:2014 ; Esslinger:2016 ; Kollar:2017 ; Leonard:2017 . Only recent experimental developments using multimode cavities Vaidya:2018 ; Lev:2019-short ; Lev:2019-long have demonstrated the tunability of the range down to scales below typical cloud’s sizes, at least for the thermal case. Here, we have shown that sign-changing interactions behave drastically different as soon as the finiteness of the interaction range is appreciable, offering generic scenarios for the phenomenon of crystallization which have not been considered so far. The proposed scheme should be feasible within the current experimental capabilities: in particular, the magnitude and the color charge of the droplets can be nondestructively monitored by imaging the amplitude and the phase of the scattered light Lev:2019-short ; Lev:2019-long . Since the droplets exist in the classical (thermal) regime, it is possible to work in a much larger temperature range and with much bigger atomic clouds, so that the interaction range is smaller than the cloud’s size. For a typical interaction scale $U_{0}$ of the order of kHz, the relevant gas temperatures can be estimated as $T\simeq 10^{3}$kHz $\approx 50\mu$K. No fine tuning is needed to observe the droplet plasma phase.

Finally, we emphasize the limits of applicability of the studied model. The approximation of the lattice gas is applicable when $T\ll V_{0}$, in this case the atoms’ positions in optical lattice cells are bound by a sphere with characteristic radius $r_{T}\simeq a\sqrt{T/V_{0}}\ll a$. The quantum effects can be neglected if $n\lambda_{T}^{3}\ll r_{T}^{3}$, where $n$ is the number of atoms in one cell and $\lambda_{T}=(2\pi\hbar^{2}/mT)^{3/2}$ is the thermal de Broglie wavelength (for simplicity we take the frequency of the potential, confining atoms to the $z=0$ plane to be the same as for the in-plane optical lattice). If $n>n_{BEC}\sim T\sqrt{ma^{2}/\hbar^{2}V_{0}}$ then the nucleation of a droplet is accompanied by its Bose-Einstein condensation. Contact repulsion of atoms qualitatively preserves the described picture. When the droplet formation begins, every optical lattice site occupied of the forming droplet is being filled up to some maximum density of particles until the contact hard-core repulsion makes unfavorable the further increase of the number of atoms at the site. If $a_{\mathrm{hard}}$ is the hard-core radius of the atoms this happens when the number of atoms in optical lattice site exceeds $n_{\mathrm{hard}}\sim(r_{T}/a_{\mathrm{hard}})^{3}=(a/a_{\mathrm{hard}})^{3}(T/V_{0})^{3/2}.$ Such a contact interaction thus will stabilize the extended droplets with respect to the single-site ones.

Summary and outlook. We introduced a novel type of droplet with the distinguishing property of an emergent color charge. The latter appears in lattice systems with sign-changing interactions (e.g., in current experiments with atoms in multimode cavities), where the phenomenon of crystallization becomes deeply connected with the droplet formation. Future explorations shall involve the study of the quantum regime and in particular the interplay of zero-point energy, particle statistics, as well as possible short-range repulsive potentials with the sign-changing interactions considered here.

Acknowledgments. We are grateful to Julian Leonard, Roderich Moessner, Farokh Mivehvar, Helmut Ritsch, and Georgi Dvali for helpful discussions. PK acknowledges the support of the Alexander von Humboldt Foundation and the Ministry of Science and Higher Education of the Russian Federation.