Superfluidity from correlations in driven boson systems

Most transport phenomena rely on the presence of particles with the capacity to move individually, as encoded in the kinetic energy term of the microscopic Hamiltonian, or the hopping term in case of lattices. Recently there has been interest in quantum many-body systems whose dynamics is not based on the kinetic energy of individual particles but rather on correlated behavior stemming from interactions. For instance, considerable attention has been paid to the Sachdev-Ye-Kitaev (SYK) model for fermions, which postulates two-body interactions that are random and long-ranged in a real-space lattice. This not only provides a route to the study of metallic behavior without quasiparticles Chowdhury et al. (2022), but has also been proposed as providing a means of modeling phenomena in quantum gravity Jafferis et al. (2022).

Both in solid state and cold atom physics, flat band systems have attracted considerable attention in recent years. In the case of quantum materials, this research has been spurred by the discovery of superconductivity in twisted bilayer graphene Cao et al. (2018) at the magic angles where a flat band is predicted to occur Bistritzer and MacDonald (2011).

Systems with a vanishing single-particle group velocity have also been investigated in the context of cold atom setups and for bosons in particular Huber and Altman (2010); Zurita et al. (2020). The origin of the single-particle flat band can be due to frustration Huber and Altman (2010), spin-orbit coupling Hui et al. (2017) or, more often, to the destructive interference between different paths intervening in the elementary hopping process, giving rise to what is termed “Aharonov-Bohm caging” Zurita et al. (2020); Hyrkäs et al. (2013); Takayoshi et al. (2013); Tovmasyan et al. (2018); Kuno et al. (2020); Pelegrí et al. (2020).

Recently, the use of Floquet engineering has been proposed to realize many-particle systems in which first-order (i.e., unassisted single-particle) hopping processes are suppressed Pieplow et al. (2018, 2019); Mateos et al. (2021), which provides an alternative path to the design of flat-band systems. The method relies on a fast time-periodic modulation of the hopping energy with a vanishing time average. When applied to the Bose-Hubbard model we have shown that this kinetic driving leads to an atypical interacting many-body system without a single-particle hopping term Pieplow et al. (2018). As a function of the driving amplitude, the system can be tuned continuously from the Mott-insulating regime to a peculiar form of superfluidity where the system shows a relatively robust cat-like ground state with branches characterized by the preferred occupation of opposite nonzero momentum eigenstates Pieplow et al. (2019). In the absence of a flux threading the ring, or in a segment delimited by hard walls, the two cat branches carry no current (here described by another exotic operator Pieplow et al. (2019)) but their different crystal momentum makes them behave very differently when allowed to expand in a larger lattice Molinero et al. (2022). The main properties of this system have been shown to be robust against variations in the driving signal and switching protocol Mateos et al. (2021).

The purpose of the present work is to explore the superfluidity of the kinetically driven Bose-Hubbard (KDBH) model, of which so far we have presented indirect evidence based on general theorems for one-dimensional systems Pieplow et al. (2018); Sowiński (2012); Cazalilla et al. (2011); Giamarchi (2003) and hints of its unusual nature Pieplow et al. (2019). As a benchmark, and using the same exact diagonalization method, we study in parallel the superfluidity of the conventional (i.e., undriven) Bose-Hubbard (CBH) model, a relatively well-understood system which however still can offer some new insights, especially when treated exactly.

Superfluidity and superconductivity have long been recognized as fundamental quantum phenomena of an essentially identical nature, both involving some form of Bose-Einstein condensation and only differing in the charge of the elementary carriers, which can be bosons or paired fermions. As emphasized by Leggett Leggett (2006), the term superconductivity actually refers to a variety of phenomena that usually come together and which roughly amount to the Meissner effect (an equilibrium property) and the metastability of supercurrents (a non-equilibrium property). A similar classification applies to neutral superfluids, albeit with nuances that are discussed in section III. In this paper we investigate both aspects of superfluidity in the KDBH model, whenever possible comparing with the CBH system ¹¹1We note that in one dimension the Mermin-Wagner theorem formally forbids the existence of Bose condensation. For convenience, however, when discussing these one dimensional systems we shall generally use the term “condensate” in preference to the more correct but verbose “quasi-condensate”.. Interestingly, a system of independent bosons is superfluid only in one of these senses and, for this reason, in our discussion we also consider the CBH model in the particular case of zero interaction.

Common to all phenomena that fit under the umbrella of superconductivity and superfluidity is the resilience of quantum flow against the presence of barriers or impurities of moderate strength. Thus, a major component of our present work is devoted to the generalization of the KDBH model to the presence of an impurity in the one-dimensional ring.

By explicitly considering the presence of an impurity and the full current-flux dependence, as well as the excitation spectrum (see below), our theoretical proof of superfluidity goes beyond the calculation of the condensate fraction or the Drude weight invoked in e.g. Refs. Hui et al. (2017); Tovmasyan et al. (2018) to claim superfluidity in flat-band boson systems.

This paper is arranged as follows. Section II discusses the central concept of quantum transport from correlations. In section III we describe the two main notions of superfluidity as addressed in the present context. Section IV presents the model Hamiltonian and current operator used in this work. In section V we present and discuss the numerical results for the Hess-Fairbank effect, which amounts to an incomplete Meissner effect due to the absence of Ampère’s law for neutral particles. Section VI deals with a discussion of metastable supercurrents and its possible decay mechanisms. It includes a calculation of the dynamic form factor. In both sections V and VI we argue that the superfluidity of the kinetically driven Bose-Hubbard system is, in many regards, qualitatively different from that of the conventional BH model. Section VII contains some concluding remarks.

In a tight-binding lattice with periodic boundary conditions, the site and momentum boson operators are related by

where the crystal momenta take values $k_{\ell}=2\pi\ell/L$, with $\ell\in\{0,...,L-1\}\subset\mathds{Z}$.

In systems where the total particle number is conserved, the fundamental requirement for the possibility of quantum transport is $\dot{\rho}_{q}\neq 0$, where $\dot{\rho}_{q}=[\rho_{q},H]/i\hbar$ is the time-derivative of the Fourier component of the density operator,

with $n_{x}=a_{x}^{\dagger}a_{x}$, and $H$ the underlying Hamiltonian.

In a typical many-body problem with two-body density-density interactions of the type

where $V_{q}$ is the Fourier transform of the two-body interaction potential $V(x-x^{\prime})$, and $\Omega\rightarrow\infty$ is the thermodynamic volume, we have

Therefore, a kinetic energy term (or hopping energy in a tight-binding picture) is needed to have $\dot{\rho}_{q}\neq 0$ and, with it, the possibility of quantum transport.

The Hamiltonians (7)-(11) and (21) which we will derive in section IV have in common that they do not commute with the density operator and thus give rise to a highly correlated form of quantum transport, qualitatively different from the conventional transport supported by a standard kinetic-energy term or, in a tight-binding picture, by the elementary process of unassisted single-particle hopping between neighboring sites.

For neutral bosons, the Hamiltonian (7) gives rise to a novel type of superfluidity, microscopically based on the correlated motion of particles and whose study is the object of this article.

Superfluidity is generally associated with the capacity of a quantum fluid to flow through barriers without a drop in the chemical potential. However, when we need to be more specific, and as noted in the Introduction, we must recognize that superfluidity actually comprises a variety of phenomena that usually (but not universally) go together. In the case of superconductors, these phenomena essentially boil down to two: the Meissner effect, an equilibrium effect under the constraint of a magnetic field, and the existence of persistent currents, a non-equilibrium property Leggett (2006).

For superfluids, the equivalent of a magnetic field is a rotation and the absence of electric charge translates into the Hess-Fairbank (HF) effect, which may be viewed as an incomplete Meissner effect where only the London equation applies and the equivalent of Ampère’s law is missing because its gravitational equivalent, the Lense-Thirring effect, is extremely weak ²²2We might identify a third phenomenon which is the flow of current through a capillary connecting two bulk reservoirs. As argued in Ref. Leggett (2006), the physics of such a setup amounts to the HF effect or the metastability of supercurrents depending on whether the total phase variation along the constriction is smaller or larger than $\pi$, respectively..

In our model system, the HF effect is studied by calculating the dependence of the space- and time-averaged current on the external flux $L\phi$ and in particular its linear dependence for small flux. This is the object of Section V, where the HF effect is studied in the presence and absence of an impurity for both interacting and independent bosons.

The existence of metastable supercurrents is explored in Section VI by investigating the possible decay mechanisms for a flow without pressure drop through a barrier or impurity without the assistance of an external flux (whose presence is essential for the HF effect). To that end we focus on the “mean-field current-carrying excited states”, which are those configurations characterized by the macroscopic occupation of a state with nonzero momentum and nonzero current.

For a single barrier in the ring, the lifetime against current reversal diverges as $\sim L$ for a single particle in the thermodynamic limit, and even more strongly for $N$ repulsively interacting bosons, with $N$ large.

The other decay mechanism is the spontaneous generation of quasiparticles when the speed of flow exceeds the Landau critical velocity. The existence of an acoustic mode guarantees superfluidity for flow speeds below the speed of sound. The Landau decay mechanism, when it applies, becomes most relevant in the large $L$ limit, because a continuum of possible momenta permits the excitation of quasiparticles of arbitrarily low momentum and energy ³³3In quasi–one-dimensional systems, phase slips can be another current decay mechanism. We do not address it here..

In Ref. Pieplow et al. (2018), the superfluidity of the KDBH system was indirectly proved by invoking general theorems from Luttinger liquid theory Pieplow et al. (2018); Sowiński (2012); Cazalilla et al. (2011); Giamarchi (2003). Here we will perform a more direct check by computing the dynamic form factor Pines and Nozières (1996), showing that the results obtained are consistent with the presence of an acoustic mode and thus a nonzero Landau critical velocity.

Both for the HF effect and the metastability of supercurrents, we include calculations for the undriven, conventional BH system to use it as a benchmark.

Since we are interested in superfluidity, we generalize the work of Refs. Pieplow et al. (2018, 2019); Mateos et al. (2021) to include an impurity and a threading flux in the BH ring, a setup which in the absence of driving can be modeled with the Hamiltonian

This is the periodic 1D Bose-Hubbard Hamiltonian (we impose $a_{x}=a_{x+L}$) with an external flux per link $\phi$ and an off-diagonal impurity parameterized by $\varepsilon\in[0,1]$. We assume $U>0$. For $\varepsilon=0$ the system is a periodic ring. As $\varepsilon$ increases, the hopping amplitude between sites $x=r$ and $x=r+1$ reduces, as schematically depicted in Fig. 1. The limit $\varepsilon=1$ corresponds to the segment or hard-wall limit, which has been investigated in Ref. Pieplow et al. (2019). In practice a continuous value of $\varepsilon$ can be realized by shining a blue-tuned laser into a small section of the ring in order to erect a repulsive barrier of tunable height.

From a numerical viewpoint, a practical advantage of the off-diagonal feature is that the system can be continuously tuned from a ring to a segment by varying $\varepsilon$ without changing the number of sites. Hereafter we set $\hbar=1$ and momenta will be measured in units of the inverse lattice spacing.

In the momentum representation, (5) acquires the form

where the function $h$ reads

The presence of a Peierls phase $\phi$ permits the inclusion of an effective total flux $L\phi$ threading the ring.

Under the effect of fast kinetic driving, $J\to J\cos(\omega t)$, and as described in Refs. Pieplow et al. (2018, 2019), an effective static Hamiltonian $H_{\rm eff}$ results which can be written in closed form only for small $\varepsilon$, as indicated below, where the result is given up to order $\varepsilon^{2}$. The hard-wall limit $\varepsilon=1$ also admits a closed form, as shown in Ref. Pieplow et al. (2019).

The smallness of $\varepsilon$ is not a major shortcoming because, to investigate superfluidity, it suffices to study the case of weak impurities. Specifically, we calculate

The zeroth-order term $H^{(0)}_{\text{eff}}$ (absence of impurity) reads

where $\mathcal{J}_{0}$ is the zeroth order Bessel function of the first kind, $\kappa=J/\omega$ is the driving parameter, and

The resulting impurity-free ring model has been studied in Refs. Pieplow et al. (2018, 2019) for $\phi=0$ and in Ref. Mateos et al. (2021) for nonzero $\phi$. Having been obtained by Floquet analysis in the high frequency limit Pieplow et al. (2018) it can be understood as the first term in an expansion in inverse powers of $\omega$, such as the Magnus or van Vleck series. In principle the system’s behaviour could be described at lower driving frequencies by calculating higher terms in the series expansion, or by applying alternative forms of Floquet engineering valid over all frequencies, such as that described in Ref. Bandyopadhyay and Thingna (2022). In this work, however, we simply confine ourselves to the $\omega\rightarrow\infty$ limit.

Within the same order in the inverse-frequency expansion, the terms of higher-order in the impurity strength are:

The HF effect refers to the failure of a superfluid to be dragged by the rotation of the ring where it resides if both are initially at rest. In the rotating frame, the HF is perceived as the establishment of a current due to the presence of a flux threading the ring. Its superconducting equivalent is described in the London equation, in which the electric current is proportional to the vector potential in the London gauge Tinkham (1996). In a neutral superfluid, that translates into a dependence of the ground-state current on the total effective flux ($L\phi$) threading the ring.

In this section, we compute the current expectation value $\langle I(\phi)\rangle$ in a variety of relevant situations and models. For small flux we find $\langle I(\phi)\rangle\sim\phi$ but, in the general case the current depends nonlinearly on the flux, very much in the way that in a superconductor, even a uniform one, the current density does not always depend linearly on the vector potential Sánchez-Cañizares et al. (2001).

The space-averaged current operator is given in Eqs. (15) and (20). For $\varepsilon=0$ and $\phi=0$ it was shown in Ref. Pieplow et al. (2019) that its expectation value is zero in the ground state and for each of the cat branches taken separately. In the following subsections we aim to understand the structure of the full flux dependence of the current expectation value.

Metastable currents in a ring with a barrier and not threaded by a flux are one of the features traditionally associated with superfluidity. In the absence of flux, there are non-stationary but long-lived configurations that carry a nonzero steady current. In the presence of some imperfection in the ring (which is usually the case), such current-carrying states can decay due to current reversal or to the spontaneous excitation of quasiparticles when the flow velocity is high enough.

Current reversal due to a single impurity is negligible in the thermodynamic limit. The resulting metastability is further enhanced by the presence of repulsive interactions, which translates into an attractive interaction in momentum space that favors the macroscopic occupation of a single momentum state Heimsoth et al. (2012); Pieplow et al. (2019).

With our small ring model we can explicitly analyze the mechanism of current reversal, which is particularly clear in the case of the conventional BH system. For small but nonzero flux we identify current-carrying excited states of the mean-field type, which are characterized by the macroscopic occupation of a opposite nonzero momenta, for instance, $\pm\pi/4$. For zero flux and zero barrier strength, these states are degenerate and genuinely stationary ⁴⁴4The case $\varepsilon=0$ with $\phi=0$ must be handled with care, since exact diagonalization may yield arbitrary states within the degenerate Kramers doublet. To avoid this problem, we work with a tiny flux that breaks the degeneracy without appreciably shifting the energy levels.. For $\phi=0$ and $0<\varepsilon\ll 1$, the weak barrier breaks the degeneracy by mixing the two states and yielding a gap that can be identified with the inverse lifetime against current reversal of the metastable current-carrying states.

Since repulsive interactions in real space favor the occupation of a single momentum state Heimsoth et al. (2012), we can expect the lifetime to grow (or the gap to decrease) with increasing $U>0$. This is what we actually find for the CBH model, as shown in Fig. 7 for different impurity strengths. The clear global trend is that the gap $\Delta$ (proportional to the inverse lifetime) decreases with increasing $U$ and decreasing $\varepsilon$. The discontinuities in $\Delta$ as a function of $U$ arise from spurious finite-size effects.

A similar study in the KDBH system is more cumbersome to perform because there are many states between the symmetric and antisymmetric combinations of the metastable, mean-field–like current-carrying states. Moreover, the global role played by $U$ [see Eqs. (7)-(11)] prevents its use as an interesting tuning parameter.

The other mechanism of metastable current decay is the spontaneous generation of quasiparticles when the flow velocity exceeds the Landau critical velocity which, in the presence of an acoustic mode, corresponds to the speed of sound. This mechanism remains relevant, and even more efficient, in the thermodynamic-limit, because of the presence of a continuum of available momenta and energies in the spectrum of low-lying excited states.

In Ref. Pieplow et al. (2018) some Luttinger-liquid theorems were invoked to indirectly prove the existence of superfluidity, which is ultimately associated with the existence of an acoustic mode. Here we choose to explore the excitation spectrum directly by calculating the dynamic form factor Pines and Nozières (1996)

where $\rho_{q}$ is given in Eq. (2) and $\omega_{n0}=E_{n}-E_{0}$, with $E_{0}$ the energy of the ground state $\ket{\Psi_{0}}$ and $E_{n}$ the energy of $n$-th excited state $\ket{\Psi_{n}}$.

The numerical results for the number-conserving excitation spectrum are shown in Fig. 8 for the CBH and the KDBH models in their respective superfluid regimes. The frequency dependence is smoothed out with a convolution procedure similar to that employed in Ref. Pieplow et al. (2018). The existence of an acoustic mode is clear in both cases. Figure 8b provides direct numerical evidence of the existence of superfluidity in the KDBH model. This superfluidity is qualitatively different from that of the CBH model in several respects which we have discussed. We identify the existence of excited states with a dispersion relation similar to that of conventional quasiparticles but which actually are excitations shared by the two branches of the cat-type ground state.

For CBH (see Fig. 8a), it is interesting to note that, for large momenta, the spectral peaks split into two. This is a system whose elementary excitations are well explained by Bogoliubov theory. In a tight-binding lattice, the acoustic relation does not hold for momenta near the boundary of the Brillouin zone. As a consequence, a given momentum of the elementary excitation can be due to configurations with different total energy, which results in a double peak in the dispersion relation.

It is interesting to analyze the dependence of the sound velocity $c$ with respect to the total particle density $n=N/L$. For CBH we expect $c\sim\sqrt{n}$, if we take the Bogoliubov approximation in the thermodynamic limit as a reference. We lack a similar a priori expectation for the KDBH system. For CBH, Fig. 8a reveals $c=1.41$ for $(N,L)=(8,8)$, while the same calculation for the case $(N,L)=(4,8)$ produces the result $c=1.16$ (not shown). This yields a ratio $1.22$, not very different from the reference value $\sqrt{2}$. A similar calculation for the KDBH model yields the ratio $0.61/0.25=2.44$. The fact that speed of sound is considerably more sensitive to the density than in the CBH case, is consistent with the results reported in section V.3 for the dependence of the total current on the boson density (see Fig. 6).

Although a standard, Bogoliubov-type theory of the elementary excitation does not apply to the KDBH system Pieplow et al. (2019), some physics insights are still possible. When a pair of total momentum $\pi$ is “created” (borrowing from the Bogoliubov mean-field language), it is not possible to distinguish from which cat branch the two atoms come, as both branches provide the same total momentum, namely, $\pi$, which is equivalent to $-\pi$. This fundamental inability to distinguish between the two cat branches lies at the root of the unusually robust cat features of the ground state.

For completeness and comparison, it is interesting to plot the entire energy spectrum as a function of the total momentum

The results are shown in Fig. 9 for both CBH and KDBH. In both cases, we provide an inset that tracks the ground-state energy of the various $Q$ sectors. It is interesting to compare those curves with the dynamic form factor, which for a given $q$ involves many excited states. For this reason, $S(q,\omega)$ peaks at energies higher than the ground-state energy of the same momentum sector.

We have investigated the nature of the superfluidity in a kinetically driven one-dimensional boson system described, in the absence of driving, by the Bose-Hubbard model. Kinetic driving is a form of Floquet engineering where the hopping energy is made to oscillate with zero time-average. The resulting effective dynamics is that of a system where first-order single-particle hopping processes are suppressed and only higher-order processes involving more than one particle are allowed. Thus, kinetic driving is an alternative route to the design of flat-band systems.

In our study, we compare the dynamics of this highly correlated system with that of the conventional (undriven) Bose-Hubbard model. In both cases, we employ exact diagonalization, which limits the system’s size and particle number but permits detailed studies. The main message is that superfluidity is possible in a flat-band system and that, in many regards, it is qualitatively different from the conventional superfluidity based on autonomous single-particle hopping.

We have paid attention to the two different facets or definitions of superfluidity, namely, the Hess-Fairbank effect and the metastability of supercurrents, with explicit inclusion of an impurity in some cases. For the HF effect, we have studied the dependence of the kinetically driven Bose-Hubbard (KDBH) ground-state current on an effective flux. When comparing with the conventional Bose-Hubbard (CBH) system, we find a number of important differences.

The superfluid current of the two branches of the cat-like ground state is identical despite their crystal momenta being opposite. Moreover, the interference between the various collective components of the ground-state wave function is essential to produce the “diamagnetic” current. The diagonal contribution of the ideal cat part to the current is zero. The analogous contributions of the other two components (shared condensate and reduction cloud) are nonzero but considerably smaller than those resulting from their interference with the ideal cat. This contrasts sharply with the CBH case, where the single condensate carries diagonally most of the flux-dependent current and its crossed contribution with the depletion cloud is zero.

Another important feature is that, in the presence of zero or weak impurities, the KDBH shows a roughly sinusoidal dependence of the current on the flux, instead of the characteristic linear behavior of the CBH system. This means that, even in the absence of impurities, the KDBH behaves like CBH does only when a weak link (strong impurity) is present in the ring.

The dependence of the diamagnetic current on the density is also quite anomalous. While CBH shows the expected linear dependence, the KDBH model displays a higher sensitivity. Similar results are obtained for the superfluid fraction, which is close to unity in the CBH case and appreciably smaller for the KDBH system.

Defining metastable supercurrents as those that are long lived in the presence of an impurity and without the help of a flux, we have studied the current reversal due to the presence of an impurity, a mechanism that can be important in small rings. For the CBH case we have explicitly shown that the inverse lifetime against current reversal increases with the impurity strength and decreases with the real-space repulsive interaction.

To investigate the robustness of the supercurrent against the spontaneous creation of excitations, we have computed the dynamic form factor and found that the KDBH shows an acoustic mode with a speed of sound that, like for CBH, is to be identified with the Landau critical velocity. This ensures the existence of metastable supercurrents in the thermodynamic limit.

One may wonder whether our conclusion on the viability of superfluidity in the KDBH model is applicable to other flat-band systems. This question cannot be addressed in general, but it seems reasonable to expect that, as long as bosons can hop in pairs between nearest neighbors, they have the potential to form a superflowing condensate. However, if that is the only elementary process in real space, such a moving condensate will typically have a momentum per particle close to zero and thus will be not so different from a conventional superfluid, at least in this regard. In any case, we note that a full theoretical claim on the superfluidity of a flat-band system must be substantiated by an explicit check of its resilience against impurities in the presence of a flux, and by arguments supporting the existence of a nonzero Landau critical velocity.

In summary, we have found that a kinetically-driven boson system exhibits a qualitatively new type of superfluidity which is entirely based on the correlated motion of strongly interacting particles. Within the framework of a tractable model, our work proves that superfluidity in a flat band system is feasible. Floquet engineering based on the driving of the hopping energy opens new avenues in quantum many-body physics that may include higher-dimensional boson and fermion systems with possibly surprising properties.