Quantum tomography of the superfluid-insulator transition
for a mesoscopic atomtronic ring

The study of quantum phase transitions has long been a cornerstone of condensed matter physics. Among these transitions, the superfluid (SF) to Mott insulator (MI) transition in the Bose-Hubbard model (BHM) stands out as a paradigmatic example [1, 2, 3, 4]. This model provides a simplified description of interacting bosonic particles confined in a lattice potential. It captures essential themes, including super-fluidity, self-trapping, and the formation of solitons.

Quantum chaos.– If the inter-particle interaction is not too strong, the ground state of the Bose-Hubbard model exhibits superfluidity, characterized by the coherent flow of particles throughout the lattice. As the density or the strength of the interactions is increased, the system undergoes the SF-to-MI transition [5, 6, 7]. In fact this transition is apparent also in the parametric evolution of the excited states. In this context, quantum chaos is a fascinating aspect that should not be overlooked [8, 9, 10, 11, 12, 13, 14, 15, 16]. It refers to the fingerprints of chaotic dynamics that characterizes the corresponding classical model, which is the discrete nonlinear Schrodinger equation (DNLSE), aka the discrete Gross-Pitaevskii equation.

The BHM chain.– The 1D version of the BHM concerns $N$ bosons that can hope along a chain that consists of $L$ sites. For ${L>2}$ it is more illuminating to assume periodic boundary conditions (ring geometry), aka atomtronic circuit. The hopping frequency is $-J$, and the on site interaction is $U$. If one regards the Hamiltonian as a classical system that consists of $L$ coupled non-linear oscillators, the generated equation of motion is the DNLSE, which contains a single dimensionless parameter $u$. Upon quantization $1/N$ plays the role of Planck constant, and a second dimensionless parameter $\gamma$ is defined. Namely,

The classical parameter $u$ controls the appearance of self-trapped states, and the stability of superflow [17, 18, 19, 13], while the quantum parameter $\gamma$ controls the SF-MI transition. In the MI phase, it becomes important whether ${\bar{n}\equiv N/L}$ is integer or not. Thus, as $u$ is increased, the ground state changes from coherent condensation in a momentum orbital, to a fragmented site-occupation. Formally speaking, the transition is abrupt only for an infinite chain, but its mesoscopic version is clearly apparent even for a two-site model (dimer), as pointed out long ago by Leggett [1], who called it a transition from Josephson-regime to Fock-regime. See also [20, 21]. It is therefore natural to ask how the transition is reflected in a mesoscopic ring, say a trimer that has ${L=3}$ sites. The new ingredient is chaos. The question arises how the SF-MI transition is mediated or affected by chaos.

Puzzle.– There is an extensive body of literature about the quantum spectrum of mesoscopic Bose-Hubbard rings, and in particular about mesoscopic superfluidity. But there is a missing bridge to themes that are related to the SF-MI transition, as $u$ is increased. In particular one wonders what is the relation between the observed quantum spectrum and the prediction of the Gutzwiller Mean Field Theory (GMFT) [22, 23, 24], aka slaved-bosons formalism. What is the nature of the low lying excitations [25, 26], and what is the border between the GMFT quasi-regular regime and the possibly ergodic quantum chaos regime.

Strategy.– We use the term quantum phase space tomography [27] in order to emphasize that the inspection of the spectrum is not limited to “level statistics”, as e.g. in [15, 16], but rather oriented to reveal the detailed relation to the underlying phase space structures, that serve as a classical skeleton for the quantum eigenstates. A useful strategy is to provide 3D images of the spectrum. Each point in such image represents an eigenstate, whose vertical position is the energy. The extra dimensions (horizontal axis/axes and/or color-code) are exploited to reveal properties of the eigenstates. Such technique has been exploited e.g. to highlight Monodromy-related features in the spectrum [28] (and further references therein). This can be complemented by inspection of representative eigenstates, using a quantum Poincare sections, as in [29]. The big picture can be summarized by a ${(u,E)}$ phase-diagram that shows the dependence of the spectrum on the the dimensionless interaction parameter $u$.

Outline.– In Sec. (I) we introduce the BHM Hamiltonian, with focus on the $L{=}2$ dimer and the $L{=}3$ trimer. The main measures for characterization of eigenstates are presented in Sec. (III). Then we present the tomography of the spectrum for the dimer in Sec. (IV), and proceed with the phase-diagram of the trimer in Sec. (V). The tomography of the spectrum and the fingerprints of chaos are discussed in Sec. (VI). The characterization of fluctuations is worked out in Sec. (VII), with associated results for the Higgs measure in Sec. (VIII). Finally we zoom into the lowest excitation bands in Sec. (IX). Further discussion of open issues is presented in the summarizing Sec. (X). Some technical sections appear as appendices.

The BHM Hamiltonian for an $L$ site system is

where $a_{j}$ and $a_{j}^{{\dagger}}$ are annihilation and creation operators, and the summation is over the site index $j$ modulo $L$. There is a possibility to consider a ring in a rotating frame [17, 12, 13]. The Sagnac phase $\Phi$ is proportional to the rotation velocity, and equals zero unless we state otherwise. The classical version of the Hamiltonian can be written in action-angle variables. Namely, with the substitution ${a_{j}\mapsto\sqrt{n_{j}}\exp(i\varphi_{j})}$ one obtains:

Due to conservation of particles, ${N=\sum n_{j}}$ is constant of motion, and therefore we have reduction to $d=(L{-}1)$ degrees of freedom. For example, for the trimer ($L=3$) we can regard ${(n_{1},n_{2})}$ as “position” space, with conjugate coordinates ${\tilde{\varphi}_{1}=\varphi_{1}-\varphi_{3}}$ and ${\tilde{\varphi}_{2}=\varphi_{2}-\varphi_{3}}$.

Optionally, one can define creation operators in momentum orbitals, namely,

where ${x_{j}=j}$, and ${k=(2\pi/L)\times\text{integer}}$ is defined modulo $2\pi$. The associated occupation operators are ${\hat{n}_{k}=b_{k}^{\dagger}b_{k}}$. The distinction between site-basis and momentum-basis coordinates is implied by the index ($i,j$ for sites, $k$ for momentum). Dropping a constant, the BHM Hamiltonian takes the following form

where the orbital energies are ${\varepsilon_{k}=-J\cos(k-(\Phi/L))}$. Note that the interaction term favors condensation in momentum orbitals, which is complementary to Eq. ​​(II) where it favors fragmented site-occupation.

The dimer Hamiltonian ($L=2$) can be written using generators of spin-rotations. The observable $S_{z}$ is defined as half the occupation difference in the site representation, namely ${S_{z}=\frac{1}{2}[a^{{\dagger}}_{1}a_{1}-a^{{\dagger}}_{2}a_{2}]}$. Then, one defines ${S_{+}=a^{{\dagger}}_{1}a_{2}}$ and ${S_{-}=S_{+}^{{\dagger}}}$, and the associated $S_{x}$ and $S_{y}$ operators. Accordingly, $S_{x}$ is identified as half the occupation difference in the momentum representation, where the momentum states are the mirror-symmetric orbitals: for the lower one is even, and the upper one is odd. With the above notations the dimer Hamiltonian takes the following form,

For $L>2$ ring we can generalize the spin-style notations. In particular, we can define jump operator as ${S_{i,j}=a_{i}^{\dagger}a_{j}}$, and associated $S_{x}$ and $S_{y}$ operators for each pair of sites.

The eigenstates of the BHM are ordered by energy and indexed by ${\nu=0,1,2,\cdots,\mathcal{N}{-}1}$, where $\mathcal{N}$ is the Fock-space dimension. For a dimer $\mathcal{N}=N{+}1$, while for a trimer

The standard representations $\Psi_{\bm{n}}=\left\langle\bm{n}\middle|\Psi\right\rangle$ is either in the site basis $\bm{n}=\{n_{j}\}$ with $j=1,\cdots,L$, or in the momentum-orbital basis $\bm{n}=\{n_{k}\}$. In this section we list a minimal set of measures that are later used in order to numerically characterize the eigenstates.

Quasi momentum.– In the presence of interaction the $n_{k}$ are not good quantum numbers, but still, without fear of confusion, we can use the notation

Still, due to translation symmetry, the manybody Bloch quasi-momentum $q=(2\pi/L)\times\text{integer}$ remains a good quantum number, and therefore the eigenstates divide into $L$ symmetry classes. For each eigenstate we can calculate the total momentum $P=\sum_{k}n_{k}k$. But $P$ is mathematically ill defined and depends on arbitrary modulo convention, whereas the quasi-momentum, unlike $P$, is rigorously a good quantum number:

The current ${I=(1/L)\sum_{k}\left\langle n_{k}\right\rangle v_{k}}$ is an optional quantity that can be calculate, where $v_{k}$ is the velocity of a particle that is placed in the $k$ orbital, see [12]. But $I$ unlike $q$ is not a good quantum number.

Condensation measures.– In the absence of interaction the BHM ground-state is a coherent state where all the particles are condensed in the zero-momentum orbital. As the interaction is increased, the ground-state get-squeezed. We use the following measures to characterize the condensation:

As explained below, we regard $\sigma_{0}^{2}$ as the measure for the amplitude fluctuations of the order parameter.

Order parameter.– We define ${S_{i,j}\equiv a_{i}^{{\dagger}}a_{j}}$. For ${i\neq j}$ we use the optional notation ${S_{i,j}\equiv S_{x}+iS_{y}}$. The expectation values $\left\langle S_{i,j}\right\rangle$ are the components of a generalized Bloch vector. The diagonal elements $\left\langle S_{j,j}\right\rangle$ are the average site occupations. The off-diagonal elements of $\left\langle S_{i,j}\right\rangle$ provide an indication for off-diagonal long range order (ODLRO). Due to the symmetry of the system under translations, it depends on the distance ${r=|i-j|}$ mod($L$), and therefore the matrix $\left\langle S_{i,j}\right\rangle$ becomes diagonal once we switch to the momentum-basis. It follows that ODLRO is related to the average occupations $n_{k}$ of the momentum orbitals, see Appendix A. At low energies we can regard $n_{0}$ as the major ODLRO measure.

Purity measure.– The one-body reduced probability matrix is ${\rho_{i,j}=(1/N)\left\langle S_{i,j}\right\rangle}$. The purity is defined as

Roughly speaking $1/\mathcal{S}$ is the minimal number of orbitals that are required in order to accommodate the particles. Purity that equals unity defines the notion of coherent state. In the BHM context, coherent state means condensation of all the particles in a single orbital. In the vicinity of the SF ground state, the deviation of $\mathcal{S}$ from unity reflects the depletion ${n_{dep}=N-n_{0}}$, see Appendix B. At the SF-to-MI transition the squeezed ground-state turns into a fragmented MI state, that has the lowest purity ($\mathcal{S}=1/L$).

The purity can also be used to detect self-trapping in a single site. Strictly speaking, due to translation symmetry such states always form “cat state” superpositions. But in practice any weak disorder breaks the translation symmetry, hence self-trapped states are formed, instead of very narrow solitonic bands. Such unavoidable symmetry breaking occur also due to numerical noise, or intentionally by introducing weak on-site potential, aka detuning.

On-site fluctuations.– We use the notation $\hat{n}_{site}$ to indicate any of the $\hat{n}_{j}$ operators. Due to translation symmetry we have $\left\langle\hat{n}_{site}\right\rangle=(N/L)$. We define

Later we also define the correlator ${C(r)=C_{i,j}=\left\langle\hat{n}_{i}\hat{n}_{j}\right\rangle}$. were $r$ is the “distance” between the sites.

The entanglement between a given site and the other sites provides an indication for the departure from the regime where GMFT applies. The definition is as follows. We use the standard basis (site occupation). Given an eigenstate $\Psi$ we define

The matrix $\rho^{(site)}_{n,n^{\prime}}$ is diagonal, because in each term of the sum we must have ${n=n^{\prime}=N-n_{2}-n_{3}}$ in order to get a non-zero result. The diagonal elements are the probabilities $\rho^{(site)}_{n,n}$. Hence $1/\mathcal{S}_{\text{ent}}$ is just another version of the on-site dispersion $\sigma_{\parallel}$.

Fluctuations of the order parameter.– We already defined $\sigma_{\parallel}$ for the characterization of the of the single-site fluctuations, and $\sigma_{0}$ for the fluctuations of the zero-orbital occupation. The latter can be regarded as a measure for the fluctuations in the amplitude of the order parameter. Additionally we can define a measure $\sigma_{\varphi}$ for the fluctuations in the phase of the order parameter. These three measures ${(\sigma_{0},\sigma_{\varphi},\sigma_{\parallel})}$ correspond to the variances of the of the ${(S_{x},S_{y},S_{z})}$ components of the order parameter. The technical details regarding the generalization of the “dimer” Bloch-vector language will be provided in later sections. What we call total fluctuation of the order parameter, denoted as $\sigma_{\perp}^{2}$, corresponds to the sum of the variances $\sigma_{0}^{2}$ (amplitude fluctuations) and $\sigma_{\varphi}^{2}$ (phase fluctuations).

Higgs measure.– In Sec. (VII) we derive a sum rule from which we can extract $\sigma_{\perp}^{2}$ given the average occupations $n_{k}$, and the on-site fluctuations $\sigma_{\parallel}^{2}$. Irrespective of that, we calculate the variance $\sigma_{0}^{2}$ that characterizes the amplitude fluctuations of the order parameter. Then we define the Higgs measure as the ratio, namely,

This measure is very small compared with unity for phase oscillations, and becomes of order unity for amplitude oscillations. Of particular interest is the identification of energy levels where the transition from the SF to the MI phase is non-monotonic, exhibiting relatively large or relatively small $\beta$ at the transition.

Ergodicity measures.– The participation number $\mathcal{M}$ tells us how many basis states participate in the superposition that forms an eiegenstate. Given a basis $\bm{n}$ it is defined as follows:

where ${p_{n}=\left|\Psi_{\bm{n}}\right|^{2}}$. An individual eigenstate is possibly not ergodic, and does not accommodate the energetically allowed space. In order to determine the volume of the allowed space, we calculate the averaged $p_{n}$ within a small energy window, and then calculate the associated participation number which we denote as $\overline{\mathcal{M}}$. The ratio $\mathcal{M}/\overline{\mathcal{M}}$ serves as a quantum ergodicity measure. For a fully chaotic system such as billiard one expects it to be somewhat less than unity due to fluctuations. In practice the value is much smaller indicating lack of ergodicity. We calculate both $\mathcal{M}_{\text{sites}}$ in the site basis, and $\mathcal{M}_{\text{orbitals}}$ in the orbital (momentum) basis, and plot

We note that for ${u\gg 1}$, where the SF-MI transition takes place, ${\mathcal{M}=\mathcal{M}_{\text{sites}}}$.

Chaos measure.– In practice it is difficult to associate with an individual eigenstate a measure that indicates whether it is supported by a chaotic sea or by a quasi-regular island. In fact it has been demonstrated in a recent study [29] that many of the eigenstates do not obey such dichotomy. Nevertheless, we are going to present an efficient method for identification of “quantum chaos” via what we call tomographic inspection of the spectrum.

Using spin language, and dropping a constant, the dimer Hamiltonian is ${\mathcal{H}_{\text{dimer}}=US_{z}^{2}-JS_{x}-\Delta S_{z}}$. For sake of discussion we have added a detuning potential $\Delta$ between the two sites. Unless stated otherwise ${\Delta=0}$. In the classical limit the dynamics is generated by Hamilton’s equations via Poission brackets that assume spin algebra. The phase-space of the motion is the Bloch sphere ${S_{x}^{2}+S_{y}^{2}+S_{z}^{2}=[(N/2){+}1](N/2)\approx(N/2)^{2}}$. The classical energy contours ${\mathcal{H}_{\text{dimer}}(S_{x},S_{y},S_{z})=E}$ feature a seperatrix if ${u>1}$. See Fig. ​​2 for demonstration. The energy of the seperatrix equals the energy of the unstable hyperbolic point at $S_{x}=-N/2$, that opposes the ground state at $S_{x}=+N/2$. Namely,

We refer to $E>E_{x}$ as the self-trapping (ST) region, while $E<E_{x}$ is the SF region. The latter is diminished and cannot accommodate a quantum eigenstates if ${u>N^{2}}$. See [20] for details and further references. We plot the eigenenergies $E_{\nu}$ as a function of $u$ in Fig. ​​2 and in Fig. ​​4a, and provide further characterization in the additional panels there. Namely, for each eigenstate we calculate the purity $\mathcal{S}$, the ergodicity measure $\mathcal{M}$, the average occupation $n_{0}$ of the ground-state orbital, and the Higgs measure $\beta$.

Borders of the MI and SF phases.– If we focus on the ground-state, the definition of the MI phase is unambiguous: it appears for ${u>N^{2}}$, because eigenstates cannot be accommodated in the SF-region. But if we look at higher energies the notion of MI-phase becomes blurred. There are in fact two relevant borders. The $u_{s}(E)$ border is determined by perturbation theory (see Appendix E), and is indicated by black line in Fig. ​​4c. As we cross this border from right to left, the MI eigenstates start to mix. But there is a second border, of the SF phase, as we go from left to right in the diagram. The latter border has to do with the SF separatrix, and is further discussed below. This border dominates the purity in Fig. ​​4b, and the depletion in Fig. ​​4d.

Spectral perspective.– From a semiclassical perspective the SF transition in the $(u,E)$ diagram is determined by the separatrix. But in a quantum context one would like to have an unbiased independent determination of the transition. For a detuned system (${\Delta\neq 0}$), WKB theory implies a minimal level spacing at the transition, where the classical oscillation frequency $\omega(E)$ vanishes. In the absence of detuning the levels in the MI phase cluster into pairs of quasi-degenerate levels with exponentially small tunnel-splitting.

Phasespace perspective.– Irrespective of the spectral aspect, the transitions from the ST ro the SF region is characterized by localization at the hyperbolic point. We demonstrate this localization by plotting Husimi functions of selected eigenstates. The procedure is defined in Appendix C, and the plots are displayed and discussed in Appendix D. The observed localization is related to the classical pendulum picture. Namely, at the separatrix energy, where the classical oscillation frequency goes to zero, the dwell time at the “top” position (the hyperbolic point ${S_{x}=-(N/2)}$) diverges.

ODLRO perspective.– Closer inspection of Fig. ​​4e reveals that the localization at the hyperbolic point happens whenever an even superposition crosses the separatrix (the indication for this localization is smaller ${n_{0}}$). This localization is associated with the appearance of a large Higgs measure $\beta$. A better quantitative inspection over the dependence of $\beta$ on $u$ is provided in Fig. ​​4.

For each value of $u$ we diagonalize the exact BHM Hamiltonian; find the eigenergies $E_{\nu}$, and calculate for each eigenstate the Purity $\mathcal{S}$ and other measures. The global ${(u,E)}$ phase-diagram for the trimer Fig. ​​5 is obtained by plotting the spectrum for a wide range of $u$ values. Each pixel represents an eigenstate, and is color-coded by its Purity $\mathcal{S}$ (panel a) or by the ergodicity measures $\mathcal{M}$ (panels b and c), or by the order parameter $n_{0}$ (panel d), or by the Higgs measure $\beta$ (panels e and f). The extracted measures help us to classify the eigenstates. In Sec. (VI) we will take a closer look at representative spectra, for selected values of $u$, that are displayed in Fig. ​​6. Some representative eigenstates are presented in Fig. ​​7.

In practice it is difficult to associate with an individual eigenstate a measure that indicates whether it is supported by a chaotic region. The common practice is to look on the level statistics. But such approach has two issues: (i) It provides numerically clear results only for rather simple systems with large $N$, such that the spectrum in the range of interest is dense enough; (ii) It does not allow classification of the eigenstates in the typical situation of underlying mixed chaotic and quasi-regular dynamics. We therefore suggest below to adopt a different strategy that we call spectrum tomography. This tomography has both quantum and corresponding classical versions.

It is pedagogically illuminating to use the semiclassical Bloch-sphere picture of the dimer in order to discuss the characterization of fluctuations. This language can be extended to any $L$ site system as discussed in Appendix A. In this section, for the purpose of providing a simplified introduction, we assume ${N\gg 1}$, and ignore relative error of order $1/N$.

One can regard an eigenstate of the dimer as a cloud of points on the Bloch sphere (technically speaking we can use a Huismi function for visualization). Any point of the cloud satisfies ${S_{x}^{2}+S_{y}^{2}+S_{z}^{2}\approx(N/2)^{2}}$, with the identification ${S_{z}\equiv n_{site}-(N/2)}$, and ${S_{x}\equiv n_{0}-(N/2)}$. Averaging over the cloud one deduces that ${\left\langle S_{x}\right\rangle^{2}+\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{z}^{2}=(N/2)^{2}}$, which assumes ${\left\langle S_{y}\right\rangle=\left\langle S_{z}\right\rangle=0}$ due to symmetry. Given $\langle\hat{n}_{0}\rangle$ and ${\sigma_{x}^{2}=\sigma_{0}^{2}}$ and ${\sigma_{z}^{2}=\sigma_{\parallel}^{2}}$ we can extract ${\sigma_{y}^{2}\equiv\sigma_{\varphi}^{2}}$. The relation can be written as

where ${n_{dep}=N-n_{0}}$ is the depletion, and ${\sigma_{\perp}^{2}=\sigma_{0}^{2}+\sigma_{\varphi}^{2}}$ is what we call total fluctuations of the order parameter. The analogous relation for the trimer is

which assumes small depletion (${n_{dep}\ll N}$). A precise version of these relations, and their generalization for an $L$ site ring, are discussed in the following subsections. They are useful for the calculation of the Higgs measure $\beta$, and for getting insights.

For visualization purpose we note that any eigenstate can be represented by an ellipsoid whose major axes are ${(\sigma_{0},\sigma_{\varphi},\sigma_{\parallel})}$. The actual phase-space distribution might look very different. This statement is clarified using the caricature of Fig. ​​11.

The Higgs measure $\beta$ of Eq. ​​(18) is the ratio of $\sigma_{0}^{2}$ to $\sigma_{\perp}^{2}$. By definition it becomes of order unity for amplitude oscillations of the ODLRO. Irrespective of that we have $\sigma_{\parallel}^{2}$ that characterizes the “diagonal” on-site fluctuations. Numerical results for the $\sigma$-s are summarized in Fig. ​​12, and the result for the Higgs measure were already displayed in Fig. ​​5. We identify that there are levels where the dependence of $\beta$ on $u$ is non-monotonic. This is further illustrated, for representative levels, in Fig. ​​13. Below we clarify analytically the numerical results for the various families of eigenstates.

Fig. ​​15 displays the lowest energies $E_{\nu}$ versus $u$ for the $L=3$ trimer and for an $L=5$ ring. The $q=0$ levels are colored in red, and in some sense “frame” the band structure. There are two limits where the structure of the spectrum is rather simple. In the MI phase we have the $q{=}0$ ground-state, and the first band of excitations that contains $L{-}1$ sub-bands, each with $L$ momentum states ${q=(2\pi/L)\times\text{integer}}$. Those states are superpositions of gapped particle-hole excitations. In the SF phase we have the $k{=}0$ ground state, the single-phonon band that contains $L$ states, and the double-phonon states that contain both $q=0$ states that are formed from $\pm k$ excitations, and $q\neq 0$ double-phonon excitations.

The non-trivial aspect is the MI-SF transition, during which there is migration of $q{\neq}0$ levels from the gapped MI band towards the $q{=}0$ ground level, leading to the formation of a Goldstone band in the SF region. This migration is caricatured in Fig. ​​15.

We now turn to provide a more detailed semiclassical description of the MI-SF transition. The lowest seperatrix defines the SF region ${E<E_{SF}}$. The energy width of this region is of order $NJ$. The quantum parameter $\gamma$ of Eq. ​​(2) tells us what is the size of Planck cell relative to the size of the SF region. Large value (${\gamma>1}$) implies that the SF region cannot accommodate quantum eigenstates. This is the MI phase. This phase features narrow bands of particle-hole excitations, see Fig. ​​15. The spacing between those bands is of order $U$. Due to $J$ they form sub-bands. As $\gamma$ becomes smaller eigenstates migrate through the separatrix into the SF region, where they are re-arranged into phononic bands. The latter can be regraded as occupation states of momentum-orbitals.

The levels that migrate into the SF region form the so-called Goldstone band. The latter has no gap from the ground state (it is like small vibrations around the ground state). As opposed to that, the MI gapped bands are formed of levels that reside above the separatrix. For large enough $J$, the SF regions expands and swallows all the eigenstates, and accordingly all the bands become phonon-type.

Let us take a closer look at a trimer that is occupied by $\bar{n}$ states in each site. The first band of excitations is spanned by the 6 basis states $|x,s\rangle$ of Eq. ​​(70), where ${x=1,2,3}$ is the “position” index of the configuration, and ${s=a,b}$ is like a sublattice index that distinguish two sets of configurations that differ by the cyclic order of particle-hole excitations. Upon translation ${D|x,\sigma\rangle=|x{+}1,\sigma\rangle}$ mod(3). The hopping terms are alternately $-(J/2)[1{+}\bar{n}]$ and $-(J/2)\bar{n}$. The eigenstates of the Hamiltonian form two bands. By Bloch theorem

with quasi-momentum ${q=2\pi/3\times\text{integer}}$, namely, ${D|q,\pm\rangle=e^{-iq}|q,\pm\rangle}$. The lowest excitation ${|q{=}0,+\rangle}$ is a zero momentum particle-hole excitation that features the outstanding large Higgs measure ${\beta\approx 11/8}$, as derived in Appendix I. Note that the excitations in this band are insensitive to the introduction of external gauge field.

The objective of the the present study was to provide characterization of the quantum eigenstates of the BHM, using a semiclassical phase-space perspective. For visualization we have introduced a tomographic approach for inspection of the spectrum. Such approach is both numerically efficient and informative. In particular it allows the identification of underlying chaos and/or mixed phase-space dynamics.

The major borders in the ${(u,E)}$ phase-diagram of the BHM are separatrices ${(E_{SB},E_{SF},E_{ST})}$ and the perturbative border $u_{s}(E)$. The ODLRO and the associated fluctuations are mapped on this diagram. In particular it was important for us to clarify the notion of MI-SF transition in the context of such diagram, not to focus merely on the ground-state. It is important to realize that the transition has several stages. As $u$ is decreased, we first approach the perturbative border $u_{s}(E)$ where fluctuations become outstanding. After crossing this border the eigenstates migrate to the SF region below the seperatrix. Whether a mobility edge is formed in the thermodynamic limit remains a matter for speculations.

The study was partially motivated by the desire to make a bridge with the field-theory perspective, notably with GMFT [23, 24], where the emphasis was on the identification of amplitude (Higgs) modes as opposed to phase modes. Let us recall how phase and amplitude modes are defined within the common perspective. It is natural to start with the familiar Bogulyubov framework, where the classical variation of the order parameter is described by

The variation of $\psi$ in the complex plane is along an ellipse that is characterized by a squeeze factor ${v/u=\tanh(\lambda)}$. Using the common gauge convention, $\psi^{(0)}$ is real and positive, and both $v$ and $u$ are real numbers. In the standard Bogulyubov framework both have the same sign, and ${|v/u|<1}$.

The GMFT suggests quantum results for $v/u$, that go beyond the Bogulyubov prediction. One considers the dynamics of a single site under the influence of a frozen mean-field of the other sites. Such approximation automatically excludes irregular chaotic motion. Nevertheless, for the regular solution it predicts the appearance of pure amplitude (Higgs) modes as the $J/U$ ratio is increased. The argument goes as follows: For $J=0$ the excitations are either particle or hole depending on the chemical potential. In the latter case it formally means $v/u=\pm\infty$. Setting ${J\neq 0}$, and considering a hole excitation, the mode changes its character as $J$ is increased, and for large enough $J$ it eventually becomes a particle-like Bogulyubov excitation with ${0<v/u<1}$. It follows that along the way we should encounter either pure phase-mode (formally $v/u=+1$) or pure amplitude mode (formally $v/u=-1$). The latter is termed Higgs mode, and constitutes an indication of a quantum interference effect that goes beyond the Bogulyubov framework.

The present work, unlike the GMFT, treats the full many-body problem. We note that a parallel study [30] aims to develop a theory for the Goldstone and Higgs excitations based on a refined cumulant expansion around the self-consistent mean field. Nevertheless, in the exact numerical treatment, it is $N$ rather than the chemical potential that is fixed. Accordingly the lowest excitations in the MI phase are correlated particle-hole excitations. This complicates the practical comparison between field-theory predictions and numerical results. Consequently, we have adopted a theoretically unbiased characterization of eigenstates that reflects the distinction between phase and amplitude modes. Specifically, we have defined an Higgs measure $\beta$ to identify eigenstates that have outstanding amplitude fluctuations.

For the dimer, the observed picture is very simple: eigenstates that feature outstanding $\beta$ appear at the MI-SF transition when an even-symmetry eigenstate crosses the separatrix. It is complementary to the localization of the eigenstate at the unstable hyperbolic fixed-point. For the trimer the picture is more complicated because the MI-SF transition is mediated by chaos. Nevertheless, we found that outstanding $\beta$ is not related to separatrix crossing, but rather can be explained within the framework of perturbation theory. Namely, it is related to special superpositions of quasi-degenerate eigenstates.

Acknowledgments — We thank Idan Wallerstein and Eytan Grosfeld for insightful discussions. Preliminary work that concerns Sec. (IX) has been carried out by Naama Harcavi within the framework of a BSc project. The research has been supported by the Israel Science Foundation, grant No.518/22.