Quantum Critical Behavior of One-Dimensional Soft Bosons in the Continuum
Stefano Rossotti Martina Teruzzi Davide Pini Davide Emilio Galli Gianluca Bertaina
cond-mat.quant-gas




Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy

International School for Advanced Studies (SISSA), Via Bonomea 265, Trieste, Italy
Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy

Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy

Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy

gianluca.bertaina@unimi.it
Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy


We consider a zero-temperature one-dimensional system of bosons interacting via the soft-shoulder potential in the continuum, typical of dressed Rydberg gases. 
We employ quantum Monte Carlo simulations, which allow for the exact calculation of imaginary-time correlations, and a stochastic analytic continuation method, to extract the dynamical structure factor.
At finite densities, in the weakly-interacting homogeneous regime, a rotonic spectrum marks the tendency to clustering. With strong interactions, we indeed observe cluster liquid phases emerging, characterized by the spectrum of a composite harmonic chain. Luttinger theory has to be adapted by changing the reference lattice density field. In both the liquid and cluster liquid phases, we find convincing evidence of a secondary mode, which becomes gapless only at the transition. In that region, we also measure the central charge and observe its increase towards c=3/2, as recently evaluated in a related extended Bose-Hubbard model, and we note a fast reduction of the Luttinger parameter. For 2-particle clusters, we then interpret such observations in terms of the compresence of a Luttinger liquid and a critical transverse Ising model, related to the instability of the reference lattice density field towards coalescence of sites, typical of potentials which are flat at short distances. Even in the absence of a true lattice, we are able to evaluate the spatial correlation function of a suitable pseudo-spin operator, which manifests ferromagnetic order in the cluster liquid phase, exponential decay in the liquid phase, and algebraic order at criticality.


Quantum Critical Behavior of One-Dimensional Soft Bosons in the Continuum
    Gianluca Bertaina
    
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Quantum phase transitions (QPT) <cit.> play an intriguing role in many-body systems, due to the possibility of unveiling new exotic phases. 
The progress in the manipulation of ultracold gases allows for the exploration of QPTs, by engineering well-controlled synthetic quantum many-body systems, confined for example by optical lattices <cit.> or in quasi-one-dimensional geometries <cit.>. Recently, Rydberg atoms <cit.> have emerged as a new route to QPTs <cit.>. These are atoms in highly-excited electronic states, with a very large electronic cloud.
In particular, theoretical <cit.> and experimental <cit.> efforts have focused on ensembles of dressed Rydberg atoms, which are superpositions of the ground state and the above mentioned excited states, coupled via a Rabi process. Their effective interaction can be a soft-shoulder potential, with a flat repulsion up to a radius R_c related to the highly excited state, and a repulsive van-der-Waals tail at large distances <cit.>. Quite interestingly, this repulsive interaction belongs to the class that has been recognized to induce cluster formation at high density in classical statistical mechanics <cit.>, thanks to the relative freedom of particles at short distances. This has opened a recent flourishing of research on quantum cluster phases: in high dimensions, coexisting cluster crystal and superfluid order have been predicted, yielding supersolid behavior <cit.>, while in one dimension (1D), cluster Luttinger liquids (CLL) have been proposed on a lattice <cit.>.



In this Letter, we investigate a prototypical system of N bosons in 1D at linear particle density n, governed by the following Hamiltonian in the continuum:

    H = -ħ^2/2m∑_i^N ∂^2/∂ x_i^2 + ∑_i<jV_0/r_ij^6+R_c^6

where x_i are the particle coordinates, r_ij=|x_i-x_j| the distances, m is the mass and V_0 and R_c are the strength and the radius of the soft-shoulder potential V(r). 
If not otherwise specified, in the following we use units of R_c for the length, E_c=ħ^2/m R_c^2 for the energy, and ħ/R_c for the momenta (See Supplemental Material [See Supplemental Material at [URL will be inserted by publisher] for details on the potential, the experimental realization, the methods, the extraction of relevant observables.]). The zero-temperature phase diagram (Fig. <ref>) thus depends on the following two dimensionless quantities: strength, U=V_0/(E_c R_c^6), and density, ρ=nR_c. 
By evaluating relevant static and dynamical properties, we show that, while for small U and moderate ρ the system is a Luttinger liquid (LL) <cit.>, although with strong correlation effects, for higher U or ρ a transition occurs towards CLL. In particular, we focus on the QPT to the dimer cluster liquid, which turns out to be of the 2D Ising universality class. A similar phenomenology has been recently studied in a 1D lattice system governed by the extended Bose-Hubbard hamiltonian <cit.>, while we observe it for the first time in the continuum, where we find that an effective spin hamiltonian emerges at the transition, even in absence of an underlying lattice.

For generic coupling and density, this system falls into the LL universality class <cit.>, characterized by a gapless bosonic mode at small momenta, with sound velocity v.
The low-energy and momentum sector of the Hilbert space is governed by the Hamiltonian H_LL=(v/2π)∫ dx [K_L (∇θ)^2+(∇ϕ)^2/K_L], where a large Luttinger parameter K_L>1 favors the fluctuations of the particle counting field ϕ(x), while small values of K_L induce crystal-like behavior, by disordering the phase field θ(x). The central charge of the associated conformal field theory (CFT) is c=1 <cit.> and, for our Galilean invariant system, v=ρπ/K_L <cit.>.

In the dilute limit, the effects of the interaction are well described by the scattering length a_1D; in particular, for U ∼ 1.09, we get a_1D∼ 0 <cit.>, corresponding to the Tonks–Girardeau (TG) model <cit.>. Conversely, at higher densities, the full shape of V(r) is relevant. Its 1D Fourier transform Ṽ(q) <cit.> features a global minimum at q_c ≃ 4.3, at which Ṽ(q_c)<0, providing a typical length b_c=2π/q_c ≃ 1.46. It has been recognized, in the context of classical physics, that such density-independent distance favors clustering, even with a completely repulsive potential <cit.>. 
Classically, one obtains a T=0 cluster crystal, which is destabilized in 1D by finite temperature, in favor of cluster-dominated liquid phases with different average occupation <cit.>. Quantum mechanics induces coherent delocalization even at T=0, rendering the cluster phase a CLL and triggering a QPT towards a LL without cluster order (Fig. <ref>).



To study the phase diagram in a non-perturbative way, we use the well-established path integral ground state (PIGS) quantum Monte Carlo method <cit.>, which represents the ground state as the imaginary-time projection exp(-τ H)|Ψ_T⟩ of a trial wavefunction. We simulate up to N=200 particles in a segment of length L=N/ρ, using periodic boundary conditions (PBC). The trial wavefunction is of the two-body Jastrow form: Ψ_T(x_1… x_N) = exp[-1/2∑_i<j(u(r_ij) + χ(r_ij))], where χ(r) accounts for long-wavelength phonons <cit.>, while exp[-u(r)/2] is the numerical solution of a two-body Schrödinger equation <cit.>, with the effective potential V_eff(r)=c_1 V(r) + c_2 ∑_l V(r-l b). To reduce projection times, it is crucial to use and optimize this mean-field potential, which accounts for the presence of nearby clusters <cit.>.
We consider excitations associated to density fluctuations, which are commonly investigated via the dynamical structure factor S(q,ω) = ∫ dt e^i ω t/2π N⟨ e^itH/ħρ_q e^-itH/ħρ_-q⟩.
The PIGS algorithm evaluates the numerically exact imaginary-time intermediate scattering function, which yields S(q,ω) via analytic continuation, through the genetic inversion via falsification of theories algorithm <cit.>. 

We now proceed to discuss our results, first in the LL, then in the CLL regimes. Finally, we discuss the QPT in between the two liquids.



  
Liquid regime– Here and in the following, k_F=πρ and E_F=k_F^2/2 are effective Fermi momentum and energy. We concentrate on interaction U ≃ 1.09, and increase the density (dot-dashed line in Fig. <ref>). 
In the low-density regime ρ≲ 0.1 (Fig. <ref>a), S(q,ω) is almost constant, at fixed q, in between the particle-hole boundaries ε_IFG(q) = | k_F q ± q^2/2|, analogously to the TG gas, which can be mapped to an ideal Fermi gas (IFG). However, within our resolution, the spectral weight has started to gather, especially at the upper boundary, similarly to what happens in the Lieb-Liniger model with decreasing coupling parameter <cit.>.
In fact, already at ρ=0.6 (panel b),

the spectrum has evolved into a main mode.

As such, it is very well described by the single-peak Feynman approximation ε_FA(q)=ε_0(q)/S(q), with ε_0(q)=q^2/2 the free-particle energy and S(q) the static structure factor. 

By further increasing ρ, we simply monitor the evolution of ε_FA(q) (Fig. <ref>c), and notice that the main excitation becomes more structured, with a roton minimum moving towards q=q_c. A standard Bogoliubov analysis <cit.> yields the dispersion ε_B(q)=√(ε_0(q)[ε_0(q)+2 ρṼ(q)]), which depends only on the combination α=ρ U. In this approximation, it is clear that the emergence of the roton minimum is allowed by the momentum dependence of Ṽ(q), which has a negative part <cit.>. The roton softens at α=α_c ≃ 20.65. While the agreement between the single-mode ε_FA(q) and ε_B(q) approximations is very good for 0.6≲ρ≲ 19, such treatments are in general not valid anymore for U ≳α_c/ρ (dashed line in Fig. <ref>), where indeed our simulations show that clustering occurs.

On increasing ρ, the pair distribution function g(r) at first gradually approaches 1 everywhere (Fig. <ref>d), as in classical soft-core fluids in the absence of clustering <cit.>. However, for very high ρ, large-amplitude slowly-decaying oscillations appear, with wavelength b_c. Again, this behavior is akin to that of classical systems, in the presence of clustering <cit.>. 
A gaussian fit of the peaks indicates, on average, N_c≃ 36 particles per cluster at ρ=24.6. In the quantum case, the oscillations of g(r) eventually decay as in a cluster liquid, a behavior that we can easily see in the more relevant cluster phases at low ρ and large U. In fact, a hamiltonian description of dressed Rydberg gases is questionable at high ρ, due to increased losses to other Rydberg levels in current experiments <cit.>.







  
Commensurate Cluster Luttinger Liquid– 
We therefore now focus on the density ρ=2/b_c≃ 1.37 (solid line in Fig. <ref>), commensurate to clusters of N_c=2 particles <cit.>. In Fig. <ref>a, the PIGS results for g(r) are shown, indicating an evolution to a cluster structure on increasing U, with peaks containing two particles. g(r) manifests long-range algebraic decay of the peaks' heights, 
which demonstrates absence of true crystal order <cit.>. To interpret these results, we employ CLL theory. In the standard bosonization approach <cit.>, the counting field fluctuates around a lattice with spacing ρ^-1: hamiltonian H_LL is then derived with the assumption that fluctuations are small. However, in a commensurate cluster liquid, clearly fluctuations are small only around a lattice of clusters, with spacing ρ^-1N_c=b_c. We follow <cit.>, and obtain the following commensurate CLL form of g(r):

    g(r)r≫ 1/ρ≃ 1-2K_L/(2πρ r)^2+∑_l=1^∞ A_l cos(2π l ρ r/N_c)/r^2K_L^' l^2

The r^-2 term is analog to the standard LL case, 

while the last term yields dominant density oscillations of wavevector 2k_F/N_c=q_c, modulated by an effective Luttinger parameter K_L^'=K_L/N_c^2. This implies that, in the CLL phase, the divergence of S(q_c)∝ N^1-2K_L^' is much stronger than what would result from K_L. We extract K_L and K_L^' from the small momentum behavior of S(q) and large distance decay of g(r), respectively <cit.>. Interestingly, K_L scales as U^-1/2 in both the LL and CLL regimes, but with different prefactors. Moreover, we verify that the number of excess particles per cluster δ=√(K_L/K_L^')-1 quickly goes to 1 for U>18 (Fig. <ref>b).

Deeply in the cluster phase, a composite harmonic chain (HC) theory can also be envisaged. We write a model hamiltonian of the type H_HC=∑_iν p^2_i,ν/2+γ∑_iνμ(x_i,ν-x_i+1,μ)^2/2, where x_i,ν is the displacement of the ν-th particle (with 1≤ν≤ N_c) from the average position of cluster i (with 1≤ i≤ N/N_c), and springs of strength γ are present only between particles in adjacent clusters, modeling the fact that V(r) is flat at short distances [PBC are implied]. We obtain center-of-mass modes, of acoustic frequencies ω_acou(k)=2√(N_cγ)sin(k b_c/2), and optical modes, of dispersionless frequency ω_opt=√(2N_cγ). The latter correspond to relative vibrations of particles in a cluster.
We relate γ to the mean-field potential felt by a particle if all the others are in a cluster crystal with spacing b, and find γ(b)=-(4π^2/b^3) ∑_j=1^∞ j^2Ṽ(2π j/b) <cit.>. It is clear that a stable structure is possible only if  Ṽ(2π/b)<0 for some b <cit.>. 

In Fig. <ref>, the spectra at ρ=1.37, with decreasing U, are shown. Panel a (U=100) is deep in the CLL phase: the main peak is in good agreement with the acoustic mode of HC theory, with b=b_c. A secondary structure appears at higher frequencies, which we interpret as the optical mode <cit.>. This is however not flat, but strongly modified by anharmonic couplings: these are clearly even more crucial at smaller U, where they induce cluster melting.





  
Ising transition– The question is now: how are the LL and CLL phases really different? Is the transition simply a crossover? 
Our data, which show Luttinger liquid behavior on both sides, exclude Berezinskii-Kosterlitz-Thouless transition to a charge-density-wave, or Peierls transition, even though, at U=18, we get K_L≃ 0.52(1) <cit.>. Moreover, the atomic-pair superfluid transition <cit.> is also excluded, since, here, formation of larger clusters is allowed

and the CLL phase manifests strong quasi-solid order (K_L<1/2).  

In fact, the physics of this relatively simple system is very rich. 
The acoustic mode of the CLL phase (Fig. <ref>, panels a-b) is gapless at q=q_c, corresponding to k_F, at this density. After the transition, to be located at U=U_c≃ 18 (panel c), this lowest excitation turns into the rotonic mode (panels d-e).
Quite interestingly, a weaker secondary mode appears not only in the cluster phase, but also in the strongly correlated liquid phase, in the form of a secondary roton, which connects to the higher-momenta main mode.

It is reasonable to associate this secondary excitation, in the LL phase, to incipient cluster formation, due to particles being preferentially localized close to either the left or the right neighbor. The crucial observation is that the gap of both such LL excitations (panels d-e), and the anharmonic optical modes of the CLL phase (a-b), vanishes at the transition (c), which implies that they proliferate at that point. 
This behavior is consistent with that of the 1D transverse Ising (TI) model <cit.> of a chain of coupled two-level systems. Its hamiltonian is H_TI=-J∑_i σ^z_iσ^z_i+1-h∑_iσ^x_i, where σ^x/z_i are Pauli matrices at site i. It contains both a ferromagnetic coupling (J>0), which forces alignment, and quantum tunneling (h>0) between the eigenstates of σ^z, favoring a paramagnetic state.
This model is exactly solvable with a Jordan-Wigner transformation and Bogoliubov diagonalization <cit.> and yields excitations of energy 

    ε_TI(q)=√(Δ^2+4Jh(sinqa/2)^2) ,

where Δ=|J-h| is the gap and a is lattice spacing, which are gapless only for h=J. This signals a QPT from the ferromagnetic to the paramagnetic state, which is dual to the 2D classical thermal Ising transition. In our case, it is natural to associate Δ to the gap of the secondary mode at q=q_c, and set a=b_c, implying that a spin should be identified every two particles. We fit Eq. (<ref>) from our spectra (Fig. <ref>a): within our accuracy, the behavior of Δ in U-U_c is linear close to the transition, consistent with the dynamical exponent z=1 <cit.>. The point at U=18 requires very long projection times: another indication of the presence of a very low-energy mode. Within our resolution, the Luttinger and critical Ising modes have the same velocity at U=U_c, which would imply low-energy supersymmetry <cit.>. 

To corroborate our interpretation, we recall that the central charge c of the critical TI model is c=1/2, so that, at the transition, the total central charge should be c=1+1/2=3/2, as calculated for the related lattice model <cit.>. We estimate c from the slope of the energy per particle ε(N)=ε_∞ - c E_F/(6 K_L N^2) versus 1/N^2, employing a standard CFT result for the dominant finite-size effects <cit.>. An increase of c is manifest in Fig. <ref>a. It is in fact delicate to extrapolate c close to U_c: higher order corrections may become relevant, and field theoretical methods should elucidate the interplay between the Luttinger and Ising fields, as done in <cit.>. It would be interesting to extract c also from the entanglement entropy, as recently introduced in PIGS <cit.>.

It is particularly appealing to investigate the microscopic realization of this effective TI model. The many-body potential surface reduces to double wells as a function of relative distances: two nearby bosons have preferred configurations if they are at r_ij=0 or r_ij≃ b_c <cit.>. Thus, for each even particle, for example, there are left ψ_L and right ψ_R preferred cluster configurations, and the potential energy is minimized when subsequent even particles choose the same clustering direction. Anharmonic terms give instead rise to delocalization. The cluster phase is then to be thought as the ferromagnetic state, where all even particles have chosen either ψ_L or ψ_R (Fig. <ref>b), while the liquid phase is made of ψ_+=(ψ_L+ψ_R)/√(2) states (Fig. <ref>a), where particles continuously hop left and right. 

This mapping can be made quantitative, by introducing a simple, but effective string representation of σ^z, inspired by <cit.>: 
first, particles are ordered by their position k, and even positions are assigned a lattice index i=k/2; then, a pseudo-spin σ_i^z=1 is assigned if |x_k-x_k-1|<|x_k-x_k+1|, or σ_i^z=-1 in the opposite case. We evaluate the spatial correlator g_σ(|i-j|)=⟨σ^z_iσ^z_j⟩ of such ℤ_2 pseudo-spin (Fig. <ref>b). It is very remarkable that g_σ behaves as expected for the TI model: in the LL (paramagnetic) phase it decays exponentially, while in the CLL (ferromagnetic) phase it manifests true long-range order, which is nonlocal <cit.>, because of the preliminary ordering of particles. At U=18, its behavior is close to an algebraic decay with exponent η≃-1/4.

This pseudo-spin mapping in a continuous system at the LL-CLL transition, as revealed by excitation spectra and a suitable spin correlator, is the key result of this Letter. Such critical regime could be probed even at finite T, given finite experimental sizes. Future work will investigate effects of non commensurability of ρ with 1/b_c, which is particularly relevant for trapped gases. Also, an open issue is the presence of quantum Potts transitions at densities commensurate to N_C≥3.



We acknowledge useful discussions with M. Dalmonte, M. Fleischhauer, C. Gross, R. Martinazzo, A. Parola, N. Prokof'ev, H. Weimer.
We acknowledge the CINECA awards IscraC-SOFTDYN (2015) and IscraC-CLUDYN (2017) for the availability of high performance computing resources and support. We acknowledge funding from the University of Milan (grant PSR2015-1716LPERI-M).










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Supplemental Material: Quantum Critical Behavior of One-Dimensional Soft Bosons in the Continuum



















1mm

We give details on the potential and relevant mean-field potentials; make some comments on the experimental realization; briefly explain the PIGS and GIFT algorithms, the optimization of the trial wavefunction, the extraction of the Luttinger parameters and of the central charge, providing also tables with the results used to produce Figs. 3 and 5 in the main text.  

Note: citations in this Supplemental Material refer to the bibliography in the main paper.



§ POTENTIAL AND EFFECTIVE MEAN-FIELD POTENTIAL




The soft-shoulder potential V(r)=U/(1+r^6) has the 1D Fourier transform (see Fig. <ref>)

    Ṽ(q)=∫_-∞^+∞e^i q r V(r)dr=
    U π e^-q/2[e^-q/2 + cos(√(3) q/2) + √(3)sin(√(3) q/2)]/3  .


When the density is ρ=2/b_c≃ 1.37, an approximation of the mean-field potential felt by a particle in the liquid phase can be obtained by letting all other particles on a lattice of spacing 1/ρ=b_c/2: V_MF^LL(r)=-V(r)+∑_i=-∞^∞ V(r - i b_c/2) (red solid line in Fig. <ref>). This lattice periodicity is clearly unstable, since a double well appears close to the origin: so the liquid phase is stable only thanks to kinetic energy. The mean-field potential felt by a particle in the cluster liquid phase can be approximated by letting all other particles on a lattice of spacing 2/ρ=b_c: V_MF^CLL(r)=-V(r)+2∑_i=-∞^∞ V(r - i b_c), which implies that pairs of particles overlap (blue solid line in Fig. <ref>). Here, a clear harmonic confinement is present at r=0, which stabilizes the cluster phase, and whose typical energy ε_h(b_c) is given in the main text.



An instructive mean-field potential can also be visualized, in the LL phase, by letting three particles free, and fixing only their center of mass at the center of a lattice of the other particles with spacing 1/ρ=b_c/2: the resulting surface, in terms of the relative distances r_12=|x_1-x_2| and r_23=|x_2-x_3|, is plotted in Fig. <ref>. The two spheres indicate the positions of the two main equivalent minima, which clearly correspond to (r_12,r_23)≃(0,b_c) and (r_12,r_23)≃(b_c,0). For particle 2, suitable orbitals centered in such minima correspond to the two states ψ_L and ψ_R introduced in the text. They define the Hilbert space governed by the Ising hamiltonian introduced in the main text.






§ CONSIDERATIONS ABOUT THE EXPERIMENTAL REALIZATION

In the context of ultracold Rydberg gases, the parameters of the 3D shoulder potential can be related to the Rabi frequency Ω_R and detuning Δ_R, by U=(C_6 m^3 |Δ_R|^2/16ħ^4)^1/3β^4 and R_c=(C_6/2ħ|Δ_R|)^1/6, where β=Ω_R/2Δ_R is the admixture of the Rydberg level with the ground-state, and C_6 is van-der-Waals coefficient [11].

Eq.(1), in the main text, is the effective 1D Hamiltonian [21],

which is relevant in an elongated quasi-1D configuration, once the transverse degrees of freedom are frozen in the ground state, due to strong trapping [5].

The transverse confinement size a_⊥ does not affect much the effective potential, if a_⊥≲ R_c. With respect to the original 3D potential, the quasi-1D interaction has reduced U and larger q_c. The condition a_⊥≲ R_c is already under reach in current experiments, where both a_⊥ and R_c can be of order of 100 nm [6].

The above condition also prevents the zig-zag instability [53]

for strong interactions. To experimentally probe the phase diagram in Fig.1 of the main text, one can vary detuning Δ_R, linear density n, mass m (by changing the atomic species), and C_6∝n̅^11, by changing the addressed Rydberg level of principal number n̅. Note that, while Rydberg levels have typical lifetimes τ_R of order of μ s, Rydberg dressed atoms with small admixture β have proportionally increased lifetimes τ=τ_R/β^2 [16,17].

Cryogenic techniques would be beneficial to reduce the detrimental black-body radiation, due to the experimental apparatus [8].

The experimental realization of hamiltonian of Eq.(1) in the main text is presently challenging in the absence of a lattice, since no short-range hard core is enforced. However, clustering effects persist even if one introduces a hard core, provided it is small when compared to R_c.



§ PATH-INTEGRAL GROUND STATE METHOD


(For completeness, we add a brief description, adapted from the Supplemental Material of Ref.[38].)


The Path Integral Ground State (PIGS) Monte Carlo method is a projector 
technique that provides direct access to ground-state expectation values
of bosonic systems, given the microscopic
Hamiltonian Ĥ [34,35].

The method is exact, within unavoidable statistical error bars, which can nevertheless be reduced by performing longer simulations, as in all Monte Carlo methods.  Observables Ô are calculated as ⟨Ô⟩ = lim_τ→∞⟨Ψ_τ|Ô|Ψ_τ⟩/⟨Ψ_τ|Ψ_τ⟩, 
where Ψ_τ = e^-τĤΨ_T is the imaginary-time projection of an initial trial wave-function Ψ_T. Provided non-orthogonality to the ground state, the quality of the wave-function only influences the projection time practically involved in the limit and the variance of the results.



The trial wavefunction is of the two-body Jastrow form: Ψ_T(x_1… x_N) = exp[-1/2∑_i<j(u(r_ij) + χ(r_ij))]. The Reatto-Chester contribution 

    χ(r)=- αlog[sin^2(π r/L)+sinh^2(πR̅/L)/ 1+sinh^2(πR̅/L)]

accounts for long-wavelength phonons [36,30]

and allows for faster convergence of the small momenta parts of the structure factors. R̅ is a variational parameter delimiting the long-range part from the short-range contribution. The latter is embodied in the f(r)=exp[-u(r)/2] factor, which we take as the numerical solution (described in detail in [30])

of a suitable two-body Schrödinger equation (with reduced mass μ=m/2), with the effective potential 

    V_eff(r)=c_1 V(r) + c_2 ∑_l=-l_C^l=l_C V(r-l b)  ,

and the boundary condition u(R̅)=0. Here, the first term is a renormalized two-body potential, while the sum accounts for a series of nearby clusters in a lattice of spacing b (l=0 is excluded and l_M=int[L/(2b)]). All parameters in the trial wave-function are variationally optimized, as described in the next Section, before projecting with the PIGS method, in order to maximize efficiency of the simulations. One finds that typically R̅∼ 2.2,  b∼ b_c, and c_1 and c_2 are usually of the same order in the cluster phase. We found it is crucial to introduce such effective cluster contributions in the two-body Jastrow factor, since the variational energy per particle is typically 30% higher than the PIGS result, if one sets c_2=0. On the contrary, by properly optimizing the parameters, we obtained discrepancies of at most a few percent of the PIGS result. A typical effective potential, and the corresponding short-range Jastrow correlation f(r), are shown in Fig. <ref>. Notice the behavior of f(r) at r≃ i b_c, with i=1,2. 

In the PIGS method, the imaginary time τ of propagation is split into M_P time steps of size δτ=τ/M_P so that a suitable short-time approximation for the propagator can be used; in this case, we employ the fourth-order pair-Suzuki approximation [35].


The timestep is selected by performing various simulations at short fixed projection time τ with increasing number of beads M_P, and monitoring the energy per particle, looking for convergence within errorbars of order 10^-4. At ρ=1.37, the resulting dτ approximatively follows the form dτ≃ 0.1 U^-1/2: this is consistent with the consideration that the imaginary-time density matrix typically varies at the scale of the amplitude l_HCO∝ U^-1/4 of the cluster vibrational modes.

Once the timestep is selected, convergence to the ground-state is obtained by selecting a sufficiently long projection time τ, which renders the potential energy flat (within errorbars) as a function of time slice. With our optimized initial wavefunctions, the typical number of time-steps for convergence is 35.
Once convergence in τ is obtained, a further projection time of typically M_F=60 time-slices is used to sample the intermediate scattering function F(q,τ).

Close to the transition, convergence in τ is very slow, due to the low-energy Ising mode. We have thus also employed a more refined effective potential, of the form

    V_eff(r)=c_1 V(r)  
    
    +c_2 ∑_l=-l_C^l=l_C∑_i=-p^p V(r-l b-iδ_l) g(iδ_l,s_l)  ,

where we introduced a gaussian kernel g(r,s_l)=exp[-r^2/(2s_l^2)]/𝒩(s_l), with 𝒩(s_l)=∑_i=-p^pexp[-(iδ_l)^2/(2s_l^2)], for better describing the clusters. The width s_l=s_0 l^γ is allowed to increase with the cluster position, and the convolution is discretized with p=10 and δ_l=3s_l/p. In the limit s_0→0 and γ→0, one recovers expression (<ref>), while in general s_0 and γ are two additional variational parameters to be optimized.

For U=18, we employed the parameters c_1=1.127, c_2=1.076, b=1.655, s_0=0.2, α=4.84, R̅=1.83, and γ=0.4. With dτ=0.03, simulations still required 100 time-slices of projection.





§ PARAMETERS OPTIMIZATION THROUGH SIMULATED ANNEALING


The parameters appearing in the trial wavefunction are optimized with the Variational Monte Carlo method (VMC), which corresponds to PIGS, when no imaginary-time projection is made. We employ a simulated annealing procedure in parameters' space, aiming at both reducing the energy associated to the trial wavefunction, and obtaining a good pair distribution function g(r). To each configuration ξ of the parameters, the following Boltzmann weight λ =  -β[E(ξ) + ζ χ^2(ξ)] is associated, with inverse temperature β, where E(ξ) is the VMC energy, and

    χ^2(ξ) = ∑_i^i_M[g_0(r_i) - g_ξ(r_i)]^2/σ_0^2(r_i) + σ_ξ^2(r_i)

is the discrepancy between the pair distribution functions obtained at a particular step g_ξ, and the one obtained with a preliminary unoptimized PIGS simulation g_0, weighted by the corresponding uncertainties σ_ξ and σ_0. This difference is evaluated only up to the i_M-th bin of the histogram of g(r), which includes the first peak of g(r) for r≠ 0. The motion of a fictitious particle is simulated in parameters' space, subjected to such Boltzmann weight, where ζ is the relative role of χ^2 with respect to E. The effective temperature is progressively decreased in order to reach the configuration having the lowest value possible of E(ξ) and χ(ξ) simultaneously, and the algorithm is stopped once the variations of the weight are of the order of errorbars. The typical used value of ζ is ∼ 10^-4, except close to the transition, where we observed that increasing this value led to shorter projection times in the subsequent PIGS simulations. Fig. <ref> shows an example of the convergence of the two parameters c_1 and c_2.



§ GENETIC INVERSION VIA FALSIFICATION OF THEORIES METHOD


(For completeness, we add a brief description, adapted from the Supplemental Material of Ref.[38].)


The relation between the intermediate scattering function F, which is evaluated within the PIGS simulations, and the dynamical structure function is

    F(q,τ) = 1/N⟨ e^τ Hρ_q e^-τ Hρ_-q⟩ = ∫_0^∞ dω e^-τω S(q,ω).


This is a Fredholm equation of the first kind and is an ill-conditioned problem, because a small variation in the imaginary-time intermediate scattering function F produces a large variation in the dynamical structure factor S. At fixed momentum q, the computed values F_j=F(q,jδτ), where j=0… M, are inherently affected by statistical uncertainties δ F_j, which hinder the possibility of deterministically infer a single S(q,ω), without any other assumption on the solution. The Genetic Inversion via Falsification of Theories method (GIFT) exploits the information contained in the uncertainties to randomly generate Q compatible instances of the scattering function F^(z), with z=1… Q, which are independently analyzed to infer Q corresponding spectra S^(z), whose average is taken to be the “solution”. This averaging procedure, which typifies the class of stochastic search methods (See the recent review [40]),

yields more accurate estimates of the spectral function than standard Maximum Entropy techniques. This method is able to resolve the first spectral peak and approximately the second one, if the errorbars of F are sufficiently small; generally speaking, the most relevant features of the spectrum are retrieved in their position and (integrated) weight. 

Given an instance z, the procedure of analytic continuation from F^(z) to S^(z) relies on a stochastic genetic evolution of a population of spectral functions of the generic type S^(z)(q,ω) = m_0∑_i=1^N_ωs_iδ(ω-ω_i), where m_0=F^(z)(q,0) and the zeroth momentum sum rule ∑_i=1^N_ωs_i=1 holds. The N_ω support frequencies ω_i are spaced linearly up to an intermediate frequency ω_m, which is of order of the maximum considered momentum squared q^2, then a logarithmic scale is used up to very high frequencies. Genetic algorithms provide an extremely efficient tool to explore a sample space by a non-local stochastic dynamics, via a survival-to-compatibility evolutionary process
mimicking the natural selection rules; such evolution aims toward increasing the fitness of the individuals, defined as

    Φ^(z)(S) =
    -∑_j=0^M 1/δ F_j^2[F^(z)_j - m_0∑_i=1^N_ω e^-j δτω_i s_i]^2
    
    
    - ϵ[m_1 - m_0∑_i=1^N_ωω_i s_i]^2  ,

where the first contribution favors adherence to the data, while the second one favors the fulfillment of the f-sum rule, with m_1=q^2/2 and ϵ a parameter to be tuned for efficiency. A step in the genetic evolution replaces the population of spectral functions with a new generation, by means of the “biological-like” processes of selection, crossover, local mutation, non-local mutation, which are described in detail in [37,40].

Moreover, the genetic evolution is tempered by an acceptance/rejection step based on a reference distribution p^(z)(S)=exp(Φ^(z)(S)/T), where the coefficient T is used as an effective temperature in a standard simulated annealing procedure.
We found that this combination is optimal in that it combines the speed of the genetic algorithm with the prevention of strong mutation-biases thanks to the simulated annealing.
Convergence is reached once |Φ^(z)(S)|< 1, and the best individual, in the sense that it does not falsify the theory represented by (<ref>), is chosen as the representative S^(z). The final spectrum is obtained by taking the average over instances S̅(q,ω)=1/Q∑_z=1^Q S^(z)(q,ω).







§ EXTRACTION OF THE LUTTINGER PARAMETER FROM THE STATIC STRUCTURE FACTOR


The behavior of S(q) for small momenta is shown in Figure <ref>. Since its Luttinger liquid behavior is S(q) ∼ qK_L/(2k_F), we fit 2 k_F S(q)/q with a constant. Clearly, for extremely low momenta, size effects become relevant, since finite imaginary-time projection has not been able to correctly reconstruct the long-distance density fluctuations, leading to a non linear behavior of the data. For this reason, the error on the fitted value of K_L has been taken to be the absolute value of the difference between the fitted K_L and the value of 2k_F S(q)/q for the smallest q at our disposal. The resulting K_L are summarized in Table <ref>.




§ EXTRACTION OF THE ANOMALOUS LUTTINGER PARAMETER FROM THE PAIR DISTRIBUTION FUNCTION


CLL theory implies that the long-range behavior of g(r) has to be of the form of Eq.(2) in the main text. The easiest way to extract K_L^' from g(r) consists in evaluating the power-law exponent relative to the cosine decay. In such procedure we have neglected the -2K_L/(2πρ r)^2 term, as well as all the cosine terms, but the first one, as their decay is usually too fast to be relevant. Values of g(r) <1 are determined by short-range effects: it is thus appropriate to fit just the upper part of the peaks of g(r). To make such procedure more precise, we have fitted the function [g(r) - 1] · (ρ r)^η, which is supposed to behave like ∼ A_1 cos(πρ r) (ρ r)^η -2K_L' for long distances, with η tuned as to have the periodic peaks at comparable heights. Thus the cosine decay is considerably softened, leading to an easier fitting of K_L^'. An example of one of our fits is shown in figure <ref>. We have not been able to apply this methodology for U<18, due to large noise-to-signal ratio, and at U=18, possibly due to the role of subleading terms in the decay. The resulting K_L^' are summarized in Table <ref>.







§ EXTRACTION OF THE CENTRAL CHARGE




We estimate the central charge c by fitting the slope of the energy per particle ε(N)=ε_∞ - c E_F/(6 K_L N^2) versus 1/N^2, where K_L is inferred from Table <ref>. The range of considered numbers of particles is typically from N=10 to N=40 (N=100 in some cases), and we progressively increase the minimal N_min used in the fit, to assess the role of higher order contributions, finding that the extracted c is generically stable if N_min=20. The errorbar comes from the uncertainty in ε(N) and K_L, and from the variance of the extracted c at different N_min>20.
Examples of extraction of c are shown in Fig. <ref>, corresponding to U=17 and to the more difficult U=18 point. Table <ref> summarizes the results.

