Quantum Fluctuations Beyond the Gutzwiller Approximation in The

PHYSICAL REVIEW RESEARCH 2, 033276 (2020)

Quantum ﬂuctuations beyond the Gutzwiller approximation in the Bose-Hubbard model

Fabio Calefﬁ ,1,2,* Massimo Capone,1,3 Chiara Menotti,2 Iacopo Carusotto,2,4 and Alessio Recati2,4,† 1International School for Advanced Studies (SISSA), I-34136 Trieste, Italy 2INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy 3CNR-IOM Democritos, I-34136 Trieste, Italy 4Trento Institute for Fundamental Physics and Applications, INFN, 38123, Trento, Italy

(Received 9 August 2019; accepted 29 July 2020; published 20 August 2020)

We develop a quantum many-body theory of the Bose-Hubbard model based on the canonical quantization of the action derived from a Gutzwiller mean-field ansatz. Our theory is a systematic generalization of the Bogoliubov theory of weakly interacting gases. The control parameter of the theory, defined as the zero point fluctuations on top of the Gutzwiller mean-field state, remains small in all regimes. The approach provides accurate results throughout the whole phase diagram, from the weakly to the strongly interacting superfluid and into the Mott insulating phase. As specific examples of application, we study the two-point correlation functions, the superfluid stiffness, and the density fluctuations, for which quantitative agreement with available quantum Monte Carlo data is found. In particular, the two different universality classes of the superfluid-insulator quantum phase transition at integer and noninteger filling are recovered.

DOI: 10.1103/PhysRevResearch.2.033276

I. INTRODUCTION On the theoretical side, a very common approach to the BH model is based on the Gutzwiller ansatz. While The Hubbard model is one of the most celebrated models many important features of both the superfluid and the in- of quantum condensed matter theory. The main reason is prob- sulating phases are accurately captured by the Gutzwiller ably the widespread belief that its two-dimensional fermionic wave function, its local, site-factorized form typically makes version holds the key to understand how high-temperature physical quantities involving off-site quantum correlations to superconductivity emerges on doping a Mott insulator [1,2]. be missed. In the weakly interacting regime, a Bogoliubov Its central feature is the competition between the kinetic approach for the fluctuations around the mean-field Gross- energy term, which favors delocalized states, and the local Pitaevskii (GP) ground state of a dilute Bose-Einstein con- Coulomb repulsion, which favors localization [3,4]. In the densate provides an accurate description of the equilibrium two-dimensional fermionic model this physics is, however, state and of the excitations of the gas [15,19], including somewhat hidden by the presence of other phases bridging quantum correlations between particles [20]. In the strongly the Mott insulator and the superconducting and the metallic interacting regime, however, the GP mean-field theory and states, including the celebrated pseudogap [5] and charge- the Bogoliubov approach based on it become clearly in- ordered [6] phases. adequate. The rich physics of the strongly interacting BH The archetypal competition between the kinetic and in- model across the Mott-superfluid transition and specifically teraction energies is found in the bosonic version of the in the insulating phase has been attacked through a number model, the so-called Bose-Hubbard (BH) model [7], where it of different approaches, ranging from semianalytical methods manifests itself as a direct quantum phase transition between such as random phase approximation (RPA) [21–23], slave- a superfluid and a Mott insulator. As a consequence of its boson representation [24,25], and time-dependent Gutzwiller paradigmatic nature, this transition has attracted an enormous approximation [26,27] to numerical techniques including experimental interest in the past few years, fostering imple- quantum Monte Carlo methods [28–31], bosonic dynamical mentations with cold atoms trapped in optical lattices [8–15] mean-field theory (B-DMFT) [32–34], and numerical renor- and, more recently, implementations with photons in the novel malization group (NRG) [35,36]. All these methods provide context of nonequilibrium phase transitions [16–18]. qualitatively concordant results on the phase diagram as well as on the spectral properties of the model. The collective phonon excitations of the Bogoliubov theory of dilute conden- *fcaleffi@sissa.it sates are replaced by a multibranch spectrum of excitations †[email protected] [23,26,27,37], containing in particular the gapless Goldstone mode and a gapped (also refereed to as Higgs) mode on Published by the American Physical Society under the terms of the the superfluid side and the particle-hole excitations in the Creative Commons Attribution 4.0 International license. Further insulating phase (see, e.g., Refs. [27,38,39]). distribution of this work must maintain attribution to the author(s) In spite of these remarkable advances, a complete, eas- and the published article’s title, journal citation, and DOI. ily tractable, and physically intuitive description of the

2643-1564/2020/2(3)/033276(11) 033276-1 Published by the American Physical Society FABIO CALEFFI et al. PHYSICAL REVIEW RESEARCH 2, 033276 (2020) collective excitations and their fingerprint on quantum ob- We use the Gutzwiller ansatz [26,41,42] servables across the whole phase diagram of the model is still lacking. In particular, the development of nonlocal cor- |G= cn(r) |n, r, (2) relations across the Mott-superfluid transition and the proper r n characterization of the strongly interacting superfluid state and of its excitation modes remain a challenging problem. where the wave function is site factorized and the complex In this paper, we combine the successful features of the amplitudes cn(r) are variational parameters with normal- | |2 = Gutzwiller and Bogoliubov approaches to develop a new strat- ization condition n cn(r) 1 to reformulate the Bose- egy to systematically quantize the time-dependent Gutzwiller Hubbard model in terms of the following Lagrangian func- ansatz. In spite of the local nature of the Gutzwiller ansatz tional: [see Eq. (2)] the accurate description of the excitations and, ∗ in particular, of their zero-point fluctuations allows us to L[c, c ] =G| i h¯ ∂t − Hˆ |G correctly reproduce the nonlocal many-body correlations in i h¯ ∗ the different phases, as well as the different critical behaviors = [c (r)˙cn(r) − c.c.] 2 n of the commensurate and incommensurate phase transitions r,n [7,38]. Time-dependent Gutzwiller approaches addressing the ∗ + J [ψ (r) ψ(s) + c.c.] − H |c (r)|2. (3) linear-response dynamics in the BH model [26] and in lattice n n , r,n Fermi systems [40] have been recently developed. The ad- r s vantage of our formalism is that it directly includes quantum In the previous equation, the dot indicates that the temporal fluctuations of the collective modes and could naturally incor- derivatives, porate those effects beyond linearized fluctuations that stem from interactions between quasiparticles. This is essential to U successfully tackle problems such as the finite quasiparticles Hn = n(n − 1) − μ n, (4) 2 lifetime via Beliaev-like nonlinear interaction processes and the quantum correlations between the products of their decay, are the on-site terms in the Hamiltonian and which will be the subject of future investigations. √ The paper is organized as follows. Section II is devoted to ψ = = ∗ (r) aˆr ncn−1(r) cn(r) (5) the derivation of the quantum Gutzwiller theory for the Bose- n Hubbard model. The original features of the method are high- lighted and its advantages and disadvantages are discussed in is the mean-field order parameter. In this formulation, the ∗ = comparison to other approaches. In Sec. III, we present the conjugate momenta of the parameters cn(r)arecn (r) predictions of the quantum Gutzwiller method for observables ∂L/∂c˙n(r). The classical Euler-Lagrange equations associ- in which local and nonlocal quantum correlations strongly ated to Lagrangian (3)arethetime-dependent Gutzwiller modify the standard mean-field picture, such as two-point equations as derived, e.g., in Refs. [21,26] and from which the correlation functions, superfluid density, and pair correlations. excitation spectrum can be determined. In the uniform system We conclude in Sec. IV with an outlook on future studies and the stationary solutions are homogeneous. The system is in a on possible extensions of the quantum formalism introduced Mott insulator (MI) state if ψ(r) = 0 and in a superfluid (SF) in this work. state otherwise. The spectrum of the collective modes ωα,k is plotted in II. MODEL AND THEORY Fig. 1 in different regions of the phase diagram shown in Fig. 1(a). In the MI phase [Fig. 1(b)], the two lowest excitation In Sec. II A, we briefly review the basic concepts of the branches are the gapped particle and hole excitations. In the Bose-Hubbard model and of the C-number Gutzwiller ansatz SF phase [Fig. 1(c)], the lowest of them becomes the gapless needed to develop the quantum Gutzwiller approach presented Goldstone mode of the broken U(1) symmetry. The other in the following Secs. II B to II D. The quantum Gutzwiller gapped excitation is often referred to as the Higgs mode method is put into perspective and compared with other [27,43–45] and is related to the fluctuations of the amplitude approaches in Sec. II E. of the order parameter in some specific region of the phase diagram [27]. A. Lagrangian formulation within the Gutzwiller ansatz The quantum phase transition from the MI to the SF phase can belong to two different universality classes [7,38] We consider the three-dimensional BH model depending on whether the transition is crossed while changing U Hˆ =−J (â† aˆ + H.c.) + nˆ (nˆ − 1) − μ nˆ , the density, the so-called commensurate-incommensurate (CI) r s 2 r r r transition [blue point in Fig. 1(a)], or it is crossed at a fixed and r,s r r (1) commensurate filling (at the tip), the so-called O(2) transition [red point in Fig. 1(a)]. At the CI transition points only one where J is the hopping amplitude, U the on-site interaction, μ mode becomes gapless (the Goldstone branch in the SF), the chemical potential, while r, s labels all pairs of nearest- whereas the other mode is gapped and related to the particle or neighboring sites. The annihilation and creation operators of the hole branch of the MI depending on the chemical potential † a bosonic particle at site r area ˆr anda ˆr , respectively, and [Fig. 1(d)]. On the other hand, at the tip critical point both = † nˆr aˆr aˆr is the local density operator. modes become gapless [Fig. 1(e)].

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1 0 1 0 1 0 1 0 1.00 (a) . . . . O(2) Transition 0.8 0.8 0.8 0.8 0.75 MI nˆ =1 Superﬂuid ] 0 6 0 6 0 6 0 6 0.50 O(2) . . . . U

[ Mott CI Transition μ/U ω 0 4 0 4 0 4 0 4 0.25 . . . . CI 0.2 0.2 0.2 0.2 0.00 SF (b) (c) (d) (e) nˆ =0 0.0 0.0 0.0 0.0 −0.25 0.0 0.1 0.2 0 π/4 π/2 3π/4 π 0 π/4 π/2 3π/4 π 0 π/4 π/2 3π/4 π 0 π/4 π/2 3π/4 π 2 dJ/U kx = ky kx = ky kx = ky kx = ky

FIG. 1. (a) Mean-ﬁeld Gutzwiller phase diagram around the nˆ=1 Mott lobe. The black points refer to the MI and SF spectra shown in panels (b) and (c), while the blue and red points indicate the CI and O(2) critical points, respectively. (b) Energy spectra of hole (solid line) and particle (dashed line) excitations in the MI phase for μ/U = 0.2and2dJ/U = 0.03. (c) Goldstone mode (solid line) and Higgs mode (dashed line) energy dispersion in the strongly correlated SF phase for μ/U = 0.2and2dJ/U = 0.2. (d) Excitation spectrum at the CI critical point corresponding to μ/U = 0.2and2dJ/U = 0.13, see the blue dot in panel (a). (e) Excitation spectrum at the O(2)-invariant critical point, see the red dot in panel (a). Both modes become gapless and present a linear dispersion.

B. The quantum Gutzwiller theory Appendix B. A suitable Bogoliubov rotation of the Gutzwiller operators In order to go beyond the Gutzwiller ansatz reviewed in the previous subsection, it is natural to consider how quantum δ ˆ = ˆ + ∗ ˆ† Cn(k) uα,k,n bα,k vα,−k,n bα,−k (11) (and thermal) effects populate the excitation modes of the α α system and to address how they affect the observable quantities. We include quantum fluctuations by building a theory recasts the quadratic form (10) into a diagonal form, of the excitations starting from Lagrangian (3) via canonical (2) † Hˆ = h¯ ωα, bˆ bˆα, , (12) quantization [46,47], namely promoting the coordinates and k α,k k α their conjugate momenta to operators and imposing equal- k ˆ† time canonical commutation relations, where each bα,k corresponds to a different many-body excita- † tion mode with frequencies ωα, , labeled by its momentum k [ˆc (r), cˆ (s)] = δ , δ , . (6) k n m r s n m and branch index α. Bosonic commutation relations between ˆ ˆ† In analogy with the Bogoliubov approximation for dilute the annihilation and creation operators bα,k and bα,k, Bose-Einstein condensates [15,19], we expand the operators † 0 ˆα, , ˆ = δ , δα,α , cˆn around their ground-state values cn, obtained by minimiz- [b k bα ,k ] k k (13) ing the energy |Hˆ | ,as G G are enforced by choosing the usual Bogoliubov normalization = ˆ 0 + δ . cˆn(r) A(r) cn cˆn(r) (7) condition ˆ δ ∗ · − ∗ · = δ , The normalization operator A(r) is a function of cˆn(r) and uα,k uβ,k vα,−k vβ,−k αβ (14) δ † † = cˆn(r) and ensures the proper normalization n cˆn(r)ˆcn(r) where the vectors u (v ) contain the components uα, , 1ˆ . By restricting to local fluctuations orthogonal to the ground α,k α,k k n δ † 0 = (vα,k,n). As a direct consequence of the spectral properties of state n cˆn(r) cn 0 one has L δ k, the fluctuation operators cˆn(r) satisfy the quasibosonic 1/2 commutation relations, ˆ = − δ † δ . A(r) 1 cˆn(r) cˆn(r) (8) † 0 0 [δcˆ (r),δcˆ (s)] = δ , δ , − c c , (15) n n m r s n m n m In a homogeneous system, it is convenient to work in the calculation of which is made explicit in Appendix C. momentum space by writing The correction term on the right-hand side of Eq. (15)serves to remove those unphysical degrees of freedom that are δ ≡ −1/2 ik·r δ ˆ . cˆn(r) N e Cn(k) (9) introduced by the local gauge invariance of the Gutzwiller k∈BZ ansatz (2), namely the arbitrariness of the phase of the local wave functions c (r) at each site r. Result (15) generalizes Inserting Eq. (9)intoG|Hˆ |G and keeping only terms up n to the quadratic order in the fluctuations, we obtain to a strongly interacting Bose system, known from the Bo- goliubov approach in the homogeneous weakly interacting δCˆ (k) (2) 1 † Bose gas [19]. Hˆ = E0 + [δCˆ (k), −δCˆ (−k)] Lˆk , 2 δCˆ †(−k) Even though in this paper we focus only on Gaussian k fluctuations, the inclusion of terms beyond second order in δcˆ (10) n arising from the quartic hopping term in Eq. (3) does not pose where E0 is the mean-field ground-state energy, the vector any fundamental difficulty. As in standard Bogoliubov theory, δCˆ (k) contains the components δCˆn(k), and Lˆk is a pseudo- higher-order terms describe interactions between collective Hermitian matrix, the explicit expression of which is given in modes (e.g., see Sec. III in Ref. [48]).

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10−1 (ii) Create the operator Oˆ [ˆc, cˆ†] by replacing the Gutzwiller parameters in O[c, c∗] by the corresponding op- † eratorsc ˆn(r) andc ˆn(r) without modifying their ordering; (iii) Expand the operator Oˆ order by order in the fluctua- −2 δ δ † 10 tions cˆn and cˆn, taking into account the dependence of the operator Aˆ on the fluctuation operators. The contribution of Aˆ may be of fundamental importance when higher orders in the F nˆ =1 fluctuations become relevant; 10−3 (iv) For the specific case considered in this work of neg- nˆ =0.6 ligible interactions between excitation modes, invoke Wick nˆ =0.4 theorem to compute the expectation value of products of nˆ =0.2 operators on Gaussian states—such as ground or thermal (2) 10−4 states obtained from H . 10−2 10−1 100 101 102 In the following, we apply this protocol to compute Oˆ , 2 dJ/U where the expectation value is intended to be evaluated on the Bogoliubov vacuum, i.e., bˆk,α|0=0. FIG. 2. Control parameter F of the theory as defined in (16) plotted as a function of 2dJ/U for different values of the lattice filling. E. Putting the method into perspective Dashed and solid lines indicate whether the system is in a Mott insulating (only for nˆ=1) or in a superfluid phase, respectively. Before proceeding with the presentation of the predictions of our theory, it is worth shortly commenting on the relation of our theory with other competing methods. C. General remarks on the accuracy of the Our approach owes much to the time-dependent Gutzwiller quantum Gutzwiller method method in Ref. [26] where the cn(r) parameters are considered The accuracy of the quantum Gutzwiller theory can be as C-numbers and not as operators. In the same way the quantitatively estimated by looking at the magnitude of the linearized Gross-Pitaevskii equation can be used together with “quantum fluctuations” around the Gutzwiller mean-field, linear response theory to obtain information on the quantum fluctuations [50], the time-dependent Gutzwiller approach would give the same results as our method for a number of F = 1 −Aˆ2= δc†(r) δc (r) , (16) n n properties where only quadratic fluctuations are important. n When only Gaussian fluctuations above the MF result which represents the small control parameter of our theory are considered, our approach to the BH model has strong [49]. As illustrated in Fig. 2, this quantity remains always very similarities to including quantum fluctuations by slave-boson small throughout the phase diagram, suggesting the overall techniques, as done, e.g., in the comprehensive work by Frérot reliability of the quantum Gutzwiller approach: A small value and Roscilde [25]. One important difference from this work is, of the quantum fluctuations is in fact a good indication that the however, the way in which the observables are calculated: In nonlinear terms that are not included in (10) are indeed small particular, we never rely on the microscopic reconstruction of and can be neglected. the original Bose fieldsa ˆr through the operatorsc ˆn’s and, from In particular, for commensurate density, the quantity F the very beginning, the dynamical variables of our approach δ δ † approaches zero both in the deep MI (dashed line) and in the are cˆn and cˆn. It is also worth mentioning that, to our deep SF regime, that is, in both limits where the Gutzwiller knowledge, there are no slave-boson calculations of the role ansatz recovers exactly the ground state of the BH model. As of quantum fluctuations on the one-body correlation function expected, its maximum is located at the transition point. and on the superfluid density. Even though for such quantities For noncommensurate densities, F tends to zero in the deep we expect the slave-boson and our approach to give the same SF regime (where again the Gutzwiller ansatz recovers exactly results, our method is technically easier and more transparent. the ground state) and eventually increases in the strongly inter- Finally, the slave-boson approach has been recently shown to acting superfluid regime for decreasing J/U → 0. Note that properly interpolate between strong-coupling and Bogoliubov for noncommensurate densities this limit does not correspond approaches in calculating the entanglement entropy [25], a to a Mott insulator and the Gutzwiller ansatz is not able to property accessible to our approach, but not to the time- fully capture the ground state. dependent Gutzwiller method. In the next section, we will show how the quantum Gutzwiller method can reproduce both local and nonlocal cor- D. Calculation of the observables relations with very high accuracy and successfully compares In this subsection we summarize the protocol that we use to to quantum Monte Carlo (QMC) calculations. Moreover, the compute physical observables within the quantum Gutzwiller study of time-dependent problems appears to be a straight- theory. The evaluation of the expectation value for any observ- forward generalization of the quantum Gutzwiller approach. ˆ †, able O(âr aˆr ) consists in applying the following four-step This is a crucial feature compared to QMC, which can hardly procedure: describe dynamical properties. ∗ (i) Determine the expression O[c, c ] =G|Oˆ|G in Dynamical properties can be instead attacked by means ∗ terms of the Gutzwiller parameters cn and cn; of the bosonic version of dynamical mean-field theories

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(B-DMFT) [32–34]: While this theory is very accurate for 1.00 1.0 the study of local quantities, it is, however, poorly reliable 0.99 for nonlocal quantities. In addition to not providing physi- 0.8 0 98 cal intuition, both QMC and B-DMFT are computationally ) . r ( 0.6 much more demanding than the present quantum Gutzwiller (1) 0.97 g approach. 0 96 . 0.4 0 95 . (a) (a ) III. CORRELATION FUNCTIONS ACROSS 0.2 0.94 THE MI-SF TRANSITION 0 1 2 3 4 0 1 2 3 4 r r After having introduced the quantum Gutzwiller theory, in the present section we apply it to the calculation of some 100 100 (b) (c) relevant correlation functions: (i) the coherence function, (ii) the current-current correlation function and superﬂuid density, 10−1 10−1 ) r

and (iii) the density-density correlation function. We compare ( 10−2 10−2 (1)

our results with the predictions of quantum Monte Carlo g calculations, when available, ﬁnding striking agreement. 10−3 10−3

A. Coherence function 10−4 10−4 0 1 2 3 4 5 0 1 2 3 4 The single-particle correlation function, referred to also as r r coherence function, is defined as √ FIG. 3. (a) First-order coherence function g(1)(r)forμ/U = aˆ† aˆ ψˆ †(r) ψˆ (0) 2 − 1and2dJ/U = 2, 3, 5, 10 going deeper into the SF phase g(1)(r) = r 0 → , (17) † ψˆ † ψˆ (from bottom to top). Solid and dashed lines refer to the quan- aˆ aˆ0 (0) (0) 0 tum Gutzwiller and Bogoliubov predictions,√ respectively. (a ) (1) where the last expression is the result of the quantization Gutzwiller predictions of g (r)forμ/U = 2 − 1and2dJ/U = . , . , . , . , . protocol outlined in the previous section. 0 2 0 4 0 6 0 8 1 0 approaching the critical point (from top to (1) μ/ = . Within our protocol the field operator reads bottom). (b) g (r) in the MI phase for U 0 2 and increasing / = . , . , . , . , . , . , / 2 dJ U 0 002 0 02 0 04 0 08 0 11 0 13 (2 dJ U )c approach- √ / ψˆ = † . ing the transition point (2 dJ U )c from inside the MI lobe (purple√ to (r) n cˆn−1(r) cˆn(r) (18) (1) μ/ = μ/ = − blue lines). (c) g (r) in the MI phase for U ( U )tip 2 n / = . , . , . , . , . , / 1 and increasing 2 dJ U 0 02 0 06 0 12 0 16 0 17 (2 dJ U )tip † toward the tip of the Mott lobe (dark brown to red to gold lines). Expandingc ˆn andc ˆn to the lowest order in the fluctuations † Dashed lines in panels (b) and (c) are the quantum Gutzwiller δcˆn(r), δcˆ (r), one obtains n predictions, and the (exponential and power-law) fits are displayed √ √ ψˆ ≈ 0 0 + 0 δ + 0 δ † as solid black lines. (r) ncn−1 cn n cn−1 cˆn(r) cn cˆn−1(r) n n 1 At the same level of approximation, the normalized zero- = ψ + √ ˆ i k·r + ˆ† −i k·r , 0 Uα,k bα,ke Vα,k bα,ke temperature one-body coherence function reads N α>0 k |ψ |2 + −1 | |2 · (19) 0 N k,α Vα,k cos (k r) g(1)(r) ≈ . (24) 2 −1 2 √ |ψ0| + N ,α |Vα,k| ψ = 0 0 k where 0 n ncn−1 cn is the order parameter in the ground state and the particle (hole) amplitudes In Figs. 3(a)–3(c) we plot the results for g(1) along the different √ lines at constant chemical potential in the phase diagram = + 0 + 0 , Uα,k n 1 cnuα,k,n+1 cn+1vα,k,n (20) shown in Fig. 1(a). n √ In the deep SF phase [Fig. 3(a)], the spectral weight is = + 0 + 0 , saturated by the Goldstone mode and our prediction for g(1)(r) Vα,k n 1 cn+1uα,k,n cnvα,k,n+1 (21) n reduces to the result for the weakly interacting gas (dashed lines). In the region 2 dJ/U 1 [Fig. 3(a)] of strongly inter- satisfy the Bogoliubov normalization [27], acting superfluid, the contribution of other excitation modes [23] starts to become relevant—and the Bogoliubov approach | |2 −| |2 = . ( Uα,k Vα,k ) 1 (22) (not shown) would give much higher asymptotic values. In α the MI phase [Figs. 3(b) and 3(c)], the quantum Gutzwiller In this way, the Bose field (19) satisfies the usual canonical method is able to capture an exponentially decreasing coher- (1) ∼ − /ξ ξ commutation relations ence g (r) exp ( r ) with a finite coherence length . A nonvanishing value of ξ provides a first drastic improve- † [ψˆ (r), ψˆ (s)] = δr,s (23) ment with respect to the mean-field Gutzwiller ansatz, whose factorized form cannot predict any off-site coherence, giving (1) = |ψ |2/ − δ + δ , = | 0|2 to second order in the fluctuations. gMF(r) ( 0 n0 )(1 r,0 ) r,0 with n0 n n cn .

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Moreover, the present quantum theory is also able to 1.0 capture the different critical behaviours of the superfluid- insulator transition depending on whether the transition is 0.8 approached at integer or noninteger filling. Approaching the fs 2 superfluid transition from the Mott phase away from the tip 0.6 ρc = |ψ0| /nˆ ξ of the Mott lobe, the correlation length of the MI grows but − ˆ remains bounded [Fig. 3(b)]. As soon as one enters the SF 0.4 Kx / (2J nˆ ) MI SF xx phase, long-range order suddenly appears as a nonvanishing |Λ (0, 0)| † = 0 2 long-distance coherence, aˆr aˆ0 r→∞ 0: Such a quantity . Bogoliubov 2 physically corresponds to the condensate density |ψ0| and continuously grows from zero as one penetrates the superfluid 0.0 phase. On the other hand, when approaching the SF phase at 10−2 10−1 100 101 the tip of the Mott lobe, the correlation length ξ diverges and 2 (1) dJ/U a power-law dependence for g is found at the critical point √ [Fig. 3(c)]. FIG. 4. Superfluid fraction fs along the μ/U = 2 − 1 line This remarkable difference is related to the distinct univer- crossing the tip of the nˆ=1 Mott lobe. The orange-shaded area sality class of the MI-SF transition at noninteger or integer indicates the MI region. Solid black line: Quantum Gutzwiller pre- filling [38]. In all critical points of the CI transition, either the diction. Blue dotted and light-pink dashed lines are the contributions ˆ xx hole or the particle excitation becomes gapless, respectively to fs from the kinetic energy Kx and the current response J , respec- the hole excitation below the tip or the particle excitation tively. Green dot-dashed line: Bogoliubov approach for the weakly |ψ |2/ above the tip. Since the nontrivial short-distance coherence interacting gas. Black dashed line: Condensate fraction 0 nˆ . of the MI is due to virtual particle-hole excitations, the (1) exponential decay of g (r) is dominated by the gap of the where particle (or hole) excitation which remains finite. Instead, at ˆ =− ψˆ † + ψˆ + the critical point of the O(2) transition both the particle and Kx (r) J[ (r ex ) (r) H.c.] (26) hole modes become gapless (before turning into the Goldstone is the local kinetic energy operator along the direction x of the and Higgs modes on the SF side), which explains the divergent phase twist [54] and coherence length [51]. ∞ It is worth noting that result (24) can be obtained within xx ,ω =− iωt ˆ , , ˆ − , J (q ) i dt e [Jx (q t ) Jx ( q 0)] (27) the time-dependent Gutzwiller formalism [26] as the parti- 0 cle response function. This amounts to determine the time- is the current-current response function for the current opera- dependent Gutzwiller wave function resulting from applying tor, as a perturbation the single-particle operatora ˆ , and identify- r ˆ = ψˆ † + ψˆ − . ing g(1) as the linear response function related to the variation Jx (r) iJ[ (r ex ) (r) H.c.] (28) † of the expectation value of the operatora ˆr (see the detailed The expectation value (27), as well as the average kinetic discussion in the Gross-Pitaevskii framework in Ref. [50]). energy Kˆx, are calculated applying the protocol outlined in However, the quantum Gutzwiller approach presented in this Sec. II D: The average kinetic energy reads section provides a simpler, more intuitive and straightforward (1) procedure, not only for the calculation of g but also for 2 1 2 Kˆ =−2 J ψ + |Vα, | cos (k ) , (29) many other observables. First, the Bogoliubov amplitudes in x 0 N k x (11) are calculated once for all and can be used to calculate α,k the expectation value for any combination of operators. Then while the first nonvanishing contribution to the response func- xx quantities like the superfluid density, that we are going to com- tion J turns out to be the fourth-order correlation, pute in Sec. III B, would require very involved calculations 2 Uα, Vβ, − Uβ, Vα, using the time-dependent Gutzwiller approach. Finally, as we xx , =− 2 k k k k 2 . J (0 0) 4 J sin (kx ) are going to show in Sec. III C, there are quantities, like the ωα, + ωβ, k,α,β k k density correlation function g(2), for which the contribution of (30) the normalization operator Aˆ is dominant, in particular close and in the Mott phase: While in the time-dependent Gutzwiller Equation (30) reveals the crucial role played by the cou- approach the inclusion of Aˆ would be at least a technically pling between different collective modes in suppressing the cumbersome task, our theory is able to account for the order superfluid stiffness and creating a sort of normal component by order expansion of the normalization operator in a natural [55]. For the sake of comparison, it is worth noting that the and systematic way. ground-state mean-field Gutzwiller theory would give Kˆx= − ψ2 xx , = 2 J 0 and a vanishing current response J (0 0) 0. This B. Superfluid density 2 leads to equal superfluid and condensate densities, ns =|ψ0| . The superfluid density ns is defined through the static limit The results of the quantum Gutzwiller method for the of the transverse current-current response function [52,53]of superfluid density are illustrated in Fig. 4 by the black thick the system, namely line for the superfluid fraction fs = ns/nˆ, defined as usual = xx = , ,ω −ˆ , as the ratio of the superfluid density ns over the total density 2Jns lim lim J (qx 0 qy ) Kx (25) qy→0 ω→0 nˆ . This quantity tends to unity in the deep SF and approaches

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zero at the critical point. Throughout the whole SF region, it is 0 2 10 always larger than the condensed fraction |ψ0| (black dashed →∞ † line), defined as usual as the r limit of aˆr aˆ0 .IntheMI SF − region the superfluid fraction fs is exactly zero, as expected 10 1 for a phase that does not exhibit superfluidity. MI Further insight is obtained by isolating the two contri- (0) 10−2 (2)

butions appearing on the right-hand side of Eq. (25). The g (a) current response defined in Eq. (30) (light-pink dashed line) exhibits a nonmonotonic behavior as a function of J/U:It 10−3 tends to zero in the deep Mott and superfluid phases, while it reaches its maximum at the MI-SF transition point. In the 10−4 strongly interacting SF regime, the Goldstone-Higgs vertex 0.01 0.1 1 10 almost saturates the sum in the current response (30) and 2 dJ/U leads to a complete suppression of ns. As expected, the kinetic energy Eq. (29) (see dotted blue line) has as expected a mono- √ = g(2) 2 tonic behavior, from zero at J 0 to the weakly interacting 1.00 mean-field value 2Jnˆ. In the MI phase, the vanishing ns results from the perfect cancellation of the short-range virtual particle-hole correlations and the kinetic energy. 0.99 For completeness, in Fig. 4 we also report the the weakly (b) interacting Bogoliubov prediction [56] (green dashed-dotted 0.98 line), which our result approaches in the limit 2dJ/U 1. g(2)(1) Since it takes into account only the gapless Goldstone mode, 0 97 such an approach leads in particular to a zero current response . xx , = J (0 0) 0 and thus to an overestimated superfluid density. MI SF 0.96 0 0 0 1 0 2 0 3 0 4 0 5 C. Density fluctuations ...... 2 dJ/U We consider the normally ordered equal-time density correlation function FIG. 5. Density-density correlation g(2)(r) as a function of / = aˆ†aˆ†aˆ aˆ 2dJ U across the nˆ 1 commensurate SF-MI transition. Orange g(2) r = r 0 0 r . (white) background identifies the MI (SF) region. (a) On-site correla- ( ) (31) nˆr nˆ0 tion function g(2)(0). Black solid line: Quantum Gutzwiller method; Applying the procedure outlined in Sec. II D, and since our black dashed line: mean-field Gutzwiller approach; green dot-dashed states are translational invariant, g(2)(r) reads line: weakly interacting Bogoliubov theory; red dotted line: quantum Monte Carlo simulation for a lattice size of 163 sites [57]; gray Dˆ −Nˆ /Nˆ 2, = , dashed line: strong-coupling perturbation theory [42]. (b) Nearest (2) [ (0) (0) ] (0) r 0 √ g (r) = (32) g(2) g(2) Nˆ Nˆ /Nˆ 2, = , and next-to-nearest density correlations, (1) and ( 2). Black (r) (0) (0) r 0 solid line: Quantum Gutzwiller approach; gray dotted line: strong- coupling approximation; red dots: quantum Monte Carlo calculation where the density Nˆ (r) and the square density Dˆ (r) operators for a 53 lattice [42]. are defined as Nˆ (r) = n cˆ†(r) cˆ (r), (33) n n where n 2 † Dˆ (r) = n cˆ (r) cˆn(r). (34) = 0 + . n Nα,k ncn(uα,k,n vα,k,n ) (38) n n The expectation values in Eqs. (32) are evaluated by expand- ing the operators up to second order in the δcˆ’s: Specifically, all second-order terms in the expansion arise directly from the normalization operator Aˆ(r). This corre- Dˆ = + 2 − δ † δ , sponds in saying that, in Eqs. (35) and (36), the mean-field (0) D0 (n D0 ) cˆn(0) cˆn(0) (35) = 2| 0|2 = | 0|2 n quantities D0 n n cn and n0 n n cn are corrected Nˆ = + − δ † δ , by the depletion of the mean-field solution itself due to quan- (0) n0 (n n0 ) cˆn(0) cˆn(0) (36) tum fluctuations, leading to a nontrivial description of local n 2 1 2 quantum correlations in the ground state. Similar features are Nˆ (r = 0) Nˆ (0)=n + |Nα, | cos (k · r) 0 N k shared by the nonlocal correlations in Eq. (37), where terms α,k to fourth order have to be taken into account. In particular, the role of these higher-order fluctuations dominates for strong + (n − n0 )(m − n0 ) n,m interactions, since the spectral amplitude Nα,k appearing in the linear expansion of the density operatorn ˆ(r) vanishes ×δ † δ δ † δ , cˆn(r) cˆn(r) cˆm(0) cˆm(0) (37) identically in the Mott phase [26,27].

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The quantum Gutzwiller result (32) for the local density terms, and to deal with more complex forms of the initial correlation g(2)(r = 0) is shown as a solid black line in ansatz, such as the cluster Gutzwiller wave function, which Fig. 5(a). On the SF side, the antibunching g(2)(0) < 1 due include at least some quantum correlations already in the to the repulsive on-site interactions well matches the weakly ground state. An important application of our quantized model interacting Bogoliubov prediction [56] in the deep SF (green for the excitations is the investigation of finite temperature dashed-dotted line) and increases, faster than the Bogoliubov and/or time-dependent problems, including nonequilibrium predictions, when moving toward the Mott transition. On the dynamics. Going beyond the quadratic expansion around MI side, while the mean-field Gutzwiller theory (black dashed the mean-field, our method can in fact naturally incorporate (2) ∝ − = line) predicts a perfect antibunching gMF(0) D0 n0 0, additional nonlinear terms describing interactions between the quantum Gutzwiller result (32) is able to account for quasiparticles [58] to describe, e.g., their temporal decay the virtual excitation of doublon-hole pairs. This leads to into entangled pairs via multibranch Beliaev decay processes g(2) ∝ J2 at low 2dJ/U, in excellent agreement with strongly [59]. Exciting long-term perspectives will be to apply our interacting perturbative calculations (gray dotted line) [42]. theoretical framework to those driven-dissipative models that Remarkably, close to and across the critical point, the quan- can now be realized in photonic systems [16–18]. tum Gutzwiller theory is in very good agreement with low- temperature quantum Monte Carlo predictions [57] (red dots). In order to compare the results of the two different models, the ACKNOWLEDGMENTS hopping parameter for the QMC data has been rescaled by a The authors thank S. Sachdev for useful discussions. factor J /JQMC to make the position of the critical point in c c C.M., I.C., and A.R. acknowledge financial support from the two theories coincide. Note that no other semianalytical the Provincia Autonoma di Trento and from the FET-Open theory is available to describe this region close to and across Grant MIR-BOSE (No. 737017) and Quantum Flagship Grant the critical point. PhoQuS (No. 820392) of the European Union. M.C. ac- The role of quantum fluctuations and the accuracy of the knowledges financial support from MIUR PRIN 2015 (Prot. method can be further explored by looking√ at the off-site 2015C5SEJJ001) and SISSA/CNR project Superconductiv- density correlations function for |r|=1 and 2. In Fig. 5(b) ity, Ferroelectricity and Magnetism in bad metals (Prot. we report√ the quantum Gutzwiller predictions for g(2)(1) 232/2015). and g(2)( 2) along the nˆ=1 filling line across the tip of the Mott lobe. These curves are successfully compared to available quantum Monte Carlo data (see Ref. [42] and ˆ references therein) and to strong-coupling perturbation theory, APPENDIX A: THE NORMALIZATION OPERATOR A(r) which shows that our theory is accurate across the whole The normalization operator phase transition and correctly interpolates between a strongly interacting Mott insulator phase and the weakly interacting 1/2 Bose gas. ˆ = − δ † δ A(r) 1 cˆn(r) cˆn(r) (A1) n IV. CONCLUSIONS entering the expansion of the Gutzwiller coordinatesc ˆ (r) = In this paper, we have introduced a simple and power- n Aˆ c0 + δcˆ (r) is introduced in order to satisfy automatically ful semianalytical tool to study the many-body physics of r n n the local constraint, quantum interacting particles on a lattice based on a canonical quantization of the fluctuations around the Gutzwiller † = 1ˆ , ground state. The power of the method has been validated cˆn(r) cˆn(r) (A2) on the archetypal case of the Bose-Hubbard model. In spite n of the locality of the initial mean-field Gutzwiller ansatz, the quantization procedure proposed in our work is able to which restricts the action of the Gutzwiller operators to the accurately capture very nonlocal physical features such as physical subspace and derives directly from the normaliza- the superfluid stiffness of the superfluid phase, the different tion condition for the original complex-valued parameters behaviours of the correlation functions at the different critical cn(r). Equation (A2) holds under the additional condition δ † 0 = points, and the spatial structure of the virtual particle-hole n cˆn(r) cn 0, namely that the ground-state eigenvector is pair excitations on top of a Mott insulator. In particular, these orthogonal the fluctuation field. This condition is assumed at last predictions are in quantitative agreement with quantum the beginning of the derivation of our theory and guaranteed a Monte Carlo results available in the literature. In addition to posteriori by the spectral properties of the pseudo-Hermitian its quantitative accuracy and to its computational simplicity, matrix Lˆk, as discussed in Appendix B. the quantum Gutzwiller method has the crucial advantage over The physical role of Aˆ(r) consists in taking into account other approaches of providing a deep physical insight on the the feedback of quantum fluctuations onto the Gutzwiller 0 equilibrium state and on the quantum dynamics of the system ground state cn in a self-consistent manner. The importance under investigation. of such renormalization can be immediately observed in the Due to its flexibility and numerical accessibility, our for- calculation of local observables as the average square density malism can be straightforwardly extended to treat inhomo- Dˆ (0)=nˆ2(0) contributing to the expression of the on-site geneous configurations, more exotic hopping and interaction pair correlation function g2(0) presented in Sec. III C.The

033276-8 QUANTUM FLUCTUATIONS BEYOND THE GUTZWILLER … PHYSICAL REVIEW RESEARCH 2, 033276 (2020) explicit calculation of Dˆ (0) reads 0.10 2 † ˆ Dˆ (0)= n cˆ (0) cˆ (0) δ2n /n0 n n 0 05 n . ˆ δ2ψ/ψ0 ∗ = n2 Aˆ c0 + δcˆ†(0) Aˆ c0 + δcˆ (0) 0 n n 0 n n 0.00 n = 2 ˆ2 02 +δ † δ n A0 cn cˆn(0) cˆn(0) −0.05 n MI SF = + 2 − δ † δ D0 (n D0 ) cˆn(0) cˆn(0) (A3) −0.10 − − n 10 2 10 1 100 101 102 which is the expected result of Eq. (35). The sum in Eq. (A3) 2 dJ/U is composed of two terms, one given by quantum fluctuations only and the other proportional to the mean-field quantity D , FIG. 6. Quantum corrections to the local density (solid line) and 0 = deriving exclusively from the normalization operator via the order parameter (dotted line) at fixed mean-field density n0 1. expectation value Aˆ2(0). This second contribution is crucial Orange (white) background indicates the MI (SF) phase. in the successful predictions for density correlations presented in Fig. 5. related to the physical invariance of the Gutzwiller ansatz (2) iϕ(r) under local phase transformations cn(r) → cn(r) e , which reflects into the presence of such spurious eigenvector of Lˆ APPENDIX B: PROPERTIES OF Lˆ k k both in the superfluid and in the insulating phases. Fixing the maximal local occupation number at nmax for computational purposes, the pseudo-Hermitian matrix Lˆk is a APPENDIX C: COMMUTATION RELATIONS 2 nmax × 2 nmax-dimensional object of the form The commutation relations for the Gutzwiller fluctuation L = Hk Kk . δ k − − (B1) operators cˆn(r) can be identified a posteriori once the right Kk Hk , Lˆ eigenvectors (uα,k vα,k )of k and the exact form of the The nmax × nmax blocks Hk and Kk are identical to the ones Bogoliubov rotation (11) are determined. Exploiting the fact ˆ ˆ† controlling the Gutzwiller dynamical equations at the linear that the excitation operators bα,k, bα,k satisfy Bose statistics, response level [26] it follows that √ √ nm =− ψ δ + δ [δcˆ (r),δcˆ† (s)] Hk J(0) 0( m n+1,m n n,m+1) n m U 1 i k·(r−s) + − − μ − ω δ = e (uα,k,n uα,k,m − vα,k,n vα,k,m ), (C1) n(n 1) n h¯ 0 n,m N 2 α,k √ √ − + + 0 0 J(k) n 1 m 1 cn+1 cm+1 where all the eigenvector components are assumed to be real. √ √ A well-known property of pseudo-Hermitian matrices as Lˆ + n mc0 c0 , (B2) k n−1 m−1 is the sum rule [19] √ √ nm =− + 0 0 Kk J(k) n 1 mcn+1 cm−1 − = δ − 0 0 √ √ (B3) (uα,k,n uα,k,m vα,k,n vα,k,m ) n,m cn cm (C2) + + 0 0 , α n m 1 cn−1 cm+1 where J(k) = 2 dJ− ε(k) where that follows from the fact that formally the ground-state 0, 0 ∗ L eigenvector (c (c ) ) is a zero-energy eigenvector of k d k and can be projected out of the spectral decomposition of ε(k) = 4 J sin2 i (B4) 2 the quadratic form (10) when considering only states with i=1 finite positive energy in the spectral decomposition (see is the energy of a free particle on a d-dimensional lattice. The Appendix B). Inserting expression (C2) into Eq. (C1), we ground-state energy, obtain the expected quasibosonic commutation relations (15). 2 U 02 h¯ω =−4 dJψ + n(n − 1) − μ n c , (B5) APPENDIX D: QUANTUM CORRECTIONS 0 0 2 n n In this Appendix we report explicitly how the order param- 0 is set by the classical evolution of the cn’s at the mean-field eter ψ and the the density n are affected by the quantum fluc- level and, shifting the diagonal elements of Lˆk, assures a tuations. Both quantities are modified due to the contribution gapless spectrum in the superfluid phase. of the normalization operator Aˆ: Among the relevant spectral properties of the pseudo- Nˆ (r)=n0 +δ2nˆ Hermitian matrix Lˆk of the quadratic form in Eq. (10), it ∗ is worth mentioning that (c0, (c0 ) ) is a right eigenvector = + − δ † δ , n0 (n n0 ) cˆn(r) cˆn(r) with zero energy for all momenta k. This fact is intimately n

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ψˆ (r)=ψ0 +δ2ψˆ noninteracting regime. The order parameter correction is also √ very small, becoming relevant only extremely close to the = ψ + ( n δcˆ† (r) δcˆ (r) 0 n−1 n transition to the MI phase, where ψ0 → 0 (notice that on the n scale of the figure the maximum correction is still only 10%). − ψ δ † δ . Notice also the change in sign of the correction. 0 cˆn(r) cˆn(r) ) Corrections to the mean-field Gutzwiller phase diagram shown in Fig. 1(a) can be obtained by means of a self- The corrections at fixed mean-field density n0 = 1 are re- consistent calculation including the back-reaction of the quan- ported in Fig. 6. The correction for the density is always very tum fluctuations onto the initial classical Gutzwiller wave small, being zero in the MI phase as well as approaching the function. This will be the subject of future work.

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