To cite this version: Benjamin Reichert, Grigori Astrakharchik, Aleksandra Petković, Zoran Ristivojevic. Exact Results for the Boundary Energy of One-Dimensional Bosons. Physical Review Letters, American Physical Society, 2019, 123 (25), pp.250602. �10.1103/PhysRevLett.123.250602�. �hal-02399446�

HAL Id: hal-02399446 https://hal.archives-ouvertes.fr/hal-02399446 Submitted on 9 Dec 2019

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Exact Results for the Boundary Energy of One-Dimensional Bosons Benjamin Reichert1 , Grigori E. Astrakharchik2 , Aleksandra Petkovi´c1 , and Zoran Ristivojevic1 2

1 Laboratoire de Physique Th´eorique, Universit´e de Toulouse, CNRS, UPS, 31062 Toulouse, France and Departamento de F´ısica, Universitat Polit`eecnica de Catalunya, Campus Nord B4-B5, 08034 Barcelona, Spain (Dated: December 3, 2019)

We study bosons in a one-dimensional hard-wall box potential. In the case of contact interaction, the system is exactly solvable by the Bethe ansatz, as first shown by Gaudin in 1971. Although contained in the exact solution, the boundary energy in the thermodynamic limit for this problem is only approximately calculated by Gaudin, who found the leading order result at weak repulsion. Here we derive an exact integral equation that enables one to calculate the boundary energy in the thermodynamic limit at an arbitrary interaction. We then solve such an equation and find the asymptotic results for the boundary energy at weak and strong interactions. The analytical results obtained from the Bethe ansatz are in agreement with the ones found by other complementary methods, including quantum Monte Carlo simulations. We study the universality of the boundary energy in the regime of a small gas parameter by making a comparison with the exact solution for the hard rod gas.

Experimental realizations of cold gases very often involve an external confining potential to localize the atom motion in certain directions. The harmonic well is a common choice for the trapping potential [1]. Recently, experiments with a flat box potential have been carried out in three [2, 3], two [4], and one [5] dimension(s). The advantage of a similar shape is that it permits us to create a uniform system with hard-wall boundaries. The finite-size effects become visible, e.g., in the lowest collective excitations [3], which are starkly different from the behavior of the lowest frequency mode of a harmonically trapped gas, which is independent [6] of the interaction. Another physical realization is a Bose gas in the presence of a single pinned impurity of infinite repulsion, which in one dimension effectively generates a similar effect to that of a hard wall. Physically, such an impurity can be a pinned atom of a different species or a laser creating a hole [7] in the density. A physical system of immense theoretical and experimental interest is the one of one-dimensional bosons with contact interaction, which is known as the Lieb-Liniger model [8]. Its remarkable realizations [1, 9–11] offer a fertile ground since many theoretical results for this model can be tested and verified with unprecedented accuracy. This includes quantum dynamics [12, 13], solitons [14], the crossover from the repulsive to attractive interaction regime [15], quantum correlations [16–18], etc. On the theoretical side, the Lieb-Liniger model is exactly solvable [8, 19, 20] by the Bethe ansatz [21]. Initially, the solution was found for periodic boundary conditions [8], but later also for zero boundary conditions [22]. The latter case corresponds to bosons in an enclosed hard-wall box imposing the nullification of the wave function at the two systems’ ends. The case with zero boundary conditions shows some important qualitative differences. In particular, it is characterized by the boundary energy EB , which represents the nonextensive part of the ground-state energy E0 in the thermodynamic limit [22, 23] E0 = N 0 + EB + O(1/N ).

(1)

Here 0 is the ground-state energy per particle, while N is the total number of bosons. Note that the bulk energy 0 is

identical for the two geometries, while the boundary energy EB is a surface effect and it exists only in the case of zero boundary conditions [22, 24]. The physical origin of EB is the increase in the system energy due to the hard-wall potential, which causes the density to be nonuniform and also increases its value in the bulk region. A node in the many-body wave function at the edge leads to its nonzero gradient, increasing the kinetic energy. The typical size of the density depletion near the boundary is on the order of the healing length ξ and thus involves ξn particles, where n is the (mean) boson density. This enables us to estimate the boundary energy as EB ∼ ¯h2 n/mξ, where m denotes the mass of bosons. The Lieb-Liniger model is characterized by two types of elementary excitations [19]. In addition to the particlelike type-I branch, the model supports holelike type-II excitations. At weak interaction, they are identified with gray soliton solutions of the mean-field Gross-Pitaevskii equation [25–28]. A gray soliton corresponds to a localized perturbation in the boson density moving at a fixed velocity. In the case of a complete local suppression of the density, the soliton is called dark; it becomes static and its density profile is quite reminiscent of the one near the boundaries in the system with the hard-wall potential. In the weakly interacting regime, the two density deeps, around the center of the dark soliton and around the boundary, are described by the same GrossPitaevskii equation. Since the energy functional is local in the latter theory, the energy of the dark soliton coincides with the total boundary energy arising from the two ends, EB . This simple reasoning leads to the result EB =

8 √ γ. 3

(2)

Here γ 1 is the dimensionless interaction strength defined below, while = h ¯ 2 n2 /2m is the natural unit of energy for our system. We notice that at γ 1 the healing length is √ ξ ∼ 1/n γ and the previous estimate of EB is consistent with Eq. (2). In Lieb’s classification [19], the dark soliton corresponds to the type-II excitation with zero velocity, i.e., the (Fermi) momentum π¯hn. In the limit of strong interaction, γ → ∞, its

2 energy can be easily found by using the dual model of free fermions [29, 30]. The type-II excitation corresponds in the fermionic picture to the excitation where a fermion is promoted from the bottom to the top of the Fermi sea. Its energy is therefore identical to the Fermi energy, π 2 . On the other hand, the ground-state energy of N free fermions in a PN 2 π2 h ¯2 hard-wall box of the size L is E0 = 2mL 2 j=1 j . Using Eq. (S1) one then finds the boundary energy EB =

π2 , 2

(3)

which is twice as small as the energy of the type-II excitation. The above simple arguments show that the dark soliton (i.e, the type-II excitation of the momentum π¯ hn) and the boundary energy are different, contrary to the indication that might have appeared when studying the γ 1 case. In Ref. [22], Gaudin derived the expression for the boundary energy of the Lieb-Liniger model in terms of an integral equation (see further below) that should be presumably valid at any interaction γ. However, he only solved it at weak interaction, finding the expression (2). In this Letter, we show that Gaudin’s expression for the boundary energy actually coincides with the energy of the type-II excitation of the momentum π¯ hn at any γ. Moreover, it differs from the exact boundary energy already at the subleading order O(γ) in Eq. (2). Furthermore, at strong interaction, Gaudin’s expression overestimates the boundary energy two times. Instead, here we derive an exact expression for EB and evaluate it analytically at strong and weak interactions. In addition, we use the Monte Carlo method as an independent check of our findings. Finally, by making a comparison with the exact solution for the gas of hard rods, we demonstrate that the behavior of the boundary energy of various systems in the regime of small densities is universal in terms of the gas parameter. We consider bosons in one dimension described by the Lieb-Liniger Hamiltonian [8, 20] N 2 2 X X ∂ h ¯ − δ(xi − xj ) . (4) H= 2 +c 2m ∂x i i=1 i6=j

The local repulsion is described by the coupling constant c in Eq. (4), while the thermodynamic properties of the system are governed by the dimensionless parameter γ = c/n, where n = N/L is the linear density. Here N is the number of bosons and L is the system size. We study the cases with periodic and zero boundary conditions corresponding, respectively, to the bosons on a ring and in a box trap. The Hamiltonian (4) can be diagonalized by the Bethe ansatz. The resulting equations for the ground state of a system with periodic boundary conditions of length 2L with 2N particles have the form [8, 20] X 2N 2N + 1 − θ(ki − kj ), 2ki L = 2π i − 2 j=1

where θ(k) = 2 arctan(k/c) and i = 1, 2, . . . , 2N . The system of equations (5) has a unique solution with distinct quasimomenta ki , where one-half of them are negative (ki < 0 for 1 ≤ i ≤ N ), while the remaining ones are positive (ki > 0 for N + 1 ≤ i ≤ 2N ). Moreover, the quasimomenta are positioned symmetrically around zero, i.e., ki = −k2N +1−i . It will be convenient to shift the indices in Eq. (5): i → i−N −1 for 1 ≤ i ≤ N and i → i − N for N + 1 ≤ i ≤ 2N , so that one has the property ki = −k−i . This enables us to eventually write N 1X 1 − [θ(ki − kj ) + θ(ki + kj )] , (6) ki L = π i − 2 2 j=1 where i = 1, 2, . . . , N . The ground state of the Hamiltonian (4) is thus characterized by the set of N positive quasimomenta obtained by solving the system (6), while the negative ones are automatically obtained from them. The ground-state 2 PN energy is then given as E (P ) (2N ) = h¯m i=1 ki2 , where the superscript denotes periodic boundary conditions. As first shown by Gaudin [22], the Hamiltonian (4) can also be diagonalized for a system in a box with zero boundary conditions imposed on the wave function. The Bethe ansatz equations for the ground state in this case, for a system of length L with N particles, are given by [22] k¯i L = π +

N X j=1 j6=i

c c arctan ¯ + arctan ¯ ki − k¯j ki + k¯j

, (7)

where i = 1, 2, . . . , N . Equation (7) allows only for k¯i > 0. Using the identity arctan x + arctan(1/x) = π sgn(x)/2 one can reexpress Eq. (7) as N θ(2k¯i ) 1 X ¯ θ(ki − k¯j ) + θ(k¯i + k¯j ) + k¯i L = πi − . 2 j=1 2

(8) The ground-state energy for this setup is given by E (Z) (N ) = PN ¯2 h ¯2 i=1 ki . Here the superscript denotes zero boundary 2m conditions. The boundary energy is the difference in the ground-state energy of the system with zero and periodic boundary conditions, EB (N ) = E (Z) (N ) − E (P ) (N ).

(9)

For the latter case, one can show that, at the same density, the energy of the systems with N and 2N particles are simply related as E (P ) (N ) = E (P ) (2N )/2 + O(1/N ) [22]. In the thermodynamic limit this yields EB = limN →∞ [E (Z) (N ) − E (P ) (2N )/2], i.e., N

(5)

¯ 2 X ¯2 h (ki − ki2 ), N →∞ 2m i=1

EB = lim

(10)

3 where the corresponding quasimomenta are the solutions of Eqs. (8) and (6). For the evaluation of the boundary energy (10) we subtract Eq. (6) from Eq. (8). Since in a long system the difference k¯i − ki = ∆ki = O(1/L) is small, we obtain

RQ where we have defined σ(k) = (¯ h2 /m) −Q dk 0 k 0 G(k, k 0 ). From Eq. (16) one finds that σ(k) satisfies Z c Q dk 0 σ(k 0 ) ¯h2 = σ(k) − k. (18) π −Q c2 + (k − k 0 )2 m

N

We have therefore reformulated the problem of finding the boundary energy to be the equivalent, but more convenient, problem of solving Eq. (18) and then evaluating EB of Eq. (17). Additional analytical results can be obtained in the GrossPitaevskii and Tonks-Girardeau regimes of weak (γ 1) and strong (γ 1) interactions, respectively. In the former case, the integral equation for the density (12) is solved to first two orders in Refs. [32, 33], enabling us to express Q in terms of γ. However, for the boundary energy we have to solve Eq. (18) within the same accuracy [34]. Using Eq. (17) we then find 3√ 8 √ γ + O(γ) , (19) EB = γ 1 − 3 16

∆ki L =

π θ(2k¯i ) 1 X 0 + − [θ (ki − kj )(∆ki − ∆kj ) 2 2 2 j=1 θ0 (ki + kj )(∆ki + ∆kj )] + O(1/N ).

(11)

In a system of length 2L with periodic boundary conditions we define the density of quasimomenta as ρ(ki ) = [2L(ki+1 − ki )]−1 . In the thermodynamic limit it satisfies the Lieb integral equation [8, 20] Z dk 0 ρ(k 0 ) 1 c Q = . (12) ρ(k) − 2 0 2 π −Q c + (k − k) 2π Here the Fermi rapidity Q is fixed by the normalization RQ condition n = −Q ρ(k)dk. Using the formal expression PN ρ(k) = i=1 [δ(k − ki ) + δ(k + ki )]/2L and the property ρ(k) = ρ(−k), we then obtain N

1+

1 X 0 [θ (k − kj ) + θ0 (k + kj )] = 2πρ(k). 2L j=1

(13)

The latter equation enables us to simplify Eq. (11). Introducing an odd function g(ki ) = Lρ(ki )∆ki , we obtain that it satisfies an integral equation Z c Q dk 0 g(k 0 ) g(k) − = r(k), (14a) π −Q c2 + (k 0 − k)2 sgn(k) arctan 2k c + . 4 2π The boundary energy can then be expressed as Z h2 Q ¯ EB = kg(k)dk. m −Q r(k) =

(14b)

(15)

Equation (14) is our main results. Together with Eq. (15) they establish the exact result for the boundary energy of the LiebLiniger model at an arbitrary interaction strength c > 0. To analyze the boundary energy, let us introduce Green’s function for the Lieb integral equation as [31] Z c Q dk 00 G(k 0 , k 00 ) G(k, k 0 ) − = δ(k − k 0 ). (16) π −Q c2 + (k − k 00 )2 One can show by the method of iterations that Green’s function is symmetric, G(k, k 0 ) = G(k 0 , k). Multiplying Eq. (16) by r(k 0 ) [see Eq. (14b)] and performing the integration over k 0 , one obtains the integral equation (14a) provided g(k) = RQ dk 0 G(k, k 0 )r(k 0 ). The boundary energy (15) then ac−Q quires the form Z Q EB = dkσ(k)r(k), (17) −Q

which agrees at the leading order with the result (2). In the opposite regime of strong interaction, the integral equations (12) and (18) can be perturbatively solved by iterations to an arbitrary order in 1/γ [35]. It yields [34] π2 4 4 4(120 + 7π 2 ) −4 EB = 1− − 2+ + O γ . 2 3γ 3γ 15γ 3 (20) In Fig. 1 we show the two asymptotic expressions and the exact data obtained by numerically evaluating Eq. (15) or, equivalently, Eq. (17). In Ref. [22], Gaudin found the integral equation of the form (14a) but with a different right-hand side, which instead was given by rG (k) = sgn(k)/2. Such expression is approximately the correct right-hand side of Eq. (14a) only at c → 0, as one can see by considering Eq. (14b) in this limit. Thus, Gaudin was able only to find the leading order expression (2) for the boundary energy at weak interaction. We notice that Gaudin’s result for rG (k) leads to a significant overestimation of the boundary energy, see Fig. 1. Interestingly, using Eq. (17) Gaudin’s formula for the boundary energy becomes RQ EB,G = 0 dkσ(k). Such expression formally coincides with the energy of Lieb’s type-II excitation in the (periodic) LiebLiniger model with the momentum π¯hn [20, 36, 37]. The asymptotic form of EB,G in the two regimes is given by [34] (8√ √ γ + O(γ) , 3 hγ 1 − 0 · i 2 EB,G = π 2 1 − γ4 + γ122 + 4(πγ 3−8) + O(γ −4 ) . (21) At weak interaction, EB,G of Eq. (21) and EB of Eq. (19) differ at the subleading O(γ) order. In other words, already in the first beyond mean-field correction to the energy, there is a difference between the dark soliton and the boundary energy. At large γ, EB,G is twice EB (see Fig. 1).

4

Figure 1. The boundary energy EB in units of as a function of the interaction strength γ. The lower (black) dots represent the exact numerically obtained results, while the two asymptotic behaviors at small and large γ are given by formulas (19) and (20). The upper (brown) dots represent the result of Gaudin [22] and coincides with the energy of Lieb’s type-II excitation with zero velocity (momentum π¯ hn) in the model with periodic boundary conditions. The (green) rectangles represent the boundary energy obtained from the Monte Carlo method for N = 41 particles, which approach the exact curve with increasing N .

Additional physical insights for the boundary energy can be obtained by using more elementary approaches than the Bethe ansatz. The weakly interacting case γ 1 can be studied using the Gross-Pitaevskii equation and the quantum corrections to it. Such procedure indeed recovers the boundary energy (19) [38]. In the opposite regime of strong interaction between bosons γ 1, one can study the model (4) using the perturbation theory on the related dual Cheon-Shigehara model of fermions of the same mass m, which interact via the attractive potential VF (x) = −(2¯ h2 /mc)δ 00 (x) [30, 39–41]. In the noninteracting limit of fermions [29] in a box one obtains the boundary energy π 2 /2, while the linear correction in VF reproduces the first correction ∝ 1/γ of Eq. (20) [34]. We also calculated the boundary energy using the diffusion Monte Carlo method. In this approach one approximates the many-body by the product QN QN wave function ψ(x1 , x2 , . . . , xN ) = i=1 f1 (xi ) i C3 , where K is the Luttinger liquid parameter. The free parameter α is fixed by minimizing the variational energy, K is taken from the Bethe ansatz solution [8], while the constants C1 , C2 , and C3 are fixed by the boundary and the continuity conditions. The diffusion Monte Carlo method is used to obtain the boundary energy at several values of γ for N = 21 and N = 41 particles. Both sets of results are in agreement

with the boundary energy obtained by numerically solving the discrete Bethe ansatz equations. The boundary energy for N = 21 particles is always slightly larger than the one for N = 41, which approaches the exact value of EB in the thermodynamic limit, see Fig. 1. The results for N = 21 are not shown because they would be hardly distinguishable from the ones of N = 41 on the resolution of Fig. 1. In the limit of low density, specific details of short-range potentials become irrelevant and a single parameter, namely the s-wave scattering length a, is sufficient to represent the potential. In order to verify the universality of the boundary energy in terms of the gas parameter na, we consider a gas of hard rods with the diameter a > 0. As noted by Girardeau [29], the wave function and the energy of such gas can be obtained from the Tonks-Girardeau gas by subtracting the excluded volume as the total accessible volume of the phase space is reduced by N a in the case of periodic boundary conditions and by (N − 1)a for zero boundary conditions. The difference in the reduced space arises from the physical difference between particles on a ring (for example, a single particle interacts with its own image) and zero boundary condition. In the thermodynamic limit, we find the boundary energy of hard rods to be 14 π2 4 4 π 2 1 + 3γ −3 = 1 − − + O γ , EBHR = 3 2 2 3γ γ2 1 + γ2 (22) where γ = −2/na < 0. By comparison with Eq. (20) derived for delta-interacting gas and γ > 0, one finds that the first two terms are universal. This provides the physical interpretation of the leading terms as arising from the excludedvolume effect. The validity of the excluded-volume correction to the Lieb-Liniger gas has been verified in Ref. [48] for the ground state and in Ref. [49] for the thermal (Yang-Yang) state. Another relevant consequence is that the boundary energy expressed in terms of the gas parameter is expected to be universal in rather different physical systems, including the gases with dipolar [50–52] and Rydberg [53] interactions as well as for bosonic 4 He [54] and fermionic 3 He [55] in the regime of low densities. Alternatively, the boundary energy in an excited super Tonks-Girardeau gas [43, 56] will follow Eq. (22) at small densities. However, γ is negative in this case and thus the boundary energy will be larger in comparison to the Tonks-Girardeau limit. Let us finally notice that the boundary energy (15) is derived in the thermodynamic limit, when the system size is much larger than the healing length, L ξ. In a finite system there is an additional regime where L < ∼ ξ, which can 2 occur only at very weak interaction that satisfies γ < ∼ 1/N . We leave this problem for a future study. We also notice that Eq. (S1) has finite-size corrections [23] that vanish in the thermodynamic limit. In conclusion, we have found the exact results for the boundary energy of the experimentally relevant Lieb-Liniger model. We derived the governing integral equation that we

5 analytically solved in the regimes of weak and strong interaction, while numerically we solved it everywhere. We showed that in the initial work [22] and the book [57] of Gaudin, the boundary energy was actually coincident with the energy of the type-II excitation with the momentum π¯hn. The latter excitation, which at weak interaction becomes the dark soliton, has always a greater energy than the true boundary energy at any repulsion, see Fig. 1. Our Letter thus corrects the old misconception, making a clear distinction between the dark soliton and the boundary energy of the Lieb-Liniger model. G. E. A. acknowledges useful discussions with L. P. Pitaevskii and V. A. Yurovky. This study has been partially supported through the EUR Grant No. NanoX ANR17-EURE-0009 in the framework of the “Programme des Investissements d’Avenir.” G. E. A. acknowledges funding from the Spanish MINECO (FIS2017-84114-C2-1-P). The Barcelona Supercomputing Center (The Spanish National Supercomputing Center - Centro Nacional de Supercomputaci´on) is acknowledged for the provided computational facilities (RES-FI-2019-2-0033).

[11]

[12] [13]

[14]

[15]

[16]

[17]

[1] Toshiya Kinoshita, Trevor Wenger, and David S. Weiss, “Observation of a one-dimensional Tonks-Girardeau gas,” Science 305, 1125 (2004). [2] Alexander L. Gaunt, Tobias F. Schmidutz, Igor Gotlibovych, Robert P. Smith, and Zoran Hadzibabic, “Bose-Einstein Condensation of Atoms in a Uniform Potential,” Physical Review Letters 110, 200406 (2013). [3] Samuel J. Garratt, Christoph Eigen, Jinyi Zhang, Patrik Turz´ak, Raphael Lopes, Robert P. Smith, Zoran Hadzibabic, and Nir Navon, “From single-particle excitations to sound waves in a box-trapped atomic Bose-Einstein condensate,” Physical Review A 99, 021601 (2019). [4] Lauriane Chomaz, Laura Corman, Tom Bienaim´e, R´emi Desbuquois, Christof Weitenberg, Sylvain Nascimb`ene, J´erˆome Beugnon, and Jean Dalibard, “Emergence of coherence via transverse condensation in a uniform quasi-two-dimensional Bose gas,” Nature Communications 6, 6162 (2015). [5] Bernhard Rauer, Sebastian Erne, Thomas Schweigler, Federica Cataldini, Mohammadamin Tajik, and J¨org Schmiedmayer, “Recurrences in an isolated quantum many-body system,” Science 360, 307 (2018). [6] Walter Kohn, “Cyclotron Resonance and de Haas-van Alphen Oscillations of an Interacting Electron Gas,” Physical Review 123, 1242 (1961). [7] C. Raman, M. K¨ohl, R. Onofrio, D. S. Durfee, C. E. Kuklewicz, Z. Hadzibabic, and W. Ketterle, “Evidence for a Critical Velocity in a Bose-Einstein Condensed Gas,” Physical Review Letters 83, 2502 (1999). [8] Elliott H. Lieb and Werner Liniger, “Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State,” Physical Review 130, 1605 (1963). [9] Bel´en Paredes, Artur Widera, Valentin Murg, Olaf Mandel, Simon F¨olling, Ignacio Cirac, Gora V. Shlyapnikov, Theodor W. H¨ansch, and Immanuel Bloch, “Tonks-Girardeau gas of ultracold atoms in an optical lattice,” Nature 429, 277 (2004). [10] P. Kr¨uger, S. Hofferberth, I. E. Mazets, I. Lesanovsky, and J. Schmiedmayer, “Weakly Interacting Bose Gas in the One-

[18]

[19] [20]

[21] [22] [23]

[24]

[25] [26]

[27]

[28] [29]

[30]

Dimensional Limit,” Physical Review Letters 105, 265302 (2010). F. Meinert, M. Panfil, M. J. Mark, K. Lauber, J.-S. Caux, and H.-C. N¨agerl, “Probing the Excitations of a Lieb-Liniger Gas from Weak to Strong Coupling,” Physical Review Letters 115, 085301 (2015). Toshiya Kinoshita, Trevor Wenger, and David S. Weiss, “A quantum Newton’s cradle,” Nature 440, 900 (2006). S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature 449, 324 (2007). Christoph Becker, Simon Stellmer, Parvis Soltan-Panahi, S¨oren D¨orscher, Mathis Baumert, Eva-Maria Richter, Jochen Kronj¨ager, Kai Bongs, and Klaus Sengstock, “Oscillations and interactions of dark and dark–bright solitons in Bose–Einstein condensates,” Nature Physics 4, 496 (2008). Elmar Haller, Mattias Gustavsson, Manfred J. Mark, Johann G. Danzl, Russell Hart, Guido Pupillo, and Hanns-Christoph N¨agerl, “Realization of an Excited, Strongly Correlated Quantum Gas Phase,” Science 325, 1224 (2009). B. Laburthe Tolra, K. M. O’Hara, J. H. Huckans, W. D. Phillips, S. L. Rolston, and J. V. Porto, “Observation of Reduced ThreeBody Recombination in a Correlated 1d Degenerate Bose Gas,” Physical Review Letters 92, 190401 (2004). J. Armijo, T. Jacqmin, K. V. Kheruntsyan, and I. Bouchoule, “Probing Three-Body Correlations in a Quantum Gas Using the Measurement of the Third Moment of Density Fluctuations,” Physical Review Letters 105, 230402 (2010). N. Fabbri, M. Panfil, D. Cl´ement, L. Fallani, M. Inguscio, C. Fort, and J.-S. Caux, “Dynamical structure factor of one-dimensional Bose gases: Experimental signatures of beyond-Luttinger-liquid physics,” Physical Review A 91, 043617 (2015). Elliott H. Lieb, “Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum,” Physical Review 130, 1616 (1963). Vladimir E Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum inverse scattering method and correlation functions (Cambridge University Press, 1993). H. Bethe, “Zur Theorie der Metalle,” Zeitschrift f¨ur Physik 71, 205 (1931). M. Gaudin, “Boundary Energy of a Bose Gas in One Dimension,” Physical Review A 4, 386 (1971). H. W. J. Bl¨ote, John L. Cardy, and M. P. Nightingale, “Conformal invariance, the central charge, and universal finite-size amplitudes at criticality,” Physical Review Letters 56, 742 (1986). M. T. Batchelor, X. W. Guan, N. Oelkers, and C. Lee, “The 1d interacting Bose gas in a hard wall box,” Journal of Physics A: Mathematical and General 38, 7787 (2005). Toshio Tsuzuki, “Nonlinear waves in the Pitaevskii-Gross equation,” Journal of Low Temperature Physics 4, 441 (1971). P. P. Kulish, S. V. Manakov, and L. D. Faddeev, “Comparison of the exact quantum and quasiclassical results for a nonlinear Schr¨odinger equation,” Theoretical and Mathematical Physics 28, 615 (1976). Masakatsu Ishikawa and Hajime Takayama, “Solitons in a OneDimensional Bose System with the Repulsive Delta-Function Interaction,” Journal of the Physical Society of Japan 49, 1242 (1980). Lev Pitaevskii and Sandro Stringari, Bose-Einstein Condensation and Superfluidity (Oxford University Press, 2018). M. Girardeau, “Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension,” Journal of Mathematical Physics 1, 516 (1960). V. I. Yukalov and M. D. Girardeau, “Fermi-Bose mapping for

6

[31] [32]

[33]

[34] [35] [36]

[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[45]

one-dimensional Bose gases,” Laser Physics Letters 2, 375 (2005). M. Takahashi, Thermodynamics of One-Dimensional Solvable Models (Cambridge University Press, 1999). V. Hutson, “The circular plate condenser at small separations,” Mathematical Proceedings of the Cambridge Philosophical Society 59, 211 (1963). V. N. Popov, “Theory of one-dimensional Bose gas with point interaction,” Theoretical and Mathematical Physics 30, 222 (1977). See Supplemental Material for the details of calculation. Zoran Ristivojevic, “Excitation Spectrum of the Lieb-Liniger Model,” Physical Review Letters 113, 015301 (2014). M. Pustilnik and K. A. Matveev, “Low-energy excitations of a one-dimensional Bose gas with weak contact repulsion,” Physical Review B 89, 100504 (2014). Aleksandra Petkovi´c and Zoran Ristivojevic, “Spectrum of Elementary Excitations in Galilean-Invariant Integrable Models,” Physical Review Letters 120, 165302 (2018). Benjamin Reichert, Aleksandra Petkovi´c, and Zoran Ristivojevic, “Fluctuation-induced potential for an impurity in a semiinfinite one-dimensional Bose gas,” arXiv:1907.02169 [Physical Review B, (to be published)]. Taksu Cheon and T. Shigehara, “Fermion-Boson Duality of One-Dimensional Quantum Particles with Generalized Contact Interactions,” Physical Review Letters 82, 2536 (1999). Diptiman Sen, “The fermionic limit of the -function Bose gas: a pseudopotential approach,” Journal of Physics A: Mathematical and General 36, 7517 (2003). M. Khodas, M. Pustilnik, A. Kamenev, and L. I. Glazman, “Dynamics of Excitations in a One-Dimensional Bose Liquid,” Physical Review Letters 99, 110405 (2007). G. E. Astrakharchik and S. Giorgini, “Correlation functions and momentum distribution of one-dimensional Bose systems,” Physical Review A 68, 031602 (2003). G. E. Astrakharchik, J. Boronat, J. Casulleras, and S. Giorgini, “Beyond the Tonks-Girardeau Gas: Strongly Correlated Regime in Quasi-One-Dimensional Bose Gases,” Physical Review Letters 95, 190407 (2005). D. S. Petrov and G. E. Astrakharchik, “Ultradilute LowDimensional Liquids,” Physical Review Letters 117, 100401 (2016). L. Parisi, G. E. Astrakharchik, and S. Giorgini, “Spin Dy-

[46]

[47] [48]

[49]

[50]

[51]

[52]

[53]

[54]

[55]

[56]

[57]

namics and Andreev-Bashkin Effect in Mixtures of OneDimensional Bose Gases,” Physical Review Letters 121, 025302 (2018). L. Parisi, G. E. Astrakharchik, and S. Giorgini, “Liquid State of One-Dimensional Bose Mixtures: A Quantum Monte Carlo Study,” Physical Review Letters 122, 105302 (2019). L. Reatto and G. V. Chester, “Phonons and the Properties of a Bose System,” Physical Review 155, 88 (1967). G. E. Astrakharchik, J. Boronat, I. L. Kurbakov, Yu. E. Lozovik, and F. Mazzanti, “Low-dimensional weakly interacting Bose gases: Nonuniversal equations of state,” Physical Review A 81, 013612 (2010). Giulia De Rosi, Pietro Massignan, Maciej Lewenstein, and Grigori E. Astrakharchik, “Beyond-Luttinger-liquid thermodynamics of a one-dimensional Bose gas with repulsive contact interactions,” Physical Review Research 1, 033083 (2019). A. S. Arkhipov, G. E. Astrakharchik, A. V. Belikov, and Yu. E. Lozovik, “Ground-state properties of a one-dimensional system of dipoles,” Journal of Experimental and Theoretical Physics Letters 82, 39 (2005). R. Citro, E. Orignac, S. De Palo, and M. L. Chiofalo, “Evidence of Luttinger-liquid behavior in one-dimensional dipolar quantum gases,” Physical Review A 75, 051602 (2007). M. D. Girardeau and G. E. Astrakharchik, “Super-TonksGirardeau State in an Attractive One-Dimensional Dipolar Gas,” Physical Review Letters 109, 235305 (2012). O. N. Osychenko, G. E. Astrakharchik, Y. Lutsyshyn, Yu. E. Lozovik, and J. Boronat, “Phase diagram of Rydberg atoms with repulsive van der Waals interaction,” Physical Review A 84, 063621 (2011). G. Bertaina, M. Motta, M. Rossi, E. Vitali, and D. E. Galli, “One-Dimensional Liquid 4 He: Dynamical Properties beyond Luttinger-Liquid Theory,” Physical Review Letters 116, 135302 (2016). G. E. Astrakharchik and J. Boronat, “Luttinger-liquid behavior of one-dimensional 3 He,” Physical Review B 90, 235439 (2014). Elmar Haller, Mattias Gustavsson, Manfred J. Mark, Johann G. Danzl, Russell Hart, Guido Pupillo, and Hanns-Christoph N¨agerl, “Realization of an excited, strongly correlated quantum gas phase,” Science 325, 1224 (2009). Michel Gaudin, The Bethe Wavefunction (Cambridge University Press, 2014).

S7

Exact Results for the Boundary Energy of One-Dimensional Bosons –Supplemental material– Benjamin Reichert1 , Grigori E. Astrakharchik2 , Aleksandra Petkovi´c1 , and Zoran Ristivojevic1 1

2

Laboratoire de Physique Th´eorique, Universit´e de Toulouse, CNRS, UPS, 31062 Toulouse, France Departamento de F´ısica, Universitat Polit`eecnica de Catalunya, Campus Nord B4-B5, 08034 Barcelona, Spain

BETHE ANSATZ EQUATIONS

If we introduce the dimensionless units and rescale all momenta by Q, the set of Bethe ansatz equations for ρ and σ of the main text, respectively, become Z %(x0 ) 1 λ 1 0 dx 2 = %(x) − , (S1) π −1 λ + (x − x0 )2 2π Z ς(x0 ) λ 1 0 dx 2 = x. (S2) ς(x) − π −1 λ + (x − x0 )2 The normalization condition is then reexpressed as Z

1

γ

dx%(x) = λ.

(S3)

−1

We notice that λ = c/Q. The boundary energy EB and the energy EDS of type II excitation with the momentum π¯ hn are given by Z 1 2x γ2 1 1 , (S4) dxς(x) + arctan EB = 2 2 λ 0 2 π λ Z γ2 1 ¯h2 n2 EDS = EB,G = 2 2 . (S5) dxς(x), = λ 0 2m Weakly interacting limit

The solution of Eq. (S1) to the first two orders is the regime of weak interaction was found by Popov [33]: √ 1+x 1 + ln 16π − x ln 1−x 1 − x2 λ √ %(x) = + O(λ). + 2πλ 4π 2 1 − x2

(S6)

Equation (S6) applies for x no too close to Fermi rapidities, i.e., it is valid at 1−x2 λ. However for our purpose this limitation turns out not to be important and thus we will integrate %(x) from −1 < x < 1. This leads to 32π √ √ 1 − ln γ γ λ= −γ + O(γ 3/2 ). (S7) 2 8π Using the approach of Ref. [33], we solved Eq. (S2) within the same accuracy. We found √ 1+x x 1 + ln 16π + (1 − 2x2 ) ln 1−x x 1 − x2 λ √ ς(x) = + + O(λ). 2λ 4π 1 − x2 Notice that the same comment for the range of x as above for %(x) applies for ς(x). We eventually obtain √ 3 γ 8 √ EB = γ 1 − + O(γ) , 3 16 8 √ √ EDS = γ [1 − 0 · γ + O(γ)] . 3

(S8)

(S9) (S10)

Therefore, at weak interaction the boundary energy differs from the energy of the dark soliton at the subleading O(γ) order.

S8 Strongly interacting limit

In the strongly interacting limit the integral equations (S1)–(S2) are systematically solved in Ref. [35], yielding 4 + O(λ−5 ), ς(x) = x 1 + 3πλ3 16π γ π 4π λ = + − 2 + 3 + O(γ −4 ). π 2 3γ 3γ

(S11) (S12)

This leads to " # π2 4(120 + 7π 2 ) 40 30 + π 2 4 4 −5 EB = − + O(γ ) , 1− − 2+ 2 3γ γ 15γ 3 9γ 4 4 12 4(π 2 − 8) 40(π 2 − 2) −5 2 − + O(γ ) . EDS = π 1 − + 2 + γ γ γ3 γ4

(S13) (S14)

The leading correction to EB is in agreement with the calculation within a dual fermionic model, as we demonstrate below. PERTURBATION THEORY FOR STRONGLY INTERACTING BOSONS

We study the strongly interacting limit of the Lieb-Liniger model using the dual Cheon-Shigehara model. It is characterized by the two-particle interaction V (x) = λδ 00 (x),

λ=−

2¯ h2 . mc

(S15)

The fermions have the same mass as bosons, m. Equation (S15) shows that the strong repulsion between the bosons, c n, corresponds to the weak attraction between fermions, which is convenient as one can calculate the ground-state energy using perturbation theory. The Hamiltonian of N weekly interacting fermions in a box of size L is H = H0 + HI where Z L ¯h2 H0 = dx(∇ψ † )(∇ψ), (S16) 2m 0 Z 1 L dxdyψ † (x)ψ † (y)V (x − y)ψ(y)ψ(x). (S17) HI = 2 0 Here ψ is the single particle operator for fermions of the mass m with the standard anti-commutation relations {ψ(x), ψ † (y)} = δ(x − y) and {ψ(x), ψ(y)} = 0. In a box of size L with the hard wall boundary conditions, the single particle operators take the form r r 2X 2X † ψ(x) = sin(kx)ak , ψ (x) = sin(kx)a†k , (S18) L L k>0

k>0

where k is quantized as k = πj/L. Here j is a positive integer. The kinetic energy then becomes H0 =

X ¯h2 k 2 k>0

2m

a†k ak ,

(S19)

while the interaction is given by HI =

λ 4L

X

a†k1 a†k2 ak3 ak4 (k2 + k3 )2 (δk1 ,k2 +k3 +k4 + δk4 ,k1 +k2 +k3 − δk1 +k4 ,k2 +k3 )

k1 ,..,k4 >0

+ (k2 − k3 )2 (δk3 ,k1 +k2 +k4 − δk1 +k2 ,k3 +k4 + δk2 ,k1 +k3 +k4 − δk1 +k3 ,k2 +k4 )].

(S20)

In the framework of perturbation theory, the ground-state energy is given by E = hΩ|H0 |Ωi + hΩ|HI |Ωi + · · · ,

(S21)

S9 where the filled Fermi sea is |Ωi =

N Y

! a†πi/L

|0i.

(S22)

i=1

Here |0i denotes the vacuum. We notice the property hΩ|a†k aq |Ωi = δk,q θH (kF − k),

(S23)

where kF = πN/L and θH is the Heaviside step function. We then obtain the average kinetic energy ¯ 2 π 2 N (1 + N )(2N + 1) h 2m 6L2 2 2 π π = N + + O(N −2 ) , 3 2N

hΩ|H0 |Ωi =

=

¯h2 n2 . 2m

(S24)

The leading interaction correction to it is π2 λn(N 2 − 1)(2N + 1) 6L2 2 4π 2π 2 1 −2 + + O(N ) . = − N γ 3 3N

hΩ|HI |Ωi =

If we express the ground-state energy as E = N 0 + EB + O(1/N ), we find π2 4 −2 0 = 1 − + O(γ ) , 3 γ 2 π 4 −2 EB = 1− + O(γ ) , 2 3γ which is in agreement with the Bethe ansatz calculation.

(S25)

(S26) (S27)