Comparing numerical and analytical approaches to strongly interacting two-component mixtures in one dimensional traps
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Comparing numerical and analytical approaches to stronglyinteracting two-component mixtures in one dimensional traps
Filipe F. Bellotti, Amin S. Dehkharghani, and Nikolaj T. Zinner
Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmarke-mail: [email protected]
Received: date / Revised version: date
Abstract.
Abstract is missing. e investigate one-dimensional harmonically trapped two-component systems for repulsive interaction strengths rang-ing from the non-interacting to the strongly interactingregime for Fermi-Fermi mixtures. A new and powerfulmapping between the interaction strength parameters froma continuous Hamiltonian and a discrete lattice Hamilto-nian is derived. As an example, we show that this mappingdoes not depend neither on the state of the system nor onthe number of particles. Energies, density profiles and cor-relation functions are obtained both numerically (DMRGand Exact diagonalization) and analytically. Since DMRGresults do not converge as the interaction strength is in-creased, analytical solutions are used as a benchmark toidentify the point where these calculations become un-stable. We use the proposed mapping to set a quantita-tive limit on the interaction parameter of a discrete latticeHamiltonian above which DMRG gives unrealistic results.
Key words. cold atoms, one-dimensional, strongly inter-acting, DMRG
One-dimensional (1D) systems are among the most widelystudied problems in physics, especially due to their invalu-able pedagogical properties and their more friendly ma-nipulation of mathematical expressions both analyticallyand numerically, which often guide us through the under-standing of interesting physical systems [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. Furthermore,1D structures such as nanotubes and nanowires, amongothers, may be highly relevant in technological applica-tions [23]. Beyond that, such low-dimensional systems canbe realized in experiments with cold atomic gases [24,25,26,27,28,29] even in the limit of few-particles [30,31,32].Combined with the tunable interaction between atoms [33, 34,35,36,37,38] they form the perfect test-bed for study-ing the quantum mechanics of few- and many-body sys-tems [39,40].A key feature of 1D systems is that analytical solu-tions of many-body physics are known, such as for in-stance those obtained via the Bethe ansatz. There are,unfortunately, not many solutions available when parti-cles are confined in external traps as is the case of recentexperiments [30]. More generally, 1D systems of many in-teracting particles have been studied numerically and apowerful method in use is the density matrix renormaliza-tion group (DMRG) [41]. DMRG codes based on matrixproduct states (MPS) (e.g. [42,43,44]) are accurate, fastand have been successfully used to study real-time evolu-tions either at zero [45,46] or finite temperature [47], toefficiently implement periodic boundary conditions [48], innumerical renormalization group applications [49], in de-velopment of infinite-system algorithms [50] and in severalother contexts.The success of DMRG [51,52,53] has led the method tobe pushed to limiting cases such as the study of continuoussystems without a lattice parameter [54]. This limit is alsoaccessed from a discrete prescription, in which case it isonly valid when the occupation level N is much less thanthe number of lattice sites L , namely N/L <<
1. Theway to respect this relation is to take the limit of large L for a fixed number of particles ( N > L → ∞ [53]. Another example isthe application of DMRG methods in the limit of stronginteractions [55].Here we are interested in how well DMRG performsin dealing with continuous systems in the strongly inter-acting limit. Interest in this setup goes beyond its exper-imental realization [27,28,29,30,33], and arises also fromthe fact that an exact solution for such systems is avail-able [56,57]. This important result serves as benchmarkfor our investigation since it provides a precise referencefor numerical calculation. We propose a practical and ef-ficient way to connect parameters from continuous and Filipe F. Bellotti et al.: Comparing numerical and analytical approaches to strongly interacting systems discrete Hamiltonians, which may be a relevant tool forstudying continuous systems through lattice Hamiltoni-ans. This new mapping allows us to set a quantitative limiton the interaction parameter of a discrete lattice Hamil-tonian above which DMRG gives unrealistic results.The structure of the paper is as follows. Sect. 2 presentour systems and the methods and in Sect. 3 we presentthe new mapping between parameters from continuousand discrete Hamiltonians. The behavior of the DMRGmethod is investigated in Sect. 4 for two-component fermionsand results for energies, density profiles and pair correla-tion functions are discussed. Concluding remarks are givenin Sect. 5.
We consider a two-component system composed of N = N a + N b particles whose dynamics is restricted to one spa-tial dimension. Each component a and b can be either afermion or a boson in a specific internal state. The N particles have mass m and components a and b are distin-guished from each other only by their internal state. Allparticles are confined to the same one-dimensional har-monic trap V ( x ) and the short-range interaction betweenpairs is taken to be only repulsive. In the following sub-sections we present three ways to describe and investigatethis system. This is the description that straightforwardly connectstheoretical and experimental results [30,58]. The one-dimensionalharmonic trap acting on each particle reads V ( x ) = mω x / δ functionas U ij ( x j − x i ) = g ij δ ( x j − x i ) with g ij ≥
0. The interac-tion strength is g ij = g if i, j are either different speciesor identical bosons allowed to interact and g ij = 0 when i and j are fermions from the same species. The continuousHamiltonian describing the system is given by H c = N X i =1 (cid:18) − ~ m ∂ ∂x i + mω x i (cid:19) + N X j
0, the eigenstate wave function of a two-componentsystem composed of N = N a + N b particles is written as ψ ( x , ..., x N ) = a Ψ A for x < ... < x N a < x N a +1 < ... < x N ( a Ψ A for x < ... < x N a +1 < x N a < ... < x N (... ... a M Ψ A for x N < ... < x N a +1 < x N a < ... < x (where x n is the coordinate of the n th particle and M = N ! / ( N a ! N b !) is the number of independent distinguishablespatial configurations or in other words M is the numberof degenerated states at 1 /g →
0. The wave function Ψ A isconstructed from the antisymmetric product of the first N eigenstates of the single-particle continuous Hamiltonian(the first term in Eq. 1) and it has energy E A which is thesum of the occupied single-particle energies.Up to linear order, the energy of the system in thislimit can be written as E = E A − K/g , where K = K ( a , ..., a M ) depend on the M coefficients of the wavefunction Eq. 3 and it is independent of g , namely K ( a , ..., a M ) = M X k,p =1 ( a k − a p ) I k,p , (4)with h ψ | ψ i = 1 and coefficients I i,j given by I i,j = Z x <... N/L ≪ 1, with N beingthe number of particles and L the number of discrete lat-tice sites. The lattice discretization of Eq. 1 will hereafterbe referred as the discrete Hamiltonian, H d , and reads H d = − t P N − j =1 (cid:16) a † j a j +1 + a † j +1 a j (cid:17) − t P N − j =1 (cid:16) b † j b j +1 + b † j +1 b j (cid:17) + U ab P Nj =1 n a,j n b,j + V h P Nj =1 ( j − L/ ( n a,j + n b,j ) , (8)where a j and b j are either bosonic or fermionic field oper-ators acting on a site j , with corresponding density oper-ators n a,j = a † j a j and n b,j = b † j b j , t is the tunnel constant, U ab , U aa and U bb are the on-site interactions. The strengthof the on-site interaction is U ab = U . If particles of kind a ( b ) are allowed to interact among them U aa ( bb ) = U , oth-erwise U aa ( bb ) = 0. The strength of the harmonic potentialis V h .The discrete Hamiltonian Eq. 8 can be solved with theDMRG method [42,43,44], and the DMRG results pre-sented in this work are obtained with the open-sourcecodes from L. D. Carr and his group [42,43] and withthe open-source code from the iTensor project [44], sincewe are interested in the behavior of the DMRG methodrather than a specific code. Results from both indepen-dently developed codes are consistent, since energies anddensities agree in all cases for intermediate values of theinteraction parameter, as discussed in Sect. 4. The inter-action parameter that define the intermediate values isfound to be the same for both codes. This implies thatresults discussed in this work are inherent to the DMRGmethod, and not artifacts of specific codes. Furthermore,results from both codes show that DMRG performs betterfor ground state than for excited states, which might notbe a surprise for specialists in this technique. We will also consider the variational approach proposedin Ref. [58] which was shown to be highly accurate in esti-mating the ground state energy of two-component fermion systems up to 6 particles ( N ↑ = 1 + N ↓ = 2 , ..., | γ i which is a su-perposition of the non-interacting wave function | γ i andthe wave function in the strongly interacting limit | γ ∞ i ,namely | γ i = α | γ i + α ∞ | γ ∞ i . (9)Using the Hamiltonian in Eq. Eq. 1, the variational energyis given by E = h γ | H c | γ i / h γ | γ i . Defining ∆E = E ∞ − E , the minimization process leads the coefficients and thevariational energy to be given by α α ∞ = ∆E −h γ | U | γ i + √ ( ∆E −h γ | U | γ i ) +4 h γ | U | γ i ∆E h γ | γ ∞ i h γ | U | γ ih γ | γ ∞ i , (10) E var = E + ∆E + h γ | U | γ i + √ ( ∆E + h γ | U | γ i ) − h γ | U | γ i ∆E (1 − λ )2(1 − λ ) , (11)with U ij | γ ∞ i = 0. In order to get the right energy be-havior at 1 /g → 0, the term h γ | γ ∞ i is replaced by λ = K h γ | U | γ i / ( g∆E ) in Eq. Eq. 11 [58], with Kdefined in equation Eq. 4.The non-interacting wave function is easily calculatedonce the statistics of each particle is known. In the otherlimit, the wave function is numerically found with exactdiagonalization [59], it may be analytically obtained forthree or four particles when similar species of particlesdo not interact with each other [64,65,66,67], or in themost general case the wave function is exactly given up to30 particles in the strongly interacting limit [68]. Here weshow that this variational method works extremely welland use it to estimate energies as function of the interac-tion strength g where numerical or analytical results arenot available. Moreover, we compare the wave function Eq.9 with the one obtained using the continuum and discretedescriptions introduced above. The beauty in being able to describe the same systemfrom several perspectives is that one can benefit from thepower of each method and also avoid their setbacks. How-ever, to exploit this power, we must relate parameters andresults from the different approaches. Although there isno trivial way to make this connection, recent efforts havesuccessfully connected parameters from some specific dis-crete to continuous Hamiltonians using spin models in thestrongly interacting case [56,57]. Also a recent study hasshown how to relate Hubbard models in the continuum totight-binding lattice models within effective field theory[69].In our case, a glance at the expressions Eq. 1 andEq. 8 shows that indeed there are many different param-eters we would have to connect between the continuousand discrete Hamiltonians. Instead of relating parametersfrom the Hamiltonians beforehand, as previous works havedone, we rather use a way of connecting results from bothmethods directly. As result, energy spectra and the inter-particle interaction strengths from continuous and discretedescriptions are related through simple mathematical ex-pressions. Filipe F. Bellotti et al.: Comparing numerical and analytical approaches to strongly interacting systems First of all, the non-interacting part of the Hamiltoni-ans Eq. 1 and Eq. 8 are related. The following procedureestablishes a connection between the energy scales of bothHamiltonians, as it connects the ground state one-bodyenergy and the difference between energy levels from bothexpressions. To illustrate the procedure, let us considera two-component fermion system with N = 3 ( N ↑ = 1+ N ↓ = 2). The first step is to find energies as functionof the interaction parameter ( g for continuous and U fordiscrete system) as a solutions of the Hamiltonians Eq.1 and Eq. 8. The results are shown in Fig. 1(a). Next,the energy calculated from the discrete Hamiltonian Eq.8, labeled E d , has to be shifted by E ~ ω = E d − N E p ~ ω d + N , (12)where E p is the one particle ground state energy and ~ ω d is the difference between energy levels, both calculated inthe discrete model Eq. 8, and N is the total number ofparticles. The shifted result is shown in Fig. 1(b). Thefirst term on the right-hand-side of equation Eq. 12 comesfrom the interaction energy, where the numerator removesthe ground state energy from the discrete model and thedenominator brings the energy levels to the units of theharmonic oscillator energy from equation Eq. 1. The sec-ond term adds back the non-interacting ground state en-ergy in the same units as the first term.The last step is to relate the continuous and discreteinteraction strength g and U . The horizontal arrow in Fig.1(b) points out the explicit relation E ( − /U ) = E ( − /g ) ≈ . 2. From this we find U = 2 . g = 0 . U δ = g , with δ = 0 . δ for any value of the energy. Therefore,by shifting horizontally the discrete curve in Fig. 1(b) by δ , the results obtained from the continuous and discreteHamiltonians Eq. 1 and Eq. 8 are the same, as shownin Fig. 1(c). For Fermi-Fermi mixtures, the interactionstrengths from continuous Eq. 1 and discrete Eq. 8 de-scriptions are related by U = 0 . g . (13)Similar relation can be obtained for Bose-Fermi and Bose-Bose mixtures. The procedure is general and works alsowell for a higher number of particles even when compo-nents have the same population, as we shall see in the nextsection. Although a solution (generally obtained from nu-merical calculations) of the continuous Hamiltonian mustbe known it may not be necessary to fully solve equationEq. 1, which demands a huge effort. Instead, a variationalmethod such as the one presented in Ref. [58] could beused as a fast and accurate alternative to full numericalcalculations. The density matrix renormalization group (DMRG) methodis a very efficient tool to solve the discrete Hamiltonian -6-5-4-3-2-101234-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 E / ¯ h ω − /U − /g -5.986-5.984-5.982-5.980-5.978-5.976-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 E / ¯ h ω − /U (a) Step 1: Numerical solu-tion of the Hamiltonians givenin equationsEq. 1 and Eq. 8.Inset shows in detail the be-havior of E d . E / ¯ h ω − /g − /U (b) Step 2: Energies of thediscrete Hamiltonian Eq. 8calculated with the DMRGmethod are shifted accord-ingly to equation Eq. 12. E / ¯ h ω − /gH d H c (c) Step 3: Parameters U and g are connected through U =0 . g . Fig. 1. Graphical representation of the steps involved in thediscrete-to-continuous mapping. Example for two-componentfermion system with N = 3 ( N ↑ = 1 and N ↓ = 2). The parame-ters of H d in equation Eq. 8 are L = 128 , V h /t = 7 · − , U aa = U bb = 0 , U ab = U, t = 1. Eq. 8. This technique has been used to solve such Hamil-tonians in the low-density limit, N/L ≪ 1, and also inthe strongly interacting limit [55]. We will see that in thislimit, DMRG calculations fail in finding some observableswhen the interaction parameter U is taken to be very largein equation Eq. 8. Investigating how DMRG behave in thestrongly interacting limit, we show that it is possible toget observables correctly, given that the parameter U inequation Eq. 8 is limited to large but not excessive values.This may be benchmarked by comparing to the analyti-cal results for strongly interacting systems [56]. Further-more, the continuous-to-discrete mapping Eq. 13 gives aquantitative meaning to the above sentence “large but notexcessive values”.For the reader interested in reproducing any of theresults, here we provide some extra information on theparameters used in the simulations. We have checked thatmost of the default settings in both Carr’s group [43] andiTensor [44] codes did not need to be changed. However,results converged better within 10 − 20 sweeps and with thebond dimension allowed to grow up to 30. The discardedweight is in general < − . Specifically, the minimumcutoff after each SVD operation in the iTensor code wasset to < − . ilipe F. Bellotti et al.: Comparing numerical and analytical approaches to strongly interacting systems 5 We define impurity systems as those in which a singleparticle of one internal state interact with a number ofparticle in a different internal state. Here we consider two-component fermions with N ↑ = 1 and N ↓ = 1 , . . . , ≤ N ≤ − /g as g → ∞ . Ingeneral, deviations from the mapped DMRG energy inequation Eq. 12 to the numerical calculated energies areless than 0 . 11% for g < 10 in all cases shown in Fig. 2 andno more than 0 . 3% at the other end, i.e. where g > E / ¯ h ω − /g DMRGNumerical Variational Fig. 2. Energy as function of the inverse of the interaction pa-rameter g ( U = 0 . g ) for a system composed for N ↑ = 1and N ↓ = 1 , ... 6. “Numerical” refers to the exact diagonal-ization of equation Eq. 1, “DMRG” stands for results fromequation Eq. 8 properly shifted by the discrete-to-continuousmapping given in equation Eq. 12 and “Variational” labels re-sults from equation Eq. 11. Results from the three methodsagree with accuracy better than 0 . H d in equation Eq. 8 are V h /t = 7 · − , U aa = U bb = 0 , U ab = U, t = 1 , L = 128 for N ↓ = 1 , , L = 256 for N ↓ = 4 , , The scenario changes drastically when we look at otherobservables such as density profiles. In this case, DMRGcalculation nicely agrees with numerical results all the wayfrom the non-interacting limit until the strongly interact-ing limit is reached in numerical calculations which hap-pens around g ≈ g even more and goingtowards the exact strongly interacting limit g → ∞ , thedensity profiles obtained from DMRG deviate completelyfrom the known analytical result and have no meaning asin Fig. 3(d). The extreme case of very large g suggeststhat DMRG calculations get stuck in a particular state(not necessarily an eigenstate of the system) as shown inFig. 3(e), which represents the spatial configurations ↓↓↑ ,while the true ground state is composed for a non-trivialcombination among the three distinguishable configura-tions ↓↓↑ , ↓↑↓ and ↑↓↓ .The state where the DMRG code is stuck seems tobe random, as slightly different parameters can lead tocompletely different results for g > 100 ( U > g → ∞ . Therefore,it is very hard for DMRG methods to identify and isolatethe correct state, leading to the results shown in Fig. 3.Since DMRG is being broadly used to solve the discretemodel Eq. 8 in the investigation of continuous systems( L → ∞ ), we point out that ground state observables,in this case, are only reliable if the interaction parameterstays within U/t . 10. This conclusion does not hold forexcited states, as we show in the following. Analytical orsemi-analytical [56,58] inputs on the strongly interactinglimit are then clearly needed and they might also be usefultools to improve DMRG codes in the future. n ( x ) a H O g=0.5U=0.05 (a) g=10U=1 (b) g=100U=10 (c) n ( x ) a H O x/a HO g=200U=20 (d) x/a HO g=500U=50 (e) DMRG component- ↑ DMRG component ↓ Exact Diag. component- ↑ Exact Diag. component- ↓ Analytic component- ↑ Analytic component- ↓ Variational component- ↑ Variational component- ↓ Fig. 3. Density profiles of a two-component fermion systemwith N ↑ = 1 and N ↓ = 2 for a broad range of interactionstrengths g ( U = 0 . g ). Comparison between DMRG, ex-act diagonalization, variational calculation and analytical re-sult in the strongly interacting limit. Panel (b) shows that thestrongly interacting regime has not being numerically achievedat g = 10 ( U = 1) yet. For g > 100 ( U > 10) results from exactdiagonalization (not shown in panels (d) and (e)) and varia-tional calculation agree with the analytical expression, whileDMRG does not perform well. The parameters of H d in equa-tion Eq. 8 are V h /t = 7 · − , U aa = U bb = 0 , U ab = U, t =1 , L = 128. N ↑ = N ↓ = 2We focus now on two-component fermions with N ↑ = N ↓ = 2 [70,71,74,75] and extend the analysis also to the Filipe F. Bellotti et al.: Comparing numerical and analytical approaches to strongly interacting systems first excited state. DMRG calculation still performs wellin finding the ground state energy of the system for anyvalue of the interaction parameter g . Results for the firstexcited state agrees with numerical calculation for smalland intermediate values of g , but accuracy is lost when thestrongly interacting limit is approached as shown in theinset of Fig. 4. This figure further shows that the discrete-to-continuous mapping of equations Eq. 12 and Eq. 13does not also depend on the state of the quantum sys-tem, given another example for its efficiency and power.Finally, we see that the first state energy calculated withthe variational method is less accurate, as it would be ex-pected. E / ¯ h ω − /g DMRGNumerical Variational Fig. 4. Energy for the ground (bottom) and first excited (top)states as function of the inverse of the interaction parameter g ( U = 0 . g ) for a system composed for N ↑ = N ↓ = 2. “Nu-merical” refers to the exact diagonalization of equation Eq.1, “DMRG” stands for results from equation Eq. 8 properlyshifted by the discrete-to-continuous mapping given in equa-tion Eq. 12 and “Variational” labels results from Eq. 11. Theparameters in H d are V h /t = 7 · − , U aa = U bb = 0 , U ab = U, t = 1 , L = 128. The density profiles of each component for both states,shown in Fig. 5, are exactly the same in the strongly inter-acting limit case where the states are said to be fermion-ized [73]. DMRG calculations are able to reproduce theoverall behavior of the profiles for both states, however aglance at Fig. 5 also shows that these results have verylimited physical meaning, since parity is broken and it ispossible to identify the different components in the mix-ture. Notice that the behavior of the densities calculatedfrom DMRG does not depend on whether the energy ofthe states is well captured or not. DMRG results for theground state energy deviate 0 . 13% from numerical exactcalculation for g = 100 ( U = 10), while the deviation fromthe first excited energy at g = 50 (and U = 50) is threetimes larger (see Fig. 4). However, Fig. 5 shows that thereis basically no difference in accuracy between the profilesfor the ground and first excited states, both are inaccu-rate.The overall agreement between densities calculated an-alytically and with DMRG becomes worse when we lookat the pair correlation function Eq. 7. This tells us howthe different species are spatially organized in the trap n ( x ) a H O x/a HO DMRG ↑ DMRG ↓ Exact Diag. ↑ , ↓ Analytic ↑ , ↓ (a) Ground state for g = 100( U = 10). n ( x ) a H O x/a HO (b) First excited state for g =50 ( U = 5). Fig. 5. Density profiles for the ground (left) and first ex-cited (right) states of a two-component fermion system with N ↑ = N ↓ = 2. Densities for different states are the same in thestrongly interacting limit. This limit is numerically achieved at g ≈ 100 for the ground state and at g ≈ 50 for the first excitedstate. Note that DMRG calculation gives slightly different re-sults for each state. The parameters of H d in equation Eq. 8are V h /t = 7 · − , U aa = U bb = 0 , U ab = U, t = 1 , L = 128. and allows us to distinguish the spatial configuration ofdifferent states, which is hard to obtain from density pro-files alone. For example, looking at Fig. 5 one might arguethat DMRG results are as good for the first excited stateas they are for the ground state. We now use the paircorrelation function to show that this is not true.Analytical results for the strongly interacting limit andDMRG results at g = 100 ( U = 10) for the ground statepair correlation function are shown in Fig. 6(a). As wehave seen for the density profiles in Fig. 5, DMRG andnumerical results agree very well until the strongly inter-acting limit is reached ( g ≈ 100 and U ≈ 10) from whereincreasing g either leads to non-physical results or getsthe code stuck in a particular state as seen in the bottompanels of Fig. 6(a). Panel (4) on Fig. 6(a) corresponds tothe particles having a spatial configuration of the form ↓↑↓↑ which is certainly not the case as the ground statecontains a mix of different configurations. (a) Ground state. (b) First excited state. Fig. 6. Pair correlation functions of a two-component fermionsystem with N ↑ = N ↓ = 2. Comparison between analyticaland numerical results for ground and first excited state in thestrongly interacting limit. The parameters of H d in equationEq. 8 are V h /t = 7 · − , U aa = U bb = 0 , U ab = U, t = 1 , L =128.ilipe F. Bellotti et al.: Comparing numerical and analytical approaches to strongly interacting systems 7 For the first excited state, densities and pair correla-tion functions resembles the strongly interacting ones at g ≈ 10 ( U ≈ g = 10 ( U = 1),the agreement is not as good as for the ground state. Fur-thermore, the density profile for g = 50 ( U = 5) in Fig.5(b) seems overall similar to the analytical result in Fig.5(a), but the pair correlation function for the same inter-action strength presented in panel (3) of Fig. 6(b) is quitedifferent from the analytical result.For the first excited state, observables calculated fromthe discrete Hamiltonian Eq. 8 in the continuous ( L → ∞ )and strongly interacting limit ( U → ∞ ) are only reliableif the interaction parameter stays within U/t . 1, whichis ten times less than the value found for the ground stateobservables. Analytical results are able to reach horizonsthat are hard to be reached numerically and are an es-sential tool in the understanding of strongly interactingtrapped system in 1D. We have studied N -body system with repulsive short-range interaction in one spatial dimension using three meth-ods to describe the system and find its solutions. Thevariational method gives accurate results at low computa-tional time and cost and it can be easily implemented onceone has the knowledge of the wave function in the stronglyinteracting limit. Exact diagonalization calculations arestandard and efficient techniques employed in the studyof continuous systems, which are however limited by thenumber of particles and demands a great effort for effec-tive implementation. DMRG techniques are arguably thestate-of-art method to study discrete systems. An efficientimplementation of this advanced technique is demanding,however there are some excellent open source codes avail-able that allow an almost straightforward access.While DMRG has been successfully implemented in al-most countless studies, we show that it can be challengedby the presence of very strong interactions. When the on-site interaction parameter is pushed to arbitrary high val-ues, DMRG may give meaningless results for both energiesand densities. We have shown that if one carefully appliesDMRG then it is still possible to obtain reasonable results.Although such limitations might be known for somespecialists, we have performed a general and detailed studywhich for the first time quantify “strong interaction” inthis context. Ground state quantities are only reliable un-til U/t ≈ 10, value that is ten times less for the first ex-cited state. The lesson here is really that DMRG cannotbe considered a black-box do-all solver for systems withvery strong interactions. The large degenerate manifold ofstates that occur in the limit makes it extremely hard fora DMRG code to find the correct solutions for the groundstate or some specific excited state. The intrinsic varia-tional nature of the DMRG algorithm makes it vulnerableto large (quasi)-degenerate spaces such as is the case forvery strong interactions. This problem highlights how important analytical knowl-edge about the strongly interacting limit is. Here we haveused the method described in [56]. This involves a map-ping to a spin model with local exchange coefficients, whichare high-dimensional integrals. Fortunately, for externalharmonic confinement results up to 30 particles have beenreported [68] and this is a sufficiently large particle numberfor most cold atomic gas experiments confined down to asingle spatial dimension. Open source codes are available[76,77] from which one may obtain the exact spin modelin the case of arbitrary potentials as well. It would be veryinteresting to try to combine DMRG with these analyticalresults so as to make DMRG much more reliable also inthe case of very strong interactions.One way of approaching this is to use the spin modelsthat you get as a starting point directly in a DMRG rou-tine that solves lattice spin models. This would allow oneto address many observables and use the fact that DMRGis accurate and can be scaled to larger system sizes forspin models than typical exact diagonalization routineswhich are exponentially slow for longer spin chains. Moregenerally, one may also consider an approach where oneexpands the Hamiltonian in a basis set [78]. In an occupa-tion number basis one may then by appropriate truncationproduce lattice models that can be solved using a DMRGroutine. It would be very interesting to compare the lat-ter method to the results of the lattice spin model for verystrong interactions in order to test how well it performsas one approaches the strongly interacting regime. This work was supported by the Danish Council for Indepen-dent Research DFF Natural Sciences, the DFF Sapere Audeprogram, and the Villum Kann Rasmussen foundation. Theauthors thank M. E. S. Andersen, N. J. S. Loft, A. S. Jensen,D. V. Fedorov, M. Valiente and U. Schollw¨ock for discussions. The analytical results in the strongly interacting limithave been obtained by F. F. Bellotti and N. T. Zinnerand A. S. Dehkharghani have performed the exact diago-nalization. Variational and DMRG calculation have beenperformed by F. F. Bellotti and A. S. Dehkharghani. Allthe authors have contributed in writing and editing themanuscript. ReferencesReferences 1. Tonks L 1936 Phys. Rev. (10) 955–963 URL http://link.aps.org/doi/10.1103/PhysRev.50.955 2. Girardeau M 1960 Journal of Mathematical Physics http://scitation.aip.org/content/aip/journal/jmp/1/6/10.1063/1.1703687 3. Bloch I, Dalibard J and Zwerger W 2008 Rev. Mod. Phys. (3) 885–964 URL http://link.aps.org/doi/10.1103/RevModPhys.80.885 4. 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