Confinement and Mott transitions of dynamical charges in 1D lattice gauge theories
Matjaž Kebri?, Luca Barbiero, Christian Reinmoser, Ulrich Schollwöck, Fabian Grusdt
CConfinement and Mott transitions of dynamical charges in 1D lattice gauge theories
Matjaˇz Kebriˇc,
1, 2
Luca Barbiero,
3, 4
Christian Reinmoser,
1, 2
Ulrich Schollw¨ock,
1, 2 and Fabian Grusdt
1, 2, ∗ Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC),Ludwig-Maximilians-Universit¨at M¨unchen, Theresienstr. 37, M¨unchen D-80333, Germany Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M¨unchen, Germany ICFO - Institut de Ci`encies Fot`oniques, The Barcelona Instituteof Science and Technology, 08860 Castelldefels (Barcelona), Spain. Center for Nonlinear Phenomena and Complex Systems,Universit´e Libre de Bruxelles, CP 231, Campus Plaine, B-1050 Brussels, Belgium. (Dated: February 17, 2021)Confinement is an ubiquitous phenomenon when matter couples to gauge fields, which manifestsitself in a linear string potential between two static charges. Although gauge fields can be integratedout in one dimension, they can mediate non-local interactions which in turn influence the paradig-matic Luttinger liquid properties. However, when the charges become dynamical and their densitiesfinite, understanding confinement becomes challenging. Here we show that confinement in 1D Z lattice gauge theories, with dynamical matter fields and arbitrary densities, is related to transla-tional symmetry breaking in a non-local basis. The exact transformation to this string-length basisleads us to an exact mapping of Luttinger parameters reminiscent of a Luther-Emery re-scaling.We include the effects of local, but beyond contact, interactions between the matter particles, andshow that confined mesons can form a Mott-insulating state when the deconfined charges cannot.While the transition to the Mott state cannot be detected in the Green’s function of the charges,we show that the metallic state is characterized by hidden off-diagonal quasi-long range order. Ourpredictions provide new insights to the physics of confinement of dynamical charges, and can beexperimentally addressed in Rydberg-dressed quantum gases in optical lattices. Introduction.–
Lattice gauge theories (LGTs), origi-nally introduced to get insights about non-perturbativeregimes in particle physics [1, 2], have become a powerfultool to tackle many-body problems in condensed mattersystems [3–5]. These theories turn out to be particu-larly rich and interesting when the matter is coupled todynamical gauge fields: For example, in some cases theconfinement-deconfinement transition [1] can be associ-ated with the appearance of topological phases with non-Abelian anyons and charge fractionalization [6]. On theother hand, when the matter acquires its own quantumdynamics the confinement problem is poorly understoodand, in this regime, a general physical description of thephenomenon is still lacking. Furthermore, the high levelof complexity of LGTs makes theoretical studies basedon standard numerical methods [7, 8] very challenging.At the same time, due to their impressive level ofcontrol and accuracy, ultracold atomic systems are es-tablishing themselves as a fundamental platform whereLGT models can be systematically studied [9–18]. In thiscontext LGTs with an Ising gauge group, i.e. Z LGTs[19–21], are particularly meaningful to explore, allowingfor instance to study their connections to strongly corre-lated electronic systems [22–25] including high- T c super-conductivity [26, 27]. Recent theoretical studies of twodimensional Z LGTs with matter-gauge coupling haverevealed a wealth of intriguing properties [28–30]. Exper-imentally, a first instance of a Z LGT with dynamical ∗ Corresponding author email: [email protected] matter has recently been realized in a mixture of ultra-cold bosons in a double well potential [15] by means ofa Floquet scheme [20]. Using an extension of this Flo-quet scheme [16, 20], or coupling superconducting qubits[19, 21], allows to study Z LGTs with dynamical mat-ter in extended geometries and higher dimensions, thuspaving the way towards a deeper understanding of suchmodels. Moreover, as it will be discussed below, in onedimension (1D) the direct implementation of Hamiltoni-ans with encoded gauge degrees of freedom [31] can alsobe employed to explore Z LGTs, see Fig. 1 (a).In this letter, we solve the confinement problem in aclass of 1D Z LGTs with dynamical charges [32] at ar-bitrary densities. This is achieved by representing the Z LGT model in the non-local basis of string lengths,where we prove that confinement is equivalent to a bro-ken translational symmetry. Our argument applies for alarger class of 1D LGTs. We also study the Mott transi-tion of Z charges, which defies conventional wisdom forat least two reasons.First we show that an exponentially decaying Z in-variant Green’s function no longer provides a unique sig-nature of the Mott state. Instead, we show that theconfined Luttinger liquid [32] is characterized by hiddenoff-diagonal quasi-long range order (HODQLRO) in thestring-length basis. This quasi-condensate of string exci-tations is destroyed at the Mott transition, see Fig. 1 (b).More formally, we derive a Luther-Emery like relation be-tween the Luttinger parameters in the original Z LGTand the effective model in the string-length basis.Second we show that the Mott insulator occurring atthe specific filling n a = 2 / a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b t t t-Jz modelGauss law string anti-string (a) string-length repr.deconf.conf.Mott (b) FIG. 1. (a) The 1D t − J z model (top) maps exactly to 1D Z LGTs (middle); blue full spheres correspond to hard-corebosons or fermions and the empty blue circles denote holes.Pairs of particles are connected with green lines, which corre-spond to Z electric strings, according to the configurationsallowed by the Gauss law (bottom). (b) Comparison betweenthe deconfined, confined and Mott states, both in the origi-nal and in the corresponding string-length representation. Asindicated, the confining phases are characterized by a brokentranslational symmetry in the string-length basis. tion of the attractive confining potential and a nearest-neighbor (NN) repulsion. If either of those terms isabsent, a gapless liquid is obtained; on the other handwhen both are sizable our numerical simulations, basedon density-matrix-renormalization-group (DMRG) algo-rithm [33–36], yield a significant charge gap. Our pre-dictions can be tested in Rydberg-dressed atomic gasesin optical lattices, where site-resolved quantum projec-tive measurements provide direct access to the non-localstring-length basis. Model.–
We consider a 1D Z LGT Hamiltonian where N hard-core bosons in a lattice with L sites are coupledto Z gauge fields,ˆ H = − t (cid:88) (cid:104) i,j (cid:105) (cid:16) ˆ a † i ˆ τ z (cid:104) i,j (cid:105) ˆ a j + h.c. (cid:17) − h (cid:88) (cid:104) i,j (cid:105) ˆ τ x (cid:104) i,j (cid:105) + V (cid:88) (cid:104) i,j (cid:105) ˆ n i ˆ n j . (1)Here ˆ a † i denotes the hard-core bosonic creation opera-tor, t describes NN tunneling processes mediated by the Z gauge field ˆ τ z (cid:104) i,j (cid:105) defined on the links (cid:104) i, j (cid:105) betweenNN sites, and V > Z electric-field termˆ τ x (cid:104) i,j (cid:105) with strength h introduces quantum fluctuations ofˆ τ z (cid:104) i,j (cid:105) . The physics remains unchanged if ˆ a ’s are replacedby fermionic operators ˆ c , as can be shown by a Jordan-Wigner transformation.The Z electric-field is subject to a Gauss law which en-sures that the former changes sign across a particle [32];i.e. pairs of particles are connected by Z electric-fields ofthe same sign, which we denote as Z electric strings andanti-strings, see Fig. 1 (a). For concreteness, we assumeopen boundary conditions with τ x (cid:104) , (cid:105) = 1 (no Z elec-tric string entering from the left). The corresponding Z gauge group is defined by the operatorˆ G i = ˆ τ x (cid:104) i − ,i (cid:105) ˆ τ x (cid:104) i,i +1 (cid:105) ( − ˆ n i , (2)which commutes with the Hamiltonian [ ˆ H , ˆ G i ] = 0 anditself [ ˆ G i , ˆ G j ] = 0. As a consequence, the effective Hilbertspace of eq. (1) is split into different sectors ˆ G i = ± G i = 1 , ∀ i , see Fig. 1. Implementation.–
In order to implement the LGTHamiltonian Eq. (1), we propose a Rydberg dressingscheme in a spin-dependent super-lattice potential withperiod 2 a , a being the lattice spacing. We require thefollowing potential, V σ,j = ( − σ ω − ( − σ δ (cid:16) − ( − j (cid:17) , (3)where the first term describes the splitting ω betweenthe two spin states. The second term realizes a staggeredmagnetic Zeeman field and can be realized by an anti-magic superlattice, e.g. using Ytterbium atoms [38–40].We propose to realize the required NN Ising interactionsby dressing the spin states independently by two Rydbergdressing lasers Ω ↑ and Ω ↓ – see [41] for details.This scheme gives an effective t − J z Hamiltonian [42,43] which is best written in a rotating frame [41],ˆ H t − J z = ˆ H t + (cid:88) (cid:104) i,j (cid:105) (cid:18) J ↑↑ ˆ n ↑ i ˆ n ↑ j + J ↑↓ (cid:0) ˆ n ↑ i ˆ n ↓ j + ˆ n ↓ i ˆ n ↑ j (cid:1) + J ↓↓ ˆ n ↓ i ˆ n ↓ j (cid:19) + δ (cid:88) j ( − j ˆ S zj , (4)where ˆ H t = − t (cid:80) (cid:104) i,j (cid:105) (cid:80) σ ˆ P (ˆ a † i,σ ˆ a j,σ + h.c.) ˆ P is the hop-ping term, projected into the subspace without doubleoccupancies. This model maps to the LGT model Eq. (1)by introducing a constraint on our Hilbert space whereopposite spins appear in alternating fashion, leading tothe Z Gauss law ˆ G i = +1 (for details see [41]). Confinement in Z LGTs.–
In order to observe theconfined and deconfined phases the Z gauge-invariantequal-time Green’s function is considered g (1) ( i − j ) = (cid:104) ˆ a † i (cid:89) i ≤ (cid:96) ≤ j ˆ τ z(cid:96) ˆ a j (cid:105) . (5)An algebraic decay of the correlator Eq. (5) signals a de-confined phase where the charges can move around freely.An exponential decay, on the other hand, signals a con-fined phase where the particles are bound in pairs [32].The Gauss law, ˆ τ x (cid:104) i − ,i (cid:105) = ˆ τ x (cid:104) i,i +1 (cid:105) ( − ˆ n i , can be suc-cessively applied to express the Z electric field as [32]ˆ τ x (cid:104) i,i +1 (cid:105) = cos (cid:0) π (cid:88) j
So far we representedbasis states in our model by hard-core boson occupationnumbers n j = 0 , Z electric strings τ x (cid:104) i,j (cid:105) = ± (cid:96) , ..., (cid:96) N +1 ≥ N = (cid:80) j n j is the total conserved boson number.If x , ..., x N denote the positions ( x j = 1 , ..., L ) of hard-core bosons, we define (cid:96) = x − , (cid:96) n = x n − x n − − , (cid:96) N +1 = L − x N . (8)This allows us to identify the corresponding Fock con-figuration | n , ..., n L (cid:105) with a bosonic Fock configuration: | n , ..., n L (cid:105) = | (cid:96) , ..., (cid:96) N +1 (cid:105) ≡ N +1 (cid:89) n =1 ( ˆΨ † n ) (cid:96) n √ (cid:96) n ! | (cid:105) . (9)In the last step we introduced bosonic operators ˆΨ † n act-ing on the string-length vacuum | (cid:105) .Physically, the integers (cid:96) n ∈ Z ≥ describe the lengthof the Z (anti-) strings connecting pairs of consecutive Z charges, up to a shift of one: the shortest possiblestring connecting charges on NN sites is counted as hav-ing no excitation, (cid:96) = 0. The total number of stringexcitations, ˜ N ≡ (cid:80) N +1 (cid:96) =1 (cid:96) n = L − N , is conserved.In the new string-length basis, we can express the Z LGT Hamiltonian asˆ H = − t (cid:88) (cid:104) m,n (cid:105) (cid:0) ˆ ρ − / m ˆΨ † m ˆΨ n ˆ ρ − / n + h.c. (cid:1) − h (cid:88) n ( − n ˆ ρ n + V (cid:88) n δ ˆ ρ n , . (10)Here δ a,b denotes the Kronecker delta and ˆ ρ n = ˆΨ † n ˆΨ n is the string-length density operator. The transformedHamiltonian (10) is purely local. It is defined on a latticeof size ˜ L = N + 1 with ˜ N excitations; i.e. the averageboson density in this model is given by ρ Ψ = ˜ N ˜ L = L − NN + 1 = 1 n a − O ( / N ) . (11) It is worth to underline that the amplitude of the hop-ping in this new basis does not carry the usual Bose-enhancement factors, however, it requires extra factorsˆ ρ − / n in the Hamiltonian. Since the latter only show upin combination with ˆΨ n , the expression vanishes and re-mains well-defined when the bosonic occupation numbersbecome zero. Field theory analysis.–
Now we analyze the model (10)from a field-theoretic perspective. By construction thesemodels are connected by a unitary transformation (thenon-local basis change), ensuring their spectra to coin-cide. At long wavelengths, distances are related as fol-lows: x in the Z LGT corresponds to a ’distance’ (par-ticle number) in the string-length basis ˜ x = n a x . As aresult we can directly relate coarse-grained densities inthe two models.This allows us to directly relate their Luttinger param-eters ˜ K and K , which can be defined via the compress-ibility [44]. An explicit calculation [41] yields: K = ( n a ) ˜ K, (12)reminiscent of the Luther-Emery re-scaling solution [44,45], except for a factor of two.Alternatively, we can relate density-density correla-tions at long-distances in the two models: we start from (cid:104) δ ˆ n ( x ) δ ˆ n (0) (cid:105) , where δ ˆ n ( x ) = ˆ n ( x ) − n a denotes localdensity-fluctuations. At long wavelengths, the densityof hard-core bosons is ˆ n ( x ) ≈ ∆ ˆ N a / ∆ x , when ∆ ˆ N a par-ticles are found per coarse-grained distance ∆ x . In thestring-length basis, ˆ ρ (˜ x ) describes the distance ∆ˆ x be-tween two hard-core bosons, minus one unit per particle(Eq. (8)), per coarse-grained number of particles ∆˜ x ; i.e.ˆ ρ (˜ x ) ≈ (∆ˆ x − ∆˜ x ) / ∆˜ x . This leads toˆ n ( x ) ≈ (cid:2) ρ (˜ x ( x )) (cid:3) − , (13)which allows us to calculate density fluctuations at longdistances, δ ˆ n ( x ) = − ( n a ) δ ˆ ρ (˜ x ) + O ( δ ˆ ρ ). Hence bothmodels share the same long-wavelength correlations: (cid:104) δ ˆ n ( x ) δ ˆ n (0) (cid:105) (cid:39) ( n a ) (cid:104) δ ˆ ρ ( n a x ) δ ˆ ρ (0) (cid:105) . (14)For the local Hamiltonian (10) we can safely applyLuttinger-liquid theory, which yields [44] (cid:104) δ ˆ ρ (˜ x ) δ ˆ ρ (0) (cid:105) (cid:39) ˜ K π x + ( ρ Ψ ) (cid:18) ˜ α ˜ x (cid:19) K cos(2 πρ Ψ ˜ x ) + ... (15)where ˜ α is a non-universal short-distance cut-off. FromEq. (14) we thus predict in the original model: (cid:104) δ ˆ n ( x ) δ ˆ n (0) (cid:105) (cid:39) ˜ K ( n a ) π x + ... == K π x + ( n a ) (cid:16) αx (cid:17) K cos(2 πn a x ) + ... (16)This result confirms the relation between Luttinger pa-rameters stated earlier, see Eq. (12).Note however that the relation (14) does not correctlypredict the power-law of the oscillatory part in the cor-relations, which involves large wavevectors 2˜ k F = 2 πρ Ψ .We believe this is directly related to the failure of naivebosonization arguments in predicting the correct long-wavelength behavior of the Green’s function. Sincecos(2 πρ Ψ ˜ x ) = cos(2 π ( x − n a x )) ≡ cos(2 πn a x ), the pe-riod of the oscillations is correctly captured however. Asshown in [41] our field-theoretic arguments are confirmedby the behavior of the density-density correlations which,for h = V = 0, we calculate by Monte-Carlo sampling ofthe resulting free fermion theory [37] in the string-lengthrepresentation and by DMRG calculations for finite h and V . The resulting fits confirm the universal Luttingerliquid behaviors (15), (16) and the predicted relation be-tween the Luttinger parameters. Confinement as translational symmetry breaking.–
Inthe string-length basis, the gauge invariant Green’s func-tion g (1) ( x ) translates to a highly non-local operator. Itsmost important effect is to shift string-length labels (cid:96) m → (cid:96) m +1 for particle numbers m between ˜ x < m < ˜ x ,where ˜ x − ˜ x = ˜ x = n a x , i.e.: g (1) ( x ) (cid:39) (cid:104) ˆ T (0 , ˜ x ) (cid:105) , (17)where we define the partial translation operator:ˆ T (˜ x , ˜ x ) | ...(cid:96) ˜ x − (cid:96) ˜ x ... (cid:96) ˜ x − (cid:96) ˜ x (cid:96) ˜ x +1 ... (cid:105) == | ...(cid:96) ˜ x − (cid:96) ˜ x (cid:96) ˜ x ... (cid:96) ˜ x − (cid:96) ˜ x +1 ... (cid:105) (18)which cyclically shifts all string-occupations by one unitbetween ’sites’ ˜ x and ˜ x .Aside from local terms around ˜ x and ˜ x , which canbe assumed to yield non-zero additional factors and werethus neglected in Eq. (17), the g (1) ( x ) function essen-tially probes translational invariance of the eigenstatesin the string-length basis. Whenever the lattice trans-lation symmetry ˜ x → ˜ x + 1 is broken throughout thesystem (spontaneously, or as in Eq. (10) by a non-zerofield h (cid:54) = 0), it follows that g (1) ( x ) (cid:39) (cid:104) ˆ T (0 , ˜ x ) (cid:105) (cid:39) e − ˜ κ ˜ x = e − ˜ κn a x , (19)i.e. the corresponding Z LGT is confining.Using this argument, it is now easy to see that the orig-inal model in Eq. (1) must be confining for any h (cid:54) = 0 . Random h (cid:104) i,j (cid:105) would similarly lead to confinement. Evenfor h = 0 it can become confining if translational sym-metry is spontaneously broken by additional interactions:this case corresponds to a Mott insulating phase. Mott transition and HODQLRO.–
Earlier studies ofthe model (1) have revealed no Mott insulating states inthe absence of the repulsive NN interaction, V = 0 [32].There, the model maps to free fermions for h = 0 [37] andthe field h (cid:54) = 0 inducing confinement is not sufficient toreach the insulating state. In the limit where h → ∞ and V = 0 the particles are bound in dimers and the effectivemodel maps exactly to a 1D Heisenberg antiferromagnet.Further analysis showed that at a special filling n a = 2 / V / t h / t c / t FIG. 2. Charge gap extrapolated in the thermodynamic limit,∆ c , at filling n a = 2 / h and V . As a guidefor the eye the violet (blue) bars denote ∆ c /t > .
05 (∆ c /t ≤ .
05) where we expect a Mott insulator (Luttinger liquid). the system is at the critical point with K = 1 /
2, rightat the transition from the Luttinger liquid to a Mottinsulating phase [32]. In this regime the string-lengthmodel features HODQLRO since ˜ K = 9 / > n a = 2 / ρ Ψ = 1 /
2. We consider the repulsive interaction V ≥ h → ∞ , infinites-imal V > SU (2) in-variant Heisenberg model following [32]. On the otherhand, for h = 0 even V → ∞ is insufficient to obtainthe gapped state. However, as we show by an explicitcalculation in [41], an infinitesimal h (cid:54) = 0 is sufficient toobtain a gapped phase when V → ∞ .For generic nonzero values h, V > c ( L, N ) = 12 (cid:2) ( E LN +2 − E LN ) − ( E LN − E LN − ) (cid:3) , (20)where E LN is the ground state energy of the original Z LGT model with chain length L and boson number N . We fixed the ratio of N and L at n a = NL = 2 / c in the thermodynamic limit byconsidering L → ∞ , see [41]. As can be seen in Fig. 2the Mott insulating state is reached only in the case whenboth parameters are nonzero h, V (cid:54) = 0 and large enough.For a fixed value of V we observe an exponential open-ing of the gap as a function of h . The precise value ofthe transition point h c is difficult to extract, but the ex-ponential behavior of the gap opening points to a BKTnature of the transition, see [41]. Discussion and outlook.–
We have solved the confine-ment problem of dynamical charges in a class of 1D Z LGTs by means of a non-local string-length repre-sentation, which has revealed an unexpected relation totranslational symmetry breaking. Our arguments shouldapply equally for other gauge groups in 1D. We foundthat, while the gauge symmetry keeps the Luttinger-liquid paradigm valid, the non-local interactions medi-ated by the gauge field must be treated with care. Inparticular, the confined gapless phase is characterized byan exponentially decaying Z invariant Green’s functionbut we found that it features HODQLRO in the string-length basis before a confined Mott state is realized.We have analyzed the Mott insulating state at filling n a = 2 / h, V (cid:54) = 0. An interesting future extension wouldbe to consider the filling n a = 1 /
2, where repulsive NNinteractions V (cid:54) = 0 can readily stabilize a Mott insulatorwhen h = 0. On the other hand, one still finds a gaplesssystem for h (cid:54) = 0 [32] and a large | h | (cid:29) t is expected todestabilize the Mott insulator. Other extensions of ourwork include generalization to spin-full systems, higher dimensions and more complicated gauge groups. Acknowledgements.–
We thank U. Borla, S. Moroz, R.Verresen, N. Goldman, C. Schweizer, M. Aidelsburger,L. Pollet, F. Horn, F. Palm and S. Mardazad for fruitfuldiscussions. This research was funded by the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) via Research Unit FOR 2414 under project num-ber 277974659, and by the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) under Ger-many’s Excellence Strategy – EXC-2111 – 390814868.M.K. acknowledges the Ad Futura Scholarship (244.javni razpis) from the Public Scholarship, Develop-ment, Disability and Maintenance Found of the Repub-lic of Slovenia. L. B. acknowledges support from Agen-cia Estatal de Investigaci´on (“Severo Ochoa” Center ofExcellence CEX2019-000910-S, Plan National FIDEUAPID2019-106901GB-I00/10.13039 / 501100011033, FPI),Fundaci´o Privada Cellex, Fundaci´o Mir-Puig, and fromGeneralitat de Catalunya (AGAUR Grant No. 2017 SGR1341, QuantumCAT U16-011424, CERCA program) andfrom Topocold ERC starting grant. [1] J. B. Kogut, Reviews of Modern Physics , 659 (1979).[2] K. G. Wilson, Physical Review D , 2445 (1974).[3] X.-G. Wen, Quantum field theory of many-body systems (Oxford University Press, 2004).[4] M. Levin and X.-G. Wen, Reviews of Modern Physics ,871 (2005).[5] P. A. Lee, N. Nagaosa, and X.-G. Wen, Reviews of Mod-ern Physics , 17 (2006).[6] A. Kitaev, Annals of Physics , 2 (2003).[7] M. Troyer and U.-J. Wiese, Physical Review Letters ,170201 (2005).[8] M. G. Alford, A. Schmitt, K. Rajagopal, and T. Sch¨afer,Reviews of Modern Physics , 1455 (2008).[9] U.-J. Wiese, Annalen der Physik , 777 (2013).[10] E. Zohar, J. I. Cirac, and B. Reznik, Physical Review A , 023617 (2013).[11] E. Zohar, J. I. Cirac, and B. Reznik, Reports on Progressin Physics , 014401 (2015).[12] J. Bender, E. Zohar, A. Farace, and J. I. Cirac, NewJournal of Physics , 093001 (2018).[13] M. Dalmonte and S. Montangero, Contemporary Physics , 388 (2016).[14] E. A. Martinez, C. A. Muschik, P. Schindler, D. Nigg,A. Erhard, M. Heyl, P. Hauke, M. Dalmonte, T. Monz,P. Zoller, and R. Blatt, Nature , 516 (2016).[15] C. Schweizer, F. Grusdt, M. Berngruber, L. Barbiero,E. Demler, N. Goldman, I. Bloch, and M. Aidelsburger,Nature Physics , 1168 (2019).[16] F. G¨org, K. Sandholzer, J. Minguzzi, R. Desbuquois,M. Messer, and T. Esslinger, Nature Physics , 1161(2019).[17] A. Mil, T. V. Zache, A. Hegde, A. Xia, R. P. Bhatt, M. K.Oberthaler, P. Hauke, J. Berges, and F. Jendrzejewski,Science , 1128 (2020).[18] B. Yang, H. Sun, R. Ott, H.-Y. Wang, T. V. Zache, J. C.Halimeh, Z.-S. Yuan, P. Hauke, and J.-W. Pan, Nature , 392 (2020). [19] E. Zohar, A. Farace, B. Reznik, and J. I. Cirac, PhysicalReview Letters , 070501 (2017).[20] L. Barbiero, C. Schweizer, M. Aidelsburger, E. Demler,N. Goldman, and F. Grusdt, Science Advances (2019),10.1126/sciadv.aav7444.[21] L. Homeier, C. Schweizer, M. Aidelsburger, A. Fedorov,and F. Grusdt, (2020), arXiv:2012.05235 [quant-ph].[22] R. Sedgewick, D. Scalapino, and R. Sugar, Physical Re-view B , 054508 (2002).[23] E. Demler, C. Nayak, H.-Y. Kee, Y. B. Kim, andT. Senthil, Physical Review B , 155103 (2002).[24] R. K. Kaul, Y. B. Kim, S. Sachdev, and T. Senthil,Nature Physics , 28 (2007).[25] S. Sachdev and D. Chowdhury, Progress of Theoreticaland Experimental Physics , 12C102 (2016).[26] T. Senthil and M. P. A. Fisher, Physical Review B ,7850 (2000).[27] P. A. Lee, Reports on Progress in Physics , 012501(2007).[28] S. Gazit, M. Randeria, and A. Vishwanath, NaturePhysics , 484 (2017).[29] U. Borla, B. Jeevanesan, F. Pollmann, and S. Moroz,(2020), arXiv:2012.08543 [cond-mat.str-el].[30] U. Borla, R. Verresen, J. Shah, and S. Moroz, (2020),arXiv:2010.00607 [cond-mat.str-el].[31] F. Grusdt and L. Pollet, Physical Review Letters ,256401 (2020).[32] U. Borla, R. Verresen, F. Grusdt, and S. Moroz, PhysicalReview Letters , 120503 (2020).[33] S. R. White, Physical Review Letters , 2863 (1992).[34] U. Schollw¨ock, Annals of Physics , 96 (2011).[35] C. Hubig, F. Lachenmaier, N.-O. Linden, T. Reinhard,L. Stenzel, A. Swoboda, M. Grundner, and S. Mardazad,“The SyTen toolkit,” .[36] C. Hubig,
Symmetry-Protected Tensor Networks , Ph.D.thesis, LMU M¨unchen (2017).[37] C. Prosko, S.-P. Lee, and J. Maciejko, Physical Review B , 205104 (2017).[38] F. Gerbier and J. Dalibard, New Journal of Physics ,033007 (2010).[39] W. Yi, A. J. Daley, G. Pupillo, and P. Zoller, New Jour-nal of Physics , 073015 (2008).[40] B. Yang, H.-N. Dai, H. Sun, A. Reingruber, Z.-S. Yuan,and J.-W. Pan, Physical Review A , 011602 (2017).[41] S. S. material, [Url will be entered by the publisher].[42] C. D. Batista and G. Ortiz, Physical Review Letters ,4755 (2000).[43] A. Montorsi, S. Fazzini, and L. Barbiero, Physical Re-view A , 043618 (2020).[44] T. Giamarchi, Quantum Physics in One Dimension (Ox-ford University Press, 2004).[45] A. Luther and V. J. Emery, Physical Review Letters ,589 (1974).[46] M. Saffman, T. G. Walker, and K. Mølmer, Reviews ofModern Physics , 2313 (2010).[47] N. Henkel, R. Nath, and T. Pohl, Physical Review Let-ters , 195302 (2010).[48] M. Ogata and H. Shiba, Physical Review B , 2326(1990).[49] T. A. Hilker, G. Salomon, F. Grusdt, A. Omran, M. Boll,E. Demler, I. Bloch, and C. Gross, Science , 484(2017).[50] A. Auerbach, Interacting electrons and quantum mag-netism (Springer-Verlag, 1994).
I. SUPPLEMENTARY MATERIALA. Realization of the Z LGT by Rydberg dressing
In the following we will show that the Hamiltonianobtained from the Rydberg dressing scheme (4) maps tothe 1D LGT model (1), provided that we restrict ourHilbert space to a specific spin configuration.Rydberg states can be used in cold atom experimentssince they provide a way to precisely tune interactionbetween particles [46]. Due to high strength of such in-teractions it is sufficient to use the dressing scheme wherethe effective Ising interactions are proportional to r + a B ,where a B is the blockade radius [47]. Due to the afore-mentioned proportionality we can consider a situationwhere the on-site interaction is sufficiently large that dou-blon formation is forbidden. This requires a b < a where a is the lattice spacing. On the other hand the interac-tion beyond NN can be neglected due to rapid decay ofthe potential, V ∝ /r .In order to implement the potential Eq. (3) two Ryd-berg dressing lasers and an anti-magic potential for theatoms is needed (see Fig. 3) as discussed in the main text.The effective t − J z type Hamiltonian obtained in theRydberg dressing scheme equals toˆ H t − J z = ˆ H t + (cid:88) (cid:104) i,j (cid:105) (cid:18) J ↑↑ | ↑ i ↑ j (cid:105)(cid:104)↑ i ↑ j | ++ J ↑↓ (cid:0) | ↑ i ↓ j (cid:105)(cid:104)↑ i ↓ j | + | ↓ i ↑ j (cid:105)(cid:104)↓ i ↑ j | (cid:1) ++ J ↓↓ | ↓ i ↓ j (cid:105)(cid:104)↓ i ↓ j | (cid:19) + δ (cid:88) j ( − j ˆ S zj . (21)We again use the projector, ˆ P which projects onto aHilbert space with zero or single occupancy on each lat-tice site and explicitly rewrite the Hamiltonianˆ H t − J z = − t (cid:88) (cid:104) i,j (cid:105) ,σ ˆ P (cid:16) ˆ c † i,σ ˆ c j,σ + h.c. (cid:17) ˆ P + (cid:88) (cid:104) i,j (cid:105) (cid:18) J ↑↑ ˆ n ↑ i ˆ n ↑ j ++ J ↑↓ (cid:0) ˆ n ↑ i ˆ n ↓ j + ˆ n ↓ i ˆ n ↑ j (cid:1) + J ↓↓ ˆ n ↓ i ˆ n ↓ j (cid:19) + δ (cid:88) j ( − j ˆ S zj , (22)where ˆ c † i,σ is the fermion creation operator on site i with spin σ and ˆ n i,σ = ˆ c † i,σ ˆ c i,σ is the fermion densityoperator with spin σ . Full density on site i is henceˆ n i = (cid:80) σ ∈{↑ , ↓} ˆ n i,σ . This Hamiltonian conserves the spinpattern since it only contains a classical Ising interaction.We now consider the so called squeezed space [48, 49]picture of our system. Squeezed space is defined as asystem from which we remove all empty sites (holes) andaccordingly relabel the lattice indices, while maintain-ing the initial order of the spin configuration. We con-strain our system in the squeezed space to N´eel states.As a result the spin configuration for zero doping is | ↑ ⟩ | ↑ ⟩ | ↓ ⟩ | ↓ ⟩ | R ↓ ⟩ | R ↓ ⟩ | R ↑ ⟩ | R ↑ ⟩ even sites odd sitesperiod a ω Δ ↑ Δ ↓ Ω ↓ Ω ↑ Ω ↓ Ω ↑ δδ Δ ↑ Δ ↓ FIG. 3. Proposed Rydberg dressing scheme. By using theanti-magic wavelength property of, e.g., Yb atoms one canachieve an effective potential Eq. (4). By using two differ-ent Rydberg dressing lasers Ω ↑ and Ω ↓ , with a large detuning∆ ↑ , ∆ ↓ (cid:29) Ω ↑ , Ω ↓ this setup ultimately maps to the NN inter-action of Eq. (1). ˆ S zj = ( − j . For non-zero doping (in real space) eachhole shifts the spins by one site, henceˆ S zj = 12 ( − j (cid:2) (cid:89) i ≤ j ( − ˆ n hi (cid:3) (1 − ˆ n hj ) . (23)This motivates us to define the non-local Z electricfield ˆ τ x (cid:104) j,j +1 (cid:105) as ˆ τ x (cid:104) j,j +1 (cid:105) = (cid:89) i ≤ j ( − ˆ n hi , (24)where the ˆ n hi = 1 − ˆ n i is the hole density operator. Bydefinition ˆ τ x (cid:104) j,j +1 (cid:105) changes sign across each hole. Hencewe obtain the Gauss law, as can be seen also from directcalculationˆ G i = ˆ τ x (cid:104) i − ,i (cid:105) ˆ τ x (cid:104) i,i +1 (cid:105) ( − ˆ n hi = ( − ˆ n hi ( − ˆ n hi = 1 (25)and the obtained gauge sectors are G j = 1 , ∀ j .Now we will use relations (23) and (24) to show thatthe Hamiltonian (22) maps to (1). We will do this termby term. The staggered term is the easiest to show aswe just have to make use of Eq. (23) and substitute theproduct (cid:81) i ≤ j ( − ˆ n hi with Eq. (24) δ (cid:88) j ( − j ˆ S zj = δ (cid:88) j ( − j
12 ( − j (cid:2) (cid:89) i ≤ j ( − ˆ n hi (cid:3) (1 − ˆ n hj )= 12 δ (cid:88) j (ˆ τ x (cid:104) j,j +1 (cid:105) − ˆ τ x (cid:104) j,j +1 (cid:105) ˆ n hj ) . (26)by rewriting the hole operator as ˆ n hi = (1 − ( − ˆ n hi ) andrewriting the Gauss law asˆ τ x (cid:104) i − ,i (cid:105) = ˆ τ x (cid:104) i,i +1 (cid:105) ( − ˆ n hi (27) we obtain the mapping δ (cid:88) j ( − j ˆ S zj → h (cid:88) (cid:104) i,j (cid:105) ˆ τ x (cid:104) i,j (cid:105) , (28)where δ = h .For the hopping term we employ the slave particle for-malism [50] where we write the annihilation operator asˆ c i,σ = ˆ h † i ˆ f i,σ , (29)with the constraint (cid:88) σ ˆ f † i,σ ˆ f i,σ + ˆ h † i ˆ h i = 1 , (30)which restricts the Hilbert space to zero or singly occu-pied lattice sites, meaning that we can drop the projec-tors ˆ P . We rewrite the hopping term in terms of slaveparticles ˆ P ˆ c † i,σ ˆ c j,σ ˆ P = ˆ h i ˆ h † j ˆ f † i,σ ˆ f j,σ = ˆ H ij ˆ Z ij,σ , (31)where we defined ˆ Z ij,σ = ˆ f † i,σ ˆ f j,σ and ˆ H ij = ˆ h i ˆ h † j . Weknow that hopping of a fermion to the right is equivalentto hopping of a hole to the left. It is therefore useful tostudy the effect of the operator (31) on the Fock stateˆ τ x (cid:104) j,j +1 (cid:105) | ψ (cid:105) = (cid:81) i ≤ j ( − ˆ n hi | ψ (cid:105) = α | ψ (cid:105) ˆ H j +1 j ˆ Z j +1 j,σ ˆ τ x (cid:104) j,j +1 (cid:105) | ψ (cid:105) = + α ˆ H j +1 j ˆ Z j +1 j,σ | ψ (cid:105) . (32)ˆ τ x (cid:104) j,j +1 (cid:105) ˆ H j +1 j ˆ Z j +1 j,σ | ψ (cid:105) = − α ˆ H j +1 j ˆ Z j +1 j,σ | ψ (cid:105) , (33)which follows from Eq. (24), meaning that a movementof charge (hole) by one site flips the gauge field betweenthe old and the new site. This leads us to the mappingˆ Z j +1 j,σ → ˆ τ z (cid:104) j +1 ,j (cid:105) , (34)and consequently − t (cid:88) (cid:104) i,j (cid:105) ,σ ˆ P (cid:16) ˆ c † i,σ ˆ c j,σ + h.c. (cid:17) ˆ P → − t (cid:88) (cid:104) i,j (cid:105) (ˆ h † i ˆ τ z (cid:104) i,j (cid:105) ˆ h j + h.c. ) . (35)Finally we need to show the mapping of the NN inter-action. We use one of the definitions of the spin operatorˆ S zi = (cid:80) σ ( − σ ˆ c † i,σ ˆ c i,σ and rewrite the density opera-tors as ˆ n i, ↑ = 12 ˆ n i + ˆ S zi , ˆ n i, ↓ = 12 ˆ n i − ˆ S zi . (36)Interaction terms in (22) can be rewritten asˆ n ↑ i ˆ n ↓ j + ˆ n ↓ i ˆ n ↑ j = 12 ˆ n i ˆ n j − S zi ˆ S zj , ˆ n ↑ i ˆ n ↑ j = 14 ˆ n i ˆ n j + 12 ˆ S zi ˆ n j + 12 ˆ n i ˆ S zj + ˆ S zi ˆ S zj , ˆ n ↓ i ˆ n ↓ j = 14 ˆ n i ˆ n j −
12 ˆ S zi ˆ n j −
12 ˆ n i ˆ S zj + ˆ S zi ˆ S zj . (37)Note that since we are considering the 1D case, we write j = i + 1. We once again use Eqs. (23) and (24) to writethe spin operator as ˆ S zj = ( − j ˆ τ x (cid:104) j,j +1 (cid:105) (1 − ˆ n hj ) and usethis to map the terms in (37)ˆ S zi ˆ S zi +1 = −
14 ˆ τ x (cid:104) i,i +1 (cid:105) ˆ τ x (cid:104) i +1 ,i +2 (cid:105) (1 − ˆ n hi − ˆ n hi +1 +ˆ n hi ˆ n hi +1 ) == −
14 (1 − ˆ n hi )(1 − ˆ n hi +1 ) , (38)ˆ S zi ˆ n i +1 + ˆ n i ˆ S zi +1 == ( − i − ˆ n hi )(1 − ˆ n hi +1 )(ˆ τ x (cid:104) i,i +1 (cid:105) − ˆ τ x (cid:104) i +1 ,i +2 (cid:105) ) == − ( − i (cid:0) τ x (cid:104) i,i +1 (cid:105) ˆ τ x (cid:104) i +1 ,i +2 (cid:105) + ˆ τ x (cid:104) i − ,i (cid:105) ˆ τ x (cid:104) i,i +1 (cid:105) ++ ˆ τ x (cid:104) i − ,i (cid:105) ˆ τ x (cid:104) i +1 ,i +2 (cid:105) (cid:1) (ˆ τ x (cid:104) i,i +1 (cid:105) − ˆ τ x (cid:104) i +1 ,i +2 (cid:105) ) = 0 , (39)where in the last lines of Eqs. (38) and (39) we used rela-tion ˆ n hi = (1 − ˆ τ x (cid:104) i − ,i (cid:105) ˆ τ x (cid:104) i,i +1 (cid:105) ) which is directly obtainedfrom the Gauss Law. Using ˆ n i ˆ n i +1 = (1 − ˆ n hi )(1 − ˆ n hi +1 )the relations (37) becomeˆ n ↑ i ˆ n ↓ j + ˆ n ↓ i ˆ n ↑ j = (1 − ˆ n hi )(1 − ˆ n hi +1 ) , ˆ n ↑ i ˆ n ↑ j = 0 , ˆ n ↓ i ˆ n ↓ j = 0 . (40)The interaction term therefore maps to (cid:88) (cid:104) i,j (cid:105) (cid:18) J ↑↑ ˆ n ↑ i ˆ n ↑ j + J ↑↓ (cid:0) ˆ n ↑ i ˆ n ↓ j + ˆ n ↓ i ˆ n ↑ j (cid:1) ++ J ↓↓ ˆ n ↓ i ˆ n ↓ j (cid:19) → (cid:88) (cid:104) i,j (cid:105) V (1 − ˆ n hi )(1 − ˆ n hj ) , (41)where V = J ↑↓ . Extra terms V and V ˆ n hi in Eq. (41)amount to a constant energy offset and contribute tochemical potential. B. Relation of Luttinger K ’s – compressibilities The Luttinger K -parameter of a one-dimensionalquantum liquid is related to its compressibility κ by [44] κ ≡ − L (cid:18) ∂L∂P (cid:19) N = 1( n a ) Kπu a (42)where n a is the corresponding density, u a denotes itsspeed of sound and P = − (cid:0) ∂E∂L (cid:1) N is the pressure with E being the energy. Applying the general equation aboveto the string-length representation of the Z LGT yieldsfor the corresponding compressibility˜ κ = 1( ρ Ψ ) ˜ Kπ ˜ u , (43) where ˜ K , ρ Ψ and ˜ u denote the respective Luttinger pa-rameters, density and speed of sound in the string-lengthbasis.We can relate the two Luttinger parameters K and ˜ K by using the geometric relations characterizing the map-ping from the Z LGT to the string-length basis. Westart by noticing that κ = − L (cid:18) ∂L∂P (cid:19) N = (cid:20) L (cid:18) ∂ E∂L (cid:19) N (cid:21) − , (44)and similarly ˜ κ = (cid:20) ˜ L (cid:18) ∂ E∂ ˜ L (cid:19) ˜ N (cid:21) − , (45)where the energy E coincides in both representationssince they are related by a unitary transformation. Inthe following we will relate ˜ κ to κ .The ’particle number’ ˜ N and ’system size’ ˜ L in thestring-length model are related to N and L in the Z LGTmodel by N ( ˜ N , ˜ L ) = ˜ L, L ( ˜
N , ˜ L ) = ˜ N + ˜ L. (46)Hence for any function A ( ˜ L, ˜ N ) (cid:18) ∂A∂ ˜ L (cid:19) ˜ N = (cid:18) ∂A∂L (cid:19) N (cid:18) ∂L∂ ˜ L (cid:19) ˜ N + (cid:18) ∂A∂N (cid:19) L (cid:18) ∂N∂ ˜ L (cid:19) ˜ N = (cid:18) ∂A∂L (cid:19) N + (cid:18) ∂A∂N (cid:19) L . (47)Applying this relation twice on the right hand side of(45) thus yields (cid:18) ∂ E∂ ˜ L (cid:19) ˜ N = (cid:18) ∂ E∂L (cid:19) N + 2 (cid:18) ∂ E∂L∂N (cid:19) + (cid:18) ∂ E∂N (cid:19) L . (48)Using the following Maxwell relations for the chemicalpotential µ = ( ∂E/∂N ) L , (cid:18) ∂µ∂N (cid:19) L = − LN (cid:18) ∂µ∂L (cid:19) N = LN (cid:18) ∂P∂N (cid:19) L = − L N (cid:18) ∂P∂L (cid:19) N , (49)we can simplify (cid:18) ∂ E∂L∂N (cid:19) = (cid:18) ∂µ∂L (cid:19) N = LN (cid:18) ∂P∂L (cid:19) N , (50) (cid:18) ∂ E∂N (cid:19) L = (cid:18) ∂µ∂N (cid:19) L = − L N (cid:18) ∂P∂L (cid:19) N , (51) (cid:18) ∂ E∂L (cid:19) N = − (cid:18) ∂P∂L (cid:19) N . (52)Combining these results we obtain (cid:18) ∂ E∂ ˜ L (cid:19) ˜ N = − (cid:18) ∂P∂L (cid:19) N (cid:20) − LN + L N (cid:21) . (53)Next we plug Eq. (53) into Eq. (45). Using the defini-tions of the densities, n a = N/L and ρ Ψ = ˜ N ˜ L = 1 n a − , (54)as well as Eq. (44) yields:˜ κ = − (cid:34) L ˜ LL (cid:18) ∂P∂L (cid:19) N (cid:18) − n a (cid:19) (cid:35) − = − L ˜ L ( ρ Ψ ) − (cid:20) L (cid:18) ∂P∂L (cid:19) N (cid:21) − = ( ρ Ψ ) − ( n a ) − κ. (55)In the last step we used that L/ ˜ L = L/N = 1 /n a . Hence:˜ κκ = 1 n a ( ρ Ψ ) . (56)We can also use the relations (42), (43) of the com-pressibilities to the Luttinger parameters to obtain˜ κκ = ˜ KK (cid:18) n a ρ Ψ (cid:19) u ˜ u . (57)Since ’distances’ ˜ x in the string-length representation arerelated to distances x in the Z LGT by ˜ x = n a x , thespeed of sound ˜ u is related to u a by˜ u = d ˜ xdt = n a dxdt = n a u. (58)Finally, combining Eqs. (56), (57) and (58) yields˜ KK = 1( n a ) , (59)as claimed in the main part of the paper. C. Density correlations: Luttinger liquid fits
Monte-Carlo sampling of free fermions.–
As a first stepto verify our Luttinger liquid calculations we comparedthe Monte-Carlo sampled free-fermion and string-lengthbasis density-density correlations. In particular we wereinterested whether the results in two different bases ad-here to Eq. (14). To this end we calculated free fermiondensity-density correlations, (cid:104) δ ˆ n ( x ) δ ˆ n (0) (cid:105) , and by usingthe mapping in Eq. (8) the string-length density-densitycorrelations, (cid:104) δ ˆ ρ (˜ x ) δ ˆ ρ (0) (cid:105) . All calculations were per-formed for n a = 2 /
3, for which we simulated 240 fermionson 360 lattice sites.The results of our calculations can be seen in Fig. 4.String-length data points were multiplied by ( n a ) andthe ˜ x had to be rescaled as ˜ x → x = ˜ x/n a for a directcomparison. The data points for larger values acquired x | n (0) n ( x ) ||( n a ) (0) ( x = n a x ) | x n (0) n ( x )( n a ) (0) ( x = n a x ) (a) (b) FIG. 4. Density-Density correlations (cid:104) δ ˆ n ( x ) δ ˆ n (0) (cid:105) (blue datapoints) together with ( n a ) (cid:104) δ ˆ ρ ( x ) δ ˆ ρ (0) (cid:105) (red data points)plotted in a linear scale (a) and in a log-log scale (b) as afunction of x . We consider free fermions, h = 0, at filling n a = 2 / ρ Ψ = 1 / substantial relative error and hence only the first 15 datapoints are shown. High oscillations in the free-fermionbasis, make the comparison a bit difficult, however a goodoverall agreement between the two bases can be seen. DMRG density-density correlations.–
For zero fieldregime, h = 0, the original 1D Z LGT model maps toa free fermion model [37]. We thus expect the Luttingerparameter K to be equal to unity. Using the DMRG (seesection D) we calculated the charge-charge correlationsfor chain length L = 210 and fitted the results usingEq. (16) with an added constant offset, see Fig. 5 (a).The resulting fit yields K = 0 . ± . , α = 0 . ± . , n = 0 . ± . A = 5 · − ± . · − ,where A is a constant offset added to Eq. (16) andthe errors are square root values of the diagonal val-ues of the covariance matrix of the fit. The parame-ters n and A were restricted to values close to 2 / − − < A < · − respectively. As can be seenin Fig. 5 (b) the amplitude of oscillations is not per-fectly captured, most probably due to high sensitivity ofEq. (16) to the value of α .We also performed DMRG simulations of the string-length Hamiltonian (10), and performed a similar fitfor the density-density correlation function. Fixing theLuttinger-liquid parameter to the expected value ˜ K =2 .
25 according to Eq. (12) and the density to ρ Ψ = 1 / α = 0 . ± .
11 and A s = − · − ± .
001 where weagain constrained the constant offset − · − < A s < · − . Due to larger value of ˜ K the oscillations are sup-pressed and are not as prominent as in the original basis.Nevertheless the slope of the fitting function agrees wellwith the data points presented in Fig 5 (d).We also made a direct comparison of the data in theoriginal and in the string-length basis for zero field, h = 0and for a nonzero field-value h/t = 0 .
1, see Fig. 6. Apartfrom large oscillations the slope of the original density-density correlator and the rescaled density-density corre-lator ( n a ) (cid:104) δ ˆ ρ ( x ) δ ˆ ρ (0) (cid:105) , ˜ x → x = ˜ x/n a are similar whichis in agreement with Eq. (14).In addition we also calculated the density-density cor-0 x |( )( x )| h / t =0DMRGLL theory x ( )( x ) h / t =0 DMRGLL theory (c) (d) x | n ( ) n ( x )| h / t =0DMRGFit x n ( ) n ( x ) h / t =0 DMRGFit (a) (b) FIG. 5. Density-density correlation in the original basis (bluedots) as a function of distance x and the corresponding fittingfunction, f ( x ) = A − K π x + ( n a ) (cid:0) αx (cid:1) K cos(2 πn a x ) (greendata points) in linear (a) and log-log scaled axes (b). Density-density correlation in the string-length basis (red dots) as afunction of distance ˜ x and the corresponding fitting function, f (˜ x ) s = A s − ˜ K π ˜ x − ( ρ Ψ ) (cid:0) α ˜ x (cid:1) K cos(2 πρ Ψ ˜ x ) (green datapoints), where we fixed the Luttinger parameter, ˜ K = 2 . h = 0, at filling n a = 2 / ρ Ψ = 1 / x h / t =0.1 | n (0) n ( x ) ||( n a ) (0) ( x = n a x ) | x h / t =0.1 n (0) n ( x )( n a ) (0) ( x = n a x ) x h / t =0 | n (0) n ( x )||( n a ) (0) ( x = n a x )| x h / t =0 n (0) n ( x )( n a ) (0) ( x = n a x ) (a) (b)(c) (d) FIG. 6. Comparison of DMRG simulations of the density-density correlations of the original Z LGT model (bluedata points) and the rescaled charge-charge correlations,( n a ) (cid:104) δ ˆ ρ ( x ) δ ˆ ρ (0) (cid:105) in the string-length basis model (red datapoints). Aside from the oscillations for the original model anice agreement can be seen for the zero field case, h = 0 in (a)and (b). Similar behaviour is found for nonzero field value, h/t = 0 . n a = 2 / ρ Ψ = 1 / relations at filling of n a = 4 /
5. We find good agreementfor h = 0 and a good initial agreement for h >
0, seeFig. 7. When we increase h the first few data points co-incide nicely, but for higher values of x the correlationsstart to deviate, see Fig. 7 (c) and (d). x h / t =0.5 | n (0) n ( x ) ||( n a ) (0) ( x = n a x ) | x h / t =0.5 n (0) n ( x )( n a ) (0) ( x = n a x ) x h / t =0 | n (0) n ( x ) ||( n a ) (0) ( x = n a x ) | x h / t =0 n (0) n ( x )( n a ) (0) ( x = n a x ) (a) (b)(c) (d) FIG. 7. Comparison of DMRG simulations of the density-density correlations of the original Z LGT model n a = 4 / n a ) (cid:104) δ ˆ ρ ( x ) δ ˆ ρ (0) (cid:105) in the string-length basis model at ρ Ψ =1 / h = 0 in (a) and (b) where in (a) the axesare scaled linearly and in (b) we plotted the same data in alog-log scale. Similar behaviour survives also for the first fewdata points for nonzero field value, h/t = 0 . This can be understood by closer examination of thescaling of the two correlation functions. We can see thatthe first terms of the correlation function Eq. (15) andEq. (16) decay as ∼ x − whereas the second term decaysas ∼ x − K and ∼ x − K respectively. The latter, os-cillatory terms, clearly deviate from the scaling (14), aswas already discussed in the main text. Moreover we seethat the value of K for the original model drops, K < h is increased. When K <
1, the oscillatory partbecomes dominant at long distances (quasi crystalliza-tion). This is noticeable in the linear plots but difficultto see in the log-log plot in Fig. 7. On the other handthe corresponding string-length exponent is considerablyhigher: E.g. for n a = 2 /
3, the Luttinger parameter is˜ K ≈ . > n , e.g., n = 4 / K = ( n a ) ˜ K ensures lower difference betweenboth Luttinger parameters. Hence the oscillations in thestring-length basis become visible already for h = 0 (seeFig. 7 (a) and (b) ) and become prominent as we increase h , since ˜ K drops closer to unity, see Fig. 7 (c) and (d).Precise fits are hard to obtain due to high sensitivityto α and ˜ α . The fits at h = 0 for the original model,Fig. 8 (a) yield K = 0 . ± .
008 which is again close tothe expected K = 1. The string-length basis was againdifficult to fit. However, we compare the DMRG datawith the curve obtained by using Eq. (15) and insertingthe expected Luttinger parameter, ˜ K = 1 .
56. Tuning thevalue of ˜ α yields convincing agreement, see Fig. 8.1 x |( )( x )| h / t =0 DMRGFit x | n ( ) n ( x )| h / t =0DMRGFit (a) (b) FIG. 8. (a) Density-density correlation in the original basis,(blue dots) at filling n a = 4 /
5, as a function of distance x and the corresponding fitting function, f ( x ) = A − K π x + ( n a ) (cid:0) αx (cid:1) K cos(2 πn a x ) (green data points). (b) Density-density correlation in the string-length basis, (red dots) at fill-ing ρ Ψ = 1 / x and the correspond-ing function, f (˜ x ) s = A s − ˜ K π ˜ x + ( ρ Ψ ) (cid:0) α ˜ x (cid:1) K cos(2 πρ Ψ ˜ x )(green data points), where we fixed the parameters to ˜ K =1 .
56 and ˜ α = 1 . D. DMRG simulations
Simulations of the Z LGT.–
DMRG calculations ofthe Z LGT model were performed by using the
SyTen toolkit created by Claudius Hubig [34–36]. The Z LGTmodel Eq. (1) was mapped to a spin-1/2 model by usingthe Gauss law, which was fixed to a sector where G i = 1on every lattice site. Using this result the Z electricfield configuration on the links between lattice sites com-pletely determines the hard-core boson configuration onthe sites, ˆ n j = 12 (1 − ˆ τ x (cid:104) i,j (cid:105) ˆ τ x (cid:104) j,k (cid:105) ) . (60)By writing the Pauli matrices in terms of the spin-1/2operators ˆ τ x (cid:104) j,j +1 (cid:105) → S xj and ˆ τ z (cid:104) j,j +1 (cid:105) → S zj the original Z LGT model becomesˆ H s = t (cid:88) i (4 ˆ S xi − ˆ S xi +1 −
1) ˆ S zi − h (cid:88) i ˆ S xi + V (cid:88) i
14 (1 − S xi − ˆ S xi )(1 − S xi ˆ S xi +1 ) + µ (cid:88) i ˆ S xi +1 ˆ S xi , (61)where the factor (4 ˆ S xi − ˆ S xi +1 −
1) in the hopping term isnecessary to obtain the same matrix elements as in theoriginal model, namely the hopping is only allowed whenthe original lattice site is occupied by a hard-core bosonand the prospective lattice site, onto which the bosoncan hop, is empty. All other configurations do not allowhopping. The density-density correlator, (cid:104) δ ˆ n ( x ) δ ˆ n (0) (cid:105) was implemented in a similar way using the Eq. (60).In order to calculate the the charge gap (see Supple-mentary E) we had to subtract the chemical potentialcontribution to the overall energy by modifying the chem-ical potential term to µ (cid:80) i ((0 . − n a ) − S xi +1 ˆ S xi ). Simulations in the string-length basis.–
Although theconsidered Z LGT expressed in terms of the stringlength basis Eq. (10) is a purely local Hamiltonian, theabsence of the usual Bose enhancement makes an ac-curate study of this model challenging. In particularbosonic systems would require to consider a maximumoccupation number equal to the total number of bosons N . Apart from the case of very small systems, this rig-orous choice would make untreatable the bosonic prob-lem with a quasi-exact method like DMRG. In order toachieve an efficient description, the usual strategy whenstudying standard bosonic models is to perform a cutoffin the local Hilbert space. This turns out to be a totallysafe and precise manner to approach such systems espe-cially in the regimes of low and intermediate densities andnon attractive interactions. On the other hand, the pe-culiar structure of the string length model makes Hilbertspace truncations very delicate. In particular we findthat if we cut the maximum occupation number to toosmall values, but still large compared to the usual choicesin ”standard” bosonic models, we do not get agreementbetween the density-density correlation decay Eqs. (14)and (15). In order to achieve agreement and thus thecorrect Luttinger parameters relation Eq. (12) we employextensive DMRG simulations by considering a maximumoccupation number of N/
2. This clearly requires a largenumber of DMRG states, that we fix = 1400, in order tokeep the truncation error on the energy < − . E. Gap Calculations
In order to calculate the opening of the Mott gap inthe thermodynamic limit the ground state energies werecalculated for different chain lengths L . The energieswere then plotted as a function of inverse chain lengthand fitted with a quadratic function, see Fig. 9 (a). Thevalue of the fitted equation at x = 1 /L = 0 was thentaken as the value of ground state in the thermodynamiclimit, L → ∞ . The filling was varied by tuning the chem-ical potential µ to obtained the correct value of N . Bychanging the chemical potential we also calculated theground state for N − N + 2 in order to calculatethe gap, ∆ c ( L, N ) using Eq. (20). We furthermore notethat the chemical potential contribution to the overallenergy was deducted. By varying the parameters h and V the diagram in Fig. 2 was obtained.The opening of the gap appears to behave similarly asin the BKT case, as is demonstrated in Fig. 9 (b). There,the transition point was estimated to be h c /t ∼ .
15 inorder to produce a fit. Although the exact transitionpoint is hard to deduce, the general behaviour appearsto agree with an exponential gap opening.2 L c / t h / t = V / t =0 h / t =1, V =0 h / t = V / t =1 h / t c / t V / t =1 (a) (b) FIG. 9. Gap extrapolation to the thermodynamic limit (a) byfitting the data points with a quadratic function: h = V = 0(blue), h/t = 1 , V = 0 (red) and h/t = V /t = 1 (green). (b)The opening of the gap as a function of h at V /t = 1 (blackdata points) and the corresponding fit with the function f ∆ = A ∆ + C ∆ e − B ∆ / √ h − h c . The critical value was set manually to h c /t = 0 .
15. The filling was set to n a = 2 / F. Particle-hole mapping
The particle-hole mapping is performed by using theunitary transformation [37]ˆ a † i → ˆ a i , ˆ a i → ˆ a † i , (62)and the Hamiltonian (1) becomesˆ H = − t (cid:88) (cid:104) i,j (cid:105) (cid:16) ˆ a i ˆ τ z (cid:104) i,j (cid:105) ˆ a † j + h.c. (cid:17) − h (cid:88) (cid:104) i,j (cid:105) ˆ τ x (cid:104) i,j (cid:105) + V (cid:88) (cid:104) i,j (cid:105) (1 − ˆ n i )(1 − ˆ n j ) , (63)where the number operator was transformed by using theEq. (62) ˆ n i = ˆ a † i ˆ a i → ˆ a i ˆ a † i = 1 − ˆ n i . (64)In the last step the commutation relations of the hard-core bosons were used. Apart from the NN interactionterm, Eq. (63) has the exact same form as the originalmodel (1).Due to Eq. (64) the Gauss law maps toˆ G i = ˆ τ x (cid:104) i − ,i (cid:105) ˆ τ x (cid:104) i,i +1 (cid:105) ( − ˆ n i → ˆ τ x (cid:104) i − ,i (cid:105) ˆ τ x (cid:104) i,i +1 (cid:105) ( − − ˆ n i = − ˆ G i . (65)We can obtain the original form by performing unitarytransformation ˆ τ x (cid:104) i,i +1 (cid:105) → ( − i ˆ τ x (cid:104) i,i +1 (cid:105) and ˆ τ y (cid:104) i,i +1 (cid:105) → ( − i ˆ τ y (cid:104) i,i +1 (cid:105) but ˆ τ z (cid:104) i,i +1 (cid:105) → ˆ τ z (cid:104) i,i +1 (cid:105) which brings theGauss law in the same form as in the original model − ˆ τ x (cid:104) i − ,i (cid:105) ˆ τ x (cid:104) i,i +1 (cid:105) ( − ˆ n i → − ( − i − ˆ τ x (cid:104) i − ,i (cid:105) ( − i ˆ τ x (cid:104) i,i +1 (cid:105) ( − ˆ n i = ˆ τ x (cid:104) i − ,i (cid:105) ˆ τ x (cid:104) i,i +1 (cid:105) ( − ˆ n i . (66) p/h mappingsqueezep/h mapping h > 0 + + + + + + + + - - - - - - - - ++ + + - -- + + - + + + hh (a)(b) (c)(d) FIG. 10. (a) Particle hole mapping. The V → ∞ limit, whichis taken in the next step, can be seen as a particle (depictedas a transparent oval) spreading to an empty site to the rightof the occupied lattice site. The periodic exchange of + and − represent the staggered field ( − j h . (b) Mapping to a”quasi” squeezed space where the empty lattice sites follow-ing the occupied lattice sites are deleted from the picture andreintroduced in the Hamiltonian (70) as a new interactionterm Eq. (69). The shift of the staggered field due to rela-belling of the lattice indices is again demonstrated with +and − bellow the lattice links. (c) Particle-hole mapping ofthe ”quasi” squeezed space introduced above. (d) Expectedground state due to the staggered term in Hamiltonian (74)for half-filled lattice. This last transformation of the gauge fields brings theHamiltonian to its final formˆ H = − t (cid:88) (cid:104) i,j (cid:105) (cid:16) ˆ a † i ˆ τ z (cid:104) i,j (cid:105) ˆ a j + h.c. (cid:17) − h (cid:88) i ( − i ˆ τ x (cid:104) i,i +1 (cid:105) + V (cid:88) (cid:104) i,j (cid:105) (1 − ˆ n i )(1 − ˆ n j ) . (67)As can be seen in Eq. (67) the hopping term is identicalto the original model whereas the field term h acquires astaggered term ( − i and there is no linear dependenceon the string length. The NN interaction transforms asexpected for a hard-core boson case. G. V → ∞ limit In this section we show that in the limit where V → ∞ and h > n a = 2 /
3. We start by performing a particle-hole map-ping of the original model as in the previous section, seeFig. 10 (a). In the particle-hole transformed picture thedensity becomes n h = 1 /
3. Next we take the limit of V → ∞ by making the particles larger and thus effec-tively occupying an extra lattice site to the right, seeFig. 10 (b). With such construction we make sure thatthere is at least one empty site between two particles.In the new picture the number of lattice sites decreasesby N and hence the density becomes n (cid:48) = NL − N = n − n which for a filling n h = 1 / n (cid:48) = 1 /
2. Another3important consequence of such construction is the factthat now one extra link variable and empty lattice siteto the right of the occupied lattice site is hidden, seeFig. 10. First of all, this means that the field in the newpicture acquires an extra term − h N (cid:88) j =1 ( − j ˆ τ x (cid:104) j,j +1 (cid:105) → − h L − N (cid:88) j =1 ( − j (cid:89) i ≤ j ( − ˆ n i ˆ τ x (cid:104) j,j +1 (cid:105) (68)where the site index j in the new picture correspondsto j + (cid:80) i 2, we obtain a band insulator withband gap ∆ = 2 hh