Effective narrow ladder model for two quantum wires on a semiconducting substrate
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Effective narrow ladder model for two quantum wires on a semiconducting substrate
Anas Abdelwahab and Eric Jeckelmann
Leibniz Universit¨at Hannover, Institut f¨ur Theoretische Physik, Appelstr. 2, 30167 Hannover, Germany (Dated: March 1, 2021)We present a theoretical study of two spinless fermion wires coupled to a three dimensionalsemiconducting substrate. We develop a mapping of wires and substrate onto a system of two coupledtwo-dimensional ladder lattices using a block Lanczos algorithm. We then approximate the resultingsystem by narrow ladder models, which can be investigated using the density-matrix renormalizationgroup method. In the absence of any direct wire-wire hopping we find that the substrate can mediatean effective wire-wire coupling so that the wires could form an effective two-leg ladder with a Mottcharge-density-wave insulating ground state for arbitrarily small nearest-neighbor repulsion. In othercases the wires remain effectively uncoupled even for strong wire-substrate hybridizations leadingto the possible stabilization of the Luttinger liquid phase at finite nearest-neighbor repulsion asfound previously for single wires on substrates. These investigations show that it may be difficult todetermine under which conditions the physics of correlated one-dimensional electrons can be realizedin arrays of atomic wires on semiconducting substrates because they seem to depend on the model(and consequently material) particulars.
I. INTRODUCTION
Systems of metallic atomic wires deposited on semi-conducting substrates attracted lots of attention in thelast two decades. One of the main issues regard-ing these systems is the existence of features relatedto one-dimensional (1D) electrons, e.g. Luttinger liq-uid phases [1–5], Peierls metal-insulator transitions andcharge-density-wave states [6–9]. However, the theoreti-cal framework of 1D correlated electrons is derived pri-marily from purely 1D models [10–14], which are thenextended to anisotropic two (2D) and three dimensional(3D) systems. These extensions are not applicable formetallic atomic wires on semiconducting substrates dueto their strong asymmetric nature, i.e. they representarrays of 1D wires coupled to a 3D reservoir. There-fore, even if we assume that the atomic wires themselvesare systems of 1D correlated electrons, it is necessary toinvestigate two aspects: firstly, the influence of the cou-pling to the 3D bulk semiconducting substrate on the 1Dfeatures; secondly, the possibility of substrate-mediatedcoupling between the wires.In a series of previous publications [15–17], we ad-dressed the first issue. We established that, indeed, thecoupling of a single metallic atomic wire to a 3D semicon-ducting substrate can stabilize the 1D nature of the wireand, in particular, can support the occurrence of Lut-tinger liquid phases. In the current article we would liketo address the second issue. We develop a method to mapmulti wires coupled to semiconducting substrates onto asystem of coupled 2D ladders (one ladder per wire). Thismethod is based on the block Lanczos algorithm [18].Then we approximate the original systems by keepingonly a few legs of each 2D ladder, i.e. by constructing anarrow ladder model (NLM) that can be investigated us-ing well-established methods for quasi-one-dimensionalcorrelated quantum systems. The original system andthe resulting NLM are depicted in Fig. 1.Using exact diagonalizations of noninteracting wires
FIG. 1. The upper panel display sketch of two atomic wires(red spheres) on a 3D substrate with four numbered shells.The lower panel ladder representation of the same systemwith the upper-most legs corresponding to the atomic wires(red circles) and the other legs (in green and light blue) rep-resenting the shells one to four. and the density-matrix renormalization group (DMRG)method [19, 20], we investigate in details two wires on asemiconducting substrate (TWSS) for spinless fermionswithout direct wire-wire coupling and compare to theknown results for a single wire on a substrate [17] andfor two-leg ladders without substrate [21]. We find thatthe substrate can mediate an effective wire-wire couplingso that the atomic wires form an effective two-leg ladderthat is known to have a Mott charge-density-wave insu-lating ground state for arbitrarily small nearest-neighborrepulsion. In other cases the wires remain effectivelyuncoupled even for strong wire-substrate hybridizations,which should result in a Luttinger liquid phase at finitenearest-neighbor repulsion as found previously for singlewires on substrates.The article is organized as follows. In the secondsection we introduce the TWSS model, the mappingof multi-wire-substrate models onto systems of coupledmulti 2D ladders and the approximation by few-leg NLM.In the third section we discuss noninteracting wires whilein the fourth section we present our results for correlatedwires. We conclude in the fifth section.
II. MODELING TWO WIRES ONSEMICONDUCTING SUBSTRATES AND THEAPPROXIMATION BY NARROW LADDERMODELS
In this section we construct a model for TWSS andapproximate it by NLM. This procedure can be easilygeneralized to more than two wires on a semiconductingsubstrate.
A. The model of two wires on semiconductingsubstrate
The substrate is described as explained in Ref. [15] butsince we focus on spinless fermion wires, we omit spindegrees of freedoms. We restrict ourselves to insulatingor semiconducting substrates, hence each substrate sitehas two orbitals, one contributing to the formation of theconduction bands and one to the valence bands. Thus,the substrate Hamiltonian in real space takes the form H s = H c + H v = X s=v , c ǫ s X r n s r − t s X h rq i (cid:0) c † s r c s q + H.c. (cid:1) . (1)The first sum runs over conduction (s=c) and valence(s=v) bands. The second sum runs over all sites of acubic lattice and the third one over all pairs h rq i ofnearest-neighbor lattice sites. The operator c † s r createsa spinless fermion on the orbital s localized on the sitewith coordinates r = ( x, y, z ), and the fermion densityoperator on each orbital is n s r = c † s r c s r . The transforma-tion to the momentum-space is done only in x -directionwhich is the alignment direction of the two wires. Theother two dimensions are irrelevant for the two wires andthey can remain in the real-space representation for thesubstrate. This allows a mixed real-space momentum-space representation r k x = ( k x , y, z ) where k x is the wavevector component in x -direction. This representation is more convenient for the purpose of ladder mapping. TheHamiltonian (1) takes the form H s = X s=v , c X k x ,y,z ǫ s ( k x ) d † s r kx d s r kx − t s X k x , h ( y ,z ) , ( y ′ ,z ′ ) i d † s r kx d s r ′ kx (2)with two single-electron dispersions ǫ s=v , c ( k x ) = ǫ s=v , c − t s=v , c cos( k x ) , (3)the sum over nearest-neighbor site pairs h ( y , z ) , ( y ′ , z ′ ) i in the yz -layer for a given k x , and r ′ k x = ( k x , y ′ , z ′ ). The(possibly indirect) gap between the bottom of the con-duction band and the top of the valence band is given by∆ s = ǫ c − ǫ v − | t v | + | t c | ) and the condition ∆ s ≥ t ab between adjacent sites in the two wires. Thetwo wires are described by the Hamiltonian H w = X w=a , b ǫ w X x n xy w − t w X x (cid:16) c † xy w c x +1 ,y w + H.c. (cid:17) + V X x n xy w n x +1 y w ! − t ab X x (cid:0) c † xy a c xy b + H.c. (cid:1) . (4)The wires w ≡ a and b of length L x are aligned in the x -direction at positions r = ( x, y a ,
0) and r = ( x, y b , y a and y b ∈ { , . . . , L y } . The sums over x runover all wire sites. The operator c † xy a creates an electronon the wire site r = ( x, y a ,
0) for w ≡ a and the operator c † xy b creates an electron on the wire site r = ( x, y b , ≡ b . Every site in the wires is exactly on topof the corresponding substrate site. t a and t b are theusual hopping terms between nearest-neighbor sites inthe corresponding wire. ǫ a and ǫ b are on-site potentials ofthe corresponding wires and V is the interaction betweenfermions on nearest-neighbor sites. The single-particlehopping t ab determines the direct coupling between thetwo wires. The Hamiltonian of the two spinless-fermionwires can be written in the mixed representation H w = X w=a , b X k x ǫ w ( k x ) d † k x y w d k x y w + VL x X k x ,k ′ x ,k ′′ x ,k ′′′ x d † k x y w d k ′ x y w d † k ′′ x y w d k ′′′ x y w δ k x − k ′ x ,k ′′′ x − k ′′ x − t ab X k x (cid:16) d † k x y a d k x y b + H.c. (cid:17) (5)with two wire single-electron dispersions ǫ w=a , b ( k ) = ǫ w=a , b − t w=a , b cos( k ) . (6)Here k x , k ′ x , k ′′ x , and k ′′′ x denote momenta in the x -direction.The hybridization between each wire and the substrateis modeled by H ws = H wa + H wb = X w=a , b;s=v , c ( − t ws ) X x (cid:0) c † s r w c xy w + H.c. (cid:1) (7)which represents a hopping between each wire site andthe nearest valence and conduction band sites at r w =( x, y w , H ws = H wa + H wb = X w=a , b;s=v , c ( − t ws ) X k x (cid:16) d † s k w d k x y w + H.c. (cid:17) (8)with k w = ( k x , y w , H = H w + H s + H ws . (9) B. Two-impurity subsystems
In this section we reformulate the Hamiltonian (9) asa set of two-impurity-host subsystems. This is first donefor the noninteracting case ( V = 0). In the mixed repre-sentation, the Hamiltonian takes the form H = X k x H k x (10)where H k x are independent sub-system Hamiltonianssuch that (cid:2) H k x , H k ′ x (cid:3) = 0 ∀ k x , k ′ x . Each sub-systemHamiltonian represents a two-impurity subsystem thattakes the form H k x = ǫ a ( k x ) d † k x y a d k x y a + ǫ b ( k x ) d † k x y b d k x y b − t ab (cid:16) d † k x y a d k x y b + H.c. (cid:17) + X s=v , c "X y,z ǫ s ( k x ) d † s r kx d s r kx − t s X h ( y ,z ) , ( y ′ ,z ′ ) i d † s r kx d s r ′ kx − X s=v , c;w=a , b (cid:16) t ws d † s k w d k x y w + H.c. (cid:17) . (11) H k x represents two non-magnetic impurities with the en-ergy levels ǫ a ( k x ) and ǫ b ( k x ) corresponding to the twowires. These energy levels are coupled to a 2D host deter-mined by a substrate ( y, z )-slice through the hybridiza-tion parameter t ws . Each ( y, z )-slice corresponds onlyto the given wave vector k x . For a noninteracting wire,each H k x is a single-particle problem and it is amenableto exact diagonalization. Each single-particle Hamilto-nian has the dimension N imp = 2 L y L z + 2. C. Ladder representation
The two-impurity subsystem can be mapped onto atwo-leg ladder system using the Block-Lanczos (BL) al-gorithm. This method has been used to investigate mul-tiple quantum impurities embedded in multi-dimensionalnoninteracting hosts [18, 22, 23] and, very recently, a sin-gle wire coupled to two multi-dimensional noninteractingleads [24]. The BL algorithm is an extension of the Lanc-zos algorithm to formulate block tridiagonal matrix start-ing from more than one basis state. The number of basisstates chosen to start the iterations determines the sizeof each single block within the resulting block tridiagonalmatrix.The BL procedure starts by forming a matrix P withdimensions N row × N col where N row refers to the numberof rows and N col refers to the number of columns. P isgiven by the matrix P = h(cid:16) d † k x y a | Φ i (cid:17) (cid:16) d † k x y b | Φ i (cid:17)i (12)where | Φ i is the vacuum state. The first column is deter-mined by the vector d † k x y a | Φ i which represents the firstimpurity site and the second column is determined bythe vector d † k x y b | Φ i which represents the second impu-rity site. Therefore, P has the dimensions N row = N imp and N col = 2. Notice that if we have more than twowires the multi-impurity subsystem will have more thantwo impurities, i.e. N col > P . The BL iteration isdefined as P l +1 T † l = H k x P l − P l E l − P l − T l − (13)where E l = P † l H k x P l , P = 0 and T = 0. The decom-position of the left-hand side of (13) into two operators P l +1 and T † l can be obtained using the QR decomposi-tion [25]. Thus P l is a column-orthogonal matrix and T l is a lower-triangular matrix, i.e. with matrix elements[ τ l ] n,n ′ = 0 for n < n ′ . We denote the matrix elementsof E l as [ e l ] n,n ′ . The BL basis P l spans the Krylov sub-space of H k x . The full implementation of the BL methodgenerates a matrix P = [ P P P ... ] , (14)which has the dimensions N row = N col = N imp . Thematrix P can be used to block-tridiagonalize the Hamil-tonian H BLk x in the form H BLk x = E T · · · T † E T · · · T † E T · · · T † E . . .... ... ... . . . . . . . (15)Since the number of impurities is two, each block in (15)is a 2 × H BLk x represents a two-leg ladderwhich is written in the form H BLk x = N imp / X l =1 2 X n,n ′ =1 [ e l ( k x )] n,n ′ f † k x ln f k x ln ′ (16)+ [ N imp / − X l =1 2 X n,n ′ =1 h [ τ l ( k x )] n,n ′ f † k x ln f k x ,l +1 ,n ′ + H.c. i where f k x , , = d k x y a (17) f k x , , = d k x y b (18) f k x ,l> ,n = X r kx (cid:16)(cid:12)(cid:12)(cid:12) h P l i n ED h P l i n (cid:12)(cid:12)(cid:12) d v r kx + (cid:12)(cid:12)(cid:12) h P l i n ED h P l i n (cid:12)(cid:12)(cid:12) d c r kx (cid:17) . (19)We distinguish two kinds of wire-substrate hybridiza-tions. The first kind is the hybridization of the two wireswith two substrate sites belonging to different sublat-tices in the substrate bipartite lattice, e.g. the wires arenearest neighbor (NN) with | y a − y b | = 1. In this casethe system is particle-hole symmetric, i.e. the Hamil-tonian is invariant under the transformation f k x ln → ( − l ( − n f † k x ln , and thus half-filling corresponds to theFermi energy ǫ F = 0. The second kind is the hybridiza-tion of the two wires with two substrate sites belongingto the same sublattice in the substrate bipartite lattice,e.g. the wires are next nearest neighbor (NNN) with | y a − y b | = 2. In this case the system is not particle-holesymmetric as long as t ab = 0 and the full wire-substratelattice is not bipartite. However, for t ab = 0 the systemis again bipartite and particle-hole symmetric.To illustrate the procedure, we calculate the param-eters [ e l ( k x )] n,n ′ and [ τ l ( k x )] n,n ′ for NN impurities cou-pled to an insulating substrate with the wire-substratemodel parameters t w = 3, t ab = 0, t s = 1 and t ws = 8,i.e. without direct coupling between the two impurities.The diagonal terms [ e l ( k x )] n,n depend of the dispersionin the wire direction and the on-site chemical potentials.For n = n ′ the inter-leg hopping terms [ e l ( k x )] n,n ′ vary as l increases as shown in Fig. 2(a). Since the initial vectorsare associated with sites belonging to different sublat-tices, the inter-rung hopping terms are [ τ l ( k x )] n,n ′ = 0for n = n ′ (the BL algorithm always enforces [ τ l ] n,n ′ = 0for n < n ′ ). For n = n ′ we found [ τ l ( k x )] , = [ τ l ( k x )] , which is shown in Fig. 2(b).We also calculate the parameters [ e l ( k x )] n,n ′ and[ τ l ( k x )] n,n ′ for 2D hosts coupled to NNN impurities.We use similar parameters as those used for NN wireswithout any direct coupling between the two impurities.Similar to the NN impurities case, the diagonal terms[ e l ( k x )] n,n depend of the dispersion in the wire directionand the on-site chemical potentials. For n = n ′ the inter-leg hopping terms must vanish, i.e. [ e l ( k x )] n,n ′ = 0, sincethey connect sites between similar sublattices. Neverthe-less, we observe finite values for these parameters after a [e l (k x )] n,n´ l (a)n=1,n´=2n=2,n´=1 0 2 4 6 8 10 12 5 10 15 20 25 [ τ l (k x )] n,n ′ l (b) n=1n=2n=2,n ′ =1 0 0.1 0.2 0.3 0.4 0.5 5 10 15 20 25 [e l (k x )] n,n´ l (c)n=1,n´=2n=2,n´=1 0 2 4 6 8 10 12 5 10 15 20 25 [ τ l (k x )] n,n ′ l (d) n=1n=2n=2,n ′ =1 FIG. 2. Hopping parameters of the two-impurity subsystemafter the Block-Lanczos transformation. The parameters aregiven in the text. (a) Intra-rung hopping terms [ e l ( k x )] n,n ′ forthe system with NN impurities. (b) Intra-leg hopping terms[ τ l ( k x )] n,n as well as inter-leg diagonal hoppings [ τ l ( k x )] , for the system with NN impurities. (c) Intra-rung hoppingterms [ e l ( k x )] n,n ′ for the system with NNN impurities. (d)Intra-leg hopping terms [ τ l ( k x )] n,n as well as inter-leg diago-nal hoppings [ τ l ( k x )] , for the system with NNN impurities.The horizontal axis represents the number of Bloc-Lanczosshell. Note that the results are indistinguishable for n=1 andn=2 (square symbols) in (a), (b) and (c) but not in (d). few BL iterations as shown in Fig. 2(c). This is due to theloss of orthogonality in the BL calculation when we ini-tiate from NNN impurities. However, we have observedthat the accuracy is better when the separation | y a − y b | between the two impurities is larger. As we mentionedbefore, the inter-rung hopping terms [ τ l ( k x )] , = 0 whilethe other values of [ τ l ( k x )] n,n ′ are shown in Fig. 2(d).We observe that the BL iterations produce accurate re-sults for 2D hosts coupled to NN impurities for relativelylarge number of iterations. Despite the less accurate re-sults for large number of BL iterations in the case of NNNimpurities we observe that at least for the minimal num-ber of iterations (with l = 3 corresponding to 6-leg lad-ders) the results are accurate enough. This will allow usto construct a minimal approximation of the full TWSS.We emphasize that this minimal approximation includesan overlap between the BL vectors generated from thetwo impurities and thus possibly an indirect substratemediated coupling between wires. D. Real-space representation
We now transform the Hamiltonian (16) back to thereal-space representation in x -direction. As the wirestates have not been modified by the mapping of themulti-impurity subsystem to the ladder representation,the two-wire Hamiltonian H w remains unchanged. Bydefining new fermion operators g † xln = 1 √ L x X k x e − ik x x f † k x ln (20)that create electrons at position x in the l -th shell andthe n -th 2D sheet, we get a new representation of the fullwire-substrate Hamiltonian H = ( N imp / X l =1 X xx ′ X nn ′ [ e l ( x − x ′ )] nn ′ g † xln g x ′ ln ′ + ( N imp / − X l =1 X xx ′ X nn ′ h [ τ l ( x − x ′ )] nn ′ g † xln g x ′ ,l +1 ,n ′ + H.c] (21)where [ e l ( x )] nn ′ = 1 L x X k x [ e l ( k x )] nn ′ exp( ik x x ) (22)are the hopping amplitudes in the wire direction withinthe same shell l (or the on-site potential for x = 0)whereas [ τ l ( x )] nn ′ = 1 L x X k x [ τ l ( k x )] nn ′ exp( ik x x ) (23)are the hopping amplitudes between sites in shells l and l + 1. Therefore, we have obtained a new representationof the Hamiltonian H with long-range hoppings on twosheets of 2D lattices of size L x × N imp / k x up to energy shifts. It followsthat the hopping terms between nearest-neighbor shellsare [ τ l ( x )] nn ′ = − [ t rung l ] nn ′ δ x, (24) with [ t rung l ] nn ′ = − [ τ l ( k x )] nn ′ . In addition, one finds that[ e l ( x )] nn ′ = − (cid:2) t leg x (cid:3) nn ′ + [ µ l ] nn ′ δ x, (25)with (cid:2) t leg x (cid:3) nn ′ = − L x X k x [ ν ( k x )] nn ′ exp( ik x x ) (26)and [ µ l ] nn ′ = [ e l ( k x )] nn ′ − [ ν ( k x )] nn ′ .At this point, we have obtained a representation ofthe wire-substrate Hamiltonian H in the form of twoladder-like sheets, such that each sheet has L x rungs and N imp / l = 1 are the two wires, in particular g † x,l =1 ,n =1 = c † ax and g † x,l =1 ,n =2 = c † bx , while legs with l = 2 , . . . , N imp / H w , the direct wire-wire coupling H ab ,the hopping terms [Γ] nn ′ (hybridization) between wiresites and sites in the first two legs ( l = 2) represent-ing the substrate, the nearest-neighbor and next-nearest-neighbor rung hoppings t rung lnn ′ between substrate legs withindices l − l , the on-site potentials and leg-leg cou-plings [ µ l ] nn ′ δ x, − t leg0 within each substrate shell, andthe same intra-leg hopping terms t leg x in every substrateleg. The latter are identical to the hopping terms in theoriginal substrate Hamiltonian H s .For substrates with dispersions of the form ǫ s ( k ) = ǫ s − t s [cos( k x ) + cos( k y ) + cos( k z )] (27)we have ν ( k x ) = − t s cos( k x ), so that the hopping withinsubstrate legs takes place between nearest-neighborsonly, t leg x = ( t s if | x | = 1 , . (28)The explicit form of the full Hamiltonian is then H = H w + H ab + X x,n,n ′ h [Γ] nn ′ g † x,l =2 ,n g x,l =1 ,n ′ + H.c. i + N imp / X l =2 X x,n [ µ l ] nn g † xln g xln + N imp / X l =2 X x,n = n ′ h [ µ l ] nn ′ g † xln g xln ′ + H.c. i − t s N imp / X l =2 X x,n h g † xln g x +1 ,l,n + H.c. i − N imp / − X l =2 X x,n,n ′ h t rung lnn ′ g † xln g x,l +1 ,n ′ + H.c. i . (29)For the TWSS model, a narrow ladder approximation(NLM) with N leg = 2 N shell legs is obtained by project-ing the full Hamiltonian (29) onto the subspace given bythe first N shell blocks of BL vectors, i.e. by substituting N shell ≤ N imp / N imp / N leg ≤ N imp for N imp ) in Eq. (29). As the intra-wire Hamiltonian H w is not affected by the mapping or the projection, wecan apply this procedure to systems of interacting wires( V = 0) to obtain interacting NLM. III. NONINTERACTING WIRES
To compute spectral properties of the full TWSS modelin the mixed representation, we use the Hamiltonian (10)with (11). The spectral function in this representation isgiven by A ( ω, k x ) = L y L z +2 X λ =1 δ ( ω − ε λk x ) (30)where ε λk x ( λ = 1 , . . . , N imp = 2 L y Lz + 2) denote theeigenvalues of the Hamiltonians (11). Similarly, to com-pute spectral properties of the effective NLM, we usethe Hamiltonian (10) with H k x in the BL representa-tions (16) with N shell ≤ N imp / N imp / A ( ω, k x ) = N leg X λ =1 δ ( ω − ε λk x ) (31)where ε λk x ( λ = 1 , . . . , N leg ) denote the eigenvalues ofthese Hamiltonians. This spectral function can be easilycalculated for any 1 ≤ N leg ≤ N imp .We compare spectral functions of the full system withthose of the NLM with various numbers of legs. Unlessotherwise stated, we use symmetric intra-wire hopping t a = t b = t w = 3 and wire-substrate hybridization t a c = t a v = t b c = t b v = t ws . The substrate parameters are t c = t v = 1 and ǫ c = − ǫ v = 7. The system sizes are L x =1000 , L y = 32 and L z = 8. These model parameterscorrespond to an indirect gap ∆ s = 2 and a constantdirect gap ∆ s ( k x ) = 6 for all k x in the substrate single-particle band structure [in the absence of wires or for avanishing wire-substrate coupling ( t ws = 0)]. Our mainaim in this investigation is to understand the influenceof the substrate on the one-dimensional physics of thewires. Therefore, we focus on systems with two wires butwithout direct wire-wire coupling, i.e. we set t ab = 0.Figure 3(a) displays the spectral function of the fullTWSS model with NN wires and a relatively strong wire-substrate hybridization t ws = 8. Despite the absence ofany direct wire-wire coupling, we clearly see two sepa-rated bands crossing the Fermi level at ω = ǫ F = 0 (forhalf filling) in the middle of the substrate band gap. Thisstructure corresponds to the dispersions found in a two-leg ladder with the separation between the (bonding andanti-boding) bands given by twice the rung hopping term -20-15-10-5 0 5 10 15 20 -3 -2 -1 0 1 2 3 ω k x -4 -3 -2 A ( k x , ω ) (a)-20-15-10-5 0 5 10 15 20 -3 -2 -1 0 1 2 3 ω k x -5 -4 -3 -2 A ( k x , ω ) (b)-20-15-10-5 0 5 10 15 20 -3 -2 -1 0 1 2 3 ω k x -5 -4 -3 -2 -1 A ( k x , ω ) (c) FIG. 3. Spectral functions of two NN wires on a semiconduct-ing substrate with t w = 3, t ws = 8 and t s = 1. (a) Full TWSSmodel. (b) NLM with N leg = 2 N shell = 54 . (b) NLM with N leg = 2 N shell = 6. t ⊥ [11]. Therefore, this observation demonstrates the ex-istence of an effective, substrate mediated coupling be-tween both wires even when the model does not includethe bare hopping term t ab .Moreover, in Fig. 3 we see two bands above the sub-strate conduction band continuum and two other bandsbelow the substrate valence band continuum. These fourbands are due to the strong wire-substrate hybridizationwhich forms energy levels like in a hexamer structurein first approximation t ws ≫ t w , t s . Each hexamer ismade of two wire sites and the four substrate orbitalsthat are strongly hybridized to these sites by the Hamil-tonian term (7). A single hexamer has six distinct energylevels and the separation increases with t ws . Finite val-ues of the hoppings t w and t s hybridize the energy levelsof the L x hexamers in the full system and form the sixbands separated from the continuum.We see a similar behavior when we use the BL represen-tation projected onto the subspace for N leg = 2 N shell =54 as displayed in Fig. 3(b). The two bands crossingthe Fermi level resemble those seen in the full systemwhile two other pairs of bands lie above and below theapproximate representation of the substrate continua, re- -20-15-10-5 0 5 10 15 20 -3 -2 -1 0 1 2 3 ω k x -4 -3 -2 A ( k x , ω ) (a)-20-15-10-5 0 5 10 15 20 -3 -2 -1 0 1 2 3 ω k x -5 -4 -3 -2 -1 A ( k x , ω ) (b) FIG. 4. Spectral functions for two NNN wires on a semicon-ducting substrate with t w = 3, t ws = 8 and t s = 1. (a) FullTWSS model. (b) NLM with N leg = 6. spectively. However, the distribution of spectral weightsin the conduction and valence bands are different fromthose of the full system due to the reduction of the num-ber of bands.We observe in Figs. 3(a) and (b) that the substrate en-ergy gap is well approximated despite the restriction to N leg < N imp . However, by investigating different num-bers of legs we find that the substrate gap increases withdecreasing N leg . This is clearly seen in Fig. 3(c) for N leg = 6. In this case, we see only six bands. Againtwo bands cross the Fermi level and are similar to thetwo central bands observed in the full system while twoother pairs of bands lie well above and below the Fermilevel, respectively. These six bands were explained usingthe hexamer limit above but it should be noticed thatin the NLM with N leg = 6 the four bands away fromthe Fermi energy are the remains of the two substratecontinua. Thus these results confirm that a 6-leg NLM(i.e. with N shell = 3) can be a good approximation ofthe full system for a pair of NN wires as long as we areconcerned with the physics occurring close to the Fermienergy ǫ F = 0 on or around the wires. This agrees withand generalizes our previous findings for a single wire ona semiconducting substrate [15].The full TWSS system with NNN wires reveals inter-esting differences in the substrate role depending on thewire positions. In Fig. 4(a) we see three isolated dis-persive features, one crossing the Fermi level inside thesubstrate band gap, one over the top of the conductionband continuum, and one below the bottom of the va-lence band continuum. Examining the central structuremore closely, as shown in Fig. 5(a), we distinguish twobands crossing the Fermi level at k x = ± π with smalldifferences in their bandwidths. These dispersions re- -3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 ω k x -3 -2 A ( k x , ω ) (a)-3-2-1 0 1 2 3 -3 -2 -1 0 1 2 3 ω k x -5 -4 -3 -2 -1 A ( k x , ω ) (b) FIG. 5. Enlarged view of the spectral functions in Fig. 4. (a)Full TWSS model. (b) NLM with N leg = 6. semble those that would be found in a pair of uncoupledone-dimensional wires with small difference in their intra-hopping terms. Thus for NNN wires we do not find anyevidence for a substrate induced hybridization of the twowires resulting in an effective two-leg ladder.As we mentioned before, the BL method suffers froma fast loss of orthogonality in systems with NNN wiresalthough it becomes more accurate for larger separationsbetween wires. Moreover, the NLM representation gen-erated with the BL method for N shell = 3 does not sufferfrom this loss of orthogonality. We can see in Figs. 4(b)and 5(b) that the bands crossing the Fermi level are re-produced even if they are somewhat smeared out. There-fore, we think that the 6-leg NLM ( N shell = 3) could bea useful approximation of the full TWSS systems withNNN wires as in the case of NN wires. Contrary to theNN wires, however, the substrate does not seem to medi-ate an effective wire-wire coupling between NNN wires.Thus we will investigate the effects of interaction inducedcorrelations for these two cases. IV. INTERACTING WIRES
In this section we investigate the 6-leg NLM approxi-mating the TWSS model for interacting spinless fermionsusing the density matrix renormalization group (DMRG)method [19, 20]. Additionally, we compare with DMRGresults for a two-leg ladder without substrate as well asfor a three-leg NLM approximating a single wire on asubstrate [17]. DMRG is a well established method forquasi-one-dimensional correlated quantum lattice mod-els [19, 20, 26, 27]. Recently, we have shown that DMRGcan be applied to NLM with relatively large widths [15–17]. In this work we compute the ground-state proper-ties of the 6-leg NLM with open boundary conditionsin the leg direction as well as in the rung direction.We always simulate an even number of rungs up to L x = 200. The finite-size DMRG algorithm is used withup to m = 1024 density-matrix eigenstates yielding dis-carded weights smaller than 10 − . We vary m and ex-trapolate the ground-state energy to the limit of vanish-ing discarded weights in order to estimate the DMRGtruncation error. We focus on half-filled systems, i.e. thenumber of spinless fermions is N = N leg × L x / t ab = 0 in our model) is expectedto be negligibly small in atomic wire systems while aLuttinger liquid can occur in the presence of interchaintwo-particle interactions [11]. Thus we restricted our in-vestigation to the 6-leg NLM without direct wire-wirecoupling, i.e. t ab = 0, and focus on two questions: (i)whether the substrate can mediate an effective couplingbetween two wires, and (ii) whether 1D physics, in par-ticular a Luttinger liquid, can occur in this system.The physics of the half-filled spinless fermion model ona two-leg ladder (i.e., our TWSS model without the sub-strate) was thoroughly investigated a few decades agousing field theoretical methods, renormalization group,and bosonization [11, 21, 28, 29] as well as exact diago-nalizations [30]. When the interaction is restricted to anearest-neighbor repulsion (i.e. V >
0) between fermionson the same leg and the only inter-leg coupling is a single-particle rung hopping t ⊥ >
0, the system does not haveany gapless phase but is a Mott insulator with a chargedensity wave, even for arbitrarily small parameters V and t ⊥ . In contrast, an uncoupled chain ( t ⊥ = 0) re-mains a gapless Luttinger liquid for 0 ≤ V ≤ t k , where t k > V > t k . The introduction of an infinitelysmall interchain hopping t ⊥ is sufficient to generate theMott gap and the long-range CDW order in analyticalstudies [11, 21]. For small t ⊥ and V , however, gap andCDW amplitudes can be too small to be identified withcertitude using DMRG due to finite-size effects.In a previous work [17] we investigated a single spin-less fermion wire on a substrate thoroughly. We foundthree phases for varying couplings V >
0: a Luttinger liq-uid phase with gapless excitations localized in the wirefor
V < V
CDW , a CDW insulating phase with excita-tions still localized in the wire for intermediate couplings V CDW < V < V BI , and a band insulator with excitationsdelocalized in the substrate for V > V BI . An importantobservation is that V CDW increases significantly with in- E p V (a)6-leg NLM t ws =0.56-leg NLM t ws =83-leg NLM t ws =0.53-leg NLM t ws =8Two-leg ladder t ⊥ =0.5 0 2 4 6 8 10 12 0 20 40 60 80 100 E p V (b)6-leg NLM t ws =0.56-leg NLM t ws =83-leg NLM t ws =0.53-leg NLM t ws =8Two-leg ladder t ⊥ =0.5 0 2 4 6 8 10 12 14 16 0 5 10 15 20 25 E p V (c) 0 2 4 6 8 10 12 14 16 0 5 10 15 20 25 E p V (d) FIG. 6. Single-particle gap E p in a 6-leg NLM as function ofthe intra-wire interaction V . The upper plots (a) and (b) showresults for two NN wires and two NNN wires, respectively.The lower figures (c) and (d) show an enlarged view of thesame results. The upper and lower triangles are results fora three-leg NLM representing a single wire on a substrate.The diamonds show results for the two-leg ladder model witha rung hopping t ⊥ = 0 . t k = 3 but nosubstrate. The ladder length is L x = 128. creasing wire-substrate hybridization t ws above the value V CDW = 2 t w for the isolated wire (i.e., in the limit t ws → A. Single-particle gap
In the light of these previous results for related sys-tems, we now discuss the gap, the CDW order param-eter, and the density distribution of excitations in the6-leg NLM representation for TWSS using DMRG. Wefirst investigate the single-particle gap which is definedas E p = E ( N + 1) + E ( N − − E ( N ) (32)where E ( N ) is the ground-state energy for a system with N fermions. Figure 6 displays this single-particle gap asfunction of the interaction V . We compare the gaps forthe 6-leg NLM with those for a 3-leg NLM describing asingle wire on a substrate using t w = 3 and the samesubstrate parameters as for the noninteracting system inthe previous section. Additionally, we show results fora two-leg ladder without substrate with a leg hopping t k = 3 and a rung hopping t ⊥ = 0 . t ⊥ as found for non-interacting systems in the previous section. Accordingto the analytical findings for two-leg ladders [11, 21],we thus expect to observe a Mott insulator with long-range CDW order in the 6-leg NLM for V >
0. For aweak wire-substrate hybridization t ws = 0 .
5, however,the effective coupling is weak and the expected smallsingle-particle gap cannot be distinguished from finite-size effects for small V . t w = 6 in the 6-leg NLM, seeFig. 6(c). This is similar to the finite-size gaps foundin both the 3-leg NLM for a single wire and the two-legladder without substrate, which are a Luttinger liquidand a Mott/CDW insulator in the thermodynamic limitfor that parameter regime, respectively. Similarly, thesingle-particle gap of the 6-leg NLM matches the Mottgap seen in the Mott/CDW phase of the 3-leg NLM fora single wire and the two-leg ladder without substratefor stronger coupling V & t w and V > t k , respectively.This agreement persists up to the points where the gapsof the 6-leg and 3-leg NLM saturate (i.e. V = V BI ≈ t ws = 0 . t ws = 8,we observe that the single-particle gaps of the 6-leg NLMand 3-leg NLM remain very similar up to the satura-tion interaction, as shown in Fig. 6(a) for t ws = 8. Forthe 3-leg NLM representing a single wire on a substratewe know that a stronger hybridization t ws results in asmaller effective interaction and thus in a larger criti-cal coupling V CDW [17]. Thus, the single-particle gapsof the 3-leg NLM seen in Figs. 6(c) for t ws = 8 and V < V
CDW ≈
19 are finite-size effects while they corre-spond to a finite Mott gap above this critical interaction.The single-particle gaps of the 6-leg NLM are not signif-icantly larger than those for the 3-leg NLM. Therefore, we cannot determine whether the single-particle gap oftwo NN wires is finite in the thermodynamic limit for all
V >
0, as expected for two-leg ladders from the abovediscussion, or whether a Luttinger liquid phase occurs atweak coupling V as in a single wire on a substrate (3-legNLM).Figures 6(b) and (d) display E p for NNN wires in the6-leg NLM. Again we observe a close agreement with theresults for a single wire represented by the 3-leg NLMfor all couplings V and t ws . For a weak wire-substratehybridization, these single-particle gaps are also close tothe Mott gap of the two-leg ladder without substrate.Thus, as for NN wires, we cannot determine whether twoNNN wires have a gapless Luttinger liquid phase at weakcoupling or are insulating for all V >
0. In summary,in the regime 0 < V < V BI , where a single wire on asubstrate exhibits 1D physics (i.e. Luttinger liquid orMott/CDW insulator), the analysis of the single-particlegap does not allow us to demonstrate distinct behaviorsbetween NN and NNN wire pairs (e.g. like two uncoupledwires or like an effective two-leg ladder) because of finite-size effects. Thus the existence and the role of an effectivesubstrate-mediated wire-wire coupling remains unclear inthat regime.Remarkably, the 3D band insulator phase reveals astriking difference between NN and NNN wires. For NNwires the effective band gap E p in the band insulatorregime (the value of E p at saturation) is significantlylower than the band gap of the 3-leg NLM for a singlewire, as seen in Fig. 6(a). For NNN wires the band gapdiffers only slightly from the value found in a single-wirerepresented by the 3-leg NLM as seen in Fig. 6(b). Theseresults demonstrate that two interacting NN wires in the6-leg NLM are coupled through the substrate while thetwo interacting NNN wires do not feel that they sharethe same substrate. Therefore, the effective substrate-mediated coupling between NN wires that we have foundfor noninteracting wires ( V = 0) in the previous section isconfirmed at least in the band insulator regime for stronginteractions ( V > V BI ). Finally, we note that the valuesof E p does not change significantly with t ws in interact-ing NLM although it increases with t ws in noninteractingNLM. B. CDW order parameter
The existence of the long-range CDW order offers an-other way to distinguish the Mott/CDW phase from theLuttinger liquid phase. In the NLM the CDW ordermanifests itself as oscillations in the ground-state localdensity in the form h g † x,l,n g x,l,n i = 12 + ( − x δ xln (33)where δ xln varies slowly with x . These oscillations breakthe particle-hole symmetry and, in principle, they canonly occur in the thermodynamic limit. However, in0 -0.4-0.2 0 0.2 0.4 0 5 10 15 20 25 30 δ l,n V(a)
NN wire, l=1, n=1NN wire, l=1, n=2NNN wire,l=1, n=1NNN wire,l=1, n=2First leg in 3-leg NLM -0.4-0.2 0 0.2 0.4 0 20 40 60 80 100 δ l,n V(b)
NN wire, l=1, n=1NN wire, l=1, n=2NNN wire,l=1, n=1NNN wire,l=1, n=2First leg in 3-leg NLM
FIG. 7. Charge-density-wave order parameters δ l,n for variouslegs ( l, n ) (see the text) in the 6-leg NLM for a pair of NN andNNN wires on a substrate. The parameters for the originalTWSS models are t w = 3, t ab = 0, t s = 1 and (a) t ws = 0 . t ws = 8. The CDW order parameters are also shown forthe first leg of a three-leg NLM representing a single wire on asubstrate as well as for the two-leg ladder model with a runghopping t ⊥ = 0 . t k = 3 but no substrate.The ladder length is L x = 128. DMRG calculations they become directly observable forfinite systems due to symmetry-breaking truncation er-rors. The CDW order parameter in each leg is thus givenby δ l,n = 1 L x X x ( − x h g † x,l,n g x,l,n i . (34)We have calculated this CDW order parameter for eachleg in the 6-leg NLM using the same parameters as in theprevious subsection. We again compare these results toDMRG results obtained for the 3-leg NLM representinga single wire on the substrate and for a two-leg ladderwithout substrate with t k = 3 and t ⊥ = 0 .
5. Figure 7displays δ l,n as a function of the interaction V .In Fig. 7(a) we compare the results for the two NN orNNN wires of the 6-leg NLM ( n = 1 , , l = 1) and thesingle wire of the 3-leg NLM with a weak wire-substratehybridization t ws = 0 . V .Moreover, they are close to the order parameter foundin the 3-leg NLM for a single wire (up to the arbitrarysign of δ l in the single wire). A small difference is visi-ble around V = 10 where the order parameters becomefinite. More importantly, however, we have previouslyestablished that the transition to the CDW phase al-ready takes place around V CDW ≈ δ l,n = 0 definitively indicates a CDW groundstate, we cannot exclude the occurrence of this CDWstate when δ l,n ≈
0. The CDW order parameter can betoo small to be detected with our approach. This can beseen also in the order parameters of the two-leg ladderwithout substrate. The order parameters in both legs areclearly finite for
V > t k = 6 but appears to be vanish-ing below this coupling, exactly as for a single spinlessfermion chain. According to analytical results [11, 21] theground state has a CDW long-range order for all V > t ⊥ = 0 . V with our method. We donot observe qualitative changes when increasing the wire-substrate hybridization up to t ws = 8, at least for theregime V < V BI , as seen in Fig. 7(b). Therefore, like forthe single-particle gap, the analysis of the CDW orderdoes not allow us to demonstrate distinct 1D behaviorsfor NN and NNN wires or to draw a conclusion aboutthe existence of an effective substrate-mediated couplingbetween wires in this parameter regime.Again a significant difference between NN and NNNwires become apparent in the band insulator phase ofthe 6-leg NLM. As seen in Figs. 7(a) and (b), the CDWorder in the two wires is out of phase, i.e. δ , = − δ , ,for V < V BI , that is before the saturation of the single-particle gap. In the two-leg ladder (without substrate),it is known exactly that the ground state has this out-of-phase configuration for all V >
0. In contrast, we mostlyobserve in-phase CDW ordering ( δ , = δ , ) for NNwires in the band insulator regime (e.g. V > V BI ≈ t ws = 0 . V > V BI ≈
50 for t ws = 8). This stablerelation between their CDW ordering indicates stronglythat each wire feels the presence of the other one in allthe above cases. However, for NNN wires in the bandinsulator regime (e.g. V > V BI ≈
21 for t ws = 0 . V > V BI ≈
70 for t ws = 8), we observe both types ofrelative ordering indifferently. This is seen as apparentlyrandom sign fluctuations of δ l,n for large V in Fig. 7(b).This observation suggests a degeneracy of the in-phaseand out-of-phase CDW ordering in the NNN wires likein two uncoupled chains with CDW order (i.e., in theground state of the two-leg ladder with V > t k and t ⊥ = 0). Therefore, the analysis of the single-particlegap and the CDW order parameter yields a qualitativelyconsistent picture for the band insulator regime only. C. Excitation density
The single-particle gap and the CDW order parametersare obvious quantities to be examined in a system thatcould have Luttinger liquid and Mott/CDW insulatingphases. We have seen in the previous two subsections,however, that they do not allow us to draw a conclusionfor intermediate interactions 0 < V < V BI . Thus we nowturn to the density distribution of low-energy excitationsto gain more information. Similar quantities have already1 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 5 10 15 20 25 30 N l,n V (a) l=1, n=1l=1, n=2-0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 5 10 15 20 25 30 N l,n V (b) l=1, n=1l=1, n=2-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0 20 40 60 80 100 N l,n V (c) l=1, n=1l=1, n=2-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0 20 40 60 80 100 N l,n V (d) l=1, n=1l=1, n=2 FIG. 8. Distribution of the single-particle excitation density N l,n defined in Eq. (35) between the two wires of a 6-legNLM with t ws = 3, t ab = 0, and t s = 1: (a) NN wires with t ws = 0 .
5, (b) NNN wires with t ws = 0 .
5, (c) NN wires with t ws = 8, and (d) NNN wires with t ws = 8. The upper andlower triangles show results for the two-leg ladder with runghopping t ⊥ = 0 . t k = 3 but no substrate.The ladder length is L x = 128. proven to be useful to understand the ground state ofinhomogeneous ladder systems [16, 17, 31, 32].The distribution of single-particle excitations in thelegs provides the clearest evidence for the difference be-tween NN and NNN wires in the 6-leg NLM. This dis-tribution is defined as the variation of the total densityin each leg of the NLM when a fermion is added to thehalf-filled system N l,n = X x h g † x,l,n g x,l,n i − L x , (35) where the expectation value is calculated for the groundstate with N = ( N leg × L x / N l,n in the wires (i.e. for l = 1) is displayed in Fig. 8for increasing interaction strength V . Note that N ,n vanishes in the band insulating regime ( V > V BI ) be-cause low-energy excitations are delocalized in (the wiresrepresenting) the substrate. Actually, this is how we candetermine V BI accurately.For V < V BI and a weak hybridization t ws = 0 . V but become localized in one wirefor stronger interactions. A different behavior is found inthe 6-leg NLM for NNN wires with the same parameters,as shown in Fig. 8(b). In that case the single-particle ex-citations are entirely localized in one wire starting fromthe smallest value of V . The difference between NN andNNN wires remains qualitatively similar in systems withstronger wire-substrate hybridization t ws . This is illus-trated in Figs. 8 (c) and (d) for t ws = 8. The mainchange for stronger t ws is that an increasing fraction ofthe density is distributed in the substrate shells. Thusthe difference between both types of excitations becomesless striking for N ,n alone.A localization of excitations in the wires (for weak t ws or in the shells around each wire for strong t ws ) is similarto our findings for a single wire on a substrate [17]. Itconfirms the 1D nature of the wires in the TWSS modelrepresented by the 6-leg NLM for V < V BI . In addition,the behavior of low-energy excitations is similar in theNN wires and in the two-leg ladder with a finite runghopping t ⊥ and thus shows that the NN wires are ef-fectively coupled. In contrast, low-energy excitations ofNNN wires behave like in two uncoupled chains, e.g. inthe two-leg ladder with t ⊥ → V is finite butnot too strong. This complements our findings for non-interacting systems ( V = 0) in the previous section andfor the band insulator regime ( V > V BI ) in the previoustwo subsections. V. CONCLUSION
In previous works we showed how to map models fora single correlated quantum wire deposited on an insu-lating substrate onto narrow ladder models (NLM) thatcan be studied with the DMRG method [15]. We usedthis approach to show that the 1D Luttinger liquid andCDW insulating phases found in isolated spinless fermionchains can survive the coupling to a substrate [17]. In thiswork we have extended this mapping to multi wires ona substrate using a block Lanczos algorithm. A minimal26-leg NLM has been used successfully to approximate asystem made of two wires on a semiconducting substrate(TWSS) but numerical errors originating from the loss oforthogonality of block Lanczos vectors could be an issuefor broader ladders.We have applied this approach to an interacting spin-less fermion model for two wires on a substrate. Study-ing the resulting 6-leg NLM without a direct couplingbetween wires, we have found that low-energy single-particle excitations are localized in or around the wiresfor weak to intermediate interactions V . Thus the TWSSrealizes an effectively 1D correlated system in agreementwith the previous detailed study of a single wire on asubstrate [17].The main result of the present study is the discoveryof the nonuniversal influence of the substrate on the ef-fective ladder system built by the two wires in the 6-legNLM. We have found that the two wires are coupled byan effective substrate-mediated hybridization when bothwires are deposited on top of nearest-neighbor (NN) sitesof the substrate lattice but not when they are positionedon top of next-nearest-neighbor (NNN) sites. More gen-erally, for noninteracting wires we observe a substrate-mediated coupling when adjacent wire sites belong todifferent sublattices of the bipartite lattice but not whenthey belong to the same sublattice.In the absence of a direct wire-wire coupling (asexpected in real atomic wire systems), the substrate-mediated effective coupling could then have a decisiveinfluence on the wire properties. According to analyticalresults for a two-leg spinless fermion ladder (without sub-strate), the 6-leg NLM with NN wires should be a Mottinsulator with long-range CDW order for any coupling V > V CDW like for uncoupled wires. Unfortunately, we could notverify directly that the single-particle gap or the CDWordering are different for the NN and NNN wires. Bothgaps and CDW amplitudes are very small for weak in- teractions V and weak wire-wire coupling. Due to thehigh cost of DMRG computations for the 6-leg NLM,we have not been able distinguish them from finite-sizeeffects. Nevertheless, the density istributions of single-particle excitations between the two wires indicate clearlyan effective coupling between NN wires but effectivelydecoupled NNN wires. Therefore, we cannot determinewhether the Luttinger liquid phase of isolated wires canexist when more than one wire is deposited on a sub-strate or the substrate-mediated coupling leads system-atically to insulating ground states like the Mott/CDWphase. We think that this question could be resolved inthe future for the 6-leg NLM model constructed in thiswork using other methods for 1D correlated systems, e.g.field-theoretical approaches. A question that we couldnot address in the present work is whether broader lad-der approximations of the wire-substrate systems couldlead to different conclusions. The previous systematicstudies of single wires on substrates suggest that the re-sults found in the minimal NLM (6 legs) should remainat least qualitatively valid for broader NLM.Finally, our investigations show that it may be dif-ficult to determine under which conditions the physicsof correlated one-dimensional electrons can be realizedin arrays of atomic wires on semiconducting substratesbecause they seem to depend on the model (and conse-quently material) particulars. ACKNOWLEDGMENTS
We would like to thank T. Shirakawa for fruitful discus-sions on the BL algorithm. This work was done as partof the Research Units Metallic nanowires on the atomicscale: Electronic and vibrational coupling in real worldsystems (FOR1700) of the German Research Foundation(DFG) and was supported by grant JE 261/1-2. TheDMRG calculations were carried out on the cluster sys-tem at the Leibniz University of Hannover. [1] C. Blumenstein, J. Sch¨afer, S. Mietke, S. Meyer,A. Dollinger, M. Lochner, X. Y. Cui, L. Patthey,R. Matzdorf, and R. Claessen, Nature Physics , 776(2011).[2] C. Blumenstein, J. Sch¨afer, S. Mietke, S. Meyer,A. Dollinger, M. Lochner, X. Y. Cui, L. Patthey,R. Matzdorf, and R. Claessen, Nature Physics , 174(2012).[3] Y. Ohtsubo, J. Kishi, K. Hagiwara, P. Le F`evre,F. Bertran, A. Taleb-Ibrahimi, H. Yamane, S. Ideta,M. Matsunami, K. Tanaka, and S. Kimura, Phys. Rev.Lett. , 256404 (2015).[4] K. Yaji, I. Mochizuki, S. Kim, Y. Takeichi, A. Harasawa,Y. Ohtsubo, P. Le F`evre, F. Bertran, A. Taleb-Ibrahimi,A. Kakizaki, and F. Komori, Phys. Rev. B , 241413(2013). [5] K. Yaji, S. Kim, I. Mochizuki, Y. Takeichi, Y. Ohtsubo,P. Le F`evre, F. Bertran, A. Taleb-Ibrahimi, S. Shin, andF. Komori, J. Phys.: Condens. Matter , 284001 (2016).[6] H. W. Yeom, S. Takeda, E. Rotenberg, I. Matsuda,K. Horikoshi, J. Schaefer, C. M. Lee, S. D. Kevan,T. Ohta, T. Nagao, and S. Hasegawa, Phys. Rev. Lett. , 4898 (1999).[7] S. Cheon, T.-H. Kim, S.-H. Lee, and H. W. Yeom, Science , 182 (2015).[8] Jin Sung Shin, Kyung-Deuk Ryang, and Han WoongYeom, Phys. Rev. B , 073401 (2012).[9] J. Aulbach, J. Sch¨afer, S. C. Erwin, S. Meyer, C. Loho,J. Settelein, and R. Claessen, Phys. Rev. Lett. ,137203 (2013).[10] K. Sch¨onhammer, Luttinger liquids: the basic concepts in D. Baeriswyl and L. Degiorgi (Eds.),
Strong Interac-tions in Low Dimension s (Kluwer Academic Publishers, Dordrecht, 2004).[11] T. Giamarchi,
Quantum Physics in One Dimension (Ox-ford University Press, Oxford, 2007).[12] J. S´olyom,
Fundamentals of the Physics of Solids, Vol-ume 3 - Normal, Broken-Symmetry, and Correlated Sys-tems (Springer, Berlin, 2010).[13] G. Gr¨uner,
Density waves in Solids (Perseus Publishing,Cambridge, 2000).[14] C.-W. Chen, J. Choe, and E. Morosan, Rep. Prog. Phys. , 084505 (2016).[15] A. Abdelwahab, E. Jeckelmann, and M. Hohenadler,Phys. Rev. B , 035445 (2017).[16] A. Abdelwahab, E. Jeckelmann, and M. Hohenadler,Phys. Rev. B , 035446 (2017).[17] A. Abdelwahab and E. Jeckelmann, Phys. Rev. B ,235138 (2018).[18] T. Shirakawa and S. Yunoki, Phys. Rev. B , 195109(2014).[19] S. R. White, Phys. Rev. Lett. , 2863 (1992).[20] S. R. White, Phys. Rev. B , 10345 (1993).[21] P. Donohue, M. Tsuchiizu, T. Giamarchi, and Y. Suzu-mura, Phys. Rev. B , 045121 (2001). [22] A. Allerdt, C. A. B¨usser, G. B. Martins, and A. E.Feiguin, Phys. Rev. B , 085101 (2015).[23] A. Allerdt, A. E. Feiguin, and S. Das Sarma, Phys. Rev.B , 104402 (2017).[24] F. Lange, S. Ejima, T. Shirakawa, S. Yunoki, and H.Fehske, Journal of the Physical Society of Japan ,044601 (2020).[25] W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C ++ . The Art of Scientific Com-puting (Cambridge University Press, Cambridge, 2002).[26] U. Schollw¨ock, Rev. Mod. Phys. , 259 (2005).[27] E. Jeckelmann, in Computational Many Particle Physics (Lecture Notes in Physics ), edited by H. Fehske, R.Schneider, and A. Weiße (Springer-Verlag, Berlin, Hei-delberg, 2008), p. 597.[28] M. Fabrizio, Phys. Rev. B , 15838 (1993).[29] H. Yoshioka and Y. Suzumura, Journal of the PhysicalSociety of Japan , 3811 (1995).[30] S. Capponi, D. Poilblanc, and E. Arrigoni, Phys. Rev. B , 6360 (1998).[31] A. Abdelwahab, E. Jeckelmann, and M. Hohenadler,Phys. Rev. B91