Incommensurate phases of a bosonic two-leg ladder under a flux

PAPER • OPEN ACCESS

Incommensurate phases of a bosonic two-leg

ladder under a flux

To cite this article: E Orignac et al 2016 New J. Phys. 18 055017

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PAPER

Incommensurate phases of a bosonic two-leg ladder under a

flux

E Orignac1_{, R Citro}2_{, M Di Dio}3_{, S De Palo}3,4_{and M-L Chiofalo}5

1 _{Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France}

2 _{Dipartimento di Fisica‘E.R. Caianiello’ and Spin-CNR, Università degli Studi di Salerno, Via Giovanni Paolo II, I-84084 Fisciano (Sa),} Italy

3 _{CNR-IOM-Democritos National Simulation Centre, UDS Via Bonomea 265, I-34136, Trieste, Italy} 4 _{Dipartimento di Fisica Teorica, Università Trieste, Trieste, Italy}

5 _{Dept. of Physics‘Enrico Fermi’ and INFN, Università di Pisa Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy}

Keywords: gaugefields, Meissner to vortex transition, incommensurations

Abstract

A boson two-leg ladder in the presence of a synthetic magnetic

flux is investigated by means of

bosonization techniques and density matrix renormalization group

(DMRG). We follow the quantum

phase transition from the commensurate Meissner to the incommensurate vortex phase with

increasing

flux at different fillings. When the applied flux is ρπ and close to it, where ρ is the filling per

rung, we

find a second incommensuration in the vortex state that affects physical observables such as

the momentum distribution, the rung

–rung correlation function and the spin–spin and charge–

charge static structure factors.

A remarkable characteristic of charged systems with broken U(1) global gauge symmetry such as

superconductors is the Meissner–Ochsenfeld effect [1]. In the Meissner phase, below the critical field Hc1, a superconductor behaves as a perfect diamagnet, i.e. it develops surface currents that fully screen the external magneticfield. In a type-II superconductor, for fields above H>Hc1, an Abrikosov vortex lattice phase is formed in the system, where the magneticfield penetrates into vortex cores. In quasi one-dimensional systems, analogues of the Meissner and Abrikosov vortex lattice have been predicted for the bosonic two-leg ladder[2–5],

the simplest system where orbital magneticfield effects are allowed. It was shown that in this model, the

quantum phase transition between the Meissner and the Vortex phase is a commensurate-incommensurate(C– IC) transition [6–8]. For ladder systems at commensurate filling, a chiral Mott insulator phase with currents

circulating in loops commensurate with the ladder was obtained[9–12]. Initially, Josephson junction arrays [13– 16] were proposed as experimental realizations of bosonic one-dimensional systems [17,18]. However,

Josephson junctions are dissipative and open systems[19–21] that cannot be described using a Hermitian

many-body Hamiltonian in a canonical formalism. Moreover, the quantum effects in the vortex phase of the Josephson ladder are weak[22]. Fortunately, with the recent advent of ultracold atomic gases, another route to realize low

dimensional strongly interacting bosonic systems has opened[23–25]. Atoms being neutral, it is necessary to

find a way to realize an artificial magnetic flux acting on the ladder. Alternatively, one can consider the mapping of the two-leg ladder bosonic model to a two-component spinor boson model in which the bosons in the upper leg become spin-up bosons and the bosons in the lower leg spin-down bosons. Under such mapping, the magneticflux of the ladder becomes a spin–orbit coupling for the spinor bosons. Theoretical proposals to realize either artificial gauge fields or artificial spin–orbit coupling have been put forward [26,27], and an artificial spin–

orbit coupling has been achieved in a cold atoms experiment[28,29]. Recently, the Meissner effect and the

formation of a vortex state have been observed for non-interacting ultracold bosonic atoms bosons on a two leg ladder in artificial gauge fields induced by laser-assisted tunneling [30]. The behavior of the chiral current as a

function of the coupling strength along the rungs of the ladder, indicates a diamagnetic phase when it reaches a saturated maximum and a vortex lattice phase when it starts to decrease. This experimental achievement has revived the theoretical interest for bosonic ladders in the presence of magneticflux and its spinor-boson equivalent in the presence of interaction, where an even richer phase diagram is expected[5,31–38].

OPEN ACCESS

RECEIVED

23 January 2016

REVISED

8 April 2016

ACCEPTED FOR PUBLICATION

28 April 2016

PUBLISHED

27 May 2016

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In the present manuscript, we study the C–IC transitions of the hard-core boson ladder with equal densities in the two legs, for varying interleg coupling andflux [39] and fixed fillings away from half-filling. We confirm

that above a threshold in the interleg coupling, the Meissner phase is stable for allfluxes [40] while below that

threshold the C–IC phase transition [3] to the vortex phase takes place at large enough flux. However, within the

vortex phase, wefind that a second incommensuration [39] appears at a flux commensurate with the filling,

which we characterize by different observables.

The paper is organized as follows. In section1, we present the model and the Hamiltonian and define the observables. In section2we describe the bosonization treatment, the Meissner state and the C–IC transition. In

section3, we discuss the second incommensuration as a function of thefilling. Finally, in the conclusion we present the phase-diagram emerging for the half-filled case.

1. Model and Hamiltonian

The lattice Hamiltonian of the bosonic ladder in aflux [2,3] reads:

å

å

= - + - W l s s ls s s ls s s s s + + - -( † † ) † ( ) H t b e b b e b b b , 1 j j j j j j j j , , i 1, 1, i , , , ,

where the operatorb_j,(†)_sdestroys(creates) a hard core boson on site j of the σ chain. We have defined σ=±1/2 as the chain index[3,39], λ as the flux in each plaquette (corresponding to a Landau gauge with the vector

potential parallel to the legs), Ω as the interchain hopping. The t eiλ σis the hopping amplitude on the chainσ. A schematic picture of the model and its relevant parameters is shown infigure1. This hard-core boson model can be mapped into a spin-ladder model with Dzialoshinskii–Moriya interactions [41,42], as detailed in appendixA. As a result of translational invariance and parity, the spectrum of the Hamiltonian(1) is even and 2π-periodic

inλ.

The leg-current operator JP( j, λ) is defined as:

å

l s l = - - = ¶ ¶ s s ls s s ls s l + + -( ) ( † † ) ( ) J j , it b_j, ei bj 1, bj 1, e i bj, H , 2 while the rung current is defined as:

= - W

-^( ) (    ) ( )

† †

J j i b b_j, j b bj, j . 3 The average densities of bosons are r =_s Ns

L, where Nσis the number of particles in chainσ and L is the

length of the chain. In the rest of the manuscript, we will be considering afixed total density r=r_+r_. In the absence of appliedflux λ the ground state of the system is a rung-Mott Insulator for ρ=1 and a superfluid for ρ<1 [43]. This situation is not changed at finite λ so that for ρ=1 Mott–Meissner and Mott–Vortex phase

[11,12,40] are obtained.

For our analysis, we are interested in the following observables: the rung-current correlator C(k)

å

= á^ ^ ñ -( ) ( ) ( ) ( ) C k J j J 0 e , 4 j kj i

the leg-symmetric density correlator Sc(k)

å

= á ñ s s s s ¢ ¢ -( ) ( ) S k n n e , 5 j j kj c , , , 0, i

Figure 1. Schematic representation of the Hamiltonian equation(1). The presence of an artificial magnetic flux λ per plaquette,

induces the hopping terms on the chain to acquire a phase that depends on the spin_{(chain). No double occupancy is allowed due to} hard-core interaction.

the leg-antisymmetric density correlator Ss(k)

å

ss = ¢á ñ s s s s ¢ ¢ -( ) ( ) S k n n e 6 j j kj s , , , 0, i

and the leg-resolved momentum distribution nσ(k)

å

= á ñ s( ) †s s - ( ) n k b b e . 7 j j, 0, ikj

The non leg-resolved momentum distribution isn k( )=n k( )+n k . The latter quantity is accessible in( )

time-of-flight spectroscopy [30].

2. Bosonization of the two-leg boson ladder

We apply Haldane’s bosonization of interacting bosons [44] to the Hamiltonian(1) assuming that Ω is a

perturbation. In the absence of interchain couplings and spin–orbit coupling, the Hamiltonian of the bosons can be written as:

ò

å

_p p f = P + ¶ s s s s s s s ( ) ( ) ( ) ⎡ ⎣⎢ ⎤ ⎦⎥ H x u K u K d 2 x , 8 0 2 2

where f[ a( )x ,Pb( )]x¢ =id dab (x- ¢x), uσis the velocity of excitations, Kσis the Tomonaga–Luttinger (TL) exponent. In the case of hard-core bosons,us=2 sint (prs0)and Kσ=1.

Introducing thefieldsqa=p

ò

xPa, we can represent[44] the boson annihilation operators as:

å

y f pr = = -s s q s s s = +¥ s ( ) ( ) ( ) ( ( ) ( ) ) ( ) b a x e A cos 2m x 2m x 9 j _x m m i 0 0

and the density operators[44] as:

å

r r p f f pr = = - ¶ + -s s s s s s s = ¥ ( ) ( ) ( ) ( ( ) ( ) ) ( ) n a x B m x m x 1 cos 2 2 . 10 j x m m 0 1 0

Here, we have introduced the lattice spacing a, while Amand Bmare non-universal coefficients. In the case of hard core bosons at halffilling, these coefficients have been found analytically [45]. From equation (9), we

deduce the bosonized expression of the interchain hopping as:

ò

q q

= -W ( - ) ( )

Hhop. A02 d cosx , 11

where we have kept only the most relevant term in the renormalization group sense[3].

For a model with equivalent up and down leg as equation(1), and in the absence of the spontaneous density

imbalance between the chains found for weak repulsion[46,47],u=u and K=K , it is convenient to

introduce the leg-symmetric and leg-antisymmetric representation:

P = 1 (P + P ) P = (P - P ) ( ) 2 1 2 , 12 c s f = 1 (f_+f_) f = (f_-f_) ( ) 2 1 2 , 13 c s

in order to rewrite(8)–(11) as:

= + ( ) H Hc Hs 14

ò

_p p f = ⎡ ( P ) + (¶ ) ( ) ⎣⎢ ⎤ ⎦⎥ H x u K u K d 2 x 15 c c c c 2 c c c 2

ò

_p p f

ò

q = ⎡ ( P) + (¶ ) - W ( ) ( ) ⎣⎢ ⎤ ⎦⎥ H x u K u K A x d 2 x d cos 2 . 16 s s s s 2 s s s 2 02 s

The Hamiltonian Hcdescribes the gapless leg-symmetric density modes, while Hs, which describes the leg-antisymmetric modes, has the form of a quantum sine-Gordon model[48–50] and is gapful for Ks>1/4. In a model of bosons with spin–orbit coupling, Hswould describe the spin modes, and Hcthe total density(i.e., the ‘charge’ in the bosonization literature) modes. Till now, we have not considered the effect of the flux λ. We now show that it can be exactly incorporated in the bosonized Hamiltonian. In the absence of interchain hoppingΩ, we can perform independent gauge transformations on the upper and the lower leg of the ladder. In particular, the gauge transformation:

=

s -ls ¯ s ( )

bj, e i jbj, 17

entirely removesλ from the Hamiltonian. We can then apply the Haldane bosonization(8) and (9) to theb ,¯j s

operators. Combining the resulting expressions with equation(17), we see that bj,σhas now a bosonized expression of the form:

å

y f pr = = -s s q -s s s s = +¥ s l ( ) ¯ ( ) ( ) ( ¯ ( ) ( ) ) ( ) b a x e A cos 2m x 2m x . 18 j _{x i} m m i 0 0 x a

The boson operators b_j,σcan be written in the form(9) with f_s=f¯ and q_s s( )x =q¯ ( )s x -sl_2axand the

Hamiltonian expressed in terms of thefields Π and f_σnow reads:

ò

å

_p p sl f = P + + ¶ s s s s s s s ( ) ( ) ⎜ ⎟ ⎡ ⎣ ⎢ ⎛_⎝ ⎞_⎠ ⎤ ⎦ ⎥ H x u K a u K d 2 x , 19 2 2

leading to a modified Hamiltonian for the leg antisymmetric modes

ò

_p p l f

ò

q = ⎡ ⎜ P + ⎟ + (¶ ) - W ( ) ( ) ⎣ ⎢ ⎛_⎝ ⎞_⎠ ⎤ ⎦ ⎥ H x u K a u K A x d 2 2 x d cos 2 . 20 s s s s 2 s s s 2 02 s

As discussed in[3], when W ¹ 0 the λ term is imposing a gradient of q-q, while the term(11) is imposing a

constant value of q-q. For sufficiently large values of λ it becomes energetically advantageous to populate the

ground state with solitons giving rise to an incommensurate(IC) phase. In the ladder language, such IC phase is the vortex lattice[3].

2.1. Gapful excitations in the Meissner state

The quantum sine-Gordon model(16) is integrable [51,52] and its spectrum is fully determined. The

Hamiltonian(16) for λ=0 has a gap D ~ u ∣aW u∣

-a

s s

K K

s 2 s

4 s 1, where a is the lattice spacing, for K_s>1/4. In its

ground state qá ñ º [s 0 p 2]. For 1/4<Ks<1/2, the excitations above the ground state are solitons and antisolitons with the relativistic dispersionE ks( ) = (u ks )2+ Ds2. The soliton and the antisoliton are

topological excitations of thefield θsthat carry a leg currentj_sz = u Ks s. In the case where one is considering

the gap between the ground state and an excited state of total spin current zero(i. e. containing at least one soliton and one antisoliton), the measured gap will be 2Δs. When Ks>1/2, the solitons [49] and the

antisolitons attract each other and can form bound states called breathers that do not carry any spin current. The measured gap between the ground state and the lowest zero current state will be the mass of the lightest breather [53] p p D = G G G -G W p -

-(

) (

)

(

)

(

)

( )

( ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ u a A u 4 sin 1 2 . 21 K K K K K K s s 1 8 2 8 2 2 4 1 1 4 1 4 02 K K s s s s s s 2 s 4 s 1

In the case of hard core bosons[54], which is the one considered in the numerical analysis here, we have

Kc=Ks=1, so D ~ Ws 2 3. In that limit, the Hamiltonian(16) has been studied in relation with spin-1/2

chain materials with staggered Dzialoshinskii-Moriya in a magneticfield [55–60]. With a weak spin-spin

repulsion logarithmic corrections[55,56] are actually obtained as a result a marginal flow,

and D ~ Ws 2 3∣lnW∣1 6.

Besides the solitons and antisolitons, there are two breathers[61–63], a light breather of mass Δsand a heavy breather of mass 3Ds.

The amplitude in equation(21) A0can be estimated for hard core bosons in the case of low density, using the continuum limit[64,65] or in the case of half-filling [45]. In the first case,A a = G( _p)n

2

3 2 2

02 4 01 2_{where G is the}

Barnes G function and n0is the number of particles per site, while in the second case, 0.588 352

A a 2

02  _{. This}

gives the estimates:

p p D = G G G G W ( ) ( ) ( ) ( ) ( ) ( ) ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ u a G a u n 2 1 6 2 3 3 4 3 2 2 1 4 , 22 s s 4 s 2 3 01 3

for low density, and:

p p D = G G G G W ( ) ( ) ( ) ( ) ( ) ⎛ ⎝ ⎜ ⎞_⎠⎟ u a C a u 2 1 6 2 3 2 3 4 1 4 , 23 s s 0 s 2 3 for half-filling.

2.2. Correlation functions in the Meissner state

As for Ks>1/4 the ground state of Hshasθslong-range ordered and the excitations above the ground state are gapped, the system described by(14) is a Luther-Emery liquid [66]. In such a phase,

~ á ñ s s q q ( ) ( ) ( ) bj, ei 2 ja ei , 24 ja s c₂

giving rise to correlations áb bjs js¢ñ ~ ∣á _∣( ( )_{- ¢}q _∣ ( )) ∣ñ

ja j j , , cos 2 K s 2

1 4 c . This behavior is a remnant of the single condensate

obtained in the non-interacting case[5,30]. Since

q = W ^ ( ) J A sin 2 ,₀2 s 25 we have áJ x^( )ñ =0and á ¢ ñ ~ x ^( )^( ) - - ¢∣ ∣ ( ) J x J x e x x , 26

as∣x- ¢  ¥x∣ . Thus, the average rung-current vanishes and itsfluctuations are short ranged and commensurate, so that C(k) takes a Lorentzian shape in the vicinity of k=0.

In the case of density–density and spin–spin correlation functions, we have:

å

r

å

p f f p r f ~ - ¶ + -s s ( ) ( ( ) ) ( ) ( ) a n B m m x m 1 2 cos 2 cos 2 27 j x m m , 0 c c 0 s and

å

s r

å

p f f p r f ~ - ¶ + -s s ( ) ( ( ) ) ( ) ( ) a n B m m x m 1 2 1 2 cos 2 sin 2 . 28 j x m m , 0 s c 0 s

Since thefield θsis long-range ordered, exponentialseibfsand derivatives¶nxfsof its dualfield fsare short-range ordered. As a result, the density correlations decay as(x- ¢x)-2_{at long distance leading toS k( )=K k∣ ∣ (2p)}

c c ,

while the spin–spin correlations are decaying exponentially giving a Lorentzian shape for Ss(k). Finally, if we consider the longitudinal spin current, the obtained bosonized expression is:

l l p = P + ( ) ⎛ ( ) ⎝ ⎜ ⎞_⎠⎟ J u K a 2 2 . 29 s s s 

In the Meissner phase, the linear behavior is obtained, witháJ( )l ñ =lu Ks s(2pa)-1. 2.3. C–IC transition

Adding the spin–orbit coupling λ in (16) gives a Hamiltonian for the spin modes:

ò

_p p l f

ò

q = ⎡ P + + (¶ ) - W ( ) ⎣ ⎢ ⎛ ⎝ ⎜ ⎞_⎠⎟ ⎤ ⎦ ⎥ H x u K a u K A x d 2 2 x d cos 2 . 30 s s s s 2 s s s 2 02 s

Expanding p( P +s l 2a)2and using pP = ¶s xqs, up to a constant shift, the spin–orbit coupling adds a term:

ò

l p q +u K ¶ ( ) a x 2 d , 31 x s s s to the Hamiltonian(16).

Now, if we call Nsis the number of sine-Gordon solitons andNs¯the number of antisolitons, we have:

ò

_p q - = ¶ -¥ ¥ ( ) ¯ N N dx 2 32 s x s s

and the contribution of the spin–orbit coupling is rewritten as

l -( ¯) ( ) u K a N N ,s 33 s s s

showing thatλ acts as a chemical potential for solitons or antisolitons. On the other hand, the energy cost of forming Nssolitons andNs¯antisolitons is Ds(Ns+N¯s). When l >l =

D

∣ ∣ a

u K

c s

s s, there is an energy gain to create

solitons(or antisolitons depending on the sign of λ) in the ground state. Because of the fermionic character of solitons[67], their density remains finite, and we obtain another Luttinger liquid. This is the C–IC transition [6– 8,68]. A detailed picture can be obtained for Ks=1/2, where solitons can be treated as non-interacting Fermions as discussed in appendixB.

In the IC phase, the Hamiltonian describing the Luttinger liquid of solitons is:

ò

_p l l p l l f = ( ) ( )( Pˆ ) + ( ) ¶ ( )( ) ( ) * * * * ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ H x u K u K d 2 s s s x , 34 2 s s s 2

with qs=qˆs-sign( ) ( )l q l x 2. The density of solitons is proportional to q(λ), whileus*( )l is the

renormalized velocity of excitations andK_s*( )l is the renormalized Luttinger exponent.

We now address the behavior of the observables in the IC phase. Near the transition[8,69], for

llc + 0,Ks*( )l 1 2,q( )l ñ µ l-lcandus*( )l µ l-lc. The expression of the spin current in

the IC phase now becomes:

l l p l l á ( )ñ = ⎜⎛ - ( ) ( )⎟ ( ) ⎝ ⎞⎠ J u K a q 2 sign , 35 s s 

namely the existence of afinite soliton density reduces the average spin current. This justifies the interpretation of these solitons as vortices letting the current toflow along the legs. For large λ, we have lq( ) ~∣ ∣ (l pa , so) that the expectation value of the spin current eventually vanishes for largeflux values.

Let us turn to the momentum distribution. In the IC phase and forfinite size L with periodic boundary conditions(PBC) one has:

d á _s sñ = s l ss p p ¢ - ¢ ¢ - ¢ +

(

)

( ) † ( )( ) ∣ ∣ ( ) ( *) ⎡ ⎣ ⎤⎦ b b e sin . 36 j j q x x L x x L K K , , i 1 4 c 1 4 s

As a result, for1 4( Kc)+1 4( Ks*)<1one has:

p = G - - + G + + G - - + s p p s l p s l p - -

-(

) (

) (

)

(

)

( ) ( ) ∣ ( ) ∣ ∣ ( ) ∣ * * * * * ⎜ ⎟ ⎛ ⎝ ⎞⎠ n k 2 L 2 1 cos 1 , 37 K K K K K K L k q K K L k q 1 ₄1 ₄1 _{8 8 8}1 ₈1 ₂ 1 8 1 8 2 K K 1 4 c 1 4 s c s c s c s c s

so that now n_σ(k) has a peak for k=σq(λ), whose height scales as_L1 1- (4Kc)-1 (4Ks*). That peak becomes a

power-law divergence in the limit ofL ¥. Comparing with the non-interacting case[30], these power-law

divergences are the remnant of the Bose condensate[5] formed at k=0 in the Meissner phase or at k=±q(λ)/

2 in the vortex phase.

Turning to the spin–current correlation function, in the IC phase we have [3,4]:

l á ¢ ñ ~ - ¢ p p ^ ^ - ¢ ( ) ( ) [ ( )( )] ( ) ∣ ∣ * ⎡ ⎣ ⎤⎦ J j J j cos q x x sin . 38 L j j L K 1 s

Since1 2K_s*1, the correlation function C(k) presents in the IC phase two cusps at k=±q(λ). Turning now to the density correlation function, we have:

å

~ - + pr s s s s _p p p ¢ ¢ ₊

( )

_{( )}

( ) ( ) ( ) * ⎡ ⎣ ⎤⎦ n n K L j 2 sin cos sin , 39 j _j L L _Lj K K , , 0, c 2 2 0 c s

å

ss¢ ~ - + pr s s s s _p p p ¢ ¢ ₊

( )

_{( )}

( ) ( ) ( ) * ⎡ ⎣ ⎤⎦ n n K L j 2 sin cos sin . 40 j j L L _Lj K K , , 0, s 2 2 0 c s

Since1Kc+ *Ks 2, wefind, after taking the Fourier transform (FT), that both Ss(k) and Sc(k) possess cusp singularities_S ( )_k ~_S (pr ))+_C ∣_k-pr( )∣K+K*

-c s c s 0 c s 0 c s 1in the vicinity ofk=πρ0in the vortex phase, with

evident notation for the subscript c/s. In the hard core boson system, with_Kc=_Ks*=_{1, the cusp singularities} become slope discontinuities.

Moreover, the behaviorsS kc( )~ 2Kc_p∣k∣and ( )~ _p

∣ ∣

S ks Ks₂k ask0 signal that both charge and spin

excitations are gapless in the vortex phase.

We performed numerical simulations for the hard-core spinless bosons on a two-leg ladder as a function of flux and interchain hopping and for different fillings by means of density matrix renormalization group (DMRG) simulations [70,71] with PBC. Simulations are performed for sizes up to L=64, keeping up to

M=1256 states during the renormalization procedure. The truncation error, that is the weight of the discarded states, is at most of order 10−6, while the error on the ground-state energy is of order 5×10−5at most.

A summary for the behavior of observables and correlation functions across the C–IC transition at two differentfillings is shown in figure2forρ=0.75 and in figure3forρ=0.5. In both cases, no spontaneous density imbalance[46,47] between the chains is present. In each panel (a) of the two figures we compare the

behavior of the FT of the rung-current correlation function C(k) in the Meissner phase and in the Vortex phase. The numerical data confirm the prediction of a structureless shape in the Meissner phase and the appearance of two cusp-like peaks in the Vortex phase, respectively at k=q(λ) and k=2π−q(λ). Since we show data in the vortex phase far from the transition, q(λ)=λ, as expected. The spin gap closure in the Vortex phase is visible also in the low-momentum behavior of the spin static structure factor Ss(k) displayed in each panel (b) of the two

figures2and3: in the Vortex phaseS ks( )=K k 2s∣ ∣ pwhile in the Meissner phase Ss(k)=Ss(0)+ak2with Ss(0)>0. In these cases Ks=1 as expected for a hard-core boson system. At large momenta the Lorentzian profile centered at k=π, characteristic of the Meissner phase, is replaced by two slope discontinuities at k=πρ and k=2π−πρ as expected in the Vortex phase for Ks=Kc=1. The same evolution can be seen in the charge static structure factors shown in the(c) panels of figures2and3. The commensurate-incommensurate transition is clearly visible in the momentum distribution shown in panels(d) of figures2and3: in the Meissner

Figure 2. First incommensuration: appearance of the standard Vortex phase. DMRG simulation for L=64 in PBC for λ=π/4 and ρ=0.75 (λ=ρπ) . FT of the correlation functions described in the text for two different values of interchain coupling Ω/t=0.0625 (black solid line) and 0.5(red solid line), respectively in the Vortex and Meissner phase. Panel (a): rung-current correlation function C(k). Panel (b): spin correlation function Ss(k) multiplied by a factor 2. Panel (c): charge correlation Sc(k) divided by a factor 2. Panel

(d): total momentum distribution n(k). See text for the corresponding definitions. The blue dashed lines in panels (b) and (c) signal the values k=πρ and k=2π−πρ. The magenta dashed lines in panel a) signals the peaks positions of C(k), k=λ and k=2π−λ. The dark-green dashed lines in panel_{(d) signals the peaks of the momentum distribution at k=±λ/2.}

phase it presents only one cusp-like peak at k=0 as expected in a bosonic TL liquid, while the Vortex phase it is characterized by two peaks with same shape, centered at k=±q(λ)/2.

For Ks=1/2, the sine-Gordon Hamiltonian(30) can be rewritten as a free Fermion Hamiltonian [48,66] allowing a more detailed treatment of the C–IC transition [3,6]. Such a treatment sheds additional light on the

physics of this C–IC transition, providing an overall alternative description considering that no differences are expected at a qualitative level away from the Ks=1/2 case. The details of such derivation are accounted for in appendixB.

3. The second incommensuration appearing at

λ ; πρ

Asλ gets close to πρ, with ρ=N/L is the density per rung and for N/L not small compared to unity, λus/a becomes of the order of the energy cutoff us/a and the form (30) for the Hamiltonian cannot be used. In order to describe the low–energy physics at λ=ρ π, it is necessary to choose a gauge with the vector potential along the rungs of the ladder, so that the interchain hopping reads:

å

= W s ps s -s

( )

† ( ) H e b b . 41 j j j j hop , i2 , , N L

Applying bosonization to(41), we obtain from (9) the following representation for the interchain hopping:

ò

p = W f_[ - (q f+ )+ - (q f- )_]+ - f_[ (q f+ )+ (q f- )_{] ( )} H a x 2 d e e e e e e . 42 hop i 2 c i 2 s s i 2 s s i 2 c i 2 s s i 2 s s

The latter can be rewritten in terms of SU(2)1Wess–Zumino–Novikov–Witten (WZNW) currents [72]:

ò

= W _[ f₍ -+ -₎+ _{] ( )}

Hhop dx iei 2 c JR JL h.c. . 43

In the case of N/L=1, the complex exponential of equation(43) is replaced [39] by a cosinecos 2f_c. At commensuratefillings, N/L is a rational number p/m with p, m mutually prime and a termcosm 2f_cis also present in the Hamiltonian. In the presence of such term, the symmetry of the U(1) charge Hamiltonian is lowered to mand a spontaneous symmetry breaking giving rise to a charge gap and a long-range orderedei 2fc

becomes possible in the presence of long ranged interactions[73]. In such case, an insulating phase with a second

incommensuration is obtained[39].

At genericfilling, or when the termcosm 2f_cis irrelevant , we have an unbroken U(1) symmetry

f_cf_c +gand qsqs+g. In such case, the Mermin–Wagner theorem [74,75] precludes long range

ordering forfc+θs. However, since the perturbation in(43) is relevant in the renormalization group sense and has non-zero conformal spin, it is still expected to give rise to IC correlations at the strong couplingfixed point. To give a qualitative picture of such incommensuration, we turn to a mean-field treatment. Compared with the half-filled case, the assumption fá ñ =_c gwould correspond to a spontaneously broken U(1) symmetry, not permitted by the Mermin–Wagner theorem. The Gaussian fluctuations of the fcmodes around the saddle point would in fact restore the U(1) symmetry that one has to assume broken to use a mean-field theory. To partially take into account the effect of thesefluctuations, we will first solve the mean-field theory for an arbitrary value of γ, and we will then average the obtained correlation functions over γ. Such averaging procedure ensures that á_ei 2fcñ =₀, and more generally that the obtained correlation functions respect the U(1) symmetry of the

Hamiltonian. Of course, that procedure is not expected to produce quantitative estimates, since thefluctations offcare underestimated. In particular, the amplitude of the incommensuration can be less than the one expected from the meanfield theory, and the decay exponents of the correlations can be larger. But the mean field treatment is providing some insight on the correlation functions that can reveal the presence of a second incommensuration at thefixed point. Assuming fá ñ =_c g, after the transformation qsqs+gand

f_cf_c +gthe Hamiltonian Hc+Hs+Hhopcan be treated in mean-field theory [76–79], After defining :

p f = Wá + ñ = Wá ñ ( ) g a J J h 8 , 8 cos 2 , 44 R y L y s MF MF c , s c c,

using aπ/ 2 rotation around the x axis,_Jny=_J˜_nz

,_Jnz = -_J˜_ny

, and applying abelian bosonization[80], we rewrite:

ò

_p p f

ò

p f = [( ˜ )P + ¶( ˜ ) ]- ¶ ˜ ( ) H dxu h x 2 2 d , 45 MF x x s s s 2 s 2 s s

which allows us to write:

å

p f p - ᶠñ = á ñ = á + ñ = -n n = ˜ _J˜ _{J J} h _{( )} u 1 2 x _{R L} 2 . 46 z Ry Ly s , s s

In turn, this allows us to solve(44) withhs~ W2andg_c ~ W3. We obtain a gap in the total density excitations,

D ~ Wc 2, while the antisymmetric modes remain gapless and develop an incommensuration. To characterize

the incommensuration, wefirst make a shift of the field f˜ f˜ + h x

u s s ₂ s s while q q ˜ ˜ s s, thus q = q˜ _{( )} sin 2 s sin 2 s 47 q = ⎛ f˜ + _{( )} ⎝ ⎜ h x⎞_⎠⎟ u cos 2 s sin 2 s 48 s s f = - q˜ _{( )} sin 2 _s cos 2 s 49 f = ⎛ f˜ + _{( )} ⎝ ⎜ h x⎞_⎠⎟ u cos 2 _s cos 2 _s s . 50 s

Since for the rung-current in the mean-field approximation we have:

p pr q g q f g pr f q f f = W - - - + + -- -^( ) [ ( ) ( ) ( ) ( ) ( )] ( ) J x a A x A A x A A

2 sin 2 2 sin 2 2 2 2 cos 2

2 sin 2 2 cos 2 51

02 s 0 1 s c s

0 1 s c s

after the rotation wefind:

p pr g f pr g q g pr q f g pr f q f = W - + - -- - + + - ¶ - + ^( ) ( ) ˜ ( ) ( ˜ ) ( ) ˜ ˜ ( ) ˜ ˜ ˜ _{( )} ⎡ ⎣ ⎢ ⎢ ⎛ ⎝ ⎜ ⎛_⎝⎜ ⎞_⎠⎟ ⎛ ⎝ ⎜ ⎞_⎠⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟⎤ ⎦ ⎥ J x a A x h x u x A A x h x u A A x A A h x u

2 sin sin 2 cos sin 2

2 cos 2 2 sin 2 cos 2

2 sin 2 2 2 sin 2 cos 2 , 52 x 02 s s s s 0 1 s s s s 0 1 s 0 1 s s s s

we will have to take the average with respect tofsandθsand with respect toγ. The latter averaging partially takes into account the restablishment of the full U(1) symmetry by fluctuations around the mean-field. Averaging over γ gives expressions that are translationally invariant. We find:

pr pr pr pr á ¢ ñ ~ W - ¢ - ¢ - ¢ + W - ¢ - ¢ + W - ¢ - ¢ - ¢ + W - ¢ - ¢ ^( )^( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∣ ∣ ( ) ∣ ∣ ( ) ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ J j J j A A j j h j j u j j A A j j j j A j j h j j u j j A j j j j 1 cos cos 2 cos 2

cos cos 1 cos . 53

2 02 12 ₂ s s 2 02 12 ₂ 2 04 s s 2 04

We therefore see that an incommensuration develops in thek;0 and k∼2 λ=2πρ component of the rung-current and density-wave correlations. In the FT peaks are located atπρ , πρ±hs/uswhile the

singularities at 2πρ and 2πρ±hs/usare discontinuities of slope. Sincehs~ W2, the incommensuration

increases with interchain hopping. One can repeat the calculation also for the Szoperator and its correlation function ( - ¢ ~) ( )l (l¢) _{- ¢}

∣ ∣

S j j cos j cos j

j j

s 1 , giving rise to a peak atk~ pr. We note that since we have

made very crude approximations to treat thefcfluctuations, we cannot make accurate predictions on the correct value of the exponents.

Regarding the calculation of the momentum distribution, since the boson annihilation operators do not correspond to primaryfields of the SU(2)1WZNW model, we cannot derive their expression in terms ofq˜sand

f˜_susing the SU(2) rotation. However, sinceeiqs_{has conformal dimensions}(1/16, 1/16) its expression in terms of

thefieldsf˜_sandq˜scan be expressed as a sum of operators of conformal dimensions(1/16, 1/16). A general

expression for the caseλ=π has been derived previously [81]. The general form on nkconsisted of three peaks centered atπ σ±hs/usandπ σ or a single broad peak π σ, depending on the value of Ω. In the present case, a broad peak centered at 2λ σ or a narrow peak at 2λ σ plus satellites centered at ls  h u2 s sare expected. As we

noted above, these results can be derived rigorously provided we are at a commensuratefilling and a charge gap is formed. At genericfilling, or when interactions in the charge sector are insufficiently repulsive, the U(1)

symmetry of the term(43) is reestablished by quantum fluctuations. In such case, the mean field treatment is

only a suggestion that ICfluctuations will be present in a fully gapless state.

Infigure4we follow the appearance of the second incommensuration in the momentum distribution for the system atρ=0.75, spanning from below the critical λc=πρ=0.75 π up to λ=π, as from panels (a) to (f). At λ=0.75π the appearance of the secondary peaks are clearly detectable.

The positions of these peaks move towards zero with increasing theflux, and disappear completely at λ=π, where the system is back in the standard Vortex phase. In the presence of the second incommensuration, the position of the peaks is no longer proportional to the appliedflux: this is apparent in figure5, where the position kmaxof the peaks in the momentum distribution is displayed as a function ofλ for the case with ρ=0.75 and Ω/ t=1.25 (red solid dots). For lower values of Ω/t, the relation q(λ)=λ is valid on a large range, and the possible deviation at criticalλ is not appreciable, as it is seen in the figure5(open black dots).

We can also follow the evolution of the momentum distribution at the criticalλc=πρ while varying the interchain hoppingΩ. In figure6we show n(k) for the system at ρ=0.5 and fixed applied flux λ=π/2 on increasing the interchain hopping. For the case withΩ/t=0.0625, represented by the dashed black line, the gap in total density is too small to be detected in the present numerical simulation at the L=64 system size. Yet, at Ω=0.5 and 0.75 the second incommensuration becomes clearly visible with the predicted appearance of the secondary peaks.

Figure 4. Second incommensuration. DMRG simulation results at L=64 in PBC. Momentum distribution n(k) at ρ=0.75 for different values of the appliedflux λ as in the legend, spanning from below to well above the threshold λ=0.75 for the appearance of the second incommensuration. The interchain hopping isfixed at the value Ω/t=1.25.

Figure 5. Second incommensuration. DMRG simulation results at L_{=64 in PBC. Position k}maxof the peaks of the momentum

As mentioned above, the second incommensuration also shows up in the correlation function for the rung-current. Infigure7we show the FT of this quantity atλ=0.75 π for different fillings. The left panel of figure7

displays the data at a small valueΩ/t=0.0625: here, the system is in the standard Vortex phase (first incommensuration) characterized by peaks located at q(λ)=0.75 π. For the ρ=0.75 the small interchain hopping leads to second incommensuration too small to be detected for the system size of the present

simulation. In allfilling cases the peaks are located at k=q(λ)=0.75 π and k=2π−q(λ)=5/4π. The right panel offigure7displays the data atΩ/t=0.75, which is instead a sufficiently large value so that the second incommensuration becomes sizable: indeed, C(k) gets the expected second incommensuration at the predicted filling ρ=0.75, while at smaller values of the filling no qualitative differences are seen with respect to the behavior shown in the left panel. Atρ=1.0 the second incommensuration appears at λ=π and the peak at k=π, and it is still detectable for this applied flux [39].

In our previous study, we found[39] a large region of stability of the second incommensuration near the

critical value ofλ. In order to see how this region evolves with the filling, we summarize in figure8the phase diagram obtained from DMRG simulations in PBC for a system size L=64 at filling ρ=0.5, in which the boundary in the transition from Meissner to Vortex phase and the extension of the region nearλ=π/2 with the second incommensuration are visible.

Figure 6. Second incommensuration. DMRG simulation results at L=64 in PBC. Momentum distribution n(k) at ρ=0.5 and fixed appliedflux λ=πρ , for different values of Ω/t as in the legend. Dashed black line: Ω/t=0.0625, where the system is in the standard vortex phase(first incommensuration). Dashed magenta line: Ω/t=1, where the system is in the Meissner phase. Red and blue solid lines:Ω/t=0.5 and Ω/t=0.75, respectively, where the occurrence of the second incommensuration is signaled by the appearance of the secondary peaks.

Figure 7. First and second incommensuration. DMRG simulation results at L=64 in PBC. FT of the rung-current correlation function C(k) at fixed applied flux λ=0.75π for different fillings as in the legend: ρ=1.0, 0.75, 0.5, 0.25 and 0.125 are represented by black, red, green, blue and magenta solid lines, respectively. Left panel: case withΩ/t=0.0625. Right panel: case with Ω/t=0.75. Data atΩ/t=0.75 and ρ=1 has been shifted to make more evident the second incommensurations peaks.

Infigure9we show the behavior of Ss(k) for different fillings in two different situations in which only the first or also the second incommensuration appears in the left panel, the behavior in the standard vortex phase is displayed, after picking small values ofλ and Ω/t. In the right panel, we show the behavior at λ=0.25 π: here, the appearance of the second incommensuration is expected atρ=0.25, characterized by two peaks develop at k=0.25π and k=2π−0.25π, and withS ks( 0)getting a sizablefinite value and a linear momentum dependence. At the otherfillings, the spin correlation function gets small finite values at k=0 and very low peaks at k=ρπ and k=2π−ρπ, apart from the case ρ=0.125 already in the Meissner phase.

We conclude this section summarizing infigure10the effects that the appearance of the second

incommensuration produces in the different observables and quantities analyzed in the text. We see that there is almost no effect on the charge static structure factor Sc(k): as shown in figures2or3, in fact no sharp slope discontinuity at k=ρπ and k=2π−ρπ emerges in the second incommensuration with respect to the first, that is the standard vortex case. We remark the difference between theρ<1 cases analyzed in the present paper and theρ=1.0 case [39], which instead corresponds to a Mott-insulator, i.e., quadratic behavior at low

momenta. The DMRG data show gapless leg-symmetric modes forρ<1 in agreement with the

Mermin–Wagner theorem [74,75] that implies no breaking of the U(1) symmetry f_cf_c+g, qsqs+g.

Figure 8. DMRG simulation results at L=64 in PBC. Phase diagram for a hard-core bosonic system on ladder as a function of flux per plaquetteλ and Ω/t, at the filling value ρ=0.5. The occurrence of the two incommensurations is evidenced as follows. The black solid line represents the phase boundary between the Meissner and thefirst incommensuration, a standard Vortex phase. The dark– green solid dots are the points where the second incommensuration appears. The dashed blue line marks the criticalλ=0.5π at which the second incommensuration is expected. For comparison, the phase boundary between the Meissner and Vortex phase for a non-interacting system is represented as well, by the red-dashed line. Notice the enhanced size of the Meissner region in the hard-core repulsive with respect to the non-interacting case.

Figure 9. DMRG simulation results at L_{=64 in PBC. Spin static structure factor S}s(k) for different fillings as in the legend: ρ=1.0,

0.75, 0.5, 0.25, and 0.125 are represented by the black, red, green, blue, and magenta solid lines, respectively. Left panel:λ=0.1875 π and the small value of interchain hoppingΩ/t=0.0625, where the system is always in the standard Vortex phase at all fillings. Right panel:λ=0.25 π and Ω/t=0.25. Notice that the system at ρ=0.125 is in the Meissner phase.

4. Conclusion

In conclusion, we have studied the C–IC transition between the Meissner and the vortex state of a two-leg bosonic ladder in an externalflux, and the formation of a second incommensuration in the vortex state when the flux is matching the particle density. The predictions of the bosonization treatment and the results of DMRG simulations on the C–IC transition from a commensurate Meissner to a standard IC Vortex phase, and the second incommensuration, have been discussed. As expected from previous results at half-filling [39], the

occurrence of a second incommensuration has been found by the DMRG simulations whenever the ratio between theflux and the filling is equal to π. The developing of the second incommensuration can be followed in the momentum distribution, that can be readily measured in experiments[30]. A qualitative picture of the

second incommensuration, based on a phase averaging of mean-field approximation of the bosonized theory has been presented. Our DMRG results have been summarized in the interchain hopping-flux phase diagram figure8at quarter-filling, displaying the Meissner phase, as well as Vortex phase and second incommensuration.

The signatures of the second incommensuration on observables and correlation functions have been summed up infigure10. Our predictions can be tested in current experiments, where observables that we have analyzed and discussed can be accessed. A few questions remain open for future investigations. For example, one could investigate whether a second incommensuration would also be observed in multi-chain systems, such as a ladder with a few legs or a two-dimensional array of bosonic chains. From the point of view of bosonization, a more rigorous derivation of the second IC in the case of incommensuratefilling would be valuable.

Appendix A. Mapping to a spin ladder

In the hard core boson case, a representation[82] in terms of (pseudo) spins 1/2 can be introduced:

= =  +  + ( ) † † b_j, S_j,1b_j, S ,_j,2 A1 = =  -  - ( ) bj, Sj,1bj, S ,j,2 A2 = + = +     ( ) † † b b S 1 b b S 2 1 2. A3 j, j, jz,1 j, j, jz,2

Figure 10. DMRG simulation results at L_{=64 in PBC. Summary of the FT observables behavior at different fillings: ρ=1.0, 0.75,} 0.5, 0.25, and 0.125 are represented by the black, red, green, blue, and magenta solid lines, respectively. The value ofλ is set to λ=πρ andΩ/t is chosen in order to make the second incommensuration visible. Panel (a): rung-current correlation function C(k). Panel (b): spin correlation function Ss(k). Panel (c): twice the charge correlation function Sc(k). Panel (d): (spin-) chain-resolved momentum

distribution nσ(k), withn-s( )k =n k . The different curve color represent different values ofs( ) Ω/t as follows: Ω/t=1.25 (black), Ω/t=1 (red), Ω/t=0.75 (green), Ω/t=0.25 (blue), and Ω/t=0.0625 (magenta).

With such mapping, we can rewrite the Hamiltonian(1) as a two-leg ladder Hamiltonian:

å

= + + - -= + + + + ( ) ( ) ( ) ( ) H J S S S S 1 D S S S S A4 j r j r x jx r j r y j r y r j r y jx r j rx j r y 1,2 , 1, , 1, , 1, , 1,

å

+ J S S^( + - +S S+ -)+J S S^ (A5) j j,1 j,2 j,1 j,2 z jz,1 jz,2

å

å

m - (S +S )+h (S -S ), (A6) j j z j z j jz jz ,1 ,2 ,1 ,2

whereJ=t cos ,l D=t sin ,l J^= W₂,J^z =Uand h=δ/2. The term D is a uniform Dzyaloshinskii–

Moriya(DM) [41,42,83,84] interaction, with the DM vector parallel to z. For d ¹ 0, the two legs of the ladder

are exposed to a different magneticfield.

Appendix B. Fermionization approach

We have:

ò

y y y y

ò

y y y y

ò

y y y y = - ( †¶ - †¶ )- ( † + † )- ( † + † ) ( ) Hs ius dx _{R x} R L x L h dx R R L L m dx R L L R , B1 wherem= Wp A a02 , = l h u a 2 s

and the Fermion annihilation operatorsy†_{R L,} are destroying the solitons. The detailed correspondence between the Fermionic and bosonic expression of the lattice operators is derived below.

The Fermionized Hamiltonian(B1) is obtained by the following correspondence with the boson operators:

y y y y q p + = -¶ ( ) † † 2 B2 R R L L x s y y y y f p - = ¶ ( ) † † 2 _B3 R R L L x s y y y y q pa + = ( ) † † cos 2 _B4 R L L R s y y y y q pa -i( _R† L- L† R)= sin 2 s (B5)

Within Fermionization approach the single-particle correlation function y yá † ñ

R L can be evaluated by the

single-particle Green’s function of the operators that diagonalize the Hamiltonian (B33)

ò

y y p j j á † ñ = dk ác†₊c +-c†_-c -ñ ( ) 2 sin cos B6 R L k k k, k, k, k,

ò

_p = + -D L ( ) k m uk m d 2 B7 h2 2 u 2 2 p = L + - ( ) ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ m v v h h m 4 ln 2 . B8 2 2 since y yá † ñ

R L is real, there is no average current between the legs of the ladder.

However, it is possible tofind fluctuations of the current as shown in equation (B38). In the commensurate

phase the correlators can be evaluated using(B33) and one obtains

ò

y y p á ¢ ñ = -+ -¥ ¥ ¢-( ) ( ) ( ) ( ) † ( ) x x k m uk m d 4 e , B9 R L k x x 2 2 i

ò

y y p á ¢ ñ = -+ -¥ ¥ ¢-( ) ( ) ( ) ( ) † ( ) x x k m uk m d 4 e , B10 L R k x x 2 2 i

ò

y y p á ¢ ñ = -+ -¥ ¥ ¢-( ) ( ) ( ) ( ) † ( ) x x k uk uk m d 4 e , B11 R R k x x 2 2 i

ò

y y p á ¢ ñ = + -¥ ¥ ¢-( ) ( ) ( ) ( ) † ( ) x x k uk uk m d 4 e , B12 L L k x x 2 2 i so that: y y y y p á †( ) ( )¢ ñ = á †( ) ( )¢ ñ = - ⎛ ∣ - ¢∣ ( ) ⎝ ⎜ ⎞_⎠⎟ x x x x m uK m x x u 2 , B13 R L L R 0

y y y y p á †( ) ( )¢ ñ = -á †( ) ( )¢ ñ = ( - ¢) ⎛ ∣ - ¢∣ ( ) ⎝ ⎜ ⎞ ⎠ ⎟ x x x x x x m uK m x x u isign 2 , B14 R R L L 1

where K0and K1are modified Bessel functions. This leads to the result (B40) and one expects an exponential decay of the rung current correlation with correlation length u/(2m).

In the IC phase, the Fermion correlation functions are expressible instead in terms of incomplete Bessel functions[85]:

ò

p = n( ) -n ( )  w z, 1 t i e d . B15 w z t t 0 sinh

Indeed, we have(for T = 0):

ò

ò

y y p p á ¢ ñ = + - + - ¢--¥ ¥ ¢-( ) ( ) ( ) ( ) ( ) † ( ) ( ) x x k m uk m k m uk m d 4 e d 4 e , B16 R L k k k x x k x x 2 2 i 2 2 i F F

ò

ò

y y p p á ¢ ñ = + + - + - ¢--¥ ¥ ¢-( ) ( ) ( ) ( ) ( ) † ⎛ ( ) ( ) ⎝ ⎜⎜ ⎞_⎠⎟⎟ x x k uk uk m k uk uk m d 4 1 e d 4 _, e , B17 R R k k k x x k x x 2 2 i 2 2 i F F

ò

ò

y y p p á ¢ ñ = -+ + + - ¢--¥ ¥ ¢-( ) ( ) ( ) ( ) ( ) † ⎛ ( ) ( ) ⎝ ⎜⎜ ⎞_⎠⎟⎟ x x k uk uk m k uk uk m d 4 1 e d 4 e , B18 L L k k k x x k x x 2 2 i 2 2 i F F

where (ukF)2+m2 =hand we have noted that yá †R( )x yL( )x¢ ñ = áy†L( )x yR( )x¢ ñ. In equation(B16), we have

two contributions, one coming from the partiallyfilled upper band, and the other from the filled lower band which was the only contribution in the commensurate case. We see that whenh +¥,kF ¥and

y y á †_R( )x L( )x¢ ñ 0uniformly. We have:

ò

_p

ò

_pq q q + = = ¢ - - - ¢ -q q q - ¢ - ¢-( ) ( ) ( ) ( ) ( ) ( )   ⎡ ⎣⎢ ⎛ ⎝ ⎜ ⎞_⎠⎟ ⎛_⎝⎜ ⎞_⎠⎟⎤ ⎦⎥ k m uk m m u m u m x x u m x x u d 4 e d 4 e i 4 , i , i B19 k k k x x 2 2 i i sinh 0 F 0 F m x x u F F F F

ò

_p

ò

_pq q q q q q + = = ¢ - - ¢ -- - ¢ - + - ¢ -q q q - ¢ - ¢-( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )     ⎡ ⎣⎢ ⎛ ⎝ ⎜ ⎞_⎠⎟ ⎛_⎝⎜ ⎞_⎠⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟⎤ ⎦⎥ k uk uk m m u m u m x x u m x x u m x x u m x x u d 4 e d 4 e sinh i 8 , i , i , i , i B20 k k k x x 2 2 i i sinh 1 F 1 F 1 F 1 F m x x u F F F F and thus: y y q q p á †( ) ( )¢ ñ = ⎡ ( ¢ - ) - - ( ¢ - ) - ∣ - ¢∣ ( ) ⎣⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟⎤ ⎦⎥ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x x m u m x x u m x x u m uK m x x u i 4 , i , i 2 B21 R L 0 F 0 F 0 y y p q q q q p y y p q q q q p á ¢ ñ = ¢ -¢ - + ¢ -- ¢ - - - ¢ -+ - ¢ - + - ¢ - ¢ á ¢ ñ = ¢ -¢ - ¢ -- ¢ - - - ¢ -+ - ¢ - - - ¢ - ¢ -( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∣ ∣ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∣ ∣ ( ) † †         ⎡ ⎣⎢ ⎛ ⎝ ⎜ ⎞_⎠⎟ ⎛ ⎝ ⎜ ⎞_⎠⎟ ⎛_⎝⎜ ⎞_⎠⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟⎤ ⎦⎥ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣⎢ ⎛ ⎝ ⎜ ⎞_⎠⎟ ⎛ ⎝ ⎜ ⎞_⎠⎟ ⎛_⎝⎜ ⎞_⎠⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟⎤ ⎦⎥ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x x k x x x x m u m x x u m x x u m x x u m x x u x x m uK m x x u x x k x x x x m u m x x u m x x u m x x u m x x u x x m uK m x x u sin 2 i 8 , i , i , i , i isign 2 sin 2 i 8 , i , i , i , i isign 2 . B22 R R L L F 1 F 1 F 1 F 1 F 1 F 1 F 1 F 1 F 1 F 1

For large distances,∣x- ¢x∣u m, we can neglect the contribution from the lower band. The

contribution from the upper band can be obtained from the asymptotic expansions given in[85] on p 146, while

the simpler derivation can be obtained from physical arguments and is presented in the main text. Indeed, in the case ukFm, we can make the approximations:

+ ( ) ( ) m uk2 m2 1 B23 + ( ) ( ) uk uk m uk m B24 2 2  giving: y y p á ¢ ñ ¢ -¢ - + - ¢ -( ) ( ) ( ) ( ) ( ) ( ) † _{x x} k x x x x O x x sin 2 B25 R L  F 3 y y p p p á ¢ ñ = ¢ -¢ - ¢ -¢ - + ¢ -¢ - + - ¢ -( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) † ⎡ ⎣⎢ ⎤ ⎦⎥ x x k x x x x uk m k x x x x k x x k x x O x x sin 2 i cos 2 sin 2 B26 R R F F F F F 2 3 y y p p p á ¢ ñ = ¢ -¢ - + ¢ -¢ - + ¢ -¢ - + - ¢ -( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) † ⎡ ⎣⎢ ⎤ ⎦⎥ x x k x x x x uk m k x x x x k x x k x x O x x sin 2 i cos 2 sin 2 . B27 L L F F F F F 2 3

Second, in the case of ukFm, we can linearize the dispersion in the upper band around the points kF.

We can then make the approximations:

  +  ( ) ( ) ( ) u k k u k k m 1 B28 F 2 F 2 2  This time, wefind:

ò

y y p p a á ¢ ñ = = + ¢ -a ¢--¥ + ¢- ¢-( ) ( ) ( ( )) ( ) † ( ) ( ( )) ( ) x x k x x e d 2 e e 2 i B29 R R k x x k x x k x x iF 0 i i F

ò

y y p p a á ¢ ñ = = - ¢ -a - ¢- +¥ - + ¢- - ¢-( ) ( ) ( ( )) ( ) † ( ) ( ( )) ( ) x x k x x e d 2 e e 2 i B30 L L k x x k x x k x x i 0 i i F F y y p á ¢ ñ = ¢ ¢ -( ) ( ) ( ) ( ) ( ) † _{x x} m uk k x x x x 2 sin . B31 R L F F

We see that the correlator yá †R( )x yL( )x¢ ñis smaller by a factorm uk( F)~m h in that limit. If we had1

instead written a bosonized Hamiltonian, we would have found that yá †R( )x yL( )x¢ ñ =0. With equation(B29), we obtain the expression for the rung-current correlator(B43).

In the Fermionic representation(B1), the Hamiltonian is readily diagonalized in the form

å

= + -= ( ( ) ) † ( ) H r u k m h c c , B32 k r k r k r , s 2 2 _, , by writing:

å

y y j j j j = - +

-( )

( ) ( ) ( ) ⎛ ⎝ ⎜ ⎞_⎠⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x x L c c 1 e cos sin sin cos , B33 R L _k kx k k k k k k i with: j = + + ( ) e uk m uk m 2i _k i

2 2. The commensurate phase[6] is obtained for∣ ∣h <∣ ∣m and the IC phase for∣ ∣h >∣ ∣m .

We can express the currents as:

l l p y y y y = - + ( ) ⎡ ( † † ) ( ) ⎣⎢ ⎤⎦⎥ J u a 2 2 R R L L , B34 s  l = y y -y y ^( ) ( † † ) ( ) J i 2m _R L L R , B35

and theq~0 component ofnj-njas:

y y y y

- ~

-  ~

(nj nj)q 0 †R R †L L. (B36)

Using(B34), one has [3] á ñ =J u_pl

a 4

s

 in the commensurate phase, and

p l l l á ñ =J u [ - - ] ( ) a 4 B37 s 2 c 2 

in the IC phase. Thefinite-size scaling of the leg current has been derived in [86]. As y yá † ñ

R L is real, the average

rung current vanishes.

However, rung-currentfluctuations are non-vanishing. Indeed, with the help of Wick’s theorem we obtain:

y y y y y y y y

áJ x J x^( )^( )¢ ñ µ á[ †_R( )x L( )x¢ ñá †R( )x¢ L( )x ñ + á †L( )x R( )x¢ ñá †L( )x¢ R( )x ñ (B38)

y y y y y y y y

In the commensurate phase, the correlators in(B38) can be evaluated using (B33). One obtains: p á^( )^( )¢ ñ µ⎛ ∣ - ¢∣ + ∣ - ¢∣ ( ) ⎝ ⎜ ⎞_⎠⎟ ⎡ ⎣ ⎢ ⎢ ⎛ ⎝ ⎜ ⎞_⎠⎟ ⎛_⎝⎜ ⎞_⎠⎟⎤ ⎦ ⎥ ⎥ J x J x m u K m x x u K m x x u 2 s , B40 2 0 s 2 1 s 2

where K0and K1are the modified Bessel functions. The exponential decay is thus recovered for - ¢ ∣x x∣u m. Taking the FT, wefind - µ - - -( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎛ ⎝ ⎜ ⎞_⎠⎟ ⎛_⎝⎜ ⎞_⎠⎟ C C k E uk m K uk m 0 2 2 , B41 2 2 2 2

where E and K are complete elliptic integrals[87]. Using the Fermion representation, we can also show that:

p - ~ - -( ) ( ) ( ) ( ) ( ) ⎛ ⎝ ⎜ ⎞_⎠⎟ S S k E uk m 0 2 . B42 s s 2 2

In the IC phase, the Fermion correlation functions are expressible instead in terms of incomplete Bessel functions[85]. For large distances, - ¢∣x x∣u m, we can neglect the contribution from the lower band. The contribution from the upper band can be obtained from the asymptotic expansions given in[85]. In the limit

ukFm, simple physical arguments give:

p á ¢ ñ ~ - ¢ - ¢ + ^( )^( ) ( ) ( ) ( ) J x J x k x x x x cos 2 4 , B43 F 2 2 

so the Fermi wavevectorkF= h2-m2 us= ( )q l 2. Taking the FT(B43), we deduce that C(k) has slope discontinuities at k=±2kF. By contrast, in that limit, wefind that á(nj-nj)(nj¢-nj¢)ñ ~(j- ¢j)2as

expected from the bosonization arguments.

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