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

0.66 MB·

by  Edmond Orignac

Incommensurate phases of a bosonic two-leg ladder under a flux
E. Orignac
Univ Lyon, Ens de Lyon, Univ Claude Bernard,
CNRS, Laboratoire de Physique, F-69342 Lyon, France
R. Citro
6 Dipartimento di Fisica ”E.R. Caianiello”, Universita` degli Studi di Salerno,
1 Via Giovanni Paolo II 132, I-84084 Fisciano (Sa), Italy and
0 Spin-CNR, Via Giovanni Paolo II, 84084 Fisciano, Italy
2

y M. Di Dio
a CNR-IOM-Democritos National Simulation Centre, UDS Via Bonomea 265, I-34136, Trieste, Italy
M
S. De Palo

3 CNR-IOM-Democritos National Simulation Centre,
2
UDS Via Bonomea 265, I-34136, Trieste, Italy and

Dipartimento di Fisica Teorica, Universita` Trieste, Trieste, Italy
]
s
a M.-L. Chiofalo
g
Dept. of Physics ”Enrico Fermi” and INFN, Universita` di Pisa Largo Bruno Pontecorvo 3 I-56127 Pisa, Italy
-
t (Dated: May 23, 2016)
n
a A boson two–leg ladder in the presence of a synthetic magnetic flux is investigated by means
u of bosonization techniques and Density Matrix Renormalization Group (DMRG). We follow the
q quantum phase transition from the commensurate Meissner to the incommensurate vortex phase
t. with increasing flux at different fillings. When the applied flux is ρπ and close to it, where ρ is
a the filling per rung, we find a second incommensuration in the vortex state that affects physical
m
observables such as the momentum distribution, the rung-rung correlation function and the spin-
- spin and charge-charge static structure factors.
d
n
o A remarkable characteristic of charged systems with broken U(1) global gauge symmetry such as superconductors
c is the Meissner-Ochsenfeld effect1. In the Meissner phase, below the critical field H , a superconductor behaves
c1
[
as a perfect diamagnet, i.e. it develops surface currents that fully screen the external magnetic field. In a type-

2 II superconductor, for fields above H > Hc1, an Abrikosov vortex lattice phase is formed in the system, where the
v magneticfieldpenetratesintovortexcores. Inquasione-dimensionalsystems,analoguesoftheMeissnerandAbrikosov
3 vortex lattice have been predicted for the bosonic two-leg ladder2–5, the simplest system where orbital magnetic field
7
effectsareallowed. Itwasshownthatinthismodel,thequantumphasetransitionbetweentheMeissnerandtheVortex
5
phase is a commensurate-incommensurate transition6–8. For ladder systems at commensurate filling, a chiral Mott
6
insulatorphasewithcurrentscirculatinginloopscommensuratewiththeladderwasobtained9–12 Initially,Josephson
0
. junction arrays13–16 were proposed as experimental realizations of bosonic one dimensional systems.17,18 However,
1
Josephsonjunctions are dissipative and open systems19,19–21 that cannot be described using a Hermitian many-body
0
Hamiltonian in a canonical formalism. Moreover, the quantum effects in the vortex phase of the Josephson ladder
6
1 are weak22. Fortunately, with the recent advent of ultracold atomic gases, another route to realize low dimensional
: strongly interacting bosonic systems has opened23–25. Atoms being neutral, it is necessary to find a way to realize
v
an artificial magnetic flux acting on the ladder. Alternatively, one can consider the mapping of the two-leg ladder
i
X bosonic model to a two-component spinor boson model in which the bosons in the upper leg become spin-up bosons
r and the bosons in the lower leg spin-down bosons. Under such mapping, the magnetic flux of the ladder becomes
a a spin-orbit coupling for the spinor bosons. Theoretical proposals to realize either artificial gauge fields or artificial
spin orbit coupling have been put forward26,27, and an artificial spin-orbit coupling has been achieved in a cold
atoms experiment28,29 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
tunneling30. 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
magnetic flux and its spinor-boson equivalent in the presence of interaction, where an even richer phase diagram is
expected5,31–38.
In the present manuscript, we study the commensurate-incommensurate transitions of the hard-core boson ladder
withequaldensitiesinthetwolegs,forvaryinginterlegcouplingandflux39 andfixedfillingsawayfromhalf-filling. We
2
confirm that above a threshold in the interleg coupling, the Meissner phase is stable for all fluxes40 while below that
threshold the commensurate-incommensurate phase transition3 to the vortex phase takes place at large enough flux.
However, within the vortex phase, we find that a second incommensuration39 appears at a flux commensurate with
thefilling,whichwecharacterizebydifferentobservables. Thepaperisorganizedasfollows. InSec.I,wepresentthe
modelandtheHamiltoniananddefinetheobservables. InSec.IIwedescribethebosonizationtreatment,theMeissner
state and the commensurate-incommensurate(C-IC) transition. In Sec. III, we discuss the secondincommensuration
as a function of the filling. Finally, in the conclusion we present the phase-diagramemerging for the half-filled case.
I. MODEL AND HAMILTONIAN
The lattice Hamiltonian of the bosonic ladder in a flux2,3 reads:
Hλ = −t(b†j,σeiλσbj+1,σ+b†j+1,σe−iλσbj,σ)−Ω b†j,σbj,−σ, (1)
j,σ j,σ
X X
wheretheoperatorb(†) destroys(creates)ahardcorebosononsitej oftheσ chain. Wehavedefinedσ = 1/2asthe
j,σ ±
chainindex3,39, λasthe fluxineachplaquette (correspondingtoaLandaugaugewiththe vectorpotentialparallelto
the legs),Ωasthe interchainhopping. Theteiλσ is the hoppingamplitude onthe chainσ. Aschematicpictureofthe
model and its relevant parameters is shown in Fig.1. This hard-core boson model can be mapped into a spin-ladder
model with Dzialoshinskii-Moriyainteractions41,42, as detailed in Appendix A. As a result of translationalinvariance
and parity, the spectrum of the Hamiltonian (1) is even and 2π-periodic in λ.
The leg-current operator J (j,λ) is defined as:
k
∂H
Jk(j,λ)= −itσ(b†j,σeiλσbj+1,σ −b†j+1,σe−iλσbj,σ)= ∂λλ, (2)
σ
X
while the rung current is defined as:
J⊥(j)=−iΩ(b†j,↑bj↓ −b†j,↓bj↑). (3)
The averagedensities of bosons are ρ = Nσ where N is the number of particles in chain σ and L is the length of
σ L σ
the chain. In the rest of the manuscript, we will be considering a fixed total density ρ = ρ +ρ . In the absence of
applied flux λ the ground state of the system is a rung-Mott Insulator for ρ = 1 and a sup↑erflui↓d for ρ < 143. This
situation is not changed at finite λ so that for ρ=1 Mott-Meissner and Mott-Vortex phase11,12,40 are obtained.
For our analysis, we are interested in the following observables: the rung-current correlatorC(k)
C(k)= J (j)J (0) e ikj, (4)
−
h ⊥ ⊥ i
j
X
the leg-symmetric density correlator S (k)
c
S (k)= n n e ikj, (5)
c j,σ 0,σ −
h ′i
jX,σ,σ′
the leg-antisymmetric density correlatorS (k)
s
S (k)= σσ n n e ikj, (6)
s ′ j,σ 0,σ −
h ′i
j,σ,σ
X′
and the leg-resolved momentum distribution n (k)
σ
nσ(k)= hb†j,σb0,σie−ikj. (7)
j
X
Thenonleg-resolvedmomentumdistributionisn(k)=n (k)+n (k). Thelatterquantityisaccessibleintime-of-flight
spectroscopy30. ↑ ↓
3
FIG.1: SchematicrepresentationoftheHamiltonianEq.1. Thepresenceofanartificialmagneticfluxλperplaquette,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.
II. BOSONIZATION OF THE TWO-LEG BOSON LADDER
We applyHaldane’sbosonizationofinteractingbosons44 totheHamiltonian(1)assumingthatΩisaperturbation.
In the absence of interchain couplings and spin-orbit coupling, the Hamiltonian of the bosons can be written as:
dx u
H = u K (πΠ )2+ σ (∂ φ )2 , (8)
0 σ σ σ x σ
2π K
σ Z (cid:20) σ (cid:21)
X
where[φ (x),Π (x)]=iδ δ(x x), u isthevelocityofexcitations,K istheTomonaga-Luttinger(TL)exponent.
α β ′ αβ ′ σ σ
In the case of hard-core bosons,−u =2tsin(πρ0) and K =1.
σ σ σ
Introducing the fields θ =π xΠ , we can represent44 the boson annihilation operators as:
α α
R +
bjσ =ψ (x)=eiθσ(x) ∞ A(σ)cos(2mφ (x) 2mπρ(0)x), (9)
√a σ m σ − σ
m=0
X
and the density operators44 as:
njσ =ρ (x)=ρ(0) 1∂ φ + ∞ B(σ)cos(2mφ (x) 2mπρ(0)x). (10)
a σ σ − π x σ m σ − σ
m=1
X
Here, we have introduced the lattice spacing a, while A and B are non-universal coefficients. In the case of hard
m m
core bosons at half filling, these coefficients have been found analytically45. From Eq.(9), we deduce the bosonized
expression of the interchain hopping as:
H = ΩA2 dxcos(θ θ ), (11)
hop. − 0 ↑− ↓
Z
where we have kept only the most relevant term in the renormalization group sense3.
For a model with equivalent up and down leg as Eq. (1), and in the absence of the spontaneous density imbalance
betweenthechainsfoundforweakrepulsion46,47,u =u andK =K ,itisconvenienttointroducetheleg-symmetric
and leg-antisymmetric representation: ↑ ↓ ↑ ↓
1 1
Π = (Π +Π ) Π = (Π Π ) (12)
c s
√2 ↑ ↓ √2 ↑− ↓
1 1
φ = (φ +φ ) φ = (φ φ ), (13)
c s
√2 ↑ ↓ √2 ↑− ↓
4
in order to rewrite (8)– (11) as:
H = H +H (14)
c s
dx u
H = u K (πΠ )2+ c (∂ φ )2 (15)
c c c c x c
2π K
Z (cid:20) c (cid:21)
dx u
H = u K (πΠ )2+ s(∂ φ )2 ΩA2 dxcos(√2θ ). (16)
s 2π s s s K x s − 0 s
Z (cid:20) s (cid:21) Z
The Hamiltonian H describes the gapless leg-symmetric density modes, while H , which describes the leg-
c s
antisymmetric modes, has the form of a quantum sine-Gordon model48–50 and is gapful for K > 1/4. In a model
s
of bosons with spin-orbit coupling, H would describe the spin modes, and H the total density (i.e. the “charge”
s c
in the bosonization litterature) modes. Till now, we haven’t 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 hopping Ω, we can
perform independent gauge transformations on the upper and the lower leg of the ladder. In particular, the gauge
transformation:
b =e iλσj¯b (17)
j,σ − j,σ
entirely removes λ from the Hamiltonian. We can then apply the Haldane bosonization (8)–(9) to the¯b operators.
j,σ
Combining the resulting expressions with Eq. (17), we see that b has now a bosonized expression of the form:
j,σ
+
√bjσa =ψσ(x)=eiθ¯σ(x)−iσλax ∞ A(mσ)cos(2mφ¯σ(x)−2mπρ(σ0)x). (18)
m=0
X
Thebosonoperatorsb canbewrittenintheform(9)withφ =φ¯ andθ (x)=θ¯ (x) σλx andtheHamiltonian
j,σ σ σ σ σ − 2a
expressed in terms of the fields Π and φ now reads:
σ
2
dx λ u
H = u K πΠ +σ + σ (∂ φ )2 , (19)
σ σ σ x σ
2π a K
σ Z " (cid:18) (cid:19) σ #
X
leading to a modified Hamiltonian for the leg antisymmetric modes,
2
dx λ u
H = u K πΠ +√2 + s (∂ φ )2 ΩA2 dxcos(√2θ ). (20)
s Z 2π " s s(cid:18) s a(cid:19) Ks x s #− 0Z s
As discussed in Ref. 3, when Ω = 0 the λ term is imposing a gradient of θ θ , while the term (11) is imposing
a constant value of θ θ . For6sufficiently large values of λ it becomes ene↑rg−etic↓ally advantageous to populate the
ground state with so↑lit−ons↓giving rise to an incommensurate phase. In the ladder language, such incommensurate
phase is the vortex lattice3.
A. Gapful excitations in the Meissner state
Thequantumsine-Gordonmodel(16)isintegrable51,52 anditsspectrumisfullydetermined. TheHamiltonian(16)
for λ = 0 has a gap ∆s ∼ uas|aΩ/us|4K2Ks−s1, where a is the lattice spacing, for Ks > 1/4. In its ground state
θ 0[π√2]. For 1/4 < K < 1/2, the excitations above the ground state are solitons and antisolitons with the
s s
h i ≡
relativistic dispersion E (k)= (u k)2+∆2. The soliton and the antisoliton are topological excitations of the field
s s s
θ that carry a leg current jz = u K . In the case where one is considering the gap between the ground state and
s s p± s s
an excited state of total spin current zero (i. e. containing at least one soliton and one antisoliton), the measured
gap will be 2∆ . When K > 1/2, the solitons49 and the antisolitons attract each other and can form bound states
s s
called breathers that do not carry any spin current. The measuredgapbetween the groundstate and the lowestzero
current state will be the mass of the lightest breather53
2Ks
∆ = 4us Γ 8K1s−2 sin 8Kπs−2 πΓ 1− 4K1s ΩA20 4Ks−1 . (21)
s a√π (cid:16) Γ (cid:17)2Ks(cid:16) (cid:17) Γ(cid:16) 1 (cid:17) 2u 
4Ks−1 4Ks
(cid:16) (cid:17)  (cid:16) (cid:17) 
5
In the case of hard core bosons54, which is the one considered in the numerical analysis here, we have K =K =1,
c s
so ∆ Ω2/3. In that limit, the Hamiltonian (16) has been studied in relation with spin-1/2 chain materials with
s
∼
staggeredDzialoshinskii-Moriyainamagneticfield55–60. Withaweakspin-spinrepulsionlogarithmiccorrections55,56
areactually obtainedas a resulta marginalflow,and∆ Ω2/3 lnΩ1/6. Besidesthe solitons andantisolitons,there
s
∼ | |
are two breathers61–63, a light breather of mass ∆ and a heavy breather of mass √3∆ .
s s
TheamplitudeA inEq.(21)canbeestimatedforhardcorebosonsinthecaseoflowdensity,usingthecontinuum
0
limit64,65 or in the case of half-filling45. In the first case,A20a = G(3/2)4n10/2 where G is the Barnes G function and n
2 √2π 0
is the number of particles per site, while in the second case, A20a 0.588352. This gives the estimates:
2 ≃
u 2Γ(1/6) √πΓ(3/4)G(3/2)4Ωa 2/3
∆ = s n1/3, (22)
s a √πΓ(2/3)(cid:18) √2Γ(1/4) us(cid:19) 0
for low density, and:
2/3
u 2Γ(1/6) 2√πΓ(3/4)C Ωa
s 0
∆ = , (23)
s
a √πΓ(2/3) Γ(1/4) u
(cid:18) s(cid:19)
for half-filling.
B. Correlation functions in the Meissner state
As for K >1/4 the ground state of H has θ long-range ordered and the excitations above the ground state are
s s s
gapped, the system described by (14) is a Luther-Emery liquid66. In such a phase,
bj,σ eiσ√2θs(ja) eiθc√(j2a), (24)
∼h i
ginivtihnegnriosne-tinotceorarrcetliantgiocnasseh5b,j3,0σ.bjS,σin′ice∼ |hco|sj(−θjs′(|j1a/)(/4K√c2))i|2. This behaviorisa remnantofthe singlecondensateobtained
J =ΩA2sin√2θ , (25)
0 s
⊥
we have J (x) =0 and
h ⊥ i
J (x)J (x′) e−|x−x′|/ξ, (26)
h ⊥ ⊥ i∼
as x x . Thus, the average rung-current vanishes and its fluctuations 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 correlationfunctions, we have:
1 √2
n ρ(0) ∂ φ + B cos(m√2φ πmρ(0)x)cos(m√2φ ), (27)
j,σ x c m c s
a ∼ − π −
σ m
X X
and
1 1
σn ρ(0) ∂ φ + B cos(m√2φ πmρ(0)x)sin(m√2φ ). (28)
j,σ x s m c s
2a ∼ − π√2 −
σ m
X X
Since the field θs is long-range ordered, exponentials eiβφs and derivatives ∂xnφs of its dual field φs are short-range
ordered. As a result, the density correlations decay as (x x) 2 at long distance leading to S (k) = K k /(2π),
′ − c c
− | |
whilethespin-spincorrelationsaredecayingexponentiallygivingaLorentzianshapeforS (k). Finally,ifweconsider
s
the longitudinal spin current, the obtained bosonized expression is:
u K λ
s s
J (λ) = Π + . (29)
s
k √2 πa√2
(cid:18) (cid:19)
In the Meissner phase, the linear behavior is obtained, with J (λ) =λu K (2πa) 1.
s s −
h k i
6
C. Commensurate-Incommensurate transition
Adding the spin-orbit coupling λ in (16) gives a Hamiltonian for the spin modes:
dx λ 2 u
H = u K πΠ + + s (∂ φ )2 ΩA2 dxcos√2θ . (30)
s Z 2π " s s(cid:18) s √2a(cid:19) Ks x s #− 0Z s
Expanding (πΠ +λ/√2a)2 and using πΠ =∂ θ , up to a constant shift, the spin orbit coupling adds a term:
s s x s
u K λ dx
s s
+ ∂ θ , (31)
x s
a√2 π
Z
to the Hamiltonian (16).
Now, if we call N is the number of sine-Gordon solitons and N the number of antisolitons, we have:
s s¯
∞ dx
N N = ∂ θ , (32)
s s¯ x s
− π√2
Z−∞
and the contribution of the spin-orbit coupling is rewritten
u K λ
s s
(N N ), (33)
s s¯
a −
showingthatλ actsasachemicalpotentialforsolitonsorantisolitons. Onthe otherhand,the energycostofforming
N solitons and N antisolitons is ∆ (N +N ). When λ > λ = a∆s , there is an energy gain to create solitons
s s¯ s s s¯ | | c usKs
(or antisolitons depending on the sign of λ) in the ground state. Because of the fermionic character of solitons67,
their density remains finite, and we obtain another Luttinger liquid. This is the commensurate-incommensurate
transition6–8,68. A detailed picture can be obtained for K = 1/2, where solitons can be treated as non-interacting
s
fermions as discussed in App.B.
In the incommensurate (IC) phase, the Hamiltonian describing the Luttinger liquid of solitons is:
dx u (λ)
H = u (λ)K (λ)(πΠˆ )2+ ∗s (∂ φ )2 , (34)
2π ∗s s∗ s K (λ) x s
Z (cid:20) s∗ (cid:21)
with θ = θˆ sign(λ)q(λ)x/√2. The density of solitons is proportional to q(λ), while u (λ) is the renormalized
s s − ∗s
velocity of excitations and K (λ) is the renormalized Luttinger exponent.
s∗
We now address the behavior of the observables in the IC phase. Near the transition8,69, for λ λ +0, K (λ)
→ c s∗ →
1/2, q(λ) √λ λ and u (λ) √λ λ . The expression of the spin current in the IC phase now becomes:
i∝ − c ∗s ∝ − c
u K λ
s s
J (λ) = sign(λ)q(λ) , (35)
h k i 2 πa −
(cid:18) (cid:19)
namely the existence of a finite soliton density reduces the average spin current. This justifies the interpretation of
these solitons as vortices letting the current to flow along the legs. For large λ, we have q(λ) λ/(πa), so that the
∼| |
expectation value of the spin current eventually vanishes for large flux values.
Letus turnto the momentumdistribution. Inthe IC phase andfor finite size L with periodic boundary conditions
one has:
eiσq(λ)(x−x′)δσσ
hb†j,σbj,σ′i= Lsin π|x−x′| 1/(4Kc)′+1/(4Ks∗). (36)
π L
h (cid:16) (cid:17)i
As a result, for 1/(4K )+1/(4K )<1 one has:
c s∗
n (k)=2 L 1−4K1c−4K1s∗ Γ 1− 4K1c − 4K1s∗ cos 8Kπc + 8Kπs∗ Γ 8K1c + 8K1s∗ + L|k−2σπq(λ)| , (37)
σ (cid:18)2π(cid:19) (cid:16) Γ(cid:17)1 (cid:16)1 1 +(cid:17) L|(cid:16)k−σq(λ)| (cid:17)
− 8Kc − 8Ks∗ 2π
(cid:16) (cid:17)
so that now nσ(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 of L . Comparing with the non-interacting case30, these power-law divergences are
the remnant of the Bose condens→ate∞5 formed at k =0 in the Meissner phase or at k= q(λ)/2 in the vortex phase.
±
7
Turning to the spin-current correlation function, in the IC phase we have3,4:
cos[q(λ)(x x)]
′
hJ⊥(j)J⊥(j′)i∼ Lsinπ|j−j−′| K1s∗ . (38)
π L
h i
Since 1/2 K 1, the correlationfunction C(k) presents in the IC phase two cusps at k= q(λ).
≤ s∗ ≤ ±
Turning now to the density correlation function, we have:
2K cos(πρ(0)j)
c
n n + , (39)
* j,σ 0,σ′+ ∼−L2sin2 πj Lsin πj Kc+Ks∗
Xσ,σ′ L π L
K(cid:0) (cid:1) (cid:2) co(cid:0)s(π(cid:1)ρ(cid:3)(0)j)
s
σσ n n + . (40)
* ′ j,σ 0,σ′+ ∼−2L2sin2 πj Lsin πj Kc+Ks∗
Xσ,σ′ L π L
(cid:0) (cid:1) (cid:2) (cid:0) (cid:1)(cid:3)
Since1 K +K 2,wefind,aftertakingtheFouriertransform,thatbothS (k)andS (k)possesscuspsingularities
≤ c s∗ ≤ s c
Sc/s(k) Sc/s(πρ0))+Cc/s k πρ(0) Kc+Ks∗−1 in the vicinity of k =πρ(0) in the vortex phase, with evident notation
∼ | − |
for the subscript c/s. In the hard core boson system, with K = K = 1, the cusp singularities become slope
c s∗
discontinuities.
Moreover, the behaviors Sc(k) ∼ 2Kπc|k| and Ss(k) ∼ K2sπ|k| as k → 0 signal that both charge and spin excitations
are gapless in the vortex phase.
We performednumericalsimulations forthe hard–corespinlessbosons onatwo-legladderas afunction offlux and
interchain hopping and for different fillings by means of DMRG simulations70,71 with Periodic Boundary Conditions
(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 5 at most.
−
×
A summary for the behavior of observables and correlation functions across the commensurate-incommensurate
transition at two different fillings is shown in Fig. 2 for ρ = 0.75 and in Fig. 3 for ρ = 0.5. In both cases, no
spontaneous density imbalance46,47 between the chains is present. In each panel a) of the two figures we compare
the behavior of the Fourier Transform (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 S (k) displayed in each panel
s
b) of the two figures 2 and 3: in the Vortex phase S (k)=K k /2π while in the Meissner phase S (k)=S (0)+ak2
s s s s
| |
with S (0) > 0. In these cases K = 1 as expected for a hard-core boson system. At large momenta the Lorentzian
s s
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 K =K =1. The same evolution can be seen in the charge static
s c
−
structure factors shown in the c) panels of Figs.2 and 3. The commensurate-incommensurate transition is clearly
visible in the momentum distribution shown in panels d) of Figs.2 and 3: in the Meissner phase it presents only one
cusp-likepeakatk =0asexpectedinabosonicTomonaga-Luttingerliquid,whiletheVortexphaseitischaracterized
by two peaks with same shape, centered at k = q(λ)/2.
For K = 1/2, the sine-Gordon Hamiltonian±(30) can be rewritten as a free fermion Hamiltonian48,66 allowing
s
a more detailed treatment of the commensurate-incommensurate transition3,6. Such a treatment sheds additional
light on the physics of this commensurate-incommensurate transition, providing an overall alternative description
considering that no differences are expected at a qualitative level away from the K = 1/2 case. The details of such
s
derivation are accounted for in Appendix B.
III. THE SECOND INCOMMENSURATION APPEARING AT λ≃πρ
As λ gets close to πρ, with ρ = N/L is the density per rung and for N/L not small compared to unity, λu /a
s
becomes of the order of the energy cutoff u /a and the form (30) for the Hamiltonian cannot be used. In order to
s
describe the low–energyphysics atλ=ρπ, itis necessaryto chooseagaugewiththe vectorpotentialalongthe rungs
of the ladder, so that the interchain hopping reads:
Hhop. =Ω ei2πσ(NL)jb†j,σbj, σ. (41)
−
j,σ
X
8
1111....5555 0000 aaaa)))) 0000....5555 1111 1111....5555 2222 1111....5555 1111166666 -----11111 ddddd))))) -----00000.....55555 00000 00000.....55555 11111 1111166666
1111122222 1111122222
C(k)C(k)C(k)C(k) 1111 1111n(k)n(k)n(k)n(k)n(k) 88888 88888
0000....5555 0000....5555 44444 44444
0000 0000 0000....5555 1111 1111....5555 2222 0000 00000-----11111 -----00000.....55555 00000 00000.....55555 11111 00000
0000 0000....5555 kkkk((((ππππ 1111)))) 1111....5555 2222 0000 0000....5555 kkkkk(((((πππππ 1111))))) 1111....5555 2222
1111 1111 1111 1111
bbbb)))) cccc))))
2 S(k)2 S(k)2 S(k)2 S(k)ssss 0000....5555 0000....5555 S(k)/2 S(k)/2 S(k)/2 S(k)/2cccc 0000....5555 0000....5555
0000 0000 0000 0000
0000 0000....5555 1111 1111....5555 2222 0000 0000....5555 1111 1111....5555 2222
kkkk((((ππππ)))) kkkk((((ππππ))))
FIG.2: Firstincommensuration: appearanceofthestandardVortexphase. DMRGsimulation forL=64inPBCfor 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)dividedbyafactor 2. Paneld): totalmomentum distributionn(k). Seetextfor thecorrespondingdefinitions. Theblue
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.
1111....5555 0000 aaaa)))) 0000....5555 1111 1111....5555 2222 1111....5555 1111166666 -----11111 ddddd))))) -----00000.....55555 00000 00000.....55555 11111 1111166666
1111122222 1111122222
C(k)C(k)C(k)C(k) 1111 1111n(k)n(k)n(k)n(k)n(k) 88888 88888
0000....5555 0000....5555 44444 44444
0000 0000 0000....5555 1111 1111....5555 2222 0000 00000-----11111 -----00000.....55555 00000 00000.....55555 11111 00000
0000 0000....5555 kkkk((((ππππ 1111)))) 1111....5555 2222 0000 0000....5555 kkkkk(((((πππππ 1111))))) 1111....5555 2222
1111 1111 1111 1111
bbbb)))) cccc))))
2 S(k)2 S(k)2 S(k)2 S(k)ssss 0000....5555 0000....5555 S(k)/2S(k)/2S(k)/2S(k)/2cccc 0000....5555 0000....5555

0000 0000 0000 0000
0000 0000....5555 1111 1111....5555 2222 0000 0000....5555 1111 1111....5555 2222
kkkk((((ππππ)))) kkkk((((ππππ))))
FIG. 3: Thesame as in Fig. 2 for ρ=0.5.
Applying bosonization to (41), we obtain from (9) the following representation for the interchain hopping:
Ω
Hhop = dxei√2φc e−i√2(θs+φs)+e−i√2(θs−φs) +e−i√2φc ei√2(θs+φs)+ei√2(θs−φs) . (42)
2πa
Z h i h i
The latter can be rewritten in terms of SU(2) Wess-Zumino-Novikov-Witten (WZNW) currents72:
1
Hhop =Ω dx iei√2φc(JR−+JL−)+H. c . (43)
Z h i
In the case of N/L = 1, the complex exponential of Eq.(43) is replaced39 by a cosine cos√2φ . At commensurate
c
fillings, N/L is a rational number p/m with p,m mutually prime and a term cosm√2φ is also present in the
c
Hamiltonian. In the presence of such term, the symmetry of the U(1) charge Hamiltonian is lowered to Z and a
m
spontaneous symmetry breaking giving rise to a charge gap and a long-range ordered ei√2φc becomes possible in
the presence of long ranged interactions73. In such case, an insulating phase with a second incommensuration is
obtained.39
9
At generic filling, or when the term cosm√2φ is irrelevant , we have an unbroken U(1) symmetry φ φ +γ
c c c
and θ θ +γ. In such case, the Mermin-Wagner theorem74,75 precludes long range ordering for φ +θ .→However,
s s c s
→
sincetheperturbationin(43)isrelevantintherenormalizationgroupsenseandhasnon-zeroconformalspin,itisstill
expected to give rise to incommensurate correlationsat the strong coupling fixed point. To give a qualitative picture
of such incommensuration, we turn to a mean-field treatment. Compared with the half-filled case, the assumption
φ =γ wouldcorrespondtoaspontaneouslybrokenU(1)symmetry,notpermittedbytheMermin-Wagnertheorem.
c
h i
The Gaussianfluctuations ofthe φ modes aroundthe saddlepointwouldinfactrestorethe U(1)symmetrythatone
c
hastoassumebrokentouseamean-fieldtheory. Topartially takeintoaccounttheeffectofthesefluctuations,wewill
firstsolvethemean-fieldtheoryforanarbitraryvalueofγ,andwewillthenaveragetheobtainedcorrelationfunctions
overγ. Suchaveragingprocedureensuresthat ei√2φc =0,andmoregenerallythattheobtainedcorrelationfunctions
h i
respect the U(1) symmetry of the Hamiltonian. Of course, that procedure is not expected to produce quantitative
estimates, since the fluctations of φ are underestimated. In particular, the amplitude of the incommensuration can
c
be less than the one expected from the mean field theory, and the decay exponents of the correlations can be larger.
But the mean field treatment is providing some insight on the correlationfunctions that can revealthe presence of a
secondincommensurationatthefixedpoint. Assuming φ =γ,afterthetransformationθ θ +γ andφ φ +γ
c s s c c
the Hamiltonian H +H +H can be treated in meahn-fiield theory76–79, After defining : → →
c s hop
g
c = 8Ω Jy +Jy ,
πa h R Lis,MF
h = 8Ω cos√2φ , (44)
s c c,MF
h i
using a π/2 rotation around the x axis, Jy =J˜z, Jz = J˜y, and applying abelian bosonization80, we rewrite:
ν ν ν − ν
dx h
HMF = u (πΠ˜ )2+(∂ φ˜ )2 s ∂ φ˜ dx, (45)
s 2π s s x s − π√2 x s
Z h i Z
which allows us to write:
1 h
∂ φ˜ = J˜z = Jy +Jy = s . (46)
− π√2h x si h νi h R Li −2πus
ν=R,L
X
In turn, this allows us to solve (44) with h Ω2 and g Ω3. We obtain a gap in the total density excitations,
s c
∆ Ω2, while the antisymmetric modes re∼main gapless∼and develop an incommensuration. To characterize the
c
inco∼mmensuration, we first make a shift of the field φ˜ φ˜ + hsx while θ˜ θ˜ , thus
s → s us√2 s → s
sin√2θ = sin√2θ˜ (47)
s s
h x
cos√2θ = sin √2φ˜ + s (48)
s s
u
(cid:18) s (cid:19)
sin√2φ = cos√2θ˜ (49)
s s
−
h x
cos√2φ = cos √2φ˜ + s . (50)
s s
u
(cid:18) s (cid:19)
Since for the rung current in the mean-field approximationwe have:
Ω
J (x) = 2A2sin(πρx √2θ γ) 2A A sin(√2θ +√2φ +2γ 2πρx)cos(√2φ )
⊥ πa 0 − s− − 0 1 s c − s
h
2A A sin(√2θ 2φ )cos(√2φ ) (51)
0 1 s c s
− −
i
after the rotation we find:
Ω h x
J (x) = [2A2 sin(πρx γ)sin(√2φ˜ + s ) cos(πρx γ)sin(√2θ˜
⊥ πa 0(cid:18) − s us − − s(cid:19)
h x
2A A cos(2γ 2πρx)sin√2θ˜ cos √2φ˜ + s
0 1 s s
− − u
(cid:18) s (cid:19)
+√2A A sin(2γ 2πρx)∂ φ˜
0 1 x s
−
h x
2A A sin√2θ˜ cos √2φ˜ + s ], (52)
0 1 s s
− u
(cid:18) s (cid:19)
10
we will have to take the averagewith respectto φ andθ and with respectto γ. The latter averagingpartially takes
s s
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:
1 h (j j )
hJ⊥(j)J⊥(j′)i ∼Ω2A20A21(j−j′)2 cos(cid:18) s u−s ′ (cid:19)cos2πρ(j−j′)+
cos2πρ(j j )
Ω2A2A2 − ′
0 1 (j j )2
− ′
h (j j ) 1 cosπρ(j j )
+Ω2A40cosπρ(j−j′)cos s u− ′ j j +Ω2A40 j j− ′ . (53)
(cid:18) s (cid:19)| − ′| | − ′|
We therefore see that an incommensuration develops in the k 0 and k 2λ = 2πρ component of the
≃ ∼
rung-current and density-wave correlations. In the Fourier transform peaks are located at πρ , πρ h /u while the
s s
±
singularities at 2πρ and 2πρ h /u are discontinuities of slope. Since h Ω2, the incommensuration increases
s s s
± ∼
with interchain hopping. One can repeat the calculation also for the S operator and its correlation function
z
S (j j ) cos(λj)cos(λj ) 1 , giving rise to a peak at k πρ. We note that since we have made very
crsude−ap′pro∼ximations to trea′t|jt−hje′|φ fluctuations, we cannot ma∼ke ±accurate predictions on the correct value of the
c
exponents.
Regarding the calculation of the momentum distribution, since the boson annihilation operators do not correspond
to primary fields of the SU(2) WZNW model, we cannot derive their expression in terms of θ˜ and φ˜ using the
1 s s
SU(2) rotation. However, since eiθs has conformal dimensions (1/16, 1/16) its expression in terms of the fields φ˜s
and θ˜ can be expressed as a sum of operators of conformal dimensions (1/16, 1/16). A general expression for the
s
case λ = π has been derived previously81. The general form on n consisted of three peaks centered at πσ h /u
k s s
±
and πσ or a single broadpeak πσ, depending onthe value of Ω. In the presentcase,a broadpeak centeredat 2λσ or
a narrow peak at 2λσ plus satellites centered at 2λσ h /u are expected. As we noted above, these results can be
s s
±
derived rigorously provided we are at a commensurate filling and a charge gap is formed. At generic filling, 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 incommensurate fluctuations
will be present in a fully gapless state.
In Fig. 4 we follow the appearance of the second incommensuration in the momentum distribution for the system
at ρ=0.75,spanning frombelow the critical λ =πρ=0.75π up to λ=π, as frompanels a) to f). At λ=0.75π the
c
appearance of the secondary peaks are clearly detectable.
-2 0 2
4 λ=0.8175 π c) 4 44 λλ==00..887755 ππ dd)) 44
n(k) nn((kk))
2 2 22 22
0 -2 0 2 0 00 00
4 λ=0.75-2 π 0 2 b) 4 4 λ=0.93--2275 π 00 22 e) 4
n(k) n(k) kk
2 2 2 2
0 0 0 0
4 λ=0.68-275 π 0 2 a) 4 -3λ=π -2 -1 0 1 2 f) 3 4
n(k) n(k)
k k
2 2 2
0 0 0 0
-2 0 2 -2 0 2
k k
FIG. 4: Second incommensuration. DMRG simulation results at L = 64 in PBC. Momentum distribution n(k) at ρ = 0.75
for different values of the applied flux λ as in the legend, spanning from below to well above the threshold λ = 0.75 for the
appearance of thesecond incommensuration. Theinterchain hoppingis fixedat thevalue Ω/t=1.25.
The positions ofthese peaksmovetowardszerowithincreasingthe flux, anddisappearcompletely atλ=π, where
the systemisbackinthe standardVortexphase. Inthe presenceofthe secondincommensuration,the positionofthe
peaks is no longerproportionalto the appliedflux: this is apparentin Fig.5, where the positionk ofthe peaks in
max
the momentum distribution is displayedas a function of λ for the case with ρ=0.75 and Ω/t=1.25 (red soliddots).