One-dimensional two-orbital SU(N) ultracold fermionic quantum gases at incommensurate filling: a low-energy approach

# One-dimensional two-orbital SU(N) ultracold fermionic quantum gases at incommensurate filling: a low-energy approach

## Abstract

We investigate the zero-temperature phase diagram of two-orbital SU() fermionic models at incommensurate filling which are directly relevant to strontium and ytterbium ultracold atoms loading into a one-dimensional optical lattice. Using a low-energy approach that takes into account explicitly the SU() symmetry, we find that a spectral gap for the nuclear-spin degrees of freedom is formed for generic interactions. Several phases with one or two gapless modes are then stabilized which describe the competition between different density instabilities. In stark contrast to the case, no dominant pairing instabilities emerge and the leading superfluid one is rather formed from bound states of fermions.

###### pacs:
71.10.Pm, 03.75.Ss

## I Introduction

Ultracold gases of alkaline-earth-like atoms have recently attracted much interest due to their striking properties. One remarquable property is the large decoupling between electronic and nuclear spin degrees of freedom for states with zero total electronic angular momentun (1); (2). It leads to collisional properties which are independent of the nuclear spin states and the emergence of an extended SU() continuous symmetry where ( being the nuclear spin) can be as large as 10 for Sr fermionic atoms. These atoms as well as Yb, Yb ones have been cooled down to reach the quantum degeneracy. (3); (4) In this respect, the Mott-insulating phase, when loading these atoms in optical lattice, and the one-dimensional (1D) Luttinger physics of these multicomponent fermions have been investigated experimentally recently. (5); (6); (7) It paves the way of the quantum simulation of SU() many-body physics with the realization of exotic phases such as SU() chiral spin liquid states or SU() symmetry-protected topological phases. (8); (9); (10); (11); (12); (13); (14); (15); (16); (17); (18)

A second important property of alkaline-earth-like atoms is the existence of a long-lived metastable excited state () coupled to the ground state () via an ultranarrow doubly-forbidden transition. This makes them ideal systems for the realization of the most precise atomic clocks (19). The existence of these two levels might also provide new experimental realization of paradigmatic models of Kondo and heavy-fermions physics (20); (21); (22); (23); (24) or two-orbital quantum magnetism such as the Kugel-Khomskii model (25); (2). In the latter case, the two electronic states and simulate the orbital degree of freedom and the spin-exchange scattering between these states has been characterized recently experimentally in fermionic Sr and Yb atoms (26); (27); (28); (29); (30). Despite the fact that and states possess no electronic angular momentum, the tuning of interorbital interactions can be performed by exploiting the existence of an orbital Feshbach resonance between two ytterbium atoms with different orbital and nuclear spin quantum numbers. (29); (30); (31) It opens an avenue for studying the interplay between the orbital and SU() nuclear spin degrees of freedom which might lead to interesting exotic many-body physics.

In this paper, we will focus on this interplay in the special 1D case by means of perturbative and non-perturbative field-theoretical techniques which keep track explicitly of the non-Abelian SU() symmetry of the problem. In this respect, several two-orbital SU() fermionic lattice models with contact interactions can be considered in the context of alkaline-earth-like atoms.(2); (14); (32); (33); (16)

A first lattice model, the model, which is directly relevant to recent experiments (26); (27); (28), is defined from the existence of four different scattering lengths that stem from the two-body collisions with and atomic states: (2)

 Hg-e=−∑m=g,etm∑iN∑α=1(c†mα,icmα,i+1+H.c.) −μ∑m=g,e∑inm,i+∑m=g,eUmm2∑inm,i(nm,i−1) +V∑ing,ine,i+Vg-eex∑i,αβc†gα,ic†eβ,icgβ,iceα,i, (1)

where denotes the fermionic creation operator on the site with nuclear spin index ( with ) and orbital index which labels the two atomic states and , respectively. In Eq. (1), the local fermion numbers of the species at the site are defined by: . The model (1) is invariant under continuous U(1) and SU() symmetries:

 cmα,i↦eiθcmα,i,cmα,i↦∑βUαβcmβ,i, (2)

being an SU() matrix. The two transformations (2) respectively refer to the conservation of the total number of atoms, that will be called U(1) charge symmetry in the following for simplicity, and the SU() symmetry in the nuclear-spin sector. On top of these obvious symmetries, the Hamiltonian is also invariant under an U(1) orbital symmetry:

 cgα,i↦eiθocgα,i,ceα,i↦e−iθoceα,i, (3)

which means that the total fermion numbers for and states are conserved separately in Eq. (1).

A second two-orbital SU() fermionic lattice model can be defined by considering only the atoms in the state as in experiments (7) and the orbital degrees of freedom are the two degenerate first-excited and states of an 2D harmonic trap. More specifically, alkaline-earth atoms are loaded into an 1D optical lattice (running in the -direction) with moderate strength of harmonic confining potential in the direction perpendicular to the chain. It is assumed that the -level of the oscillator is fully occupied while the -levels of the oscillator are partially filled. The resulting lattice model reads as follows in the tight-binding approximation: (32); (33); (14)

 Hp-band=−t∑i,mα(c†mα,icmα,i+1+H.c.)−μ∑ini (4) + U1+U24∑in2i+∑i[2U2(Txi)2+(U1−U2)(Tzi)2],

where is a creation fermionic operator with orbital index and nuclear spin components on the th site of the optical lattice. In Eq. (4), describes the density operator on the th site and a pseudo-spin operator for the orbital degrees of freedom has been defined

 Tai = 12c†mα,iσamncnα,i, (5)

being the Pauli matrices and a summation over repeated indices (except thoses which label the lattice sites) is implied in the following. In stark contrast to the model (1), the -band model (4) has a hopping term which does not depend on the orbital state. The use of harmonic potential in the direction implies a constraint of the two coupling constants in Eq. (4): (14). As we will see later, the harmonic line plays a special role for as the result of the competition between several different instabilities. It is then interesting to consider the generalized -band model (4), which can be realized by introducing a quartic confinement potential, to fully reveal the physics along the harmonic line. As it can be easily seen, the -band model (4) enjoys an continuous symmetry which is defined by Eq. (2). Along the harmonic line , it displays an additional U(1) symmetry which is a rotation along the y-axis in the orbital subspace. Model (4) exhibits also this extended U(1) symmetry when or where it becomes equivalent to two decoupled single-orbital SU() Hubbard chain model. In remaining cases, the U(1) continuous symmetry in the orbital sector is explicitly broken in contrast to the model.

In this paper, we investigate the low-energy properties of the two-orbital SU() models (1, 4) at incommensurate filling by means of one-loop renormalization group (RG) approach and non-Abelian bosonization techniques (34); (35); (36); (37). The latter approach is crucial to fully take into account the presence of the SU() symmetry in this problem of alkaline-earth cold atoms. The half-filled case of these models has already been analysed by complementary techniques in Refs. (12); (14); (17). Several interesting phases, including symmetry-protected topologically phases, have been found due to the interplay between orbital and nuclear spin degrees of freedom. The specific case at incommensurate filling has been recently studied in Ref. (38) where the competition between different dominant superconducting pairing instabilities has been revealed. As we will show here, the physics of two-orbital SU() models turns out to be very different when . We find that the zero-temperature phase diagram of these models is characterized by competiting density instabilities. In stark contrast to the conclusion of Ref. (39), no dominant superconducting pairing instabilities can appear in an SU() spin-gap phase when since they are not singlet under the SU() symmetry when . In this respect, we find that the leading superfluid instability is rather formed from bound states of fermions giving rise to a molecular Luttinger liquid behavior at sufficiently small density. (40); (41)

The rest of the paper is organized as follows. In Sec. II, we perform the continuum limit of the two models (1, 4) in terms of left-right moving Dirac fermions. The effective low-energy Hamiltonian is then described in a basis where the SU() nuclear-spin symmetry is made explicit. The one-loop RG analysis of the continuum model is presented in Sec. III. We then map out in Sec. IV the zero-temperature phase diagram of models (1, 4). Finally, Sec. V contains our concluding remarks.

## Ii Continuum limit

In this section, the continuum limit of the two-orbital SU() Hubbard models (1, 4) at incommensurate filling is determined.

### ii.1 p-band model case

Let us first consider the weak-coupling approach to the -band model (4) at incommensurate filling. The latter model has two degenerate Fermi points . The starting point of the continuum-limit procedure is the linearization of the non-interacting energy spectrum in the vicinity of the Fermi points and the introduction of left-right moving Dirac fermions (42); (34):

 cmα,i→√a0(Lmαe−ikFx+RmαeikFx), (6)

with , , and , being the lattice spacing. The non-interacting Hamiltonian density is equivalent to that of left-right moving Dirac fermions:

 H0=−ivF(:R†mα∂xRmα:−:L†mα∂xLmα:), (7)

where is the Fermi velocity and denotes the standard normal ordering of an operator . The continuum limit of the -band model is then achieved by replacing (6) into the interacting part of the lattice model Hamiltonian (4) and keeping only non-oscillating contributions.

At incommensurate filling, there is no umklapp process which couples charge and other orbital or SU() degrees of freedom. Model (4) enjoys then a ”spin-charge” separation in the low-energy limit which is the hallmark of 1D conductors (42); (34):

 Hp-band=Hc+Hso, (8)

with . The physical properties of the charge degrees of freedom are governed by and describes the interplay between SU() nuclear spins and orbital degrees of freedom.

The charge Hamiltonian takes the form of a Tomonaga-Luttinger model with Hamiltonian density:

 Hc=vc2[1Kc(∂xΦc)2+Kc(∂xΘc)2], (9)

which accounts for metallic properties in the Luttinger liquid universality class (42); (34). In this low-energy approach, the charge excitations are described by the bosonic field and its dual field . The explicit form of the Luttinger parameters and in the weak-coupling regime can be extracted from the continuum limit and we find:

 Kc = 1√1+gc/πvF vc = vF√1+gc/πvF, (10)

with .

We now consider the continuum description of the Hamiltonian in Eq. (8). To this end, we need to introduce several chiral fermionic bilinear terms which will be useful to perform a one-loop RG analysis of . These quantities can be identified by exploiting the continuous symmetry of the lattice model (4). The non-interacting model (7) enjoys an U(2) U(2) symmetry which results from its invariance under independent unitary transformations on the left and right Dirac fermions. Its massless properties are then governed by a conformal field theory (CFT) U(2)= U(1) SU() based on this U(2) symmetry. (37) Since the -band model displays an extended global SU() symmetry, we need to decompose the SU(2) CFT, with bosonic gapless modes, into a CFT which is directly related to the SU() symmetry. The resulting decomposition is similar to the one which occurs in the multichannel Kondo problem with the use of the conformal embedding (43): U(2) U(1) SU() SU(). In this respect, we introduce the currents which generate the SU() SU() CFT of the problem:

 JaL=L†nαTaαβLnβSU(N)2 (nuclear) spin currentsjiL=12L†mασimnLnαSU(2)N orbital currentsJa,iL=1√2L†mαTaαβσimnLnβmixed currents, (11)

where () and () are respectively the Pauli matrices and SU() generators in the fundamental representation of SU() normalized such that: . The combination with defines SU(2) left currents of the non-interacting theory (7).

With all these definitions at hand, we are able to derive the continuum limit of the -band model (4). We will neglect all chiral contributions which account for velocity anisotropies and focus on the interacting Hamiltonian. One can then derive the continuum limit of in terms of the currents (11) and similar expressions for the right-moving ones. After standard calculations, we get the interacting part of :

 Hintso =g1JaLJaR+g2Ja,xLJa,xR+g3Ja,zLJa,zR+g4Ja,yLJa,yR (12) +g5jxLjxR+g6jzLjzR+g7jyLjyR,

with the following identification for the coupling constants:

 g1=−a0(U2+U1)g2=−4a0U2g3=2a0(U2−U1)g4=g7=0g5=a04(N−1)NU2g6=a02(N−1)N(U1−U2). (13)

In the interacting Hamiltonian (12), we have included additional perturbations with coupling constants which will be generated at one-loop order within the RG approach as we will see in Sec. III.

### ii.2 g−e model case

We now turn to the continuum limit of the model (1) with and so that the only two Fermi points we get do not depend on the orbital state as in the -band model, see Eq. (6). The continuum Hamiltonian of the model (1) at incommensurate filling also takes the general form (8) where the charge degrees of freedom are governed by the Tomonaga-Luttinger Hamiltonian density (9). The Luttinger parameters are still given by Eq. (10) with .

The low-energy Hamiltonian , which accounts for the interaction between orbital and nuclear spin degrees of freedom, displays the same structure as in Eq. (12). The main difference stems from the fact that we have now the constraints and in Eq. (12) since the model (1) is invariant under the U(1) orbital symmetry. After straightforward calculations, we get the following identification for the coupling constants:

 g1=−a0(U+Vg-eex)g2=g4=−2a0Vg3=−2a0(U−Vg-eex)g5=g7=2a0(Vg-eex−V/N)g6=2a0N((N−1)U+Vg-eex−NV). (14)

The anisotropic case and of the model (1) is directly related to ytterbium and strontium atoms since the scattering lengths corresponding to the collisions between and states are different experimentally. (26); (27); (28) In this anisotropic case, the non-interacting energy spectrum can have now four different Fermi points which depend on the orbital state. The linearization of the spectrum around these Fermi points is then described by:

 cmα,i→√a0(Lmαe−ikmFx+RmαeikmFx), (15)

with and . One important modification of the continuum-limit procedure is the impossibility to use the basis (11) of the low-energy approach which assumes an invariance between the orbital states to define the SU() current . In this respect, we introduce U(1) and SU() left-moving currents for each orbital state :

 JmL = :L†mαLmα:=√Nπ∂xΦmL JamL = L†mαTaαβLmβ, (16)

with and similar definitions for the right-movers. The bosonic field describes the U(1) density fluctuations in the orbital subspace.

At incommensurate filling, the continuum Hamiltonian separates into the U(1) U(1) part which stands for and orbital density fluctuations, and a non-Abelian part corresponding to the symmetry breaking SU() SU() SU() when the interactions are switched on. The Abelian part can be expressed in terms of the bosonic fields for each orbital:

 HU(1)×U(1) = ∑m=g,evm2[1Km(∂xΦm)2+Km(∂xΘm)2] (17) + a0(NV−Vg-eex)π∂xΦg∂xΦe,

where is the dual field to and the Luttinger parameters are:

 Km = (1+a0(N−1)UmmπvmF)−1/2 vm = vmF(1+a0(N−1)UmmπvmF)1/2, (18)

being the Fermi velocity for the band .

The Hamiltonian (17) can then be diagonalized by introducing the charge and orbital fields: (39)

 Φc = 1√vg+ve⎛⎝√vgKgΦg+√veKeΦe⎞⎠ Φo = 1√vg+ve⎛⎝√vgKgΦg−√veKeΦe⎞⎠, (19)

so that it takes the form of two decoupled Tomonaga-Luttinger liquid Hamiltonian densities:

 HU(1)×U(1) = vc2[1Kc(∂xΦc)2+Kc(∂xΘc)2] (20) + vo2[1Ko(∂xΦo)2+Ko(∂xΘo)2],

where and (respectively and ) are Luttinger parameters (respectively velocities) for respectively the charge and orbital degrees of freedom:

 Kc,o = ⎛⎜⎝1±a0(NV−Vg-eex)√KgKeπ√vgve⎞⎟⎠−1/2 vc,o = ¯v⎛⎜⎝1±a0(NV−Vg-eex)√KgKeπ√vgve⎞⎟⎠1/2, (21)

being the average velocity.

The interacting part of the non-Abelian sector takes the form of an SU() current-current interaction after neglecting chiral contributions as before:

 HintSU(N)=∑m=g,eλmJamLJamR+λ(JagLJaeR+JaeLJagR), (22)

with , , and .

## Iii One-loop renormalization group analysis

The one-loop RG calculation enables one to deduce the infrared (IR) properties of model (12) and thus to map out the zero-temperature phase diagram of the model (1) and -band model (4).

### iii.1 RG approach for the p-band model

We start with the -band model case. To this end, it is convenient for the RG analysis to introduce the following rescaled coupling constants:

 g1 = +πNf1 g2,3,4 = 2πNf2,3,4 g5,6,7 = 2πN2f5,6,7. (23)

After cumbersome calculations, we find the following one-loop RG equations:

 ˙f1 = 14(f21+f22+f23+f24) ˙f2 = 12f1f2+N2−42N2f3f4+1N2(f4f6+f3f7) ˙f3 = 12f1f3+N2−42N2f2f4+1N2(f2f7+f4f5) ˙f4 = 12f1f4+N2−42N2f2f3+1N2(f2f6+f3f5) ˙f5 = N2−1N2f3f4+1N2f6f7 ˙f6 = N2−1N2f2f4+1N2f5f7 ˙f7 = N2−1N2f2f3+1N2f5f6. (24)

The analysis of the RG Eqs. (24) for has been done in Ref. (38). The latter case is very special since the one-loop RG Eqs. (24) contains several terms which vanish. When case, Eqs. (24) have less symmetries and the analysis is more involved. We performed the numerical analysis of the RG Eqs. (24) by a Runge-Kutta procedure. The RG flow in the case turns out to be very slow. In this respect, the vicinity of the transition lines of Fig. 1 in the repulsive regime are difficult to be resolved numerically for . When , the analysis is much simpler and it leads to the results presented in Fig. 1.

As depicted in Fig. 1, our numerical results reveal that the RG flow goes in the strong-coupling regime in the far IR along three asymptotic lines:

 I :f2=f3=f4=f5=f6=f7=f1=f∗>0 (25) II :f2=−f3=f4=f5=−f6=f7=−f1=−f∗ III :f2=−f3=−f4=f5=−f6=−f7=f1=f∗.

These lines give rise to the three different regions of Fig. 1 that we analyse in the next section.

### iii.2 RG approach for the g−e model

We now look at the model. We first consider the isotropic case with and so that the one-loop RG equations are still given by Eqs. (24) with the initial conditions (14). The numerical analysis is presented in Fig. 2 depending on the sign of .

On top of the asymptotes (I) and (II) of Eq. (25) which occur in the -band model, we have a new one which defines the region IV in Fig. 2:

 IV :f1=−f3=f∗>0,f2,4,5,6,7f1→0. (26)

In the orbital anisotropic case and , the analysis is much simpler. Indeed, model (22) describes three commuting marginal SU() current-current interactions with one-loop RG equation of the general form: . The fate of these perturbations in the far IR limit depend then only on the sign of the coupling constants and . The different scattering lengths of two atoms collision have been determined recently experimentally for strontium and ytterbium atoms. (26); (27); (28) The interactions and are always repulsive so that for and , i.e. , for ytterbium atoms. (27); (28) Yet in the following, for completeness, we will consider the two cases and as in Fig. 2. When , the interactions in the effective low-energy Hamiltonian (22) are then marginal irrelevant and a fully-gapless -component Luttinger liquid behavior with central charge emerges in the IR limit. In contrast, when , i.e. , the last interaction in Eq. (22) are marginal relevant and opens a a spin gap for the nuclear-spin degrees of freedom. The resulting phase is then a gapless phase which stems from the two-component Luttinger behavior (20) of the charge and orbital degrees of freedom.

## Iv Phase diagrams

In this section, we investigate the nature of the dominant electronic instability when in each of the corresponding regions of Figs. 1,2. The zero-temperature phase diagrams of the model (1) and the -band model (4) can then be deduced from this analysis.

### iv.1 Region I

In region I of Figs. 1, 2, the one-loop RG equations flow along a special ray where (). The RG Eqs. (24) becomes degenerate: which signals the presence of an enlarged symmetry. In fact, in region I of Fig. 1, the Hamiltonian (12) enjoys a dynamical symmetry enlargement (44); (45) in the far IR with the emergence of an higher SU(2) symmetry which unifies orbital and nuclear-spin degrees of freedom:

 Hint∗so=f∗JALJAR, (27)

where are left chiral SU(2) currents with . The IR Hamiltonian takes the form of an SU(2) Gross-Neveu (GN) model (46) which is an integrable massive field theory when (47). The orbital and nuclear spin degrees of freedom are thus fully gapped and a critical phase is formed which stems from the gapless charge degrees of freedom described by the bosonic field in Eq. (9). Using the general duality approach to 1D interacting fermions of Ref. (48), one expects the emergence of a gapless charge-density wave (CDW) phase due to this SU(2) symmetry enlargement. Its lattice order parameter is defined through: , with , and or respectively for the and -band models. In the continuum limit (6) we get:

 n2kF=R†mαLmα. (28)

Since this operator is an SU() singlet, it is worth expressing it in the non-interacting U(1) SU() CFT basis. In this respect, we use the so-called non-Abelian bosonization which enables one to find a bosonic description for fermionic bilinears in such a basis (35); (36); (37) :

 L†lαRpβ∼exp(i√2π/NΦc)gpβ,lα, (29)

where is the SU() primary field with scaling dimension which transforms in the fundamental representation of SU(). (49) The IR physics which results from the strong-coupling regime of the SU() GN model (27) can then be inferred from a simple semiclassical analysis. In particular, the interacting part of the SU() GN model can be expressed in terms of : , so that in the ground state of that phase since . Using the correspondence (29), we thus obtain in region I

 n2kF∼exp(−i√2πKc/NΦc), (30)

where we have rescaled the charge bosonic field by its Luttinger parameter. The equal-time density-density correlation can then easily be computed as follows:

 ⟨n(x)n(0)⟩≃Acos(2kFx)x−Kc/N−NKcπ2x2, (31)

where is the continuum description of the lattice density operator and is some non-universal amplitude.

Due to the existence of the SU() nuclear-spin symmetry, there is no pairing instability which competes with the CDW in region I. Indeed, a general superconducting pairing operator is not a singlet under the SU() symmetry when . In the continuum limit, its nuclear-spin part cannot sustain a non-zero expectation value in the gapful SU(2) invariant model (27). All pairing instabilities are thus fluctuating orders with short-ranged correlation functions in stark contrast to the conclusion of Ref. (39). However, we may consider a molecular superfluid (MS) instability made of fermions as in 1D cold fermionic atoms with higher spins (40); (41); (50); (51): for the model or for the -band model. In stark contrast to fermionic pairings, is a singlet under the SU(2) symmetry. The equal-time correlation function of this MS order parameter has been determined in Refs. (40); (51) in a phase where the SU() symmetry is restored in the far IR limit as in Eq. (27):

 ⟨M†(x)M(0)⟩∼x−N/Kc. (32)

We thus see that CDW and MS instabilities compete. In particular, a dominant MS instability requires and thus a fairly large value of which, with only short-range interaction, is not guaranteed. As shown in Refs. (40); (41), in the low-density regime and for attractive interactions, such value of the Luttinger parameter can be reached for onsite interactions which signals the emergence of a molecular Luttinger phase. Appart from that case, we expect a dominant CDW in region I.

### iv.2 Region II

In region II of Figs. 1,2, the one-loop RG equations flow along the special ray (II) of Eq. (25). The resulting interacting IR Hamiltonian takes the form of the SU(2) GN model (27) when a duality transformation on the left currents of Eq. (12) is performed:

 Ja,(x,y)L \lx@stackrelD1→ −Ja,(x,y)L Ja,zL \lx@stackrelD1→ Ja,zL j(x,y)L \lx@stackrelD1→ −j(x,y)L jzL \lx@stackrelD1→ jzL JaL \lx@stackrelD1→ JaL, (33)

whereas all right-moving currents remain invariant. This duality symmetry leads to in Eq. (12) which is indeed a symmetry of the one-loop RG equations (24). The duality transformation has a simple interpretation on the left-moving Dirac fermions (the right ones being untouched):

 L1α \lx@stackrelD1→ L1α L2α \lx@stackrelD1→ −L2α, (34)

where the orbital index is denoted by and for the and -band models respectively. Since the SU(2) GN model (27) is a massive field theory, we deduce that orbital and nuclear-spin low-lying excitations are fully gapped in region II. Taking account the charge degrees of freedom (9), we conclude that a Luttinger-liquid phase with central charge is stabilized. The nature of the dominant electronic instability of that phase can be deduced from the duality transformation (34) on the -CDW order parameter (28), which governs the IR physics in region I:

 n2kF=R†lαLlα\lx@stackrelD1→O2kFODWz = R†1αL1α−R†2αL2α (35) = R†lασzlmLmα.

The underlying lattice order parameter is the component of an orbital density wave (ODW) along the -axis in the orbital subspace:

 OODWz(i)=c†lα,iσzlmcmα,i, (36)

with and for respectively the model (1) and the -band model (4). Using the result (31) in region I, the leading asymptotics of the equal-time ODW correlation function is:

 Missing or unrecognized delimiter for \right (37)

Again as in region I, this instability competes with the MS one (32).

### iv.3 Region III

Asymptote (III) of the one-loop RG flow occurs only in the region III of the -band model (see Fig. 1). Along this special ray, the interacting IR Hamiltonian is again equivalent to the SU(2) GN model (27) when a duality transformation on the left currents of Eq. (12) is performed:

 Ja,(z,y)L \lx@stackrelD2→ −Ja,(z,y)L Ja,xL \lx@stackrelD2→ Ja,xL j(z,y)L \lx@stackrelD2→ −j(z,y)L jxL \lx@stackrelD2→ jxL JaL \lx@stackrelD2→ JaL, (38)

whereas all right-moving currents remain invariant. This duality symmetry leads to which is indeed a symmetry of the one-loop RG equations (24) and expresses as follows in terms of the left-moving Dirac fermions:

 L1α \lx@stackrelD2→ L2α L2α \lx@stackrelD2→ L1α, (39)

the right-moving Dirac fermions being invariant under the transformation. We have again a gapless phase where orbital and nuclear spin degrees of freedom are fully gapped. The dominant instability of phase III is obtained from the -CDW order parameter (28) by performing the duality symmetry (39):

 n2kF=R†lαLlα\lx@stackrelD2→O2kFODWx = R†1αL2α+R†2αL1α (40) = R†lασxlmLmα.

The latter being related to the continuum description of the