A Group theory conventions

# Competing Orders in One-Dimensional Half-Integer Fermionic Cold Atoms: A Conformal Field Theory Approach

## Abstract

The physical properties of arbitrary half-integer spins fermionic cold atoms loaded into a one-dimensional optical lattice are investigated by means of a conformal field theory approach. We show that for attractive interactions two different superfluid phases emerge for : A BCS pairing phase, and a molecular superfluid phase which is formed from bound-states made of fermions. In the low-energy approach, the competition between these instabilities and charge-density waves is described in terms of parafermionic degrees of freedom. The quantum phase transition for is universal and shown to belong to the Ising and three-state Potts universality classes respectively. In contrast, for , the transition is non-universal. For a filling of one atom per site, a Mott transition occurs and the nature of the possible Mott-insulating phases are determined.

Cold fermionic atoms; Conformal field theory; Bosonization; Parafermions; Exotic superfluidity
###### pacs:
71.10.Pm; 75.10.Pq; 03.75.Ss

## I Introduction

Ultracold atomic physics has attracted a lot of interest in recent years with the opportunity to study strongly correlated effects, such as high-temperature superconductivity, in a new context (1). In particular, loading cold atomic gases into an optical lattice allows for the realization of bosonic and fermionic lattice models and the experimental study of exotic quantum phases (2). A prominent example is the toolbox to engineer the three-dimensional Bose-Hubbard model (3) and the observation of its Mott insulator – superfluid quantum phase transition with cold bosonic atoms in an optical lattice (4).

Ultracold atomic systems also offer an opportunity to investigate the effect of spin degeneracy since the atomic total angular momentum , which includes both electron and nuclear spins, can be larger than resulting in hyperfine states. In magnetic traps, these components are split, while in optical traps these (hyperfine) spin degrees of freedom are degenerate and novel interesting phases might emerge. For instance, Bose-Einstein condensates of bosonic atoms with a nonzero total spin are expected to display rich interesting structures in spin space like in superfluid He. In this respect, various exotic superfluid condensates (including nematic ones), Mott insulating phases, and non-trivial vortex structures have been predicted recently in spinor bosonic atoms with (5); (6); (7); (9); (8); (10); (11). These theoretical predictions might be checked in the context of Bose-Einstein condensates of sodium, rubidium atoms (12) and in spin-3 atom of Cr (13). The spin-degeneracy in fermionic atoms, like Li, K or Yb, is also expected to give rise to some interesting superfluid phases (14); (15); (16); (17); (18). In particular, a molecular superfluid (MS) phase might be stabilized in multicomponent attractive Fermi gas where more than two fermions form a bound state. Such a non-trivial superfluid behavior has already been found in different contexts. In nuclear physics, a four-particle condensate like the particle is favored over deuteron condensation at low density (19) and it may have implications for light nuclei and asymmetric matter in nuclear stars (20); (21). This quartet condensation can occur in the field of semiconductors with the formation of biexciton (22). A quartetting phase, which stems from the pairing of Cooper pairs, also appears in a model of one-dimensional Josephson junctions (23). A possible experimental observation of quartets might be found in superconducting quantum interference devices with (100)/(110) interfaces of two d-wave superconductors (24). In particular, the periodicity of the critical current with applied magnetic flux has been interpreted as the formation of quartets with total charge (25). Finally, a bipairing mechanism has also been predicted in four-leg Hubbard ladders (26).

Recently, the emergence of quartet and triplet (three-body bound states) has been proposed to occur in the context of ultracold fermionic atoms (27); (28); (29); (30); (31); (32). Much of these studies have been restricted to the special case of an SU() symmetry between the hyperfine states of a -component Fermi gas. In this paper, we investigate the generic physical features of half-integer spins fermionic cold atoms with s-wave scattering interactions loaded into a one-dimensional optical lattice. The low-energy physical properties of -component fermions with contact interactions are known to be described by a Hubbard-like Hamiltonian (5):

 H=−t∑i,α[c†α,icα,i+1+H.c.]−μ∑ini+∑i,JUJJ∑M=−JP†JM,iPJM,i, (1)

where () is the fermion creation operator corresponding to the atomic states and is the density operator on site . The pairing operators in Eq. (1) are defined through the Clebsch-Gordan coefficients for forming a total spin from two spin- fermions: . The interactions are SU(2) spin-conserving and depend on parameters corresponding to the total spin which takes only even integers value due to Pauli’s principle: . Even in this simple scheme the interaction pattern is still involved since there are coupling constants in the general spin- case. In this paper, we shall consider a two coupling-constant version of model (1) with which incorporates the relevant physics of higher-spin degeneracy with respect to the formation of an exotic MS phase. To this end, it is enlightening to express this model in terms of the density and the BCS singlet-pairing operator for spin- which is defined by:

 P†00,i=1√2N∑α,βc†α,iJαβc†β,i=1√2N∑α(−1)2F+αc†α,ic†2N+1−α,i, (2)

where the matrix is the natural generalization of the familiar antisymmetric tensor to spin . Such a singlet-pairing operator has been extensively studied in the context of two-dimensional frustrated spin-1/2 quantum magnets (33). Using the relation , model (1) with reads then as follows:

 H=−t∑i,α[c†α,icα,i+1+H.c.]−~μ∑ini+U2∑in2i+V∑iP†00,iP00,i, (3)

with , , and .

When , i.e. , model (3) corresponds to the Hubbard model with an SU() spin symmetry: , being a SU() matrix. The Hamiltonian (3) for still displays an extended symmetry. Indeed, since the BCS singlet-pairing operator (2) is invariant under the Sp() group which consists of unitary matrices that satisfy , model (3) acquires an Sp() symmetry (34). In the spin case, model (3) reduces to the SU(2) Hubbard chain since SU(2) Sp(2). In the case, i.e. , there is no fine-tuning and models (1) and (3) have thus an exact Sp(4) SO(5) symmetry (35). For , the Sp() symmetry is only present for the two coupling-constant model (3). The Sp() symmetry simplifies the problem but may appear rather artificial at this point. However, one reason to consider this model stems from the structure of the interaction involving the density and the BCS singlet-pairing term (2) which are independent for 1. The competition between these two operators gives rise to interesting physics. Indeed, at and large , a quasi-long-range BCS phase, which is a singlet under Sp(), should emerge. In contrast, for and , we expect the stabilization of a charge-density wave (CDW) or a MS instability made of fermions which is an SU() singlet. We thus expect the existence of several distinct phases in the Sp() model (3). As it will be revealed in the following, the competition between these instabilities relies on the existence of a symmetry which is also a symmetry of the original SU(2) model (1). This symmetry can be defined by considering the coset between the center of the SU() group 2 and the center of the Sp() or SU(2): = / with

 Z2N:c†α,i → einπ/Nc†α,i,n=0,....,2N−1 (4) ZN:c†α,i → einπ/Nc†α,i,n=0,....,N−1, (5)

the symmetry being . The symmetry (5) provides an important physical ingredient which is not present in the case and stems from the higher-spin degeneracy. The stabilization of a quasi-long-range BCS phase for requires the spontaneous breaking of this symmetry since the singlet pairing (2) is not invariant under this symmetry. In contrast, if is not broken, a MS phase, which is a singlet under the symmetry, might emerge. In the following, the delicate competition between these superfluid instabilities will be investigated with a special emphasis on this symmetry. In particular, in the low-energy approach, this discrete symmetry is described by the parafermionic conformal field theory (CFT) which captures the universal properties of two-dimensional generalized Ising models (36). This approach will enable us to determine the main features of the zero temperature phase diagram of model (3). A brief summary of our main results have already been published as a letter (29).

The rest of the paper is organized as follows. The low-energy effective theory corresponding to model (3) is developed in section II. The nature of the different phases is then deduced from a renormalization group (RG) analysis. In section III, we present the phase diagram of the model (3) at zero temperature for incommensurate filling and we discuss the main physical properties of the phases as well as the nature of the quantum phase transitions. The Mott-insulating phases for the case of one atom per site are also investigated. Finally, we conclude in section IV and some technical details are presented in three appendices.

## Ii Low-energy approach

In this section, we present the low-energy description of the spin- Hubbard model with a Sp() symmetry (3) which will enable us to determine its phase diagram at zero temperature in the next section.

### ii.1 Continuum limit

Let us first discuss the continuum limit of the lattice model (3). Its low-energy effective field theory can be derived from the continuum description of the lattice fermionic operators in terms of right and left-moving Dirac fermions (for a review, see for instance the books (37); (38); (39)):

 cα,i/√a0→RαeikFx+Lαe−ikFx, (6)

with ( being the lattice spacing) and is the Fermi momentum. The left (right)-moving fermions are holomorphic (antiholomorphic) fields of the complex coordinate ( being the imaginary time and is the Fermi velocity): . These fields obey the following operator product expansion (OPE):

 R†α(¯z)Rβ(¯ω) ∼ δαβ2π(¯z−¯ω) L†α(z)Lβ(ω) ∼ δαβ2π(z−ω). (7)

In this continuum limit, the non-interacting part of the Hamiltonian (3) corresponds to the Hamiltonian density of free relativistic massless fermions:

 H0=−ivF(:R†α∂xRα:−:L†α∂xLα:), (8)

where and the normal ordering with respect to the Fermi sea is assumed as well as a summation over repeated indices. The continuous symmetry of the non-interacting part of the Hamiltonian (3) is enlarged, in the continuum limit, to U(2) U(2) under independent unitary transformations on the Dirac fermions.

The next step of the approach is to use the non-Abelian bosonization (40); (41); (42) to investigate the effect of the Sp(2) symmetry of the lattice model. To this end, we introduce a SU(2) Wess-Zumino-Novikov-Witten (WZNW) primary field to represent the spin degrees of freedom and an additional U(1) charge boson field . The free-fermion theory (8) is then equivalent to the CFT with an U(1) SU(2) symmetry. The right and left-moving fermions can be written as a product of spin and charge operators:

 Rα ∼ :exp(i√2π/NΦcR):gαR L†α ∼ :exp(i√2π/NΦcL):gαL, (9)

where and are respectively the right and left parts of the SU(2) primary field with scaling dimension which transforms in the fundamental representation of SU(2). In Eq. (9), denote the chiral parts of the charge boson field: . As is well known, the free-Hamiltonian (8) can be decomposed into charge and spin pieces:

 H0=vF2[(∂xΦc)2+(∂xΘc)2]+2πvF2N+1[:IARIAR:+:IALIAL:], (10)

where is the dual charge boson field and are the currents which generate the SU(2) conformal symmetry with central charge . They express simply as bilinears of the Dirac fermions:

 IAR=:R†αTAαβRβ:,IAL=:L†αTAαβLβ:, (11)

being the generators of SU() in the fundamental representation normalized such that: (see Appendix A). These currents satisfy the SU(2) current algebra (38); (43):

 IAL(z)IBL(ω) ∼ δAB8π2(z−ω)2+ifABC2π(z−ω)ICL(ω) IAR(¯z)IBR(¯ω) ∼ δAB8π2(¯z−¯ω)2+ifABC2π(¯z−¯ω)ICR(¯ω), (12)

where are the structure constants of the SU(2) group.

In the presence of interactions, the spin-charge separation (10) still holds away from half-filling (i.e. atoms per site). In that case, the low-energy Hamiltonian of model (3) separates into two commuting charge and spin pieces:

 H=Hc+Hs,[Hc,Hs]=0. (13)

Let us first consider the charge degrees of freedom which are described by the bosonic field .

#### Charge degrees of freedom

Using the continuum description (6) and the decomposition (9), we find that the charge excitations of model (3) are captured by the following Hamiltonian:

 Hc=vF2[(∂xΦc)2+(∂xΘc)2]+a02V+UN(2N−1)2Nπ(∂xΦc)2. (14)

It can be recast into the form of the Tomonaga-Luttinger free Hamiltonian:

 Hc=vc2[1Kc(∂xΦc)2+Kc(∂xΘc)2], (15)

where the charge velocity and the Luttinger parameter read as follows:

 vc = vF[1+a0(2V+UN(2N−1))/(NπvF)]1/2 Kc = [1+a0(2V+UN(2N−1))/(NπvF)]−1/2. (16)

The conserved quantities in this U(1) charge sector are the total particle number and current :

 N = ∫dx:R†αRα+L†αLα:=√2N/π∫dx∂xΦc I = ∫dx:L†αLα−R†αRα:=√2N/π∫dx∂xΘc. (17)

For generic fillings, no umklapp terms appear and the charge degrees of freedom display metallic (gapless) properties in the Luttinger liquid universality class (38); (39). However, as it will be discussed in section III, the existence of an umklapp process, for a commensurate filling of one atom per site, is responsible for the formation of a charge gap and the emergence of different Mott insulating phases.

#### Spin-degrees of freedom

All non-trivial physics corresponding to the spin degeneracy is encoded in the spin part of the Hamiltonian (13). Its continuum expression can be obtained by decomposing the SU() currents of Eq. (11) into and parts with respect to the Sp() symmetry of the lattice model (3). The currents generate the Sp() CFT symmetry with central charge . They can be simply expressed in terms of the chiral Dirac fermions:

 Ia∥R=:R†αTaαβRβ:,Ia∥L=:L†αTaαβLβ:, (18)

where are the generators of Sp() in the fundamental representation and normalized such that: (see Appendix A). These currents verify the Sp() Kac-Moody algebra given by Eqs. (12) with the Sp() structure constants. The remaining SU() currents () are written as:

 Ii⊥R=:R†αTiαβRβ:,Ii⊥L=:L†αTiαβLβ:. (19)

With these definitions, we can now derive the continuum description of the spin degrees of freedom using Eq. (6) and Eqs. (66, 72) of Appendix A. After some cumbersome calculations, we find that the low-energy Hamiltonian in the spin sector, , can be expressed in terms of the currents only, and displays a marginal current-current interaction:

 Hs=2πvs∥2N+1[:Ia∥RIa∥R:+:Ia∥LIa∥L:]+2πvs⊥2N+1[:Ii⊥RIi⊥R:+:Ii⊥LIi⊥L:]+g∥Ia∥RIa∥L+g⊥Ii⊥RIi⊥L, (20)

with , and are the spin velocities: , . In the simplest case, we have Sp(2) SU(2) and the Hamiltonian (20) with describes the spin-sector of the continuum limit of the spin-1/2 Hubbard chain which can be found for instance in the book (38) written with the same notations used in this work. In the , i.e. the case, one can express model (20) in a more transparent basis. Indeed, there is a simple free-field representation of the unperturbed SU(4) SO(6) CFT in terms of six real (Majorana) fermions which has been used in the context of the two-leg spin-1/2 ladder with four-spin exchange interactions (44). Introducing six real fermions and to describe respectively the , i.e. Ising, and SO(5) Sp(4) CFTs, the currents of Eq. (20) can be written locally in terms of these fermions:

 Ii⊥R,L ∼ −i√2ξ0R,LξiR,L Ia∥R,L ∼ −i√2ξiR,LξjR,L, (21)

where where . The Majorana fermions are normalized as the Dirac fermions of Eq. (7) to reproduce faithfully the Kac-Moody algebra (12). The current-current model of Eq. (20) can then be expressed in terms of these real fermions:

 Hs=−iv2[:ξiR∂xξiR:−:ξiL∂xξiL:]−iv02[:ξ0R∂xξ0R:−:ξ0L∂xξ0L:]+λ∥(ξiRξiL)2+λ⊥ξ0Rξ0LξiRξiL, (22)

with , , and the spin velocities: , . In absence of the spin-velocity anisotropy, model (22) turns out to be exactly solvable and has also been studied in the context of a SO(5) symmetric two-leg ladder (45).

### ii.2 RG analysis

In the general case, model (20) is not integrable and the main effect of the current-current interaction can be elucidated by means of a RG analysis. The one-loop RG equations of model (20) are given by (see Appendix B):

 ˙g∥ = N+14πg2∥+N−14πg2⊥ ˙g⊥ = N2πg⊥g∥, (23)

where , being the RG time. In Eq. (23), we have neglected the spin-velocity anisotropy and have absorbed in a redefinition of the coupling constants: . We have also obtained the two-loop RG equations of model (20) and the results are presented in Appendix B. The one-loop RG equations (23) can be solved and in particular the RG invariant flow which parametrizes the RG lines reads as follows:

 K=g−(N+1)/N⊥(g2∥−g2⊥). (24)

The RG flow emerging from Eqs. (23) is rich and consists of three different phases (see Fig. 1).

In region I, where both and are positive, all couplings go to zero in the infrared (IR) limit and the interaction is marginal irrelevant. The symmetry of the IR fixed point is SU(2) (up to a velocity anisotropy) leading to gapless spin excitations. The current-current interaction of Eq. (20) leads to logarithmic corrections to physical quantities (46); (47). The low-energy properties of this phase are very similar to that of the repulsive SU(2) Hubbard chain which have been studied in Refs. (48); (49). In contrast, a spin gap is opened by the interaction in the two remaining phases. In phase II, defined by and , the RG flow in the far IR limit is attracted along a special symmetric ray where the interacting part of the Hamiltonian (20) can be rewritten in a manifest SU(2) invariant form:

 H∗s,int=g∗(Ia∥RIa∥L+Ii⊥RIi⊥L)=g∗IARIAL. (25)

This IR Hamiltonian thus takes the form of the SU(2) Gross-Neveu (GN) model (50) which is an integrable massive field theory (51). The development of the strong-coupling regime in the SU(2) GN model leads to the generation of a spin gap. The low-energy properties of the spin sector of phase II can be extracted from the integrability of the SU() GN model (25). Its low-energy spectrum consists into branches with masses: that transform in the SU(2) representation with Young tableau with one column and boxes () (51). The corresponding eigenstates are labelled by quantum numbers associated with the conserved quantities of the SU() low-energy symmetry (the Cartan basis): , and , . Due to the Sp() symmetry of model (3), the numbers are conserved whereas the charges are only good quantum numbers at low energy. This is an example of a dynamical symmetry enlargement which corresponds to a situation where a Hamiltonian is attracted under a RG flow to a manifold possessing a higher symmetry than indicated by the original microscopic theory. This phenomenon occurs in a large variety of models with marginal interactions in the scaling limit (52) as, for instance, in the half-filled two-leg Hubbard model (53) and in the SU(4) Hubbard model at half filling (54), where a SO(8) symmetry emerges at low energy.

In the second spin-gapped phase (III) of Fig. 1, defined by and , the RG flow is now attracted along the asymptote: . In that case, the interacting part of the IR Hamiltonian becomes

 H∗s,int=g∗(Ia∥RIa∥L−Ii⊥RIi⊥L), (26)

which can be recast as a SU() GN model (25) by means of a duality transformation on the fermions: with

 ~Rα=JαβR†β,~Lα=Lα. (27)

Using Eqs. (18, 19), we observe that this transformation acts on the currents as: and so that indeed maps (26) onto (25). Besides the opening of a spectral gap, we thus find that phase III possesses a hidden symmetry at low energy i.e. a symmetry generated by the dual currents . The spin spectrum in phase III can be obtained from the duality symmetry and consists into the branches which transform in the representations of the dual group . The dual quantum numbers are now given by: and . We thus observe that the low-lying excitations in phase III carry quantized spin currents in the “” direction. In this sense, the phase III might be viewed as a partially spin-superfluid phase. In summary, the existence of these two distinct spin-gapped phases is a non-trivial consequence of higher-spin degeneracy and does not occur in the case.

### ii.3 Conformal embedding

The crucial point of a non-perturbative analysis is often the identification of a good basis that describe low-lying excitations of the phase. Much insight on this problem can be gained from the symmetries of the model and the use of the non-Abelian bosonization approach for 1D systems. Such an approach has been extremely powerful in the past as in quantum impurity problems (55) and in spin chains (42); (38). So far, we have used an U(2) U(1) SU(2) CFT approach to determine the low-energy properties of model (3). However, this description is not adequate to give a full understanding of the two spin-gapped phases found in the RG approach. In particular, the physical origin of the formation of the spin gaps is not clear at this point. What is the nature of the discrete symmetry which is spontaneously broken in phases II and III? A second weak point of the previous analysis is the determination of the quantum phase transition between the two spin-gapped phases which is rather unclear within the preceding description. At least, this transition should occur in the manifold which is invariant under the duality symmetry (27) i.e. the self-dual manifold defined by: . The low-energy Hamiltonian which describes the phase transition is thus (neglecting the spin-velocity anisotropy):

 HSDs=2πvs2N+1[:Ia∥RIa∥R:+:Ia∥LIa∥L:+:Ii⊥RIi⊥R:+:Ii⊥LIi⊥L:]+gSD∥Ia∥RIa∥L, (28)

with so that the marginally relevant current-current interaction opens a mass gap in the Sp(2) sector. However, one cannot conclude on the occurrence of the first-order phase transition since the non-interacting Hamiltonian of Eq. (28) contains more degrees of freedom than the Sp(2) one. They remain massless and control the quantum phase transition. The nature of these decoupled degrees of freedom is not clear at this point.

Prompted by all these questions, it is important to fully exploit the existence of the Sp(2) symmetry of the lattice model (3) and to consider the following conformal embedding: U(2) U(1) Sp(2) [SU(2)/Sp(2)]. The coset SU(2)/Sp(2) CFT has central charge which is that of the parafermionic CFT (36). In fact, as shown by Altschuler (56), it turns out that the SU(2N)/Sp(2N) CFT is indeed equivalent to the parafermionic CFT which describes self-dual critical points of two-dimensional Ising models (see Appendix C). The conformal embedding U(2) U(1) Sp(2) provides us with a non-perturbative basis to express any physical operator in terms of its charge and spin degrees of freedom which are described respectively by the U(1) and Sp(2) CFTs. Since Sp(2) SU(2), this basis for accounts for the well-known spin-charge separation which is the hallmark of 1D spin-1/2 electronic systems (38); (39). In this respect, in the general half-odd integer spin case, the symmetry plays its trick and provides a new important ingredient not present for . In the low-energy approach, the spin degrees of freedom corresponding to this symmetry are captured by an effective 2D Ising model. As in the case, these Ising models exhibit two gapped phases described by order and disorder parameters and () which are dual to each other by means of the Kramers-Wannier (KW) duality symmetry. This duality transformation maps the symmetry, spontaneously broken in the low-temperature phase ( and ), onto a symmetry which is broken in the high-temperature phase where and . At the critical point, the theory is self-dual with a symmetry and its universal properties are captured by the parafermionic CFT with becoming conformal fields with scaling dimension (36). This CFT is generated by right and left parafermionic currents (, ) with scaling dimension which are the generalization of the Majorana fermions of the Ising model. Under the symmetry, (respectively ) carries a (respectively ) charge which means:

 ΨkL,R → ei2πmk/NΨkL,RunderZN ΨkL,R → e±i2πmk/NΨkL,Runder~ZN, (29)

with . As it is discussed in Appendix C, there is a faithful representation of the first parafermionic currents in terms of the Dirac fermions and the charge bosonic field :

 L†αJαβL†β ≃ √Nπ:exp(i√8π/NΦcL):Ψ1L R†αJαβR†β ≃ √Nπ:exp(−i√8π/NΦcR):Ψ1R. (30)

We deduce, from this identification, that the duality symmetry (27) of the RG analysis together with the Gaussian duality () give and which is nothing but the KW duality transformation on the first parafermionic current (see Eq. (83) of Appendix C). It is thus tempting to interpret the formation of the spin-gapped phases of Fig. 1 as the result of the spontaneous breaking of the and discrete symmetries. In this respect, from the correspondence (30) and Eq. (29), we observe that under the symmetry, the left and right-moving fermions transform as:

 Lα→e−iπm/NLα,Rα→e−iπm/NRα, (31)

while under we have:

 Lα→e−iπm/NLα,Rα→eiπm/NRα. (32)

Using the continuum representation of the fermions (6), we find that the lattice symmetry (5) corresponds to the symmetry of an effective 2D Ising model. In contrast, the symmetry has no simple local lattice representation.

We need now to fully identify phases II and III of Fig. 1 with the high and low-temperature phases of the Ising model. To this end, we use the fact that the SU() WZNW primary field of Eq. (9) can be expressed in terms of the Sp(2) basis using the conformal embedding (see Eq. (90) of Appendix C):

 :exp(−i√2π/NΦc):L†αRα∼Trg∼μ1Trϕ(1), (33)

where is the Sp() primary field with scaling dimension . We deduce, from the identification (33), that has the same behavior as under the symmetry (Eq. (82) of Appendix C):

 Trg → TrgunderZN Trg → ei2πm/NTrgunder~ZN. (34)

The next step of the approach is to note that the interacting part of the Hamiltonian (25), which controls the strong coupling behavior of the RG flow in phase II, can be expressed in terms of : . The ground state of this phase displays long-range order associated with the order parameter : . According to Eq. (34), we then deduce that the symmetry is spontaneously broken while the symmetry remains unbroken in phase II. The Ising model thus belongs to its high-temperature phase and a spectral gap is formed. Using the KW duality symmetry or the transformation (27) on the Dirac fermions, one can conclude that phase III corresponds to the low-temperature phase of the Ising model where the symmetry is spontaneously broken.

In summary, the existence of the two spin-gapped phases of Fig. 1 is a non-trivial consequence of higher-spin degeneracy and does not occur in the case. The emergence of the spin-gap stems from the spontaneous breakdown of the or discrete symmetries. As we shall see now, these symmetries are central to the striking physical properties displayed by these phases.

## Iii Phase diagram

In this section, we discuss the phase diagram at zero temperature of the lattice model (3) for incommensurate filling and for a commensurate filling of one atom per site. The nature of the quantum phase transitions will also be investigated. Let us start with the incommensurate filling case.

### iii.1 Phase diagram for incommensurate filling

We shall now determine the nature of the dominant electronic instabilities of the different phases of Fig. 1 for incommensurate filling. The RG analysis of the preceding section reveals the existence of three different phases.

#### Critical phase

In phase I of Fig. 2, the interaction in the spin sector (Eq. (20)) is marginally irrelevant when and are both positive so that the phase displays an extended quantum critical behavior. Up to a spin-velocity anisotropy, the low-energy properties of this phase are very similar to that of the repulsive SU() Hubbard chain with one gapless charge mode and gapless spin excitations (48); (49). All correlation functions in this phase display a power-law behavior. The leading instabilities are those which have the slowest decaying correlations at long distance. For phase I, the dominant instabilities are the CDW and generalized spin-density wave (SDW) order parameters which read as follows in terms of the Dirac fermions:

 ρ2kF=L†αRα,SA2kF=L†αTAαβRβ. (35)

Using the representation (90), we obtain the leading asymptotics of the CDW correlation function:

 ⟨ρ†2kF(x,τ)ρ2kF(0,0)⟩∼(x2+v2cτ2)−Kc/2N(x2+v2τ2)−(2N+1)/2(N+2)(x2+v20τ2)−(N−1)/N(N+2), (36)

where and are respectively the spin-velocity in the Sp() and sectors. We have obtained a similar estimate for the correlations which involve the SDW operator. At this point, these two correlation functions have the same power-law decay. The logarithmic corrections will lift this degeneracy and we expect that the SDW operator will be the dominant instability as in the case (38); (39).

#### Confined phase

Let us now consider the first spin-gapped phase, i.e. phase II of Fig. 1, which occurs when and in the weak-coupling limit. In contrast to phase I, it has only one gapless mode which stems from the criticality of the charge degrees of freedom. The existence of a spin gap leads us to expect the emergence of a quasi-long-range BCS phase with the pairing of fermions, i.e a Luther-Emery liquid phase, as in the case (38); (39). However, this is not the case for due to the existence of the symmetry (5) which remains unbroken in phase II. Indeed, it costs a finite energy gap to excite states that break this symmetry and the dominant instabilities must thus be neutral under . In particular, there is no dominant BCS instability in phase II since, as already stated in the introduction, the lattice singlet-pairing operator (2) is not invariant under the symmetry (5). Another way to see this is to use the low-energy description of the BCS operator obtained in Appendix C (Eq. (91)):

 P†00∼L†αJαβR†β∼:exp(i√2π/NΘc):σ1Trϕ(1), (37)

where takes a non-zero expectation value in phase II. Since the symmetry of the underlying Ising model is not broken, the Ising spin variable has zero expectation value and short range correlations. Consequently, the equal-time correlation function of the BCS instability (37) has a short-range behavior: . The BCS singlet pairing is completely suppressed in this phase.

In contrast, the dominant instabilities in phase II are expected to be:

 ρ2kF=L†αRα,M0=ϵα1…βNR†α1…R†αNL†β1…L†βN, (38)

which are respectively the CDW and the uniform component of the lattice SU()-singlet superconducting instability made of fermions, i.e. the MS instability, . Both order parameters (38) are neutral under the symmetry (31) and we notice that is also invariant under the symmetry (32). The long-distance behavior of the correlation functions of these operators can be determined using the identifications (90) and (94, 95) of Appendix C:

 ρ2kF ∼ :exp(i√2π/NΦc):μ1Trϕ(1) (39) M0 ∼ :exp(i√2πNΘc):N/2∑p=0apϵN/2−pTrϕ(2p), (40)

where we have assumed, for the sake of simplicity, that is even for the representation of the MS instability. The fields in the Sp() sector, which occur in these expressions, have non-zero expectation values in phase II so that:

 ρ2kF ∼ :exp(i√2π/NΦc): (41) M0 ∼ :exp(i√2πNΘc