Discussion

energetics selection of the pairing and just mention possibilities likep-wave (p_x +ip_y) or f-wave that can be nicely placed on the triangular lattice.

The properties of the resulting phase are as follows. The f↓ species are gapless with Fermi surface, so we expect metal-like specific heat C = γT; note that this is the full result since the gauge field is Higgsed out by thef↑ pairing. We also expect constant spin susceptibility atT → 0since bothf↑ andf↓ systems are compressible, the former due to the pair-condensate and the latter by virtue of finite density of states at the Fermi level.

Because of thef↓ Fermi surface, we expect hS^z(r)S^z(0)ito show 2k_F↓ oscillations with 1/r³ power law. On the other hand, hS⁺(r)S⁻(0)i will show either a full gap if the f↑

pairing is fully gapped as in the case ofp_x+ip_y pairing, or a pseudogap if thef↑ pairing has gapless parts as in the case off-wave pairing. Note that this does not contradict the finite susceptibility since the f↑-pair condensate can readily accommodate ∆N↑ = ±2 changes. Related to this, spin-nematic correlations are gapless and show1/r³ power law at zero wavevector (in the mean field calculation). Interestingly, the gap or pseudo-gap to spin-1 operators would have consequences for NMR experiments done with¹H or ¹³C that are both spin-¹₂ nuclei and relax only by spin-1 excitations. From such measurements, this phase might appear gapped, but it actually has a gapless Fermi surface of one species.

(In the context of 1D models exhibiting spin-nematic phases, consequences for the NMR relaxation rate were discussed in detail, e.g., in [89].)

Finally, let us consider the analog of the situation in Chapter 4.4.2, where bothf↑ and f↓ become paired, with possibly different pairing ∆^↑_rr0, ∆^↓_rr0. In this case, S^z and S⁺ correlations are both gapped (or pseudo-gapped), while spin-nematic correlation shows long-range order. Specifically, in the mean field,

S⁺(r)S⁺(r⁰)

mf = ∆^↑∗_rr0∆^↓_rr0 . (4.85) Note that this nematic order resides on the bonds of the lattice and details depend on the

∆^↑ and∆^↓. For example, if we take ∆^↑ and ∆^↓ to have the same pattern, this will give ferro-nematic state. Curiously, if we take ∆^↑ ∼ p_x +ip_y and ∆^↓ ∼ p_x −ip_y, we get q = 0antiferromagnetic nematic order on the Kagome lattice formed by the bonds of the

triangular lattice. We emphasize that we have not discussed any energetics that may be selecting among such states. Whether something like this can appear in realistic models on the triangular lattice is an interesting open question.

Chapter 5

Insulating phases of electrons on a

zigzag strip in the orbital magnetic field

This chapter complements our earlier discussing in chapter 4 on the effects of Zeeman field on a SBM phase [29]. Here we consider the orbital magnetic field on the electronic two-leg triangular ladder.

Previous studies of ladders with orbital field were done on a square two-leg case and mainly focused on generic density (see [96, 97, 98, 99] and citations therein), while the triangular two-leg case has not been considered so far. In the context of Mott insulators at half-filling, microscopic orbital fields were shown to give rise to interesting scalar chirality terms operating on triangles in the effective spin Hamiltonian [100, 101, 102, 35]. On the other hand, it was also argued [103, 104, 105] that if a Mott insulator develops a noncopla- nar magnetic order with nontrivial chiralities, this can imply spontaneous orbital electronic currents.

In this chapter, we focus on the simplest ladder model with triangles, the zigzag strip, and discuss instabilities due to existence of orbital magnetic field and properties of the resulting phases. Our main findings are presented as follows. In Chapter 5.1, we deter- mine the electron dispersion in the orbital field and perform weak coupling renormalization group (RG) analysis in a two-band regime [29, 31, 65, 64]. Unlike the case with no field, we find that there is a four-fermion Umklapp interaction which is always relevant for re- pulsively interacting electrons and provides a mechanism to drive the metal-insulator tran- sition. This Umklapp gaps out all charge modes and produces a C0S2 state. In Chapter 5.2

A B A B A B A B A B A B

Figure 5.1: Top: Zigzag strip with uniform flux Φ penetrating each triangular plaquette.

Bottom: Convenient representation of the model as a 1D chain with first- and second- neighbor hoppings. We choose a gauge such thattx,x+1 =t1andtx,x+2 =t2eiΦ cos (πx). The unit cell consists of two sites labelledAandB.

we describe physical observables in this phase, and in Chapter 5.3 we analyze possible further instabilities in the spin sector and properties of the resulting phases. We conclude with discussion of the orbital field effects in the context of the spin Bose-metal phase of [29] where the Mott insulator is first produced by an eight-fermion Umklapp and the new four-fermion Umklapp appears as a residual interaction.

5.1 Weak coupling approach to electrons on a zigzag strip with orbital field

Let us apply weak coupling renormalization group (RG) to study effects of electronic inter- actions in the presence of the orbital magnetic field. We start with free electrons hopping on the triangular strip with uniform fluxΦpassing through each triangle. Figure 5.1 illustrates our gauge choice,

t_x,x+1 = t₁, (5.1)

t_x,x+2 = t₂eiΦ cos (πx) . (5.2)

Here and throughout, we refer to sites by their 1D chain coordinate x. Since the second- neighbor hopping depends on whetherxis even or odd, the unit cell has two sites which we labelAandB. The Hamiltonian for such an interacting electron system isH =H0+HV,

with

H0 = −X

x;α

t1c^†_α(x)cα(x+ 1) + H.c.

(5.3)

− X

x∈A;α

h

t₂e^−iΦc^†_Aα(x)c_Aα(x+ 2) + H.c.i

(5.4)

− X

x∈B;α

h

t₂e^iΦc^†_Bα(x)c_Bα(x+ 2) + H.c.i

, (5.5)

HV = 1 2

X

x,x⁰

V(x−x⁰)n(x)n(x⁰). (5.6)

In the first and last lines, we suppressed the sublattice labels, andn(x)≡P

αc^†_α(x)c_α(x).

We assume thatHV is small and treat it as a perturbation toH0. The free electron dispersion is

ξ(k) = ±2p

[t1cos (k)]²+ [t2sin (Φ) sin (2k)]²

−2t2cos (Φ) cos (2k)−µ . (5.7) We are focusing on the regime with two partially filled bands as shown in Fig. 5.2. For small flux, this regime appears when t₂/t₁ > 0.5. We denote Fermi wavevectors for the right-moving electrons ask_F1 andk_F2and the corresponding Fermi velocities asv₁andv₂. The half-filling condition readsk_F1+k_F2 =π/2.

The electron operators are expanded in terms of continuum fields, c_Mα(x) = X

P,a

e^{iP kF ax}U_{P a}^Mc_{P aα}, (5.8)

whereP =R/L = +/−denotes the right and left movers,a= 1,2denotes the two Fermi seas, and M = A orB denotes the sublattices. In the specific gauge, the wavefunctions

R1 R2 L2 L1

Π

-

^Π 2 2

k

-3 -2 -1 1 2

Ξ H k L

Figure 5.2: Free electron spectrum in the presence of the orbital field, see Fig. 5.1. Here ξ(k)is given by Eq. (5.7) with two branches and we focus on the regime when both bands are partially populated; we take t₁ = 1, t₂ = 1, and Φ = π/100 for illustration. The half-filling condition requiresk_F1+k_F2 =π/2.

U_{P a}^M are

U_R1^A = cos_θ

kF1

2

, U_L1^A = sin_θ

kF1

2

, U_R2^A =−sin_θ

kF2

2

, U_L2^A = cos_θ

kF2

2

, U_R1^B = sin_θ

kF1

2

, U_L1^B = cos_θ

kF1

2

, U_R2^B = cos_θ

kF2

2

, U_L2^B =−sin_θ

kF2

2

,

(5.9)

with

{sin(θk), cos(θk)} ∝ {t1cos(k), t2sin(Φ) sin(2k)}. (5.10) Note thatkbelongs to the reduced Brillouin zone[−π/2, π/2].

Few words about physical symmetries. The present problem has SU(2) spin rotation symmetry (R) but lacks time reversal because of the orbital field. It also lacks inver- sion symmetry and translation by one lattice spacing. However, the system is invari- ant under combined transformations such as inversion plus complex conjugation (I^∗ : x → −x, i → −i) and translation by one lattice spacing plus complex conjugation (T₁^∗ : x → x+ 1, i → −i). Table 5.1 lists transformation properties of the continuum fields under these two discrete transformations and under the SU(2) spin rotation. Since the symmetries are reduced compared to the case without the orbital field [31, 65, 64], we need to scrutinize interactions allowed in the continuum field theory.

Table 5.1: Transformation properties of the continuum fields underI^∗(inversion plus com- plex conjugation),T₁^∗(translation by one lattice spacing plus complex conjugation), andR (SU(2) spin rotation about arbitrary axis~n by an angleφ). We also show transformation properties of bilinearsE_1,2defined in Eqs. (5.41)–(5.42).

R I^∗ T₁^∗ c_{P aα} →

e^{−iφ2~n·~σ}

αβ

c_{P aβ} c_{P aα} e^{iP kF a}c_−P,aα

Ej → Ej Ej −iE_j^†

E_j^† → E_j^† E_j^† iE_j

Using symmetry considerations, we can write down the general form of the four- fermion interactions which mix the right- and left-moving fields:

H^ρ = X

a,b

w_ab^ρJ_RabJ_Lab+λ^ρ_abJ_RaaJ_Lbb

, (5.11)

H^σ = −X

a,b

w_ab^σJ~_Rab·J~_Lab+λ^σ_abJ~_Raa·J~_Lbb

, (5.12)

H^u = u₄ c^†_R2↑c^†_R2↓cL1↑cL1↓−c^†_L2↑c^†_L2↓cR1↑cR1↓

+ H.c.

, (5.13)

where we defined

J_{P ab} ≡ c^†_{P aα}c_{P bα} , (5.14)

J~_{P ab} ≡ 1

2c^†_{P aα}~σ_αβc_{P bβ} . (5.15) Note that besides the familiar momentum-conserving four-fermion interactionsH^ρandH^σ, there is also an Umklapp-type interactionH^u.

Using the symmetries of the problem, we can check that all couplings are real and satisfyw₁₂ = w₂₁ andλ₁₂ =λ₂₁, and we also use conventionw₁₁ =w₂₂ = 0. Thus there are 9 independent couplings:w^ρ/σ₁₂ , λ^ρ/σ₁₁ , λ^ρ/σ₂₂ ,λ^ρ/σ₁₂ , andu4.

With all terms defined above, we can derive weak-coupling RG equations:

λ˙^ρ₁₁ =− 1 2πv₂

(w^ρ₁₂)²+ 3

16(w₁₂^σ )²

, (5.16)

λ˙^ρ₂₂ =− 1 2πv₁

(w^ρ₁₂)²+ 3

16(w₁₂^σ )²

, (5.17)

λ˙^ρ₁₂ = 1 π(v₁+v₂)

(w^ρ₁₂)²+ 3

16(w₁₂^σ )²+ (u4)²

, (5.18)

λ˙^σ₁₁ =− 1 2πv1

(λ^σ₁₁)²− 1 4πv2

(w^σ₁₂)²+ 4w^ρ₁₂w^σ₁₂

, (5.19)

λ˙^σ₂₂ =− 1

2πv₂ (λ^σ₂₂)²− 1 4πv₁

(w^σ₁₂)²+ 4w^ρ₁₂w^σ₁₂

, (5.20)

λ˙^σ₁₂ =− 1 π(v₁+v₂)

(

(λ^σ₁₂)²+(w^σ₁₂)²−4w₁₂^ρ w₁₂^σ 2

)

, (5.21)

˙

w^ρ₁₂=−Λ^ρw^ρ₁₂− 3

16Λ^σw₁₂^σ , (5.22)

˙

w^σ₁₂=−Λ^σw^ρ₁₂−

Λ^ρ+Λ^σ

2 + 2λ^σ₁₂ π(v₁+v₂)

w^σ₁₂, (5.23)

˙

u4 = 4λ^ρ₁₂u₄

π(v₁ +v₂) . (5.24)

HereO˙ ≡∂O/∂`, where`is logarithm of the length scale. We have also defined

Λ^ρ/σ = λ^ρ/σ₁₁ 2πv1

+ λ^ρ/σ₂₂ 2πv2

− 2λ^ρ/σ₁₂

π(v1+v2) . (5.25)

We can obtain bare values of the couplings for any electronic interactions by expanding in terms of the continuum fields. In the case of small flux, the couplings λ^ρ/σ and w^ρ/σ in Eqs. (5.11) and (5.12) are only modified slightly and can be treated as the same as in [31] with extended repulsion. For the couplingu₄ in Eq. (5.13), the bare value ofu₄ in the small flux limit isP

x⁰V(x−x⁰)e^iπ2(x−x0)× ^t_t²

1[sin(k_F1) + sin(k_F2)]Φ ∝ Φ, wherexand x⁰ belong to the same sublattice (A or B). Therefore, we can see that the parameter u₄ which measures the strength of the umklapp process is linearly proportional to the flux and goes to zero if we gradually switch off the flux. For repulsive interactions, we generally expect positive λ^ρ (see, e.g., [31] with extended repulsion). Then according to the RG Eq. (5.24), positive initialλ^ρ₁₂ will driveu₄ to increase exponentially. Thus we conclude

that the starting two-band metallic phase is unstable due to the new Umklapp term.

To analyze the resulting phase(s), we use bosonization to rewrite fermionic fields in terms of bosonic fields,

c_{P aα}∼η_aαe^{i(ϕaα+P θaα)}, (5.26) with canonically conjugate boson fields:

[ϕ_aα(x), ϕ_bβ(x⁰)] = [θ_aα(x), θ_bβ(x⁰)] = 0, (5.27) [ϕ_aα(x), θ_bβ(x⁰)] = iπδ_abδ_αβΘ(x−x⁰), (5.28) whereΘ(x)is the heaviside step function. Here we use Majorana fermions{ηaα, ηbβ} = 2δ_abδ_αβ as Klein factors, which assure that the fermion fields with different flavors anti- commute.

It is convenient to introduce new variables θρ± ≡ 1

2[θ1↑+θ1↓±(θ2↑+θ2↓)], (5.29) θ_aσ ≡ 1

√2(θ_a↑−θ_a↓), a= 1 or 2, (5.30) θσ± ≡ 1

√2(θ_1σ±θ_2σ), (5.31)

and similarly forϕvariables. We can then write compactly all nonlinear potentials obtained

upon bosonization of the four-fermion interactions:

H^u = 4u4Γ sin(2ϕˆ ρ−) sin(2θρ+), (5.32) W ≡(w₁₂^ρ JR12JL12−w^σ₁₂J~R12·J~L12) + H.c.= (5.33)

= cos(2ϕ_ρ−) (

4w₁₂^ρ h

cos(2ϕ_σ−)−Γ cos(2θˆ _σ−)i

−w^σ₁₂h

cos(2ϕσ−) + ˆΓ cos(2θσ−) + 2ˆΓ cos(2θ_σ+)i )

, (5.34)

V⊥ ≡ −X

a

λ^σ_aa

2 J_Raa⁺ J_Laa⁻ +J_Raa⁻ J_Laa⁺

(5.35)

−λ^σ₁₂ 2

J_R11⁺ J_L22⁻ +J_R11⁻ J_L22⁺ + (R↔L)

(5.36)

=X

a

λ^σ_aacos (2√

2θ_aσ) (5.37)

+2λ^σ₁₂Γ cos (2θˆ σ+) cos (2ϕσ−), (5.38) where

Γˆ ≡η_1↑η_1↓η_2↑η_2↓. (5.39)

We will not analyze the RG flows in all cases. Our main interest is in exploring the orbital magnetic field effects on the C2S2 metallic phase and nearby C1[ρ−]S2 spin liquid.

Therefore we consider the situation where in the absence of theu4 term we have the stable C2S2 phase described by RG flows such that λ^ρ_ab reach some fixed point values, w₁₂^ρ/σ are irrelevant, andλ^σ_ab are marginally irrelevant—this is realized, for example, in [31] for sufficiently long-ranged repulsion.

As we have already discussed, for repulsive interactions we expect λ^ρ₁₂ > 0and hence any nonzerou₄will increase quickly. In this setting it is then natural to focus on the effects of theH^u first. From the bosonized form Eq. (5.32), we see that it pins

sin(2θ_ρ+) =−sign(u₄) sin(2ϕρ−) = ±1. (5.40) Thus, both “ρ−” and “ρ+” modes become gapped and the system is an insulator. This insulator arises because of the combined localizing effects of the orbital field and repulsive

interactions.

Having concluded that u₄ becomes large, if we were to continue using the weak cou- pling RG Eqs. (5.16)–(5.24), we would find thatu₄drivesλ^ρ₁₂to large positive value, which in turn drivesΛ^ρto negative values and destabilizes couplingsw^ρ/σ₁₂ , and all couplings even- tually diverge. If we do not make finer distinctions as to which couplings diverge faster, we would conclude that the ultimate outcome is a fully gapped C0S0. We will analyze different C0S0 phases arising from the combined effects ofu₄ andλ^σ later. Here we only note that the bosonized theory suggests that a C0S2 phase can in principle be stable. In- deed, once we pin ϕρ− to satisfy Eq. (5.40), the W interaction vanishes leaving only the effectiveλ^σ couplings in the spin sector. The stability in the spin sector is then determined by the signs of theλ^σcouplings. Ifλ^σ_ab>0, the spin sector is stable and we have the C0S2 phase. In what follows, we will identify all interesting physical observables in this phase and will use it as a starting point for analysis of possible further instabilities and features of the resulting phases.

5.2 Observables in the Mott-insulating phase in orbital