**mean-field approximation** **39**
weak-coupling regime, carried out in collaboration with E. Szirmai F. Gebhard, J. S´olyom,
and R. Noack [70]. This reexamination includes the renormalization of the coupling constants
within the mean-field approximation, which was not considered previously. The revised
ana-lytical results conclude that there is no bond-order phase. In this chapter, we first give a
de-tailed discussion (Sec. 2.2) of the revised bosonization treatment, also considering the effect
*of an additional frustrating exchange J*_{2}, which allows us to explicitly induce the bond-order
phase and to make contact with the known phase diagram of the frustrated Heisenberg chain
*at large U. We then present the results of the DMRG calculations, (see Sec. 2.3). Both the*
revised bosonization and the DMRG calculations indicate that a BOW phase is not present for
*J*_{2} = *0; the system is in a SDW phase for all positive J*_{1} *and U. We show that a BOW phase*
*can be induced by turning on J*2 *positively, with the critical value required depending on U*
*and J*_{1}*. At larger values of J*_{2}, we find additional phases, including a spin-gapped metallic
phase which we identify as a Luther-Emery phase. Finally, we conclude in Sec. 2.4.

**2.2** **Weak-coupling theory: bosonization, renormalization**

fields, φ*c,*± = (φ_{↑},±+φ_{↓},±)/2 and φ*s,*± = (φ_{↑},±− φ_{↓},±)/2, correspond to the collective charge
and spin modes, respectively. In order to bosonize the non-local processes, one must expand
*the fermion fields with respect to the lattice constant. The bosonized form of the g-ology*
Hamiltonian density corresponding to Hamiltonian (2.1), up to first order in the expansion
with respect to the lattice constant, is

H*(x)*= 1
2π

X

*r=*±

h*v*ρ(∂* _{x}*φ

*c,r*)

^{2}+

*v*σ(∂

*φ*

_{x}*s,r*)

^{2}i +

*g*ρ

2π^{2}(∂* _{x}*φ

*c,+*)(∂

*φ*

_{x}*c,*−)−

*g*

_{c}2π^{2}cos(2φ* _{c}*)

− *g*σ

2π^{2}(∂* _{x}*φ

*s,+*)(∂

*φ*

_{x}*s,*−)+

*g*

_{s}2π^{2} cos(2φ* _{s}*)−

*g*

_{cs}2π^{2}cos(2φ* _{c}*) cos(2φ

*) +*

_{s}*g*

*cσ*

2π^{2}(∂* _{x}*φ

*s,+*)(∂

*φ*

_{x}*s,*−) cos(2φ

*)−*

_{c}*g*ρs

2π^{2}(∂* _{x}*φ

*c,+*)(∂

*φ*

_{x}*c,*−) cos(2φ

*) +*

_{s}*g*ρσ

2π^{2}(∂* _{x}*φ

*)(∂*

_{c,+}*φ*

_{x}

_{c,}_{−})(∂

*φ*

_{x}*)(∂*

_{s,+}*φ*

_{x}

_{s,}_{−}), (2.3) where φ

*c/s*= φ

*c/s,+*+φ

*c/s,*−

*are the total phase fields. The renormalized velocities are v*ρ =

*2t*+

*(g*

_{4}

_{k}+

*g*

_{4}

_{⊥}−

*g*

_{1}

_{k})/2π

*and v*σ =

*2t*+

*(g*

_{4}

_{k}−

*g*

_{4}

_{⊥}−

*g*

_{1}

_{k})/2π. The Luttinger couplings of the

*charge and spin sectors are given by g*ρ =

*g*2⊥+

*g*2k−

*g*1k

*and g*σ =

*g*2⊥−

*g*2k+

*g*1k, respectively.

*The couplings g*_{c}*and g** _{s}*correspond to the Umklapp and the backward scattering of opposite

*spins, respectively, given by g*

*c*=

*g*3⊥

*and g*

*s*=

*g*1⊥

*, while the coupling g*

*cs*is also Umklapp

*scattering, but of parallel spins, given by g*

*=*

_{cs}*g*

_{3k}. The other coupling constants are given by

*g*

*=*

_{cσ}*g*

_{ρs}=

*g*

_{ρσ}=−

*J*

_{1}/2+

*J*

_{2}/2. The coupling g

_{ρs}

*and g*

_{ρσ}come from the backward scattering

*with opposite and parallel spins, respectively, while the coupling g*

*cσ*is related to Umklapp scattering with opposite spins. Here and in the following, we use the lattice constant as the unit for the coupling constants as well as for the Fermi velocities. The SU(2) symmetry of the

*spin sector assures g*

*=*

_{s}*g*σ

*, g*

*=*

_{cs}*g*

*cσ*

*, and g*ρs =

*g*ρσ. Therefore, there are five independent

*couplings which we choose to be g*ρ

*, g*

*c*

*, g*

*s*

*, g*

*cs*

*, and g*ρs. We note that the renormalization of the Fermi velocities, which is a secondary effect, will not be taken into account in the following.

The HamiltonianH*(x) (2.3) cannot be solved exactly. However, a renormalization group*
(RG) analysis permits the investigation of the relative importance of the various couplings. In
the RG procedure, the couplings are considered to be a function of some scaling parameter
*y, e.g., the logarithm of the effective bandwidth. As the scaling parameter is taken to infinity,*
the flow of the couplings shows which of them are important and which can be ignored,
depending on whether or not they tend to zero, to a finite value, or to infinity. For example,
*when all couplings but the forward scattering terms tend to zero, the Hamiltonian H describes*
a Luttinger liquid with freely propagating charge and spin degrees of freedom.

**mean-field approximation** **41**
*The one-loop RG equations for our five dimensionless running coupling constants ˜g*_{x}*(y)*≡
*g**x**(y)/4πt read [68, 73]*

*d ˜g*ρ*(y)*

*dy* =*2 ˜g*^{2}* _{c}*+

*˜g*

^{2}

*+*

_{cs}*˜g*

_{s}*˜g*ρs, (2.4a)

*d ˜g*

_{c}*(y)*

*dy* =*2 ˜g*ρ*˜g**c*− *˜g**s**˜g**cs*− *˜g**cs**˜g*ρs, (2.4b)
*d ˜g*_{s}*(y)*

*dy* = −*2 ˜g*^{2}* _{s}*−

*˜g*

_{c}*˜g*

*−*

_{cs}*˜g*

^{2}

*, (2.4c)*

_{cs}*d ˜g*

_{cs}*(y)*

*dy* = −*2 ˜g** _{cs}*+

*2 ˜g*ρ

*˜g*

*−*

_{cs}*4 ˜g*

_{s}*˜g*

*−*

_{cs}*2 ˜g*

_{c}*˜g*

_{s}−*2 ˜g*_{c}*˜g*ρs−*4 ˜g*_{cs}*˜g*ρs, (2.4d)
*d ˜g*ρs*(y)*

*dy* = −*2 ˜g*ρs+*2 ˜g*ρ*˜g** _{s}*−

*4 ˜g*

_{c}*˜g*

*−*

_{cs}*4 ˜g*

^{2}

_{cs}−*4 ˜g**s**˜g*ρs, (2.4e)

*with initial values ˜g*_{x}*(y* = 0) = *g** _{x}*/4πt. From these equations, it follows that there is only

*a single line of weak-coupling fixed points, namely g*

*=*

_{c}*g*

*=*

_{s}*g*

*=*

_{cs}*g*

_{ρs}= 0. In order to show this, we note that we have started our analysis assuming that there is neither a charge

*gap nor a spin gap. This implies that a weak-coupling fixed point corresponds to g*

*=*

_{c}*g*

*=*

_{s}*0. Equations (2.4) immediately imply that g*

*=*

_{cs}*g*

_{ρs}=

*0 also, and that only g*

_{ρ}remains undetermined.

A linear stability analysis of the fixed-point line shows that it is stable against small
*per-turbations g*_{cs}*and g*ρs*, that it is marginally stable against small perturbations g*_{s}*and g*ρ, and
*that its stability with respect to perturbations g** _{c}* depends on the sign of the fixed-point value

*g*

_{ρ}

*(stable for g*

_{ρ}<

*0, unstable for g*

_{ρ}>0). Therefore, in order to determine the weak-coupling regime, it is convenient and sufficient to consider the RG equations without the spin-charge

*coupling terms, i.e., we may consider the RG equations for ˜g*

*=*

_{cs}*˜g*ρs =0. We thus arrive at

*d ˜g*ρ*(y)*

*dy* =*2 ˜g*^{2}* _{c}*, (2.5a)

*d ˜g*_{c}*(y)*

*dy* =*2 ˜g*ρ*˜g** _{c}*, (2.5b)

*d ˜g*_{s}*(y)*

*dy* = −*2 ˜g*^{2}* _{s}* (2.5c)

in the vicinity of the weak-coupling fixed-point line.

*This simpler problem is readily analyzed. The trajectory for the spin coupling ˜g*_{s}*(y) flows*
*to infinity if g**s*< *0. In this case, a gap opens in the spin spectrum. If g**s*> 0, this coupling is
*marginally irrelevant, i.e., the spin mode remains soft. In the charge sector, g*ρ = *g** _{c}* initially,
and this relation remains valid under the RG flow. Therefore, it is sufficient to consider

*Eq. (2.5a). It is seen that for g*

*>*

_{c}*0 the charge mode becomes gapped because ˜g*

_{c}*(y) flows to*infinity, otherwise the charge excitations remain gapless.

*The simplified equations show that a fully gapless Luttinger-liquid phase, g** _{c}* =

*g*

*= 0, is*

_{s}*not possible for our model. The initial couplings would have to fulfill g*

*<*

_{c}*0 and g*

*> 0 which*

_{s}*requires J*2 >

*2U/3*+

*J*1

*for g*

*c*<

*0 and J*2 <

*(2U*−

*J*1)/3 for g

*> 0. These two conditions*

_{s}*cannot be fulfilled simultaneously with positive bare couplings U, J*

_{1}

*, and J*

_{2}. Consequently, we must redo our RG analysis under the assumption that at least one of the two modes is gapped.

When one of the fields is gapped, the spin-charge coupling processes become relevant [10, 73]. Their contribution will be considered on the mean-field level. In this picture, the gapped field is locked to a value which optimizes the interaction energy.

When there is a gap in the charge sector, the charge field φ* _{c}* is locked at φ

*= 0 modπ*

_{c}*because the initial value of the coupling g*

*is positive. Neglecting the fluctuations of the fieldφ*

_{c}

_{c}*in the g*

*cσ*

*term of the Hamiltonian (2.3), the terms proportional to g*ρs

*and g*ρσ do not contribute, and cos(2φ

*) can be replaced by its weak-coupling mean-field value, cos(2φ*

_{c}*)= 1.*

_{c}*Due to this substitution, the interaction terms proportional to g*_{cs}*and g**cσ* become marginal
*because their scaling dimensions reduce to x** _{cs}* =

*x*

*cσ*= 2. On the mean-field level, the

*spin-coupling term proportional to g*

*is of the same form as the interaction term proportional to*

_{cs}*g*

*. Therefore, the spin field φ*

_{s}

_{s}*fluctuates in the modified potential g*

^{∗}

*cos(2φ*

_{s}*) with the new*

_{s}*coupling g*

^{∗}

*,*

_{s}*g*^{∗}* _{s}* =

*g*

*−*

_{s}*g*

*=*

_{cs}*U*−

*2J*

_{2}. (2.6)

*Analogously, the interaction term proportional to g**cσ* combines with the interaction term
*pro-portional to g*σ*to produce the new coupling g*^{∗}_{σ}, with

*g*^{∗}_{σ}= *g*σ−*g**cσ* =*U* −*2J*_{2}. (2.7)

This equation shows that the SU(2) spin symmetry is preserved on the mean-field level.

In the presence of a charge gap and the SU(2) spin symmetry, we only have to analyze a

**mean-field approximation** **43**
*single equation for ˜g** _{s}*instead of the five RG equations (2.4), namely

*d ˜g**s**(y)*

*dy* =−*2 ˜g*^{2}* _{s}*, (2.8)

*with the initial value ˜g**s**(y*= 0)=*g*^{∗}* _{s}*/4πt. It is readily seen that the spin mode becomes gapped

*if g*

^{∗}

*<*

_{s}*0, i.e., J*

_{2}>

*U/2, independently of the value of the nearest-neighbor interaction J*

_{1}.

When there is a gap in the spin sector, the spin fieldφ* _{s}*is locked atφ

*= 0 modπbecause*

_{s}*the initial value of the coupling g*

*is negative. Neglecting the fluctuations of the fieldφ*

_{s}*in the*

_{s}*g*ρs

*term of the Hamiltonian (2.3), the terms proportional to g*ρσ

*and g*

*cσ*do not contribute and cos(2φ

*s*) can be substituted by its weak-coupling mean-field value, cos(2φ

*s*) = 1. Due to this

*substitution, the interaction terms proportional to g*

_{cs}*and g*ρs become marginal because their

*scaling dimensions reduce to x*

*cs*=

*x*ρs=2. On the mean-field level, the charge-coupling term

*proportional to g*

_{cs}*is of the same form as the interaction term proportional to g*

*. Therefore, the charge fieldφ*

_{c}

_{c}*fluctuates in the modified potential g*

^{∗}

*cos(2φ*

_{c}

_{c}*) with the new coupling g*

^{∗}

*,*

_{c}*g*^{∗}* _{c}* =

*g*

*+*

_{c}*g*

*=*

_{cs}*U*+

*J*

_{1}−

*J*

_{2}. (2.9)

*Using similar reasoning, the new coupling g*

^{∗}

_{ρ}becomes

*g*^{∗}_{ρ} =*g*ρ−*g*ρs= *U*+*2J*_{1}−*2J*_{2}. (2.10)
Note that these new initial couplings arenot equal, so we must analyze the two-dimensional
scaling curves defined by the equations

*d ˜g*ρ*(y)*

*dy* =*2 ˜g*^{2}* _{c}*, (2.11a)

*d ˜g*_{c}*(y)*

*dy* =*2 ˜g*ρ*˜g** _{c}*, (2.11b)

*given the initial values ˜g*_{c}*(y* = 0) = *g*^{∗}* _{c}*/4πt and ˜gρ

*(y*= 0) =

*g*

^{∗}

_{ρ}/4πt. The flow diagram is

*shown in Fig. 2.1. The conditions for a gapped charge mode are either g*

^{∗}

_{ρ}>

*0 or (g*

^{∗}

_{ρ}< 0 and

|*g*^{∗}* _{c}*|> |

*g*

^{∗}

_{ρ}|

*). This leads to the result that a gapped charge mode exists if J*

_{2}<

*2U/3*+

*J*

_{1}. In general, we find three regions where either the charge gap or the spin gap or both are finite. It is interesting to analyze the dominant correlations in the various gapped phases. The order parameters for density waves in the charge (CDW), spin (SDW), bond-charge (BCDW),

*g*

### ~

_{c}

*g*

### ~

_{ρ}

*Figure 2.1: Scaling curves for the charge-coupling parameters ˜g*_{c}*and ˜g*ρ in the presence of a
spin gap.

and bond-spin (BSDW) require the calculation of correlation functions using the operators

O* ^{i,CDW}* = (−1)

^{i}*(n*

*i,*↑+

*n*

*i,*↓), (2.12a)

O* ^{i,SDW}* = (−1)

^{i}*(n*

*i,*↑−

*n*

*i,*↓), (2.12b)

O* ^{i,BCDW}* = (−1)

^{i}*(c*

^{†}

_{i,}_{↑}

*c*

*i+1,*↑+

*c*

^{†}

_{i,}_{↓}

*c*

*i+1,*↓+

*h.c.),*(2.12c) O

*= (−1)*

^{i,BSDW}

^{i}*(c*

^{†}

_{i,}_{↑}

*c*

*i+1,*↑−

*c*

^{†}

_{i,}_{↓}

*c*

*i+1,*↓+

*h.c.)*, (2.12d) written in terms of the lattice fermions. These order parameters become

O^{CDW}*(x)*∝sinφ_{c}*(x) cos*φ_{s}*(x),* (2.13a)
O^{SDW}*(x)*∝cosφ_{c}*(x) sin*φ_{s}*(x),* (2.13b)
O^{BCDW}*(x)*∝cosφ_{c}*(x) cos*φ_{s}*(x),* (2.13c)
OBSDW*(x)*∝sinφ_{c}*(x) sin*φ_{s}*(x)* (2.13d)
in bosonized form. When the charge mode is gapped, the field φ* _{c}* is locked atφ

*= 0 modπ.*

_{c}When the spin mode is gapped, the fieldφ* _{s}*is locked atφ

*=0 modπ. Therefore, in the regime where both of the fields are gapped, we find that the BCDW order parameter is maximal. The model thus describes a phase with bond ordering (BOW) for∆*

_{s}*, 0 and∆*

_{c}*,0.*

_{s}When only the charge mode is gapped, the spin field is a free field. However, upon
*in-creasing the scaling parameter (y) of the renormalization group procedure, the initially *

**neg-mean-field approximation** **45**

∆ = 0*s*

∆ = 0*c* ∆ = 0*s*
*J**2*

*J**2*

*J*_{1}

*J*_{1}

*J**2**= U+J*_{1}*J**2* *J*_{1}

*J**2* *J*_{1}

∆ = 0*c*

∆ = 0*c*

∆ = 0*s /*

∆ = 0*s /*

∆ = 0*c*

∆ = 0*c*

∆ = 0*s /*

*U/2*

*U=0*

*U>0*

*U*

*=*

*= 2U/3+*

*/*

*/*
*/*

*/*

*SDW* *LE*

*BOW* *LE*

*BOW*

*Figure 2.2: Field-theory prediction for the half-filled t-U-J*1*-J*2 model. The solid lines give
the phase boundaries between the fully gapped regime (bond-order-wave, BOW) and the
semi-gapped regimes (spin-density wave, SDW; Luther-Emery, LE). The dashed line shows
the border between dominantly charge-density-wave and bond-order-wave correlations in the
Luther-Emery phase.

ative spin coupling grows and tends to zero, and the spin field oscillates aroundπ/2 (mod
π). Therefore, for small couplings, the dominating ordering is SDW for∆* _{c}* , 0 and∆

*= 0.*

_{s}Note that the SU(2) spin symmetry is not spontaneously broken, i.e., the spin correlations are critical without true long-range order.

Similarly, when the spin mode is gapped and the charge mode is gapless, there is no true long-range charge order. Therefore, we call this phase the Luther-Emery (LE) phase.

*The charge coupling g** _{c}* tends to zero, either from positive values or from negative values.

Depending on the sign of the charge coupling,φ* _{c}*fluctuates aroundπ/2 or around zero.
Cor-respondingly, the dominating correlations are either CDW or BCDW for∆

*= 0 and∆*

_{c}*, 0.*

_{s}The line which separates the dominant BCDW critical correlation and the dominant CDW correlations in the LE phase is indicated in Fig. 2.2 by a dashed line.

*J*

_{1}### ∆ = 0 *c /*

### ∆ = 0 *s*

*U* *SDW*

*Figure 2.3: Field-theory prediction for the half-filled t-U-J*_{1}*model. For all J*_{1} >0, the ground
state is a spin-density-wave (SDW) phase with a finite charge gap, zero spin gap and critical
spin correlations.

*The resulting phase diagram of the t-U-J*1*-J*2model at weak coupling is shown in Fig. 2.2.

*For U* = *0, the spin gap is always finite for J*_{2} > *0. For J*_{2} < *J*_{1}, the charge gap is also finite,
*and the ground state is characterized by a bond-order wave. The charge gap closes at J*2 = *J*1

and the system goes into a LE phase with no long-range charge or spin ordering but critical charge-density-wave correlations.

*For U* > *0, J*_{1} > *0, and J*_{2} < *U/2, the ground state is analogous to the spin-density-wave*
(SDW) phase of the one-dimensional Hubbard model, i.e., the charge gap is finite, the spin
*gap is zero, and the spin correlations are critical. For 2U/3*+ *J*1 > *J*2 > *U/2, both the spin*
gap and the charge gap are finite. The ground state is a BOW with long-range order in the
*bond-charge-density-wave correlations. For J*_{2} > *2U/3*+ *J*_{1}, the charge gap closes and the
system goes over to the LE phase with a finite spin gap but no charge long-range order. For
*2U/3*+ *J*_{1} < *J*_{2} < *U*+*J*_{1}, the bond-charge-density-wave fluctuations dominate, whereas, for
*J*_{2} >*U*+ *J*_{1}, the fluctuations in the charge-density-wave order parameter are maximal.

*In order to make contact with earlier work, we display the phase diagram of the t-U-J*_{1}
model separately in Fig. 2.3. In contrast to previous results [67, 68, 69], we do not find any
*signature of a BOW phase. For all J*_{1} > 0, the ground state is SDW, just as is the ground
*state of the half-filled Hubbard model for U* > 0. This result is corroborated by our numerical
DMRG data, which we present in the next section.