Bosonization for bulk and boundary operators

The model (1) around the SU(4) point ( $J_1\simeq J_2\simeq K/4$) can be bosonized from the SU(4) Hubbard model at quarter filling[1] :

$$H = \sum_i (-t c_{i+1 a\sigma}^{\dagger}c_{ia\sigma}+ {\rm H.c.})$$

$$+~\frac{U}{2} \sum_{iab\sigma\sigma^{'}} n_{ia\sigma}n_{ib\sigma^{'}} (1-\delta_{ab}\delta_{\sigma\sigma^{'}}) \ ,$$ (2)

by introducing the left and right movers for low energy degrees of freedom around the Fermi points ( $k_F=\pi/{4a_0}$) :

$$\frac{c_{ia\sigma}}{\sqrt{a_o}}\simeq R_{a\sigma}(x)\exp(ik_Fx) +L_{a\sigma}(x)\exp(-ik_Fx) \ .$$

At this point, we can bosonize the above slowly varying fields as usual through introducing four chiral bosonic fields $\Phi_{a\sigma R/L}$ using the Abelian bosonization formula[15] :

$$R_{a\sigma} = \frac{\kappa_{a\sigma}}{\sqrt{2\pi a_0}} \exp(i\sqrt{4\pi}\Phi_{a\sigma R}) \ ,$$

$$L_{a\sigma} = \frac{\kappa_{a\sigma}}{\sqrt{2\pi a_0}} \exp(-i\sqrt{4\pi}\Phi_{a\sigma L}) \ ,$$ (3)

where the bosonic fields satisfy the commutation relation $[\Phi_{a\sigma R},\Phi_{b\sigma^{'} L}]=\frac{i}{4}\delta_{ab}\delta_{\sigma\sigma^{'}}$, and the Klein factors $\kappa_{a\sigma}$ introduced here are used to insure the anticommutation relations between different flavors of fermions, which satisfies the following anticommutation rule $\{\kappa_{a\sigma},\kappa_{b\sigma^{'}}\}=2\delta_{ab}\delta_{\sigma\sigma^{'}}$. The physical properties of the system can be made more transparent by changing to a new basis :

$$\Phi_c=\frac{1}{2}(\Phi_{1\uparrow}+\Phi_{1\downarrow}+\Phi_{2\uparrow} +\Phi_{2\downarrow}) \ ,$$

$$\Phi_s=\frac{1}{2}(\Phi_{1\uparrow}-\Phi_{1\downarrow}+\Phi_{2\uparrow} -\Phi_{2\downarrow}) \ ,$$

$$\Phi_f=\frac{1}{2}(\Phi_{1\uparrow}+\Phi_{1\downarrow}-\Phi_{2\uparrow} -\Phi_{2\downarrow}) \ ,$$

$$\Phi_{sf}=\frac{1}{2}(\Phi_{1\uparrow}-\Phi_{1\downarrow}-\Phi_{2\uparrow} +\Phi_{2\downarrow}) \ .$$ (4)

Umklapp scatterings arising at higher order perturbation theory will result in a Mott transition at finite value of $U=U_c$, therefore for $U>>U_c$, the charge field $\Phi_c$ has a large gap, and only the spin-orbital part are left in the low energy sector. The remaining bosonized Hamiltonian can be further simplified by refermionization through the introduction of six Majorana fermions $\xi^a, a=1\ldots6$ :

$$(\xi^1+i\xi^2)_{R(L)} = \frac{\eta_1}{\sqrt{\pi a_0}}\exp(\pm i\sqrt{4\pi}\Phi_{sR(L)}) \ ,$$

$$(\xi^3+i\xi^4)_{R(L)} = \frac{\eta_2}{\sqrt{\pi a_0}}\exp(\pm i\sqrt{4\pi}\Phi_{fR(L)})\ ,$$

$$(\xi^5+i\xi^6)_{R(L)} = \frac{\eta_3}{\sqrt{\pi a_0}}\exp(\pm i\sqrt{4\pi} \Phi_{sfR(L)}) \ ,$$ (5)

where $\eta_i$ are Klein factors. The resulting Hamiltonian can then be written as[3] :

$${\cal H} = -\frac{i u_s}{2}({\bm\xi}_{sR}\partial_x {\bm\xi}_{sR}-{\bm\xi}_{sL} \partial_x {\bm\xi}_{sL})$$

$$+~(G_1+G_3)(\kappa_1+\kappa_2+\kappa_6)^2$$

$$-~\frac{i u_t}{2}({\bm\xi}_{tR}\partial_x{\bm\xi}_{tR}- {\bm\xi}_{tL}\partial_x {\bm\xi}_{tL})$$

$$+~(G_2+G_3)(\kappa_3+\kappa_4+\kappa_5)^2$$

$$+~2G_3~(\kappa_1+\kappa_2+\kappa_6)(\kappa_3+\kappa_4+\kappa_5) \ ,$$ (6)

where the spin and orbital triplets are defined as : ${\bm\xi}_s= (\xi^2, \xi^1, \xi^6)~$and$~{\bm\xi}_t=(\xi^4, \xi^3, \xi^5)$, $\kappa_a$ is defined as $\xi^a_R\xi^a_L$. $G_1~$and$~G_2$ measure the deviation from the SU(4) point, i.e. $J_1=\frac{K}{4}+G_1$ and $J_2=\frac{K}{4}+G_2$. $G_3<0$ is a nonuniversal parameter that could be extracted from the exact solution, and the two velocities $u_s, u_t$ are in general not equal to each other.

It was shown in Ref.[2] and [3] that the Hamintonian (6) contains several phases. Especially, there exist an extensive region where the system is gapless and the the low energy fixed point is governed by a SU(2) $_{2,s}\otimes$SU(2)$_{2,t}$ WZNW model :

$${\cal H} = {\frac{1}{2\pi}}\left\{\frac{u_s^*}{k_s+2}:{\bf J_s}\cdot{\bf J_s}:+ \frac{u_t^*}{k_t+2}:{\bf J_t}\cdot{\bf J_t}:\right\} \ ,$$ (7)

where $k_s=k_t=2$, $u^*_s$ and $u^*_t$ are renormalized velocities of the fixed point theory for spin and orbital sectors, respectively. $J^a_{s,t}(x)~(a=1, 2, 3)$ are current operators for level two SU(2) WZNW model, whose Fourier modes obey the Kac-Moody algebra :

$$[J_n^a,J_m^b] = i\epsilon^{abc}J_{n+m}^c+\frac{1}{2}kn\delta_{n+m,0} \ .$$

The spin and orbital density operators have the following general forms:

$${\bm S}_i \sim {\bm J}_{sR}+{\bm J}_{sL}+~\left(e^{i\pi x/2a_0}{\bf {\cal N}}_s+\mbox{H.c.}\right)$$

$$+~(-1)^{x/a_0}{\vec n}_s ,$$

$${\bm T}_i \sim {\bm J}_{tR}+{\bm J}_{tL}+~\left(e^{i\pi x/2a_0}{\bf {\cal N}}_t+\mbox{H.c.}\right)$$

$$+~(-1)^{x/a_0}{\bm n}_t ,$$ (8)

here ${\bm J}_{s,t}$ are the smooth $(k\sim 0)$ parts of the spin( orbital) density, while ${\bf {\cal N}}_{s,t}$ and ${\bm n}_{s,t}$ are the $2k_F=\pi/2a_0$ and $4k_F=\pi/a_0$ parts.

The current operators can be expressed in terms of Majorana fermions :

$${\bm J}_{sR(L)} = -\frac{i}{2}{\bm\xi}_{sR(L)}\wedge {\mbox{\bm$\xi$}}_{sR(L)} \ ,$$

$${\bm J}_{tR(L)} = -\frac{i}{2}{\bm\xi}_{tR(L)}\wedge {\bm\xi}_{tR(L)} \ .$$ (9)

The boson representations for $2k_F$ components ${\bf {\cal N}}_{s,t}$ are :

$${\cal N}_s^z \propto \exp\{i\sqrt{\pi}(\Phi_s+\Phi_f+\Phi_{sf})\}$$

$$-\exp\{i\sqrt{\pi}(-\Phi_s+\Phi_f-\Phi_{sf})\}$$

$$+\exp\{i\sqrt{\pi}(\Phi_s-\Phi_f-\Phi_{sf})\}$$

$$-\exp\{i\sqrt{\pi}(-\Phi_s-\Phi_f+\Phi_{sf})\} \ ,$$

$${\cal N}_t^z \propto \exp\{i\sqrt{\pi}(\Phi_s+\Phi_f+\Phi_{sf})\}$$

$$-\exp\{i\sqrt{\pi}(\Phi_s-\Phi_f-\Phi_{sf})\}$$

$$+\exp\{i\sqrt{\pi}(-\Phi_s+\Phi_f-\Phi_{sf})\}$$

$$-\exp\{i\sqrt{\pi}(-\Phi_s-\Phi_f+\Phi_{sf})\} \ ,$$

$${\cal N}_s^+ \propto \exp\{i\sqrt{\pi}(\Theta_s+\Phi_f+\Theta_{sf})\}$$

$$-\exp\{i\sqrt{\pi}(\Theta_s-\Phi_f-\Theta_{sf})\} \ ,$$

$${\cal N}_t^+ \propto \exp\{i\sqrt{\pi}(\Phi_s+\Theta_f+\Theta_{sf})\}$$

$$+\exp\{i\sqrt{\pi}(-\Phi_s+\Theta_f-\Theta_{sf})\} \ ,$$ (10)

where $\Theta_a$ are the dual fields of $\Phi_a$, and satisfy $[\Phi_a(x), \Theta_b(y)]= i\delta_{ab}\Theta (y-x)$.

The $2k_F$ components can be written in a more compact way by noting that the six Majorana fermions could be associated with six critical Ising models. Then using the order and disorder operators, $\sigma_a~$and$~\mu_a$, of the Ising models, they can be expressed as follows :

$${\cal N}^z_s \propto i\mu_1\mu_2\sigma_3\sigma_4\sigma_5\sigma_6 +\sigma_1\sigma_2\mu_3\mu_4\mu_5\mu_6 \ ,$$

$${\cal N}^z_t \propto i\sigma_1\sigma_2\mu_3\mu_4\sigma_5\sigma_6 +\mu_1\mu_2\sigma_3\sigma_4\mu_5\mu_6 \ ,$$

$${\cal N}^{+}_s \propto \left(\sigma_1\mu_2+i\mu_1\sigma_2\right) \left(\sigma_3\sigma_4\sigma_5\mu_6+\mu_3\mu_4\mu_5\sigma_6\right) \ ,$$

$${\cal N}^{+}_t \propto \left(-\mu_3\sigma_4+i\sigma_3\mu_4\right)\left(\mu_1\mu_2 \sigma_5\mu_6-\sigma_1\sigma_2\mu_5\sigma_6\right) \ .$$ (11)

The $4k_F$ part of spin and orbital operators, generated from higher harmonics of bosonization due to interactions, can be written down by noting that these operators should transform as vectors under SO(3)$_{s,t}$ and carry no chirality[1] :

$${\bm n}_s \propto i{\bm\xi}_{sR}\wedge{\bm\xi}_{sL} \ ,$$

$${\bm n}_t \propto i{\bm\xi}_{tR}\wedge{\bm\xi}_{tL} \ .$$ (12)

Since the fixed point is governed by a SU(2) $_{2,s}\otimes$SU(2)$_{2,t}$ WZNW theory, it is better to write the above operators in a way which makes the symmetry properties more transparent. This can be done by noting that each components of ${\bm S}_i$ and ${\bm T}_i$ should transform as a vector under spin and orbital SU(2) rotations, respectively. This means that each components of ${\bm S}_i$ and ${\bm T}_i$ should be primary fields of the SU(2)$_2$ WZNW model[16]. It can then be immediately seen that ${\bm{J}}_{s,t}$ are just the current operators of SU(2)$_{2s,t}$ WZNW models, and the $2k_F$ components correspond to the spin $1/2$ primary fields of it. The latter can be made evident from Eq. (11) by using the equivalence between a SU(2)$_2$ WZNW theory and three critical Ising models[17]. Especially, the spin $1/2$ primary field can be expressed as the product of three order or disorder operators of the corresponding three Ising models. We then have :

$${\cal N}^a_s \sim g^{(2)}_{sa}g^{(2)}_{t0}-ig^{(1)}_{sa}g^{(1)}_{to} \ ,$$

$${\cal N}^a_t \sim g^{(2)}_{s0}g^{(2)}_{ta}-ig^{(1)}_{s0}g^{(1)}_{ta} \ ,$$ (13)

where $a=1, 2, 3$ and $g^{(1,2)}_{s,t\alpha}$ are defined as:

$$g = \tau^{\alpha}\!\!\left(g_{\alpha}^{(1)}+ig_{\alpha}^{(2)}\right) \ .$$

Here $\tau^{\alpha}$ are Pauli matrices for $\alpha=1, 2, 3$, $\tau^0$ is the identity matrix, and $g_{s,t}$ are spin one-half primary fields of SU(2)$_{2s,t}$ WZNW theory. The remaining primary fields with spin one ${\bf\Phi}_{s,t}$ just correspond to the $4k_F$ components ${\bm n}_{s,t}$.

After completing discussions about the fixed point theory and its operator contents, we turn to the bosonized forms of the above operators in open boundary condition. The open chain boundary condition introduces the following boundary conditions on the left- and right- moving fermion fields[14] :

$$R_{a\sigma}(0)+L_{a\sigma}(0)=0 \ ,$$

when transformed into boson language, it becomes

$$\Phi_{a\sigma R}(0)+\Phi_{a\sigma L}(0)=-\frac{\sqrt\pi}{2} \ .$$

We can then analytically continue the right-moving fields to left-moving fields by $\Phi_{a\sigma R}(x,t)=-\frac{\sqrt\pi}{2}-\Phi_{a\sigma L}(-x,t)$. In this way, we arrive at a description of the system in terms of chiral fields only.

With the above relations, we find for the boundary fields, we have :

$$\Phi_{a\sigma}(x,t) = -\frac{\sqrt\pi}{2}+\Phi_{a\sigma L}(x,t)-\Phi_{a\sigma L}(-x,t)$$

$$\Rightarrow \Phi_{a\sigma}(0,t)=-\frac{\sqrt\pi}{2} \ ,$$

$$\Theta_{a\sigma}(x,t) = \frac{\sqrt\pi}{2}+\Phi_{a\sigma L}(x,t)+\Phi_{a\sigma L} (-x,t)$$

$$\Rightarrow \Theta_{a\sigma}(0,t)=\frac{\sqrt\pi}{2}+2\Phi_{a\sigma L} (0,t) \ ,$$ (14)

or in terms of Majorana fermions :

$$\xi^a_{R}(x,t)=\xi^a_{L}(-x,t) \ .$$ (15)

Substituting Eq. (14), Eq. (15) into Eq. (9), Eq. (10) and Eq. (12), it is easy to see that all components of spin (orbital) operators are propotional to the current operators, i.e. :

$${\bm S}_{boundary}\propto {\bm J}_{sL}(0) \ , \ \ \ {\bm T}_{boundary}\propto {\bm J}_{tL}(0) \ .$$ (16)

This completes our discussions about the bosonization formulas.