Spin-charge separation for paired Dirac fermions in $(1+1)$ dimensions

Here, we review kinematics of the Dirac equation.The gamma matrices we use in our (1+1)d model are standard ones. With the time-like signature $g^{\mu\nu}=\mathrm{diag}\left(1,-1\right)$ and using the $2\times 2$ gamma matrices defined in terms of the Pauli matrices, we have

which satisfy the Dirac algebra

In addition, charge conjugation can be defined as $\psi_{C}\equiv\gamma^{5}\psi^{*}$, where $\psi_{C}$ is the charge-conjugated spinor wavefunction determined by the condition $C\gamma^{\mu}C^{-1}=\pm(\gamma^{\mu})^{T}$, and the Dirac adjoint is defined in the usual way by $\bar{\psi}=\psi^{\dagger}\gamma^{0}$. Varying the free-particle action associated with the Lagrangian density

with respect to $\bar{\psi}$, gives the Dirac equation for two-dimensional spinor solutions $\psi=(\psi_{1},\psi_{2})^{T}$

This set of coupled equations has plane-wave solutions

The associated single-particle momentum space Dirac operator for finite mass and chemical potential is

which gives the Dirac equation

for spinor solutions

The Dirac operator can be reparametrized to the hyperbolic form

where $\mathrm{cosh}\,\eta\equiv(E+\mu)/m$, $\mathrm{sinh}\,\eta\equiv p/m$, and $\mathrm{tanh}\,\eta=p/(E+\mu)$. Here, $\eta$ is the usual rapidity associated with generators of the $(1+1)$d Lorentz transformations $\Lambda$

Thus, the relativistic limit of the in-medium (finite $\mu$, small $m$, large Fermi surface) Dirac operator amounts to an ordinary Lorentz boost through elements of the hyperbolic subgroup of $SL(2,\mathbb{R})$. It is straightforward to show that solutions of Eq. (12) have the form

Alternatively, one could choose a parametrization such that $\mathrm{cos}\,\phi\equiv m/(E+\mu)$, $\mathrm{sin}\,\phi\equiv p/(E+\mu)$, and $\mathrm{tan}\,\phi=p/m$, which gives

which has solutions of the form

One may also obtain a complex form by choosing the reduced gamma matrices to be

With this choice the (linear) Dirac equation is

which has the plane-wave solutions

The momentum space Dirac operator now takes the form

which can be reparamtrized to the trigonometric form

where $\mathrm{cos}(\phi)\equiv m/(E-\mu)$, $\mathrm{sin}(\phi)\equiv p/(E-\mu)$, and $\mathrm{tan}(\phi)=p/m$. Solutions then have the form

Note that, in this complex form, Lorentz transformations are now given by

In $(1+1)$ dimensions the interactions that give rise to pairing fields are those associated with scalar, pseudoscalar, vector, axial vector, difermion, and density interactions which can be expanded, respectively, as

where $\beta$ is the relative phase between the upper and lower spin components. This phase is superfluous to spinors in $(1+1)$d (the overall phase is not) and emerges naturally out of the complex difermion field. If we set $\beta=0$, for instance, the interaction types reduce considerably to

from which, without loss of generality, we may focus on the following pairing fields

Incorporating these three pairing fields, our basic starting point will be the Lagrangian density for relativistic fermions with finite mass and chemical potential interacting through scalar meson and superconducting channels given by

The physical parameters $m$, $\mu$, $g_{s}$ and $g_{d}$ are the mass, chemical potential, scalar and difermion couplings, respectively, and $C$ is the charge conjugation operator. In addition, we might consider somewhat separately the dynamical equations for both real chiral and complex difermion fields

In particular, the equations of motion for the difermion density and phase are

with the conserved current in the right equation acting as a source for density fluctuations in the left one. The low-temperature mean-field limit of Eq. (62) allows for several non-vanishing condensates

which are respectively, the average density, chiral, difermion and conjugate difermion condensates. In the mean-field approximation, Eq. (62) leads to the modified Dirac equation

It is important to recall that mean-field descriptions are justified at low temperatures and weak interactions or if one assume a standard large-$N$ limit description. To obtain plane wave solutions of Eq. (66) one could solve the $4\times 4$ system for the Fourier components

from which one obtains the dispersion through the condition

The Fourier components, generally complex, are then given by

Instead of taking this approach, one may choose a solution of the form $\xi\psi$ where $\psi$ satisfies the vacuum Dirac equation and $\xi$ contains background effects. For a uniform background (London limit), the modified plane-wave solutions have the form

with the modified boost parameters given by

and where we have defined $E_{\pm}=E+\mu\pm p$. Dependence on the difermion field $\Delta$ resides in $\zeta$, with $\eta$ depending purely on kinematic variables. As a consistency check it is instructive to express the parametrization of single-particle states in Eq. (45) purely in terms of the condensate parameters (no kinematic variables) by first expressing the latter as

then inverting. Without loss of generality, we may take $\phi\to i\phi$, ($\phi,\Delta\in\mathbb{R}$), whereby we obtain

We see that in the limit $p\to 0$, $\zeta\to\zeta^{\prime}$ since here $E_{+}=E_{-}\sim\rho$.

The emergence of SCS is momentum dependent as can be seen from Eq. (88). Let us examine the form of Eq. (88) in the deep material region where $\mu$ and $\bar{\Delta}$ are large and $m$ is small. We consider the two limits of large versus small single-particle momenta. For large $p$ and keeping only lowest nonzero contributions, $E+\mu\to p\,\Rightarrow\,E_{-}\to 0$ and $E_{+}\gg|\bar{\Delta}|\,\Rightarrow\,\beta\to\mathrm{tan}^{-1}\!\left(Im\bar{\Delta}/Re\bar{\Delta}\right)\!/2=\mathrm{arg}\!\left(\bar{\Delta}\right)\!/2$ and

In this regime the phase of $\bar{\Delta}$ amounts to a global $U(1)$ transformation of the elementary spinor field since it no longer depends on $p$. The magnitude $|\bar{\Delta}|$ replaces the factor of $E_{-}$ that appears in the numerator (denominator) of the upper (lower) component of spin acting to counter large boosts in the elementary spinor field. Thus, at large momentum we recover the original vacuum spin structure but with the density $|\bar{\Delta}|$ in the role of particle mass $m$. In this limit the current associated with the internal phase vanishes leaving only propagating bound spin and fermion number charge.

In contrast, for small $p$, to leading order we find that

where we have expanded the difermion density and phase, $\bar{\Delta}$ and $\beta_{\Delta}$, and the charge density $\rho$ and phase $\theta_{N}$ in terms of their respective momenta: $|\bar{\Delta}|\simeq d_{1}q_{\Delta}+\dots,\;\beta_{\Delta}\simeq a_{1}q_{\beta}+\dots,\;\rho\simeq\rho_{0}+\dots,\;\theta_{N}\simeq c_{1}p+\dots\;$. It is interesting that by including fluctuations in $\bar{\Delta}$ with momentum $q_{\Delta}$ the prefactor in Eq. (96) takes the form of a propagating spin degree of freedom. The density $\rho$ and phase $\theta_{N}$ together define charge propagation at momentum $p$. There is the appearance of an additional (internal) phase that propagates with characteristic momentum $q_{\beta}$ (we will identify this as a gauge potential). Generally then at the two extreme limits that contrast phase and background density momenta, i.e., $p,q_{\beta}\gg q_{\Delta}$ and $p,q_{\beta}\ll q_{\Delta}$, the system will be described by elementary spinors propagating over a background in the former case, versus decomposed spin and charge plus an additional gauge structure propagating independently in the latter case:

Note that large fluctuations in phase describes strong depletion of the difermion condensate and dissociation of difermion bound states, in contrast to the case of pure difermion density fluctuations.

We can further characterize the splitting into spin and charge degrees of freedom by contrasting the group structures in the elementary fermion and spin-charge formulations. First note that Eq. (66) is symmetric under parity when parity transformations are defined in the usual way through $\mathcal{P}:\psi(x,t)\rightarrow\gamma^{0}\psi(-x,t)$, since $\bar{\Delta}$ is odd and $\sigma$ and $m$ are even under $\mathcal{P}$. Invariance of Eq. (66) under time-reversal however, i.e., $\mathcal{T}:\psi(x,t)\rightarrow\gamma^{0}\psi^{*}(x,-t)$, holds only if $\bar{\Delta}=-\bar{\Delta}^{*}\Rightarrow Re\bar{\Delta}=0$, in the chiral limit $m,\sigma=0$. Recall that in $(1+1)$d our choice for the gamma matrices forces us to work with an anti-Hermitian matrix, $\gamma^{1}\in SO^{+}(2)<SL^{+}(2,\mathbb{R})$, yet we are able to render it Hermitian through the action of $\gamma^{0}\in SO^{-}(2)<SL^{-}(2,\mathbb{R})$. Since we only have one spatial gamma matrix to contend with in the present dimensionally reduced problem, we may attempt to define parity inversion using $\gamma^{1}$ instead of $\gamma^{0}$, i.e., $\mathcal{P}:\psi(x,t)\rightarrow\gamma^{1}\psi(-x,t)$. One finds that under this definition of parity, $\bar{\Delta}$ and $\sigma$ are both odd so that Eq. (66) is invariant when the fermion mass is purely generated by the scalar condensate.

We can identify the total charge field in Eq. (88) as the two-component composite object whose elements decompose as a product of group factors acting on the superposition of basis vectors $n_{1}=(1,\,0)^{T}$, $n_{2}=(0,\,1)^{T}\to\Phi=g\,(n_{1}+n_{2})^{T}/\sqrt{2}$, where $g\in U(1)\times SL(2,\mathbb{C})$ and $\Phi\cong U(2)$. Explicitly,

The group factors on the right are associated, respectively, with fermion number charge $N$ and the remaining three combine to form a representation of the hyperbolic ($A\cong\mathbb{C}_{>0}$) subgroup $SL(2,\mathbb{R})_{\mathrm{boost}}\times SL(2,\mathbb{R})_{\Delta}\times SU(2)_{\mathrm{\Delta}}<SL(2,\mathbb{C})_{\mathrm{boost}+\Delta}$, which follows from the standard Iwasawa $K\cdot A\cdot N$ decomposition. The physical distinction between boosts of elementary spinors and background effects induced by the difermion field naturally decompose the hyperbolic subgroup into a product of associated elements: two subgroups of $SL(2,\mathbb{R})$ and one of $SU(2)$. We choose to absorb the boost matrix for the elementary spinors into $\Phi$ since we are interested in the deep material regime in Eq. (96) where vacuum spinors are no longer appropriate descriptions. The spin field in this regime is then just the remaining factor in Eq. (88) associated with the background driven quasi-boosts in Eq. (96). We can express it in terms of group elements as $\chi=a\,(n_{1}+n_{2})^{T}/\sqrt{2}$, where $a\in SL(2,\mathbb{R})$:

The full in-medium wavefunction is then expressed as

Returning to the concept of parity, it is significant that $\gamma^{1}$ is anti-idempotent under multiplication and the group that it generates $\mathcal{G}=\left\{\mathbb{1},\,\gamma^{1},-\mathbb{1},\,-\gamma^{1}\right\}$ is isomorphic to the fourth roots of unity $\mathcal{U}=\left\{i^{0},\,i^{1},\,i^{2},\,i^{3}\right\}$ generated by $e^{i\pi/2}$ under multiplication equivalent to $\mathbb{Z}_{4}$, the cyclic group of order four. The definition of parity that uses $\gamma^{1}$ can be interpreted as a rotation by $180^{\mathrm{o}}$ in 2D about the y axis by noting that

Here $\phi$ is the polar angle that parametrizes spin rotations in the $x,z$-plane which are isomorphic to the $K$ subgroup in the Iwasawa decomposition. Such rotations map to parity inversion when restricted to $\phi=\pi$. Note that the other gamma matrix $\gamma^{0}$ offers no such interpretation. Thus, even though there are no rotations in 1D, two parity inversions defined in this way result in the characteristic overall factor of $-1$. Since the part of Eq. (108) parameterized by $\eta$ solves the vacuum Dirac equation, it must transform in the same way under parity acquiring an overall factor of $-1$ by itself. We have seen that parity inversion in Eq. (88) amounts to flipping the signs of both $\bar{\Delta}$ and $p$, and one may expand $\bar{\Delta}\simeq d_{1}q_{\Delta}+d_{3}q_{\Delta}^{3}+\dots$ in the difermion momentum $q_{\Delta}$ in Eq. (96) to recover a (quasi) spin structure when $p\to 0$. In this way, a hand off of spin from the elementary fermions to the background fluctuations takes place through a crossover region (as discussed in Sec. III.1).

Having identified $\chi$ as the carrier of spin, one may consider a Maxwellian $U(1)_{M}$ gauge transformation that sends $\psi\to\exp(i\theta_{M})\,\psi$. Since $\chi\in SL(2,\mathbb{R})$, it cannot change under such gauge transformations but $\Phi\in U(2)$ has room within its $U(1)_{N}$ factor, hence $\Phi$ is the natural carrier of charge. An additional internal symmetry can be identified in the decomposition $\chi\Phi$ if one treats $\chi$ as an embedding $SL(2,\mathbb{R})\hookrightarrow SL(2,\mathbb{C})\cong SL(2,\mathbb{R})\times SU(2)$ by incorporating half of the internal phase $\beta$ into $\chi$. A local internal $SU(2)$ symmetry can then be identified that sends $\chi\to\chi\,g$ and $\Phi\to g^{-1}\Phi$ with $g\in SU(2)$. This internal symmetry is then broken by the presence of local fluctuations in $\beta$ through phase (versus density) fluctuations in $\bar{\Delta}$. One may summarize the group structure for the dynamical degrees of freedom for each regime through the map

Let us turn our attention to the phase of $\bar{\Delta}$ to show that it gives rise to a gauge structure. For clarity, we first focus on the fermion kinetic, mass, and chemical potential terms in the Lagrange density $\mathcal{L}_{\psi}$. Expanded, these give

Let us now introduce the spin-charge decomposition in Eq. (91) and take a general approach for breaking the internal $U(1)\leq SU(2)$ symmetry by making the substitutions

where $\phi_{1,2}$ are quasiparticle spin components and where $\alpha_{1,2}$, $\beta$ and $\zeta$ contain the background. This decomposition gives the spin-charge separated Lagrange density

where we use the short hand notation $\partial\equiv\partial_{t}+\partial_{x}$, $\bar{\partial}\equiv\partial_{t}-\partial_{x}$ and have reabsorbed the $\zeta$ factors in order to isolate the internal phase. Thus, allowing for the phases to vary independently and locally gives a model with explicit symmetry breaking at the classical level. Then taking $\alpha_{1}=\alpha_{2}=\alpha$ to eliminate redundancy, we can identify $\alpha=\theta_{N}$ and $\beta=\beta_{\Delta}$ (the background fermion and difermion phase, respectively), which leads to the form