Interacting 1D Chiral Fermions with Pairing: Transition from Integrable to Chaotic

We consider a 1D model with two flavors of chiral complex fermions $c_{s}$ ($s=\uparrow,\downarrow$), which has an action:

The Lagrangian density takes the form

where the fermion fields satisfy the commutation relations

and the Hamiltonian density can be divided into three local terms:

The spin index $s=\uparrow,\downarrow$ here need not be the physical spin, but can be a pseudospin or any flavor index. The first term in the Hamiltonian density is a charge conserving free term of two chiral complex fermions:

where the velocities $v_{s}>0$ are real, and the matrix $M_{ss^{\prime}}$ is Hermitian. The second term is a generic superconducting pairing term:

By a proper gauge choice, we can set the parameter $J_{s}$ to be real here. Lastly, there is a local (delta-function) interaction term between the local densities of the two fermion flavors:

The total Hamiltonian is given by $H=\int\mathcal{H}dx$. Ignoring all the irrelevant terms, this is the most generic translationally invariant interacting model for two flavors of chiral complex fermions, up to unitary transformations.

Equivalently, one can rewrite the model of Eq. (5) in terms four flavors of chiral Majorana fermions $\psi_{i}$ ($i=1,2,3,4$) defined by:

The Lagrangian density under the Majorana fermion representation can be shown to take the form

where the Hamiltonian density

where the velocities are

and the matrix $A_{ij}$ is real antisymmetric, given by

where $a.s.$ stands for anti-symmetrization.

The model can also be rewritten by a bosonization mapping. We define the scalar boson fields $\phi_{s}$ by

where the boson fields satisfy the commutation relation

with $\text{sgn}(x)$ being the sign of $x$. This allows us to calculate the mapping of all operators between fermions and bosons. For instance, here we will need the mappings $c^{\dagger}_{s}c_{s}=\frac{\partial_{x}\phi_{s}}{2\pi}$, $-ic^{\dagger}_{s}\partial_{x}c_{s}=\frac{(\partial_{x}\phi_{s})^{2}}{4\pi}$, and $-ic_{s}\partial_{x}c_{s}=2\pi e^{2i\phi_{s}}$ ($s=\uparrow,\downarrow$) lian2019 ; hu2021chiral . As a result, our model can be mapped into a chiral boson representation

with the Hamiltonian density

where the velocity coefficients $V_{ss^{\prime}}$ is given by

When the fermion interaction vanishes, namely,

one simply has free fermions. In the Majorana fermion representation, we define the momentum space Majorana fermions

where $L$ is the spatial length of the system. Without bulk flux insertion, the fermions satisfy anti-periodic boundary conditions, thus the single-fermion momentum $k\in\frac{2\pi}{L}(\mathbb{Z}+\frac{1}{2})$. By defining $\psi_{k}=(\psi_{1,k},\psi_{2,k},\psi_{3,k},\psi_{4,k})^{T}$, one can then rewrite the Hamiltonian in Eq. (11) with $U=0$ as

where the $4\times 4$ matrix $h(k)$ is given by

Diagonalizing the matrix $h(k)$ then gives the single-fermion energy spectrum $\epsilon_{n}(k)$ ($1\leq n\leq 4$) of the model. We emphasize that for the system to have a lower energy bound and be stable, the parameters have to satisfy $v_{j}\geq 0$ ($1\leq j\leq 4$).

The other solvable point is the chiral Luttinger liquid point, which is when each of the fermion spin flavor has a U(1) charge symmetry, namely, when the system has a total global symmetry U(1)${}_{\uparrow}\times$U(1)_↓. This requires the vanishing of the following parameters:

Therefore, there is no superconductivity pairing, i.e., $\mathcal{H}_{P}=0$. By Eq. (17), the bosonized Hamiltonian becomes a free boson Hamiltonian with terms no higher than the second order of boson fields $\phi_{s}$, although the fermion form of the Hamiltonian is interacting. The boson fields $\phi_{s}$ ($s=\uparrow,\downarrow$) can be expanded in modes as

where $L$ is the spatial length, and $a_{s,k}$ and $a_{s,k}^{\dagger}$ are the annihilation and creation operators of the normal boson modes. Besides,

is the U(1) charge (or number of fermions) of the spin $s$ (where $:\mathrel{O}:$ stands for normal ordering of operator $O$), and it satisfies the commutation relation $[\frac{2\pi N_{s}}{L},\phi_{0,s}]=i$ with the constant piece $\phi_{0,s}$. If we impose the anti-periodic boundary condition for the fermions, the bosons will satisfy periodic boundary condition, and their momenta take values $k\in\frac{2\pi}{L}\mathbb{Z}$. This leads to a free boson Hamiltonian

where we have defined $v_{\eta}$ ($\eta=\pm$) as the eigenvalues of the matrix $V_{ss^{\prime}}$ in Eq. (18), and new boson eigenmodes $b_{\eta,k}$:

This is known as the chiral Luttinger liquid, where the model reduces to two free chiral boson modes with velocities $v_{\pm}$. We note that the stability of the system requires $v_{\pm}\geq 0$, which avoids infinite negative energy states.

We note that if $v_{\uparrow}=v_{\downarrow}$, one can relax the condition in Eq. (23) to allow nonzero $M_{\uparrow\downarrow}$, and still gets free chiral bosons. This is because in this case, both the fermionic velocity kinetic term $-i\sum_{s}v_{s}c^{\dagger}_{s}\partial_{x}c_{s}$ and the interaction term $Uc^{\dagger}_{\uparrow}c_{\uparrow}c^{\dagger}_{\downarrow}c_{\downarrow}$ are invariant under any SU(2) fermion basis rotation. One can therefore rotate the fermion basis $(c_{\uparrow},c_{\downarrow})^{T}$ to a new basis $(c_{\uparrow}^{\prime},c_{\downarrow}^{\prime})^{T}=\mathcal{U}(c_{\uparrow},c_{\downarrow})^{T}$ which diagonalizes the $M$ matrix. In this new basis, one again satisfies condition (23), and can thus bosonize the model into free chiral bosons.

A well-known diagnostics of quantum chaos is the many-body level spacing statistics (LSS) in a conserved symmetry charge sector. In this paper, by the symmetry charges we refer to those of the global symmetries of the model.

In particular, the generic model in Eq. (5) always has two conserved symmetry charges: the total fermion parity $(-1)^{N_{\uparrow}+N_{\downarrow}}$, and the total many-body momentum $K_{\text{tot}}$ from the translation symmetry. Since we imposed anti-periodic boundary condition, all the single-fermion momenta are half-odd integers (Eq. (28)), and thus the two conserved charges are not independent:

Therefore, it is sufficient to keep only the total momentum $K_{\text{tot}}$ for the above two conserved charges.

Assume the $n$-th many-body energy level (sorted from the lowest to the highest) in a conserved charge sector $Q$ (which includes momentum $K_{\text{tot}}$) is $E_{n}(Q)$. One can define the level spacing $\delta_{E,n}=E_{n+1}(Q)-E_{n}(Q)$, and examine the statistical probability distribution $p_{LS}(\delta_{E})$ of $\delta_{E,n}$, which is known as the LSS. There are generically two situations:

(i) If the system is quantum integrable (exactly solvable), or if there are still hidden (quasi)local conserved quantities in the conserved charge sectors $Q$, the LSS in sector $Q$ will resemble the Poisson distribution (characterizing independent random variables) Berry1977 :

where the constant $s_{0}\geq 0$ (see Fig. 1(a)). This indicates there is no repulsion between neighboring energy levels.

(ii) If the system is fully quantum chaotic within a conserved charge sector $Q$, the LSS in sector $Q$ will resemble the LSS of random Hermitian matrices $H$, which is known as the Wigner-Dyson distribution Bohigas1984 ; Dyson1970 ; Wigner1967 . Depending on symmetry classes of the system, the Wigner-Dyson distribution function is given by

where $s_{0}>0$. The integer $m=1,2,4$ for the Hamiltonian $H$ in the real (spinless time-reversal (TR) invariant), complex (without TR invariance) and symplectic (spinful TR invariant with spin-orbit coupling), respectively. Note that $p_{LS}(0)=0$ in this case, indicating that the neighboring energy levels repulse each other.

We will also numerically compute the zero-temperature spectral weight $A_{s}(\omega,k)=2\text{Im}G_{R,s}(\omega,k)$ of fermions $c_{s}$, where $G_{R,s}(\omega,k)$ is the retarded Green’s function of fermion $c_{s}$ in the energy-momentum space. If $|k,j\rangle$ denotes the $j$-th many-body eigenstate with total momentum $K_{\text{tot}}=k$ and energy $E_{k,j}$, where $1\leq j\leq N_{k}$ with Hilbert space dimension $N_{k}$ ($N_{k}>0$ if and only if $k>0$), and $|0\rangle$ denotes the zero-particle vacuum state, we can numerically calculate the spectral weight as

As is clear from this expression, the spectral weight characterizes the single-fermion density of states. In practical calculations, to avoid numerical divergences, we relax the delta function into a Lorentzian function

and take $\eta=0.3$.

We first examine the numerical LSS at the 2 solvable points of free fermions and free bosons we discussed in Sec. III. As free models, they are many-body quantum integrable, since all the many-body states are Fock states of the single-particle eigenstates. Therefore, one expects the LSS of their many-body energy spectrum in each charge sector to show Poisson distributions.

The free fermion case. In this case with the interaction $U=0$ in Eq. (8), and all the other parameters nonzero (Sec. III.1), there are only the total fermion parity $\mathbb{Z}_{2}$ symmetry and the translational symmetry, the conserved charges of which are the parity $(-1)^{N_{\uparrow}+N_{\downarrow}}$ and the total many-body momentum $K_{\text{tot}}$. As shown in Eq. (29), these two conserved charges are not independent, and we can label each symmetry charge sector by $K_{\text{tot}}$.

Fig. 2(a) and (c) shows the many-body density of states (DOS) and LSS of the free fermion case ($U=0$) in the sector of total momentum $K_{\text{tot}}=\frac{27}{2}$. The other parameters are listed in the caption of Fig. 2, which are chosen sufficiently arbitrary so that there are no additional global symmetries. As one can easily see, the LSS shows a Poisson statistics, due to the many-body integrable nature of free fermions.

Fig. 2(b) shows the zero-temperature spectral weights of $c_{\uparrow}$ (red thick line) and $c_{\downarrow}$ (blue thin line) at momentum $k=K_{\text{tot}}=\frac{27}{2}$, which are defined in Eq. (32). As expected, they show delta function peaks at the single-particle energies of the free chiral Majorana fermions (eigenvalues of Eq. (22)).

The free boson case. As shown in Sec. III.2, when $J_{s}=0$, $\Delta=0$, and $M_{\uparrow\downarrow}=0$, while the interaction $U\neq 0$, the model has a global symmetry U(1)${}_{\uparrow}\times$U(1)_↓, and is solvable as two flavors of free chiral bosons. The conserved symmetry charges are therefore $N_{\uparrow}$ and $N_{\downarrow}$.

Fig. 3 shows the ED results of such a free-boson example (parameters given in the caption). In Fig. 3(a), the unfilled line is the total DOS of the total momentum $K_{\text{tot}}=\frac{27}{2}$ sector; while the line filled with red color is the DOS of the finer symmetry sector with quantum numbers $(K_{\text{tot}}=\frac{27}{2},N_{\uparrow}=1,N_{\downarrow}=0)$. Fig. 3(c) shows the LSS of this symmetry sector $(K_{\text{tot}}=\frac{27}{2},N_{\uparrow}=1,N_{\downarrow}=0)$, which is almost a delta function at zero. This is because the free bosons’ linear dispersion leads to a large number of many-body level degeneracy. Nonetheless, the LSS can be viewed as a Poisson distribution with a large slope (i.e., small $s_{0}$ in Eq. (30)).

Fig. 3(b) shows the spectral weights of the spin up (red thick line) and down (blue thin line) fermions in this free boson case, respectively. Analytically, by refermionization, one can derive the spectal weights of the spin $s$ fermion as lian2019 ; hu2021chiral ; hu2021integrability (up to energy shifts induced by the chemical potentials $M_{\uparrow\uparrow}$ and $M_{\downarrow\downarrow}$)

where $\zeta_{s\eta}$ are the coefficients in Eq. (27). This agrees well with the numerical results in Fig. 3(b).

We now consider the case of adding a nonzero hopping $M_{\uparrow\downarrow}$ to the solvable free-boson point, namely,

Due to the nonzero term $M_{\uparrow\downarrow}$, the model only has a global U(1) symmetry. Thus, the conserved symmetry charges are the total fermion charge $N=N_{\uparrow}+N_{\downarrow}$ and the total momentum $K_{\text{tot}}$.

In addition, in the special case when $M_{\uparrow\uparrow}=M_{\downarrow\downarrow}$, the model Hamiltonian obeys a simple transformation under the particle-hole transformation $P$ that flips the U(1) fermion charge $N$:

where $\theta_{s}=s(\arg(M_{\uparrow\downarrow})+\frac{\pi}{2})$ for $s=\pm$ (corresponding to $s=\uparrow,\downarrow$). In this case (namely, $M_{\uparrow\uparrow}=M_{\downarrow\downarrow}$), the $N=0$ charge sector will have an additional symmetry $P$ and thus an additional conserved charge, the eigenvalue $\eta_{P}=\pm 1$ of operator $P$. The $N\neq 0$ sectors do not have this additional conserved charge.

Although the model takes a simple form, it cannot be solved as free bosons or free fermions. In the fermion representation, the interaction $U$ makes it not free. In the bosonized representation, the $M_{\uparrow\downarrow}$ term is bosonized into a nonlinear term

making the bosons not free, either. It is not yet known if the model is exactly solvable by certain many-body techniques. Therefore, instead, we examine the ED results of the model in this case.

Fig. 4 shows the numerical results for a set of arbitrarily chosen parameters (given in Fig. 4 caption) in this case. Panel (a) shows the DOS of the $K_{\text{tot}}=\frac{27}{2}$ sector (the unfilled line), and its subsector with symmetry charges with $(K_{\text{tot}}=\frac{27}{2},N=1)$ (line filled with red color), which is much smoother compared to the free boson case (Fig. 3(a)). The fermion spectral weights in Fig. 4(b) show irregular shapes, which is an indication of the absence of free fermion or free boson picture.

Intriguingly, the LSS in the finest symmetry sector in this case still shows Poisson statistics among the parameter space we have explored. Fig. 4(c) shows the LSS in the symmetry sector of $(K_{\text{tot}}=\frac{27}{2},N=1)$ (i.e., the red part of DOS in Fig. 4(a)). We have examined the LSS in different symmetry sectors for more sets of parameters satisfying Eq. (35), all of which show no level repulsions. This indicates the existence of hidden (quasi)local conserved quantities lian2022conserv beyond those of the global symmetries, which has not been theoretically understood yet. Moreover, the model in this case may be even totally quantum integrable, which we leave for the future studies.

Another similar case with different symmetries will be presented in Sec. IV.4 below. Lastly, we note that when we set $U=0$ in this case, the model will become free fermions with nonlinear dispersions (due to the nonzero $M_{ss^{\prime}}$). However, the behavior of the LSS with $U\neq 0$ here (Poisson) is completely different from that of generic nonlinear dispersion fermions with interaction (which is Wigner-Dyson, see Sec. V).

In this subsection, we investigate the case with a superconducting pairing: starting from the free boson case with global symmetry U(1)${}_{\uparrow}\times$U(1)_↓, we turn on the pairing within the spin down flavor, reducing the global symmetry into U(1)${}^{(\uparrow)}\times\mathbb{Z}_{2}^{(\downarrow)}$. The parameters thus satisfy

Accordingly, the conserved charges are $K_{\text{tot}}$ and $N_{\uparrow}$. The parity $(-1)^{N_{\downarrow}}$ is dependent on $K_{\text{tot}}$ and $N_{\uparrow}$, as we showed in Eq. (29).

A special case within the parameter space of Eq. (38) is when $M_{\downarrow\downarrow}=0$, for which the model transforms simply under a particle-hole transformation $P$:

Accordingly, when $M_{\downarrow\downarrow}=0$, the $N_{\uparrow}=0$ charge sector has an additional symmetry $P$, and thus gains an additional conserved charge, the eigenvalue $\eta_{P}=\pm 1$ of the operator $P$. This additional charge does not exist in all the $N_{\uparrow}\neq 0$ sectors.

Such a U(1)${}^{(\uparrow)}\times\mathbb{Z}_{2}^{(\downarrow)}$ symmetry constraint may seem unphysical, that different spins have different symmetries. However, if one regard the spin solely as a fermion flavor index, the model can have its physical context. For instance, it is shown in Ref. hu2021integrability that the interacting chiral edge states of the $4/3$ filling FQH state are equivalent to the interacting fermion model with U(1)${}^{(\uparrow)}\times\mathbb{Z}_{2}^{(\downarrow)}$ here, where the spin $\uparrow$ fermion carries an irrational electric charge $\frac{2e}{\sqrt{3}}$, while the spin $\downarrow$ fermion is charge neutral. Thus, with charge conservation, pairing is allowed for spin $\downarrow$ fermions, but not allowed for spin $\uparrow$ fermions.

With the pairing term $J_{\downarrow}$, the model has a nonlinear term

in the bosonized representation. Therefore, the model is neither a free fermion nor a free boson model. Intriguingly, as studied in Ref. hu2021integrability , this model with U(1)${}^{(\uparrow)}\times\mathbb{Z}_{2}^{(\downarrow)}$ symmetry, i.e., with parameters satisfying Eq. (38), shows Poisson LSS in each global symmetry charge sector. Fig. 5 shows the ED results of an example, with the parameters as given in the Fig. 5 caption. The unfilled line and red-filled line in Fig. 5(a) are the DOS of the entire $K_{\text{tot}}=\frac{27}{2}$ sector and the DOS of the subsector with $(K_{\text{tot}}=\frac{27}{2},N_{\uparrow}=0)$, respectively. As expected, the fermion spectral weights in Fig. 5(b) are different from those in the free fermion or free boson cases. Fig. 5(c) shows the LSS in the symmetry sector $(K_{\text{tot}}=\frac{27}{2},N_{\uparrow}=0)$, which is a clear Poisson distribution. More symmetry sectors are examined in Ref. hu2021integrability , all of which shows a Poisson LSS.

Therefore, similar to the U(1) symmetry case we discussed in Sec. IV.3, the Poisson distribution indicates the existence of hidden (quasi)local conserved quantities lian2022conserv , and moreover, the model may be fully quantum integrable. Identifying such hidden conserved quantities is an interesting task for the future studies.

We comment on an observation, that in both the U(1) symmetric case in Sec. IV.3 and the U(1)${}^{(\uparrow)}\times\mathbb{Z}_{2}^{(\downarrow)}$ case here, the bosonized representation of the model has only one nonlinear sine or cosine term in the boson fields $\phi_{s}$: the $M_{\uparrow\downarrow}$ term in the former case, and the $J_{\downarrow}$ term in the later case. This may be intrinsically related to their Poisson LSS and potential quantum integrability. As we will see in the next few subsections, if one has two or more nonlinear sine or cosine terms, the LSS in each symmetry sector will show Wigner-Dyson statistics.

We now turn on more pairing terms in our model, and examine its many-body LSS. In this subsection, we consider parameters satisfying

where both spin up and spin down have a nonzero p-wave pairing $J_{s}$. It is straightforward to see that each spin $s$ has a fermion parity symmetry $\mathbb{Z}_{2}^{(s)}$, with conserved charges $(-1)^{N_{s}}$, respectively. Moreover, the fact that all the mass terms $\Delta$, $M_{ss^{\prime}}$ are zero leads to another implicit parity symmetry, which we call $\mathbb{Z}_{2}^{(++)}$. The parity charge of this $\mathbb{Z}_{2}^{(++)}$ is most easily seen in terms of the Majorana basis defined in Eq. (9), which reads

Similarly, the above three parities and the total momentum $K_{\text{tot}}$ are not independent, as Eq. (29) implies. Therefore, a complete set of independent symmetry charges is $K_{\text{tot}}$, $N_{\uparrow}$ and $P_{++}$.

In the bosonized representation, the model now has two nonlinear sine or cosine terms given by $J_{\uparrow}$ and $J_{\downarrow}$:

Therefore, the model is more “nonlinear” compared to the cases in Secs. IV.3 and IV.4.

As shown in Fig. 6, the LSS (Fig. 6(c)) in a fixed symmetry sector $(K_{\text{tot}}=\frac{27}{2},(-1)^{N_{\uparrow}}=+1,P_{++}=+1)$ (the red part of DOS in Fig. 6(a)) in this case becomes (GOE) Wigner-Dyson statistics. Therefore, we conclude the model in this case is quantum chaotic. The linear ramp at small $\delta_{E}$ indicates it resembles a GOE distribution (with $m=1$ in Eq. (31)). Indeed, the Hamiltonian of the model in this case is a real matrix in the momentum space, as protected by a an anti-unitary PT symmetry, where $P$ is the spatial inversion and $T$ is the spinless time-reversal. Intriguingly, the spectral weights of the model in this case (Fig. 6(b)) is close to that of the free-boson case, despite being a quantum chaotic model.

The global symmetry of the model can be further lowered down to only $\mathbb{Z}_{2}^{(\uparrow)}\times\mathbb{Z}_{2}^{(\downarrow)}$ if the parameters satisfy

Compared to the case in Eq. (41), here the presence of nonzero $M_{\uparrow\uparrow}$ or $M_{\downarrow\downarrow}$ breaks the conservation of the parity charge in Eq. (42). Therefore, the independent symmetry charges in this case are $K_{\text{tot}}$ and $(-1)^{N_{\uparrow}}$.

The ED results for this case is shown in Fig. 7. Fig. 7(a) shows the DOS of the total sector of momentum $K_{\text{tot}}=\frac{27}{2}$ and the symmetry subsector with $K_{\text{tot}}=\frac{27}{2},(-1)^{N_{\uparrow}}=+1$. The LSS of this symmetry subsector is given in Fig. 7(c), which shows a GOE (linear at small $\delta_{E}$) Wigner-Dyson shape. Therefore, similar to the case in Sec. IV.5, the model here with $\mathbb{Z}_{2}^{(\uparrow)}\times\mathbb{Z}_{2}^{(\downarrow)}$ symmetry is also quantum chaotic. The GOE distribution is also due to the presence of a PT symmetry, which restricts the Hamiltonian in the momentum space to be real. The spectral weights show clear deviations from that in the free-boson case.

In the last case, if we do not impose any constraints on the parameters, the model only has a global fermion parity $\mathbb{Z}_{2}$ symmetry, the symmetry charge of which is $(-1)^{N}=(-1)^{N_{\uparrow}+N_{\downarrow}}=(-1)^{2K_{\text{tot}}}$ (Eq. (29)). Therefore, the only independent conserved charge is $K_{\text{tot}}$.

Fig. 8 shows a ED calculation for parameters (see the caption) in this generic case. In the symmetry sector of total momentum $K_{\text{tot}}=\frac{27}{2}$, the DOS distribution (Fig. 8(a)) is much smoother than all the higher symmetry cases we discussed earlier. The fermion spectral weights in Fig. 8(b) also shows less discretized peaks., indicating a higher randomness in the energy spectrum. Fig. 8(c) shows the LSS of the $K_{\text{tot}}=\frac{27}{2}$ sector, which is quadratic at small $\delta_{E}$, and thus resembles the GUE Wigner-Dyson distribution (namely, $m=2$ in Eq. (31)). This is because, with the parameters $M_{\uparrow\downarrow}$ and $\Delta$ being generically complex, the model does not have a PT symmetry or other anti-unitary symmetry, and the Hamiltonian is generically in the complex class. Therefore, one expects a GUE LSS in each symmetry charge sector. The model with only a $\mathbb{Z}_{2}$ symmetry is therefore many-body quantum chaotic.