---
title: Bicolored Bosonic Models
url: https://www.emergentmind.com/topics/bicolored-bosonic-models
type: topic
---

# Bicolored Bosonic Models

Bicolored bosonic models are bosonic systems organized by a binary structure, but the literature does not use the phrase in a single uniform sense. In one line of work, the binary label is a genuine internal component or species index, as in two-species \(a,b\) bosons or two-component hard-core bosons. In another, it is a two-sublattice or bipartite lattice structure, such as a bosonic Su-Schrieffer-Heeger chain. In a third, it is a two-mode or two-well organization of a single bosonic species with flavour-changing interactions. There is also a more specialized use in which the “two colors” are not two bosonic species at all, but a parity-induced decomposition into two Fock-space sectors. Taken together, the term is best understood as an umbrella for bosonic models with a fundamental \(\mathbb Z_2\)-type, two-component, two-sublattice, two-mode, or two-sector architecture rather than as the name of one standard Hamiltonian class [1803.05812].

## 1. Scope of the binary structure

The most important distinction is between **genuine two-component bosons** and **effective two-sector bosons**. Genuine two-component models include two-species \(p\)-wave pairing models with bosonic operators \(a_{\mathbf k},a^\dagger_{\mathbf k}\) and \(b_{\mathbf k},b^\dagger_{\mathbf k}\), two-component quantum Hall models with spin-up and spin-down bosons in species-dependent gauge fields, and two-component hard-core bosons \(a,b\) on a lattice [1908.02821]. By contrast, a bosonic SSH chain is “bicolored” because each unit cell contains two sublattice modes \(a_{1,i}\) and \(a_{2,i}\), and a spin-boson model can become “bicolored” only after symmetry reduction to two parity/spin fibers on the same bosonic Fock space [2103.15200].

A second distinction concerns whether the binary structure is **microscopic** or **emergent**. In a two-mode model with flavour-changing interaction, the two “colors” are two orthogonal single-particle modes, such as the two inequivalent minima \(X_\pm\) of the second band of a bipartite two-dimensional optical lattice [1901.08412]. In a two-well multi-state Josephson model, the internal labels \(j=1,2\) may be read as two colors or two internal levels coexisting in each well, with tunnelling amplitudes \(\Omega_{jk}\) that need not preserve the label [1309.2956]. In the parity-reduced spin-boson case, however, there is only one bosonic field species, and the binary structure comes from the qubit and the bosonic parity operator \(\Gamma(-1)=(-1)^N\) [1803.05812].

| Binary structure | Representative realization | Example |
|---|---|---|
| Two species/components | Internal labels \(a,b\) or \(\uparrow,\downarrow\) | Interspecies \(p\)-wave pairing; BIQH lattices |
| Two sublattices/bonds | Bipartite unit cell or alternating couplings | Bosonic SSH; higher-order spin models |
| Two modes/sectors | Two orbitals, two wells, or two fibers | Flavour-changing two-mode bosons; parity fibers |

This suggests that “bicolored bosonic model” is most precise when the relevant binary organization is stated explicitly: two-species, two-sublattice, two-mode, or parity-bicolored.

## 2. Two-species and two-component bosons

A genuinely two-species bosonic model is given by the exactly solvable \(p\)-wave pairing Hamiltonian
\[
H = \sum_k \eta_k^2\left(a_k^\dagger a_k+b_k^\dagger b_k\right) -\frac{G}{4}\sum_{k,k'}\eta_k\eta_{k'} \left(b_k^\dagger a_{-k}^\dagger-a_k^\dagger b_{-k}^\dagger\right) \left(b_{k'}a_{-k'}-a_{k'}b_{-k'}\right),
\]
with \(\eta_k=\sin(k/2)\) in the one-dimensional presentation. Here the binary label is an internal species index, interpreted as a pseudospin-\(\tfrac12\) boson, and the relevant pairing operator is an interspecies singlet. The model is solvable through a hyperbolic Richardson-Gaudin \(su(1,1)\) construction, with exact eigenstates
\[
|\Phi_{M,\nu}\rangle = \prod_{\alpha=1}^{M} \left( \sum_{\mathbf k>0} \frac{\eta_{\mathbf k}}{\eta_{\mathbf k}^2-e_\alpha}\, K_{\mathbf k}^{+} \right) |\nu\rangle.
\]
Its central physical result is a transition from a **gapless fragmented singlet pair condensate** to a **pair Bose superfluid (PBS)**, with a **third-order quantum phase transition** at
\[
G_c=\frac{2}{2L+M-1},\qquad g_c=\frac{2}{2+\rho}
\]
in the thermodynamic scaling used in the paper [1908.02821].

A different two-component structure appears in bosonic atoms with two internal states in **species-dependent artificial gauge potentials**. The single-particle Hamiltonian is
\[
H_0=\frac{1}{2M} \begin{pmatrix} \left(\mathbf p-e^\uparrow \mathbf A^\uparrow\right)^2 & 0\\ 0 & \left(\mathbf p-e^\downarrow \mathbf A^\downarrow\right)^2 \end{pmatrix},
\]
with the case of interest given by \(e^\uparrow B^\uparrow\) and \(e^\downarrow B^\downarrow\) of the same sign but different magnitudes. The proposed composite-fermion states are
\[
\Psi_{[f_\uparrow,f_\downarrow]} \sim \chi^\uparrow_{f_\uparrow}\, \left[\chi^\downarrow_{f_\downarrow}\right]^*\, \chi_1,
\]
and the paper studies \([1,1]\), \([2,1]\), \([1,2]\), and \([2,2]\). In exact diagonalization, \([1,1]\) and \([2,1]\) are described as plausibly gapped in the thermodynamic limit, while the line \(f_\downarrow=f_\uparrow+1\) is argued to be likely gapless and superfluid-like [1508.05139].

A lattice realization of two-component bosons is the bosonic integer quantum Hall construction on the square lattice with Hamiltonian
\[
\hat H=\hat H_0+\hat H_1+\hat H_\lambda ,
\]
where the two hard-core boson components \(a\) and \(b\) experience opposite Peierls phases in the interwire hopping and correlated hopping in \(\hat H_0\). The coupled-wire analysis yields the \(K\)-matrix
\[
K= \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix},
\]
the Hall conductance
\[
\sigma_{xy}=q^T K^{-1}q=\pm 2,
\]
and a nonchiral edge with two counter-propagating gapless modes, one charge mode and one pseudospin mode. Numerically, infinite DMRG on cylinders confirms quantized flux-induced charge pumping and entanglement spectra consistent with the BIQH edge theory [1712.01212].

These models share a common feature: the binary label is a true microscopic degree of freedom, not a bookkeeping device introduced after diagonalization.

## 3. Bipartite, sublattice, and higher-order topological bosons

In bipartite bosonic systems the binary structure is not species but sublattice. A canonical example is the bosonic SSH chain
\[
H=\sum_i\Big[t_1a_{1,i}^\dag a_{2,i}+t_2a_{2,i}^\dag a_{1,i+1}+\text{H.c.}\Big],
\]
with two sublattices per unit cell and alternating intracell/intercell couplings \(t_1,t_2\). In the quadratic-boson formalism, the relevant object is the dynamic matrix for closed systems or the effective Hamiltonian \(X\) for open Lindbladian systems. For the symmetry-preserving open SSH model with
\[
L_j=\gamma(a_{1,j}+a_{2,j}),
\]
the effective Hamiltonian is
\[
X=\left(\begin{array}{cc} 0 & t_1+t_2 e^{i k}\\ t_1+t_2 e^{-i k} & 0 \end{array}\right) +\frac{i}{2}\gamma^2\left(\begin{array}{cc} 1 & 1\\ 1 & 1 \end{array}\right),
\]
and after subtracting a constant imaginary shift one has
\[
\sigma_3 X'(k) \sigma_3=-X'(k),
\]
placing the model in class AIII. The non-Hermitian winding number is
\[
W=\frac{1}{2\pi}\int_0^{2\pi}d k\, q_1^{-1}\frac{d q_1}{d k},
\]
with the SSH criterion
\[
W=\left\{ \begin{array}{ll} 1, & t_1<t_2,\\ 0, & t_1>t_2. \end{array} \right.
\]
If the reservoir coupling is sublattice-selective, for example \(L_j=\gamma a_{2,j}\), chiral symmetry is broken in \(X\) and the open system falls into class A instead [2103.15200].

A different binary organization appears in higher-order bosonic topological phases built from spin-\(\tfrac12\) variables. One construction starts from the \(1\)D \(\mathbb Z_2\times\mathbb Z_2\) cluster chain with two \(\mathbb Z_2\) variables per unit cell, \(a\) and \(b\),
\[
H_{\mathbb{Z}_2\times\mathbb{Z}_2} = -\sum_{i=1}^{N-1} \left( Z^a_i X^b_i Z^a_{i+1} + Z^b_i X^a_{i+1} Z^b_{i+1} \right),
\]
and then stacks and couples such chains to obtain a two-dimensional bosonic higher-order topological phase with four spin-\(\tfrac12\) objects per unit cell. A complementary construction uses a dimerized antiferromagnetic XY model
\[
H_{XY}=\sum_{\langle ij\rangle}J_{i,j}\sum_{a=x,y}\sigma^a_i \sigma^a_j,
\]
with strong inter-cell coupling \(\lambda\) and weak intra-cell coupling \(t\). In both cases the bulk is gapped, the edges are gapped but topological, and the corners carry protected modes transforming projectively under \(\mathbb Z_2\times\mathbb Z_2\) [1807.09781].

The same theme extends to a two-sector bosonic semimetal built from two momentum-separated copies of the \(O(4)\) nonlinear sigma model,
\[
S = \int d^3x\, \frac{1}{2g}\mathrm{tr}\!\left[ \partial^\mu U_A^\dagger \partial_\mu U_A +\partial^\mu U_B^\dagger \partial_\mu U_B \right] -\pi S_\theta[U_A]+\pi S_\theta[U_B],
\]
with \(\theta_A=+\pi\) and \(\theta_B=-\pi\), located at momenta \(\pm \mathbf k_0\). This is a two-copy bosonic analogue of a Dirac semimetal rather than a two-species boson model, but it is still a binary-organized bosonic phase in which the \(A\) and \(B\) sectors are exchanged by \(\mathcal T\) and \(\mathcal I\) [1601.02632].

## 4. Parity-bicolored spin-boson models

A distinct use of “bicolored” arises in spin-boson type models with a parity symmetry. The Hilbert space is
\[
\mathbb C^2\otimes \mathcal F_b(H),
\]
and the full Hamiltonian is
\[
H_{\eta}(\alpha,f,\omega):=\eta \sigma_z\otimes 1+1\otimes d\Gamma(\omega)+\sum_{i=1}^{2n}\alpha_i(\sigma_x\otimes \varphi(f_i))^i.
\]
Here the bosonic interaction is not restricted to a linear field coupling; it contains higher-order powers
\[
\alpha_i(\sigma_x\otimes\varphi(f_i))^i,\qquad i=1,\dots,2n,
\]
with semiboundedness ensured by an even-polynomial dominance condition on the leading terms [1803.05812].

The binary structure is produced by the spin-parity symmetry. The bosonic parity operator is
\[
\Gamma(-1)=(-1)^N,
\]
and the paper constructs an explicit selfadjoint unitary \(U\) such that
\[
U H_\eta(\alpha,f,\omega) U^*=F_\eta(\alpha,f,\omega)\oplus F_{-\eta}(\alpha,f,\omega),
\]
where the fiber operators on Fock space are
\[
F_{\eta}(\alpha,f,\omega)=\eta\Gamma(-1)+d\Gamma(\omega)+\sum_{i=1}^{2n}\alpha_i \varphi(f_i)^i.
\]
Thus the model is “bicolored” only after symmetry reduction: the two colors are the two parity/spin sectors labeled by \(+\eta\) and \(-\eta\).

This decomposition is not merely formal. The paper proves that under its hypotheses the operators \(F_\eta\) and \(H_\eta\) are closed on natural domains, are selfadjoint and semibounded when \((\alpha,\eta)\in\mathbb R^{2n+1}\), and satisfy HVZ-type spectral statements. The ground-state identification is especially sharp:
\[
E_{-|\eta|}=E_\eta,
\]
and if \(\eta\neq 0\) and \(E_\eta\) is an eigenvalue of \(H_\eta\), then the ground state lies in the \(F_{-|\eta|}\) fiber. By contrast, eigenvalues of \(F_{|\eta|}\) correspond to excited states. The paper also shows that in the massless case \(m=0\) with \(\eta\neq 0\),
\[
F_{|\eta|}\ \text{has no ground state}.
\]
In this sense the two parity colors have different spectral roles: one carries the bottom of the spectrum, the other encodes excited states [1803.05812].

A plausible implication is that parity-bicoloring is not an alternative to genuine two-species bosons but a different mechanism by which a binary bosonic organization can emerge from symmetry.

## 5. Two-mode, two-well, and ladder realizations

A minimal two-mode bosonic model with an explicit flavour structure is the Hamiltonian
\[
\begin{aligned}
H &= \frac{\varepsilon}{2}(n_2-n_1) + \frac{g}{2}\left[\rho_{0,1}\, n_1(n_1-1)+\rho_{0,2}\, n_2(n_2-1)\right] \\
&\quad + 2g\,\rho_1\, n_1 n_2 + \frac{g}{2}\left(\rho_2^*\, a_1^\dagger a_1^\dagger a_2 a_2 + \rho_2\, a_2^\dagger a_2^\dagger a_1 a_1\right),
\end{aligned}
\]
which describes \(N\) bosons populating two orthogonal single-particle modes with a **flavour-changing contact interaction**. The pair-conversion term
\[
a_1^\dagger a_1^\dagger a_2 a_2 + a_2^\dagger a_2^\dagger a_1 a_1
\]
converts two bosons of one flavour into two bosons of the other. At mean-field level the model has three phases and selects a relative phase \(\phi=\pm \pi/2\), so the mixed phase carries finite angular momentum and breaks time-reversal symmetry. In the exact quantum treatment, the zero-temperature ground state is well approximated by a coherent superposition of the two quasi-classical states, while at low temperature it becomes a mixture. One phase exhibits single-particle fragmentation together with a finite pair order parameter, and the non-equilibrium dynamics shows a sharp transition between self-trapping and pair tunneling [1901.08412].

A related but spatially resolved two-color model is the exactly solvable two-well, two-on-well-state Hamiltonian
\[
\begin{aligned}
H &= \cdots
- \Omega_{11}(a_{1}^{\dagger}b_{1} + b_{1}^{\dagger}a_{1}) - \Omega_{12}(a_{1}^{\dagger}b_{2} + b_{2}^{\dagger}a_{1}) \\
&\quad-  \Omega_{21}(a_{2}^{\dagger}b_{1} + b_{1}^{\dagger}a_{2}) - \Omega_{22}(a_{2}^{\dagger}b_{2} + b_{2}^{\dagger}a_{2}),
\end{aligned}
\]
with bosonic modes \(a_j,a_j^\dagger,b_j,b_j^\dagger\), \(j=1,2\). Here the labels \(j=1,2\) are naturally interpreted as two colors or two internal states, while \(a,b\) denote the two wells. The distinctive feature is that the tunnelling matrix \(\Omega_{jk}\) need not be diagonal, so the model includes both color-preserving and color-changing interwell tunnelling. In the integrable realization,
\[
\Omega_{jk}=s_j t_k,
\]
and the model is solved by the algebraic Bethe ansatz through a new multi-state bosonic Lax operator [1309.2956].

The bosonic Kitaev-Hubbard chain and two-leg ladder provide a further set of binary structures. The chain Hamiltonian
\[
\begin{aligned}
H=& \sum_{r=1}^{L} \left(-te^{i \theta}b_r^{\dagger}b_{r+1} - \Delta e^{i \varphi}b_r b_{r+1}+ \text{H.c.}\right) - \sum_{r=1}^{L} \mu n_r \\
&+\sum_{r=1}^{L} \frac{U}{2}n_r(n_r-1) + U_3 \sum_{r} n_r (n_{r} -1) (n_{r}-2)
\end{aligned}
\]
contains both hopping and nearest-neighbor pairing on every bond. In the hard-core limit it maps exactly to an anisotropic XY chain with a Dzyaloshinskii-Moriya term,
\[
\begin{aligned}
H_{S}=&-\frac{1}{2}\sum_{r=1}^{L} \left[ (t \cos \theta + \Delta) \sigma_{r}^{x} \sigma_{r+1}^{x} + (t \cos \theta - \Delta) \sigma_{r}^{y} \sigma_{r+1}^{y} \right] \\
& + \frac{1}{2}\sum_{r=1}^{L} t \sin \theta (\sigma_{r}^{x} \sigma_{r+1}^{y}-\sigma_{r}^{y} \sigma_{r+1}^{x}) -\frac{1}{2} \sum_{r=1}^{L} \mu (1-\sigma_{r}^{z}),
\end{aligned}
\]
and at \(t=\Delta\), \(\theta=\pi/2\), \(\mu=0\) it becomes a 1D quantum compass model. For free bosons the bosonic BdG matrix can exhibit a non-Hermitian skin effect, but in the hard-core limit that effect disappears exactly because the model is transformed to a spin-\(\tfrac12\) system and then to a Hermitian fermionic Kitaev chain. The ladder Hamiltonian
\[
\begin{aligned}
H_L =& -\sum_{l=1,2;\,r=1}^{L} \left( t b_{l,r}^{\dagger}b_{l,r+1}+\Delta b_{l,r}b_{l,r+1} + \text{H.c.} \right) \\
& -\sum_{r=1}^{L}(t_{\perp} b_{1,r}^{\dagger}b_{2,r} + \text{H.c.}) - \sum_{l=1,2;\,r=1}^{L}\mu n_{l,r} \\
& +\sum_{l=1,2;\,r=1}^{L} \frac{U}{2}n_{l,r}(n_{l,r} - 1)
\end{aligned}
\]
is a direct two-leg binary bosonic system with inequivalent leg and rung bonds [2206.11827].

## 6. Classification, exact methods, and conceptual limits

Quadratic bosonic models with a binary organization are often analyzed through effective first-quantized structures rather than through the many-body Hamiltonian alone. One classification program studies the general quadratic bosonic Hamiltonian
\[
\mathcal{H}=\sum_{k}\phi^{\dagger}\left(k\right)H\left(k\right)\phi\left(k\right),\qquad
\phi^{\dagger}\left(k\right)=\left(\begin{array}{cc} \boldsymbol{a}^{\dagger}\left(k\right) & \boldsymbol{b}\left(-k\right) \end{array}\right),
\]
with commutation metric
\[
\tau=\left(\begin{array}{cc} \mathbb{I}_{N} & \\ & -\mathbb{I}_{N'} \end{array}\right),
\]
and proves that each stable gapped quadratic bosonic system is homotopic to a direct sum of two effective single-particle subsystems,
\[
H\left(k\right)\approx h_{0}\left(k\right)\oplus h_{0}^{\prime}\left(k\right).
\]
In this sense binary organization is intrinsic to the quadratic bosonic formalism itself: the positive and negative \(\tau\)-sectors define a natural two-sector structure even before one asks whether the microscopic system is two-species or bipartite [1905.07989].

A more restrictive conclusion is reached in the “threefold way” analysis of stable quadratic bosons. There the relevant matrix is
\[
H_\tau=\tau_3 H_b,
\]
and the paper proves three no-go theorems for stable gapped bosonic mean-field systems: the absence of parity switches, the absence of symmetry-protected-topological quantum phases, and the absence of localized bosonic zero modes under open boundary conditions. It also stresses that among the standard classifying symmetries only time-reversal is a real symmetry of the many-boson system, while particle-hole and chiral are effective single-particle constraints. For bicolored bosonic models this is a major limitation: a two-sublattice or chiral-looking bosonic Hamiltonian does not inherit fermionic zero-mode protection simply from its binary structure [2005.05986].

Exactly solvable partitioned bosonic models furnish a different methodological strand. A broad class is defined by invariant subspaces
\[
\mathcal{H}=\bigoplus_N \mathcal{H}_N,\qquad \mathcal{H}_N=\mathrm{Span}\{|\Psi^{(N)}_0\rangle,\ldots,|\Psi^{(N)}_N\rangle\},
\]
and a ladder operator
\[
\hat H=\hat A+\hat A^\dagger
\]
with
\[
\hat A|\Psi_{n+1}\rangle=\sqrt{\beta_n}\,|\Psi_n\rangle,\qquad
\hat A^\dagger|\Psi_n\rangle=\sqrt{\beta_n}\,|\Psi_{n+1}\rangle.
\]
This includes two-mode conversion models
\[
\hat H=(\hat a^\dagger)^m\hat b^k+\hat a^m(\hat b^\dagger)^k
\]
and multi-mode extensions such as
\[
\hat H=\hat a^\dagger \hat b_1\hat b_2+\hat a \hat b_1^\dagger \hat b_2^\dagger.
\]
Here the two-sector interpretation is pump versus output, or one mode versus another mode or group of modes. Exact propagators, characteristic polynomials, continued fractions, and Jacobi-matrix constructions then follow from the sequence \(\beta_n\) [2510.20046].

Taken together, these results define both the power and the limits of the topic. Bicolored bosonic models are widespread, but the binary structure can mean fundamentally different things: species, sublattice, mode, leg, or symmetry sector. This suggests that any precise use of the term should specify which binary degree of freedom is being treated as the relevant “color.”

Source: https://www.emergentmind.com/topics/bicolored-bosonic-models