Bicolored Bosonic Models

• Bicolored bosonic models are defined by binary organizations, including two-species, bipartite lattices, two-mode systems, or parity sectors, that yield diverse quantum behaviors.
• They employ both microscopic and emergent binary structures to study phenomena such as p-wave pairing, topological phases, and symmetry-driven spectral transitions.
• Exact solutions and effective theoretical frameworks in these models enhance our understanding of quantum phase transitions, superfluidity, and topological edge states.

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 (Dam et al., 2018).

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 (Lerma-Hernandez et al., 2019). 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 (He et al., 2021).

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 (Hemmerich, 2019). 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 $\mathbb Z_2$0 that need not preserve the label (Santos et al., 2013). 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 $\mathbb Z_2$1 (Dam et al., 2018).

Binary structure	Representative realization	Example
Two species/components	Internal labels $\mathbb Z_2$2 or $\mathbb Z_2$3	Interspecies $\mathbb Z_2$4-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 $\mathbb Z_2$5-wave pairing Hamiltonian

$\mathbb Z_2$6

with $\mathbb Z_2$7 in the one-dimensional presentation. Here the binary label is an internal species index, interpreted as a pseudospin-$\mathbb Z_2$8 boson, and the relevant pairing operator is an interspecies singlet. The model is solvable through a hyperbolic Richardson-Gaudin $\mathbb Z_2$9 construction, with exact eigenstates

$p$0

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

$p$1

in the thermodynamic scaling used in the paper (Lerma-Hernandez et al., 2019).

A different two-component structure appears in bosonic atoms with two internal states in species-dependent artificial gauge potentials. The single-particle Hamiltonian is

$p$2

with the case of interest given by $p$3 and $p$4 of the same sign but different magnitudes. The proposed composite-fermion states are

$p$5

and the paper studies $p$6, $p$7, $p$8, and $p$9. In exact diagonalization, $a_{\mathbf k},a^\dagger_{\mathbf k}$0 and $a_{\mathbf k},a^\dagger_{\mathbf k}$1 are described as plausibly gapped in the thermodynamic limit, while the line $a_{\mathbf k},a^\dagger_{\mathbf k}$2 is argued to be likely gapless and superfluid-like (Wu et al., 2015).

A lattice realization of two-component bosons is the bosonic integer quantum Hall construction on the square lattice with Hamiltonian

$a_{\mathbf k},a^\dagger_{\mathbf k}$3

where the two hard-core boson components $a_{\mathbf k},a^\dagger_{\mathbf k}$4 and $a_{\mathbf k},a^\dagger_{\mathbf k}$5 experience opposite Peierls phases in the interwire hopping and correlated hopping in $a_{\mathbf k},a^\dagger_{\mathbf k}$6. The coupled-wire analysis yields the $a_{\mathbf k},a^\dagger_{\mathbf k}$7-matrix

$a_{\mathbf k},a^\dagger_{\mathbf k}$8

the Hall conductance

$a_{\mathbf k},a^\dagger_{\mathbf k}$9

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 (Liu et al., 2017).

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

$b_{\mathbf k},b^\dagger_{\mathbf k}$0

with two sublattices per unit cell and alternating intracell/intercell couplings $b_{\mathbf k},b^\dagger_{\mathbf k}$1. In the quadratic-boson formalism, the relevant object is the dynamic matrix for closed systems or the effective Hamiltonian $b_{\mathbf k},b^\dagger_{\mathbf k}$2 for open Lindbladian systems. For the symmetry-preserving open SSH model with

$b_{\mathbf k},b^\dagger_{\mathbf k}$3

the effective Hamiltonian is

$b_{\mathbf k},b^\dagger_{\mathbf k}$4

and after subtracting a constant imaginary shift one has

$b_{\mathbf k},b^\dagger_{\mathbf k}$5

placing the model in class AIII. The non-Hermitian winding number is

$b_{\mathbf k},b^\dagger_{\mathbf k}$6

with the SSH criterion

$b_{\mathbf k},b^\dagger_{\mathbf k}$7

If the reservoir coupling is sublattice-selective, for example $b_{\mathbf k},b^\dagger_{\mathbf k}$8, chiral symmetry is broken in $b_{\mathbf k},b^\dagger_{\mathbf k}$9 and the open system falls into class A instead (He et al., 2021).

A different binary organization appears in higher-order bosonic topological phases built from spin-$a,b$0 variables. One construction starts from the $a,b$1D $a,b$2 cluster chain with two $a,b$3 variables per unit cell, $a,b$4 and $a,b$5,

$a,b$6

and then stacks and couples such chains to obtain a two-dimensional bosonic higher-order topological phase with four spin-$a,b$7 objects per unit cell. A complementary construction uses a dimerized antiferromagnetic XY model

$a,b$8

with strong inter-cell coupling $a,b$9 and weak intra-cell coupling $a_{1,i}$0. In both cases the bulk is gapped, the edges are gapped but topological, and the corners carry protected modes transforming projectively under $a_{1,i}$1 (Dubinkin et al., 2018).

The same theme extends to a two-sector bosonic semimetal built from two momentum-separated copies of the $a_{1,i}$2 nonlinear sigma model,

$a_{1,i}$3

with $a_{1,i}$4 and $a_{1,i}$5, located at momenta $a_{1,i}$6. 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_{1,i}$7 and $a_{1,i}$8 sectors are exchanged by $a_{1,i}$9 and $a_{2,i}$0 (Lapa et al., 2016).

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

$a_{2,i}$1

and the full Hamiltonian is

$a_{2,i}$2

Here the bosonic interaction is not restricted to a linear field coupling; it contains higher-order powers

$a_{2,i}$3

with semiboundedness ensured by an even-polynomial dominance condition on the leading terms (Dam et al., 2018).

The binary structure is produced by the spin-parity symmetry. The bosonic parity operator is

$a_{2,i}$4

and the paper constructs an explicit selfadjoint unitary $a_{2,i}$5 such that

$a_{2,i}$6

where the fiber operators on Fock space are

$a_{2,i}$7

Thus the model is “bicolored” only after symmetry reduction: the two colors are the two parity/spin sectors labeled by $a_{2,i}$8 and $a_{2,i}$9.

This decomposition is not merely formal. The paper proves that under its hypotheses the operators $X_\pm$0 and $X_\pm$1 are closed on natural domains, are selfadjoint and semibounded when $X_\pm$2, and satisfy HVZ-type spectral statements. The ground-state identification is especially sharp: $X_\pm$3 and if $X_\pm$4 and $X_\pm$5 is an eigenvalue of $X_\pm$6, then the ground state lies in the $X_\pm$7 fiber. By contrast, eigenvalues of $X_\pm$8 correspond to excited states. The paper also shows that in the massless case $X_\pm$9 with $j=1,2$0,

$j=1,2$1

In this sense the two parity colors have different spectral roles: one carries the bottom of the spectrum, the other encodes excited states (Dam et al., 2018).

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

$j=1,2$2

which describes $j=1,2$3 bosons populating two orthogonal single-particle modes with a flavour-changing contact interaction. The pair-conversion term

$j=1,2$4

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 $j=1,2$5, 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 (Hemmerich, 2019).

A related but spatially resolved two-color model is the exactly solvable two-well, two-on-well-state Hamiltonian

$j=1,2$6

with bosonic modes $j=1,2$7, $j=1,2$8. Here the labels $j=1,2$9 are naturally interpreted as two colors or two internal states, while $\mathbb Z_2$00 denote the two wells. The distinctive feature is that the tunnelling matrix $\mathbb Z_2$01 need not be diagonal, so the model includes both color-preserving and color-changing interwell tunnelling. In the integrable realization,

$\mathbb Z_2$02

and the model is solved by the algebraic Bethe ansatz through a new multi-state bosonic Lax operator (Santos et al., 2013).

The bosonic Kitaev-Hubbard chain and two-leg ladder provide a further set of binary structures. The chain Hamiltonian

$\mathbb Z_2$03

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,

$\mathbb Z_2$04

and at $\mathbb Z_2$05, $\mathbb Z_2$06, $\mathbb Z_2$07 it becomes a 1D quantum compass model. For free bosons the bosonic BdG matrix can exhibit a non-Hermitian skin effect, but in the hard-core limit that effect disappears exactly because the model is transformed to a spin-$\mathbb Z_2$08 system and then to a Hermitian fermionic Kitaev chain. The ladder Hamiltonian

$\mathbb Z_2$09

is a direct two-leg binary bosonic system with inequivalent leg and rung bonds (Wang et al., 2022).

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

$\mathbb Z_2$10

with commutation metric

$\mathbb Z_2$11

and proves that each stable gapped quadratic bosonic system is homotopic to a direct sum of two effective single-particle subsystems,

$\mathbb Z_2$12

In this sense binary organization is intrinsic to the quadratic bosonic formalism itself: the positive and negative $\mathbb Z_2$13-sectors define a natural two-sector structure even before one asks whether the microscopic system is two-species or bipartite (Zhou et al., 2019).

A more restrictive conclusion is reached in the “threefold way” analysis of stable quadratic bosons. There the relevant matrix is

$\mathbb Z_2$14

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 (Xu et al., 2020).

Exactly solvable partitioned bosonic models furnish a different methodological strand. A broad class is defined by invariant subspaces

$\mathbb Z_2$15

and a ladder operator

$\mathbb Z_2$16

with

$\mathbb Z_2$17

This includes two-mode conversion models

$\mathbb Z_2$18

and multi-mode extensions such as

$\mathbb Z_2$19

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 $\mathbb Z_2$20 (Shchesnovich, 22 Oct 2025).

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.”