Asymmetric Bose-Hubbard Model

• The asymmetric Bose-Hubbard model is a family of lattice boson Hamiltonians where key symmetries are broken in kinetic terms, component structures, or interaction processes.
• It employs varied setups—ranging from unequal tunnelling in dimers and unidirectional hopping chains to models with artificial magnetic fields—to study nontrivial quantum phase transitions and perturbative degeneracies.
• Techniques like Bethe ansatz, perturbation theory, and effective spin mappings reveal how asymmetric interactions control transitions between Mott-insulator, superfluid, and thermalized states.

The asymmetric Bose-Hubbard model denotes a family of Bose-Hubbard-type lattice boson Hamiltonians in which a symmetry of the standard model is explicitly broken in the kinetic sector, in the component structure, in the well-to-well couplings, or in the local conversion processes. In the literature, this includes two-component mixtures whose particle and hole sectors have different perturbative degeneracy structures, dimers and four-well systems with unequal tunnelling amplitudes, one-dimensional chains with strictly unidirectional hopping, models with an asymmetric breakdown interaction that turns one boson into two between adjacent sites, and two-dimensional lattices in artificial magnetic fields or with hopping anisotropy (Wang et al., 2014, Filho, 2016, Chianca et al., 2011, Zheng et al., 2023, Hu et al., 2024, Jaberi et al., 16 Jul 2025).

1. Taxonomy of asymmetry

The term does not refer to a single canonical Hamiltonian. Rather, it labels several inequivalent deformations of Bose-Hubbard physics in which the asymmetry is introduced at different structural levels.

Setting	Hamiltonian feature	Asymmetry
Two-component mixture	$H=H_a+H_b+U_{ab}\sum_i n_i^a n_i^b$	species-resolved fillings and particle/hole sectors
Four-well model	strong $J$, weak $\omega$	two tunnelling scales
Generalized two-site model	$\Omega_{12}\neq\Omega_{21}$	asymmetric tunnelling
Unidirectional chain	only $b_j^\dagger b_{j+1}$	one-way hopping
Breakdown Hubbard model	$J(\hat a_{m+1}^{\dagger})^2\hat a_m+\text{h.c.}$	directional multiplication $1\to 2$
2D asymmetric reference model	$t$ along $\hat x$, $t/2$ along $J$0	hopping anisotropy

In the two-component model, the asymmetry appears in the perturbative structure of particle and hole doping sectors rather than in a bare left-right hopping imbalance. In the four-well and generalized two-site models, it is built directly into unequal tunnelling amplitudes. In the unidirectional chain, the asymmetry is non-Hermitian because the Hamiltonian contains only the hopping operators $J$1, and not the reverse terms $J$2. In the breakdown model, the asymmetry is a nonlinear conversion process: one boson on site $J$3 becomes two bosons on site $J$4. In the magnetic-field problem, the paper stresses that the artificial magnetic field case is not merely an anisotropy in hopping magnitudes; the Peierls phases frustrate hopping around a plaquette and explicitly break translational symmetry at the single-particle level (Wang et al., 2014, Chianca et al., 2011, Filho, 2016, Zheng et al., 2023, Hu et al., 2024, Jaberi et al., 16 Jul 2025).

This suggests that “asymmetric Bose-Hubbard model” is best understood as a class of bosonic lattice models in which the departure from the standard symmetric Bose-Hubbard Hamiltonian is physically meaningful but not unique.

2. Two-component mixtures, graph connectivity, and particle-hole asymmetry

A particularly influential formulation is the two-component Bose-Hubbard model for species $J$5 and $J$6,

$J$7

with

$J$8

The coefficients $J$9 define the weighted adjacency matrix of the lattice graph $\omega$0. In the strongly interacting regime,

$\omega$1

the Hamiltonian is rewritten as $\omega$2, and the analysis is performed in the fixed-particle-number sector

$\omega$3

The unperturbed ground states of $\omega$4 are highly degenerate whenever at least one of $\omega$5 is nonzero. The central result is that the degeneracy properties of the exact ground-state energy $\omega$6 and of the first-order correction $\omega$7 are governed by the connectivity properties of the lattice graph and by the combinatorics of the extra particles or holes (Wang et al., 2014).

The key technical device is a notion of connectedness between basis states: two Fock states are connected by an operator $\omega$8 if one can reach one from the other by a finite chain of basis states with nonzero matrix elements of $\omega$9 between successive states. Combined with the Perron–Frobenius theorem for real symmetric matrices with nonpositive off-diagonal entries, this yields a sharp statement for the full Hamiltonian $\Omega_{12}\neq\Omega_{21}$0: if $\Omega_{12}\neq\Omega_{21}$1 is connected, then the exact ground-state energy $\Omega_{12}\neq\Omega_{21}$2 is nondegenerate, and the ground-state vector can be chosen positive in the Fock basis. Conversely, if $\Omega_{12}\neq\Omega_{21}$3 is disconnected, the basis splits into sectors that cannot communicate via hopping.

The first-order correction $\Omega_{12}\neq\Omega_{21}$4 is more delicate. If one species is exactly commensurate and the other is doped, such as $\Omega_{12}\neq\Omega_{21}$5, $\Omega_{12}\neq\Omega_{21}$6, then $\Omega_{12}\neq\Omega_{21}$7 is nondegenerate on any connected lattice. If $\Omega_{12}\neq\Omega_{21}$8 and $\Omega_{12}\neq\Omega_{21}$9, connectedness of $b_j^\dagger b_{j+1}$0 is not enough; one needs $b_j^\dagger b_{j+1}$1 to be 2-connected. For generic cases with $b_j^\dagger b_{j+1}$2 or $b_j^\dagger b_{j+1}$3, $b_j^\dagger b_{j+1}$4 is nondegenerate if $b_j^\dagger b_{j+1}$5 is 2-connected and not a simple circle with five or more sites. The case $b_j^\dagger b_{j+1}$6 is exceptional: all matrix elements in the reduced perturbation matrix vanish, so

$b_j^\dagger b_{j+1}$7

and $b_j^\dagger b_{j+1}$8 is degenerate independently of lattice connectivity.

The case $b_j^\dagger b_{j+1}$9 is the main instance in which standard degenerate perturbation theory fails completely. There the first-order matrix $J(\hat a_{m+1}^{\dagger})^2\hat a_m+\text{h.c.}$0 is fully degenerate, and the zeroth-order state cannot be fixed by diagonalizing $J(\hat a_{m+1}^{\dagger})^2\hat a_m+\text{h.c.}$1 inside the degenerate manifold. The paper resolves this by using lattice automorphisms $J(\hat a_{m+1}^{\dagger})^2\hat a_m+\text{h.c.}$2, which induce unitary operators $J(\hat a_{m+1}^{\dagger})^2\hat a_m+\text{h.c.}$3 commuting with $J(\hat a_{m+1}^{\dagger})^2\hat a_m+\text{h.c.}$4. If the lattice automorphism can map any site to any other site, then for $J(\hat a_{m+1}^{\dagger})^2\hat a_m+\text{h.c.}$5,

$J(\hat a_{m+1}^{\dagger})^2\hat a_m+\text{h.c.}$6

This symmetry-based construction is then tied directly to the asymmetric character of the Mott-insulator to superfluid transition between the particle and hole side: the particle and hole sectors have different combinatorial structures, different connectedness classes in the degenerate manifold, and therefore different perturbative responses.

3. Asymmetric tunnelling in dimers and four-well systems

A second major strand concerns finite-site Hermitian models with unequal tunnelling amplitudes. The generalized two-site Bose-Hubbard model is defined by

$J(\hat a_{m+1}^{\dagger})^2\hat a_m+\text{h.c.}$7

with conserved total number $J(\hat a_{m+1}^{\dagger})^2\hat a_m+\text{h.c.}$8. Its asymmetry is explicit: $J(\hat a_{m+1}^{\dagger})^2\hat a_m+\text{h.c.}$9 The model remains integrable because it can be embedded into a $1\to 2$0-invariant Yang–Baxter framework through a suitably parametrized bosonic Lax operator. The transfer matrix generates commuting conserved quantities, and the exact Bethe ansatz equations yield the spectrum. In the no interaction limit, the Bethe ansatz equations reduce to

$1\to 2$1

which is the equation of an $1\to 2$2 sphere in $1\to 2$3 when the roots are real. The asymmetry is therefore compatible with exact solvability rather than opposed to it (Filho, 2016).

The four-well Bose-Hubbard model realizes a different asymmetry: two pairs of wells are coupled by two processes with two different rates. In the effective Hamiltonian,

$1\to 2$4

the paper always takes $1\to 2$5 and then sets $1\to 2$6 to define units. This generates two strongly coupled subsystems weakly coupled to each other. The truncated Wigner equations inherit the same two-rate structure, with fast intra-dimer oscillations tied to $1\to 2$7 and slower inter-dimer exchange tied to $1\to 2$8. The study emphasizes that the quantum dynamics can differ dramatically from mean-field predictions even with $1\to 2$9 atoms, that coherent-state and Fock-state initial conditions lead to qualitatively different evolutions, and that the system equilibrates to a maximum entropy state. The “pseudo entropy”

$t$0

approaches its maximal value $t$1 when the two wells in a subsystem are equally populated, although not monotonically. In this sense the model functions as a controlled asymmetric Bose-Hubbard testbed for quantum thermalisation (Chianca et al., 2011).

4. Strictly unidirectional hopping and non-Hermitian integrability

A qualitatively different asymmetric Bose-Hubbard model is the one-dimensional chain with strictly unidirectional hopping,

$t$2

The asymmetry is exact: particles move only in one lattice direction, and the kinetic term is not Hermitian-conjugate symmetric. The paper emphasizes that this is not a mere non-Hermitian deformation of a Hermitian integrable model; it is a new integrable model with no Hermitian counterpart. Integrability is established by the quantum inverse scattering method. The transfer matrix satisfies $t$3, the Hamiltonian is recovered as a combination of the conserved charges, and the algebraic Bethe ansatz gives eigenstates

$t$4

with Bethe roots $t$5 obeying

$t$6

and energy

$t$7

The exact spectrum is therefore obtained by solving the Bethe equations and summing the roots (Zheng et al., 2023).

The Bethe roots furnish a direct characterization of the ground-state superfluid–Mott insulator transition at integer filling. For $t$8 and $t$9, the paper defines

$\hat x$0

In the thermodynamic limit, $\hat x$1 and $\hat x$2. If $\hat x$3, then $\hat x$4 and the phase is superfluid; if $\hat x$5, then $\hat x$6, indicating a Mott gap. A second signature is a peak or divergence in

$\hat x$7

with finite-size scaling

$\hat x$8

The limiting critical values are

$\hat x$9

and

$t/2$0

The same model exhibits a genuine interacting non-Hermitian skin effect when $t/2$1. Multiplying all Bethe equations gives

$t/2$2

which implies

$t/2$3

Hence amplitudes decay for $t/2$4 and grow for $t/2$5, producing boundary accumulation. The paper also shows that in the Mott insulating regime the skin effect is completely suppressed in the thermodynamic limit. Using

$t/2$6

finite-size scaling gives $t/2$7 for $t/2$8, while $t/2$9 remains nonzero for $J$00. This establishes an interacting non-Hermitian setting in which asymmetry survives in the superfluid regime but is erased in the Mott phase.

5. Breakdown interactions, exponential symmetry, and false vacua

The bosonic quantum breakdown Hubbard model generalizes the Bose-Hubbard Hamiltonian by adding an asymmetric breakdown interaction,

$J$01

At $J$02, one recovers the standard Bose-Hubbard model. At $J$03, however, the model acquires a global exponential $J$04 symmetry under

$J$05

with conserved charge

$J$06

A structural hallmark is

$J$07

so the conserved charge does not commute with translation (Hu et al., 2024).

The ground-state phase structure differs sharply from the standard model. At $J$08, the ground state is a Mott insulator with integer filling. At $J$09, the model has the usual Bose-Hubbard structure: MI at small $J$10, superfluid at larger $J$11, and a continuous second-order transition. Along the $J$12 line, by contrast, the paper finds isolated MI domes with $J$13 at small $J$14, and a breakdown condensate with $J$15 at large $J$16, where

$J$17

The phase angles satisfy

$J$18

A key result is that the MI $J$19 breakdown-condensate transition is first order everywhere along the $J$20 line, and for generic $J$21 and $J$22 the boundary between translationally invariant phases is also first order.

The most striking claim is that the spontaneous symmetry breaking breakdown condensate has no bulk gapless Goldstone mode. For $J$23, the bulk spectrum is gapped; when $J$24, the only zero-energy mode is an edge mode with

$J$25

After the similarity transformation $J$26, the continuum effective Lagrangian becomes spatially inhomogeneous,

$J$27

leading to a Hatano–Nelson-type non-Hermitian problem with real gapped bulk spectrum

$J$28

The paper argues that, because the would-be Goldstone mode is absent, the usual Mermin-Wagner/Hohenberg reasoning does not apply in the standard way, allowing stable spontaneous symmetry breaking of exponential $J$29 in 1D.

The model also supports a dynamical dielectric-to-breakdown transition. Starting from the Mott state

$J$30

one adds one boson at the left edge,

$J$31

and monitors long-time averages $J$32. The transition is identified by the sign change of

$J$33

The approximate threshold is obtained from

$J$34

The dynamical breakdown threshold lies at larger $J$35 than the ground-state MI $J$36 breakdown-condensate transition, producing an intermediate regime in which the ground state is already the breakdown condensate while the initial MI excitation remains dynamically stable. The paper interprets this regime as a false vacuum.

6. Artificial magnetic fields, frustrated hopping, and ordered phases

In two dimensions, asymmetry is also used as a benchmark for understanding frustration induced by artificial magnetic fields. The Bose-Hubbard Hamiltonian with an artificial magnetic field is written as

$J$37

For flux $J$38, the magnetic unit cell is $J$39. The paper first studies an asymmetric Bose-Hubbard reference model,

$J$40

with hopping amplitude $J$41 along $J$42 and $J$43 along $J$44. Compared with the symmetric standard Bose-Hubbard model, this asymmetry weakens coherence, localizes bosons, and enlarges the Mott-insulator lobes. The reference model is introduced precisely to isolate the effect of frustrating hopping on localization (Jaberi et al., 16 Jul 2025).

The artificial magnetic field problem is more subtle than simple anisotropy. The Peierls phases make hopping around a plaquette accumulate a nontrivial phase and explicitly break translational symmetry at the single-particle level. For $J$45, the single-particle Hamiltonian becomes a $J$46 matrix whose spectrum is the Hofstadter butterfly. The spectrum is symmetric around

$J$47

which is why the analysis focuses on $J$48. The authors use single-site Gutzwiller mean-field theory, with

$J$49

and local observables

$J$50

The standard phase identification is retained: $J$51 for MI or DW, $J$52 for SF or SS.

With nearest-neighbor repulsion

$J$53

the extended model supports density-wave and supersolid phases. The main conclusion is that frustrated hopping localizes bosons and enlarges insulating and ordered regions: MI lobes expand in the Bose-Hubbard model, while DW and SS borders increase in the extended model, with the largest increments near $J$54. In the Mott insulator phase, translational symmetry is effectively restored because $J$55 becomes uniform and gauge-invariant. In the superfluid and supersolid phases, by contrast, the field induces real-space modulations with $J$56 periodicity, and the bosonic current

$J$57

develops vortex-like patterns whose configurations depend on the commensuration between magnetic field and lattice. The checkerboard filling pattern persists in the density-wave and supersolid phases regardless of field strength.

The paper also studies thermal fluctuations. At finite temperature, MI, DW, and SS regions shrink, plateaus soften, and compressible phases appear near former boundaries. Nevertheless, the ordered phases are reported to be robust up to temperatures comparable to the interaction energy $J$58, which supports the feasibility of observing such phases in experiments.

7. Effective spin descriptions and broader theoretical context

A related line of work shows that strong-coupling reductions can encode nontrivial microscopic structure into effective low-energy anisotropies. For a one-dimensional Bose-Hubbard chain at half-integer filling,

$J$59

the low-energy sector with occupations $J$60 and $J$61 maps to a spin-$J$62 XXZ chain. After a Glazek-Wilson similarity renormalization and bosonization, the final effective Hamiltonian is

$J$63

with explicitly renormalized $J$64 and $J$65. The finite-$J$66 effective theory agrees closely with DMRG: the paper reports agreement up to $J$67, average relative errors of about $J$68 for density-density correlations and $J$69–$J$70 for off-diagonal correlations at $J$71, $J$72, $J$73, whereas the naive $J$74 XXZ limit gives much worse agreement (Giuliano et al., 2012).

The paper is not explicitly about an asymmetric Bose-Hubbard model in the sense of unequal hoppings or unequal interaction strengths on alternating bonds or sites, but it draws a broader lesson directly relevant here: a careful finite-$J$75 elimination of high-energy states can generate renormalized spin couplings, anisotropies, and effective fields that encode microscopic asymmetries in a compact XXZ-like description. In that sense, the asymmetric Bose-Hubbard problem is not only a question of writing down unequal hopping amplitudes. It is also a question of how asymmetry survives, is renormalized, or reappears in effective descriptions of low-energy bosonic matter.