Density-Dependent Hopping in Quantum Lattices

Updated 21 April 2026

Density-dependent hopping is a mechanism where tunneling amplitudes vary with the local occupancy of lattice sites, fundamentally altering kinetic and interaction-driven physics.
Advanced analytical and numerical techniques, including exact diagonalization, DMRG, and quantum Monte Carlo, are employed to map its complex phase diagrams.
This effect underpins emergent quantum phases such as chiral spin liquids, supersolid regimes, and enables simulations of lattice gauge theories via occupation-modulated hopping.

Density-dependent hopping refers to the modification of single-particle or correlated hopping amplitudes by the occupation numbers of one or both sites involved in the process. Unlike standard lattice models where the tunneling term is independent of local densities, density-dependent hopping (also: correlated hopping, bond-charge tunneling, or density-dependent Peierls phase) introduces an explicit dependence on site occupation, often due to interaction-mediated processes or constraints. The resulting modification of kinetic terms profoundly alters the many-body physics, stability of phases, excitations, and emergent topological order in various lattice systems.

1. Microscopic Origins and General Operator Structure

Density-dependent hopping arises in a wide range of settings, including effective Hamiltonians derived from strong-coupling expansions, spin-orbit coupled systems, dipolar interactions, polaronic models, driven-dissipative photonic systems, and dynamically engineered Floquet lattices. The general form of a density-dependent hopping term between sites $i$ and $j$ is

$H_{\text{hop}} = -\sum_{\langle ij\rangle} J_{ij}(n_i, n_j) \, a_i^\dagger a_j + \text{h.c.}$

where $a_i^{(\dagger)}$ are creation/annihilation operators (bosonic or fermionic), and $J_{ij}(n_i, n_j)$ is a function of the local occupations. Typical forms include:

Linear: $J_{ij} = t + T(n_i + n_j)$ with $t$ the bare hopping and $T$ a density-dependent factor (Maik et al., 2013, Chaviguri et al., 2017, Segura et al., 2023).
Nonlinear/Projective: $J_{ij}\propto e^{i \phi(n_k)}$ or $J_{ij} \sim 0$ if an intermediate or adjacent site $j$ 0 is occupied (projector structure) (Ohler et al., 2022, Lienhard et al., 2020).

Microscopically, density dependence often reflects:

Virtual processes involving occupation constraints, e.g., second-order hopping (Peierls phase) blocked if an intermediate site is occupied (Ohler et al., 2022, Lienhard et al., 2020).
Interaction-induced modifications of local potential barriers or orbital overlap (Maik et al., 2013, Chaviguri et al., 2017).
Floquet engineering via fast periodic modulation of onsite interactions or synthetic gauge fields, leading to effective Peierls phases or amplitude renormalization depending on site occupation (Greschner et al., 2013, Greschner et al., 2015).
Constraints as in the $j$ 1– $j$ 2 or Gutzwiller-projected models, where the amplitude for hopping vanishes with double-occupancy (Kuboki, 2023).

2. Density-dependent Peierls Phases and Synthetic Gauge Fields

A striking realization of density-dependent hopping is the emergence of occupation-dependent Peierls phases. In Rydberg atom arrays with spin-orbit coupled dipolar exchange, the hopping amplitude between sites $j$ 3 and $j$ 4 acquires a phase: $j$ 5 where $j$ 6 is the site bridging $j$ 7 and $j$ 8; the phase is picked up only if $j$ 9 is unoccupied (Ohler et al., 2022, Lienhard et al., 2020). This mechanism explicitly entangles matter and gauge degrees of freedom: motion of an excitation is accompanied by a correlated dynamical flux determined by local density. Experimentally, this was demonstrated in triangular Rydberg arrays, yielding direct observable signatures such as chiral transport and anyonic statistics (Lienhard et al., 2020).

Floquet approaches and Raman-assisted tunneling in cold atoms further enable the realization of density-dependent gauge fields, as occupation-modulated interactions or multi-step hopping protocols generate effective terms: $H_{\text{hop}} = -\sum_{\langle ij\rangle} J_{ij}(n_i, n_j) \, a_i^\dagger a_j + \text{h.c.}$ 0 with $H_{\text{hop}} = -\sum_{\langle ij\rangle} J_{ij}(n_i, n_j) \, a_i^\dagger a_j + \text{h.c.}$ 1 the strength of the interaction modulation. Gauge-inequivalent choices (e.g., $H_{\text{hop}} = -\sum_{\langle ij\rangle} J_{ij}(n_i, n_j) \, a_i^\dagger a_j + \text{h.c.}$ 2 vs. $H_{\text{hop}} = -\sum_{\langle ij\rangle} J_{ij}(n_i, n_j) \, a_i^\dagger a_j + \text{h.c.}$ 3) lead to unique density-dependent analogs of uniform and staggered magnetic fields (Greschner et al., 2013, Greschner et al., 2015, Rodriguez et al., 2016).

3. Impact on Correlated Quantum Phases

Density-dependent hopping induces dramatically new quantum phases and modifies standard phase boundaries. Notable phenomena include:

Quantum Spin Liquids and Topological Order: In honeycomb Rydberg models, density-dependent complex NNN hopping drives the system into a chiral, disordered quantum spin-liquid regime characterized by finite spin gap, large scalar spin chirality, and nontrivial many-body Chern number $H_{\text{hop}} = -\sum_{\langle ij\rangle} J_{ij}(n_i, n_j) \, a_i^\dagger a_j + \text{h.c.}$ 4—distinct from static Haldane or conventional Bose-Hubbard models. The quantum critical regime is stabilized near $H_{\text{hop}} = -\sum_{\langle ij\rangle} J_{ij}(n_i, n_j) \, a_i^\dagger a_j + \text{h.c.}$ 5 (Ohler et al., 2022).
Active-absorbing State Transitions: In one-dimensional assisted hopping models, mobility of particles is determined by having enough neighbors within range $H_{\text{hop}} = -\sum_{\langle ij\rangle} J_{ij}(n_i, n_j) \, a_i^\dagger a_j + \text{h.c.}$ 6; below a density threshold $H_{\text{hop}} = -\sum_{\langle ij\rangle} J_{ij}(n_i, n_j) \, a_i^\dagger a_j + \text{h.c.}$ 7, dynamics freeze into absorbing states, while above, particle activity scales as $H_{\text{hop}} = -\sum_{\langle ij\rangle} J_{ij}(n_i, n_j) \, a_i^\dagger a_j + \text{h.c.}$ 8 (Dandekar et al., 2013).
Fractional Mott Insulator and Superfluid-Mott Boundaries: Density-dependent Peierls phases and amplitude renormalization found in Floquet-driven or interaction-modulated cold atom models lead to novel Mott-insulating plateaux at both integer and half-integer filling, including a fractional Mott insulator at $H_{\text{hop}} = -\sum_{\langle ij\rangle} J_{ij}(n_i, n_j) \, a_i^\dagger a_j + \text{h.c.}$ 9 (Greschner et al., 2013).
Supersolid and Charge-Density-Wave Regimes: In the extended Bose-Hubbard model with dipolar interactions, the inclusion of bond-charge (density-dependent) tunneling shifts the boundaries between superfluid, supersolid, phase-separated, and charge-ordered phases. The sign of the density-dependent hopping relative to the bare tunneling is critical; positive $a_i^{(\dagger)}$ 0 extends superfluidity while negative $a_i^{(\dagger)}$ 1 enhances phase separation and CDW order (Maik et al., 2013).
Ferrimagnetic Chains and Spin Systems: Holstein-Primakoff bosonizations of alternating spin chains reveal that density-dependent magnon hopping (arising at subleading order in $a_i^{(\dagger)}$ 2) is essential for capturing bulk and edge properties, plateau transitions, and correct edge state occupation (Silva et al., 2021).

System	Origin of Density-dependence	Key Effect
Rydberg honeycomb	Virtual spin-flip blocked by occupancy	QSL/BIQH phase, chiral order, $a_i^{(\dagger)}$ 3 phase (Ohler et al., 2022)
EBHM (dipolar bosons)	Polar molecule overlap, interaction-induced	Phase boundary shifts, enlarged SS region (Maik et al., 2013)
Floquet BH/optical lattice	Modulated $a_i^{(\dagger)}$ 4, synthetic Peierls phase	Momentum shifts, fractional MI, new MI lobe (Greschner et al., 2013)
Spin-(1/2, S) chain	HP expansion, spin reduction per magnon	Accurate bulk, edge states, plateau fields (Silva et al., 2021)
Ionic Hubbard model	Schrieffer-Wolff/Floquet/interaction	Expanded SDI region, topological transitions (Bas et al., 2023)

4. Analytical and Numerical Methodologies

Density-dependent hopping significantly increases the complexity of model analysis, requiring advanced numerical and analytical techniques:

Exact Diagonalization (ED) is essential for uncovering phase boundaries and topological indices (e.g., many-body Chern number under twisted boundary conditions, fidelity metric peaks indicating phase transitions) in finite-size cluster geometries (Ohler et al., 2022).
Gutzwiller and Mean-field Ansatz enable analytic boundaries for Mott lobes and identification of triple points in extended BH models with density-dependent hopping (Chaviguri et al., 2017).
Quantum Monte Carlo, Stochastic Series Expansion, and composite-boson mean field provide access to large-scale phase diagrams for systems with strong density-dependent hopping (Maik et al., 2013, Greschner et al., 2015).
DMRG establishes precise magnetization curves, bulk and edge local densities in spin chains with correlated hopping (Silva et al., 2021).
Berry phase and level-crossing methods are employed to map charge and spin transitions in correlated electron models with density-dependent hopping, quantitatively extracting phase boundaries and topological phase transitions (Bas et al., 2023).

5. Experimental Realizations and Observables

Multiple platforms have realized or proposed direct observation of density-dependent hopping:

Rydberg Atom Arrays: Chiral propagation, anyonic exchange statistics, and dynamical reversal of chiral motion via magnetic field inversion directly probe the density-dependent Peierls phase. Population trajectories provide evidence for anyonic statistics (Lienhard et al., 2020).
Ultracold Polar Molecules: Strong dipolar interactions induce measurable shifts in the phase diagram as bond-charge tunneling becomes comparable to direct tunneling (Maik et al., 2013).
Driven-dissipative Photonic Systems: Coupled nonlinear microcavities enable direct interferometric measurement of interaction-induced, density-dependent hopping phases on dynamical steady-state branches (Rodriguez et al., 2016).
Optical Lattices with Raman-Assisted/Bond-Selective Tunneling: Density-dependent synthetic magnetism (DDSM) is probed by dynamic expansion measurements of doublons/holons, as effective fluxes depend on occupation and manifest in distinctive expansion rates (Greschner et al., 2015).

Other anticipated signatures include:

Shifts and broadening of momentum peaks in time-of-flight;
Density plateaux and transition lines in in-situ density scans;
Plateaux in magnetization or chiral observables as a function of external fields.

6. Phase Diagrams and Universal Features

Density-dependent hopping commonly fosters enlarged parameter regimes of exotic or topologically nontrivial order compared to their density-independent counterparts:

In Rydberg honeycomb models, four regimes emerge as a function of $a_i^{(\dagger)}$ 5: trivial BEC ( $a_i^{(\dagger)}$ 6), spin-gapped QSL/BIQH ( $a_i^{(\dagger)}$ 7), classical spiral ( $a_i^{(\dagger)}$ 8), and a second, gapless chiral regime at large negative $a_i^{(\dagger)}$ 9 (Ohler et al., 2022).
In the ionic Hubbard model with electron-hole symmetric density-dependent hopping, the spontaneously dimerized insulator region is substantially enlarged as correlated hopping is increased, avoiding spin-gap closures that would otherwise disrupt topological pumping (Bas et al., 2023).
In Bose-Hubbard ladders and 2D lattices with DDSM, Meissner-vortex superfluid transitions and chiral/nonchiral superfluid boundaries shift to higher filling or larger tunneling, depending on density and flux (Greschner et al., 2015).

7. Theoretical Implications and Future Directions

Density-dependent hopping represents a fundamental bridge between kinetic energy and interaction-driven physics in quantum lattice models. Its capacity to dynamically entangle matter and gauge fields opens avenues for simulating lattice gauge theories with dynamical matter, anyon-Hubbard models, symmetry-protected topological states, and non-Abelian gauge field analogs (Ohler et al., 2022, Lienhard et al., 2020, Greschner et al., 2013). The ongoing development of cold-atom, Rydberg, and photonic platforms capable of engineering and controlling density-dependent tunneling holds promise for exploring quantum phases and transitions beyond reach of standard Hubbard-like descriptions.

Key papers referenced:

"Quantum spin liquids of Rydberg excitations in a honeycomb lattice induced by density-dependent Peierls phases" (Ohler et al., 2022)
"A class of exactly solved assisted hopping models of active-absorbing state transitions on a line" (Dandekar et al., 2013)
"Density dependent tunneling in the extended Bose-Hubbard model" (Maik et al., 2013)
"Realization of a density-dependent Peierls phase in a synthetic, spin-orbit coupled Rydberg system" (Lienhard et al., 2020)
"Interaction-induced hopping phase in driven-dissipative coupled photonic microcavities" (Rodriguez et al., 2016)
"Density-dependent hopping for ultracold atoms immersed in a Bose-Einstein-condensate vortex lattice" (Chaviguri et al., 2017)
"Density-Dependent Synthetic Gauge Fields Using Periodically Modulated Interactions" (Greschner et al., 2013)
"The role of density-dependent magnon hopping and magnon-magnon repulsion in ferrimagnetic spin-(1/2, S) chains in a magnetic field" (Silva et al., 2021)
"Charge and spin gaps of the ionic Hubbard model with density-dependent hopping" (Segura et al., 2023)
"Phase diagram of the ionic Hubbard model with density-dependent hopping" (Bas et al., 2023)
"Phase Separation Induced by Density-Dependent Hopping Terms" (Kuboki, 2023)
"Density-dependent synthetic magnetism for ultracold atoms in optical lattices" (Greschner et al., 2015)