---
title: Correlated Hopping in Lattice Systems
url: https://www.emergentmind.com/topics/correlated-hopping
type: topic
---

# Correlated Hopping in Lattice Systems

Searching arXiv for recent and foundational papers on correlated hopping across lattice, transport, and quantum-simulation contexts.
Correlated hopping is an interaction-induced modification of kinetic motion in which a hopping matrix element depends explicitly on site or bond occupancies. In the literature it also appears as assisted hopping, density-dependent hopping, or charge-bond interaction. In second-quantized lattice models, the defining feature is that a term which would be a one-body tunneling process in the Hubbard or Bose–Hubbard limit is multiplied by occupation operators such as \(n_i\), \(n_j\), or \(n_{i\bar\sigma}\), thereby coupling transport directly to the instantaneous many-body configuration. This structure occurs in fermionic and bosonic models, in real-space and synthetic lattices, and in mesoscopic transport setups, where it breaks particle-hole symmetry, renormalizes superexchange, induces clustering, and can even generate interaction-driven topology [1310.7959] [2107.06576] [2411.19850] [2503.00589].

## 1. Definition and canonical forms

Correlated hopping is introduced by replacing a bare amplitude \(t\) with an occupation-dependent amplitude. A standard nearest-neighbour fermionic form is
\[
H = -\sum_{\langle ij\rangle,\sigma} [\,t + X\,(n_{i,\bar\sigma} + n_{j,\bar\sigma}) + Y\,n_{i,\bar\sigma}n_{j,\bar\sigma}\,]\,c^\dagger_{i\sigma}c_{j\sigma}
+ U \sum_i n_{i\uparrow}n_{i\downarrow} + V \sum_{\langle ij\rangle} n_i n_j + \dots,
\]
with \(X\) the two-body correlated-hopping amplitude and \(Y\) a three-body amplitude [1310.7959]. In the periodically modulated Feshbach realization, the effective kinetic term is written as
\[
H_J = -t \sum_{\langle ij\rangle,\sigma} (1 - X[n_{i,\bar\sigma}+n_{j,\bar\sigma} -2n_{i,\bar\sigma}n_{j,\bar\sigma}])\,c^\dagger_{i\sigma}c_{j\sigma},
\]
with \(Y=-2X\) to leading order [1310.7959].

In quantum-dot transport, the same idea appears in tunneling to leads. The extended Anderson model uses the composite operator
\[
D_\sigma \equiv d_\sigma(1-x\,n_{\bar\sigma}),
\]
so that tunneling onto or off the dot has amplitude \(V_{\lambda k\sigma}\) when the opposite spin is absent and \(V_{\lambda k\sigma}(1-x)\) when it is present [2411.19850] [2103.08985]. Equivalently, the correlated-hopping term is
\[
H_{CH} = -\,x\sum_{\lambda k\sigma} V_{\lambda k\sigma}\,\bigl[c^\dagger_{\lambda k\sigma}\,d_\sigma\,n_{\bar\sigma} + d^\dagger_\sigma\,c_{\lambda k\sigma}\,n_{\bar\sigma}\bigr].
\]
This form makes explicit that a nominally one-electron transfer is conditioned by local occupancy [2411.19850].

In bosonic systems, correlated hopping is often extended or bridge-mediated. In the one-dimensional zig-zag Bose–Hubbard geometry,
\[
H = -t\sum_{i=1}^L\bigl(b_i^\dagger b_{i+2} + b_{i+2}^\dagger b_i\bigr)
-t_c\sum_{i=1}^L\bigl(n_{i+1}\,b_i^\dagger b_{i+2}+b_{i+2}^\dagger b_i\,n_{i+1}\bigr)
+\frac{U}{2}\sum_{i=1}^L n_i(n_i-1)-\mu\sum_{i=1}^L n_i,
\]
so that hopping from \(i\) to \(i+2\) depends on the occupation of the intervening site \(i+1\) [2010.14459]. In the bosonic scar model, density-assisted hopping appears as terms such as
\[
H_1 = -J\sum_{j=1}^L [\,n_{j-1}\,b_j^\dagger b_{j-1} + b_j^\dagger b_{j+1}\,n_j\,],
\]
which provide a “soft” kinetic constraint [1910.09526].

Multi-orbital versions generalize the dependence to orbital occupations. In molecular nanomagnets the hopping sector is written as
\[
H_1  =  \sum_{\langle i,j\rangle,\,m,m',\sigma}  t_{m m'}^{i j}  [ 1  -  a ( n_{i m}  +  n_{j m'}  - 1 ) ]
c_{i m \sigma}^\dagger  c_{j m' \sigma},
\]
where \(a\) is the correlated-hopping parameter [2107.06576]. In such formulations, the modulation is not a phenomenological correction to \(t\) but part of the microscopic Hamiltonian.

## 2. Microscopic origin and formal derivations

The common microscopic origin is nonlocal Coulomb structure in a localized basis. In the Anderson-dot language, the dimensionless parameter \(x\) is related to the Coulomb matrix element \(K_{ij}=\langle ii|V|ij\rangle\) via \(K_{\lambda k\sigma}=-x\,V_{\lambda k\sigma}\) [2411.19850]. In semiconductor quantum dots or molecular junctions, the data state that \(U\) is often tens of meV, \(K\) can be a few meV, and since the bare hybridization \(V\sim\Gamma^{1/2}\) is also of order a meV, one expects \(x\approx K/V\) to be of order \(0.1\ldots 0.3\) [2411.19850].

In the extended Falicov–Kimball model, correlated hopping follows from nonlocal Coulomb matrix elements \(\langle i_f,j_d|V|i_d,j_f\rangle\), yielding
\[
H_{\rm corr}
= -\,t'_d\sum_{\langle i j\rangle}d_i^\dagger d_j\,(n^f_i+n^f_j),
\]
which modifies \(d\)-electron motion according to the \(f\)-occupation pattern [1510.06562]. In generalized Hubbard models with orbital degeneracy, the non-diagonal Coulomb matrix elements \(J_{ijkl}\) with \(i=k\neq j=l\) produce occupation-dependent hoppings \(t_{ij}(n)=t_{ij}(1-\tau_1 n)\) and \(T'_{ij}=t_{ij}\tau_2\), breaking the usual electron-hole symmetry [2101.07051].

Floquet engineering provides a controlled derivation. Starting from
\[
H(t) = H_J + [U + U_1\cos(\omega t)] \sum_i n_{i\uparrow}n_{i\downarrow},
\]
the high-frequency effective Hamiltonian becomes
\[
H_{\rm eff} = -t \sum_{\langle ij\rangle,\sigma} c^\dagger_{i\sigma}c_{j\sigma}\;J_0[K(n_{i,\bar\sigma}-n_{j,\bar\sigma})] + U \sum_i n_{i\uparrow}n_{i\downarrow},
\]
with \(K=U_1/(\hbar\omega)\). Expanding the Bessel function gives the correlated-hopping structure and defines \(X=1-J_0(K)\) [1310.7959]. This realization is notable because it directly links a drive parameter to an occupation-dependent kinetic coefficient.

Cavity QED yields a different derivation. In a synthetic ladder of ground-state Zeeman levels, far-detuned cavity-mediated interactions and two weak drives produce an atom-only effective Hamiltonian whose leading nontrivial process is
\[
H_{\rm corr}  =  J  \sum_{i\neq j}  T_i^+\,T_j^-,
\qquad
J  \simeq  \chi\sum_\nu  |\Omega_\nu|^2/\Delta_\nu^2,
\]
with \(T^+ = \sum_m C_m^{+1}C_{m+2}^{-1} a_{m+2}^\dagger a_m\). In Schwinger-boson form this gives
\[
H_{\rm corr}  \simeq  \sum_{m,m'}  J_{mm'}\,a_{m+2}^\dagger\,a_m\,a_{m'-2}^\dagger\,a_{m'}.
\]
The mechanism relies on collective cavity-mediated dressing of excited levels and interference between two drives to suppress undesired single-particle shifts while retaining correlated hopping [2208.01896].

## 3. Effective-Hamiltonian consequences

A central effect of correlated hopping is the renormalization of low-energy exchange. In the multi-orbital Hubbard description of \(Cr_8\), second-order Schrieffer–Wolff projection yields
\[
H_{\rm eff} = \frac12 \sum_{\langle i,j\rangle} \Gamma_{ij}\,S_i\cdot S_j,
\]
with \(\Gamma_{ij}=\Gamma_{CE}^{ij}+\Gamma_{SE}^{ij}\) and
\[
\Gamma_{SE}^{(a)}=(1-a)^2\Gamma_0^{(a=0)}+\Delta\Gamma.
\]
Correlated hopping reduces only the triplet–triplet virtual processes \(\Gamma_0\) by \((1-a)^2\), leaving \(\Delta\Gamma\) unchanged [2107.06576]. For \(Cr_8\), using \(U=5.98\) eV, \(J=0.26\) eV, and corrected hopping \(t_{11}^{ij}=-0.131\) eV, the reported values are \(\Gamma_{PT}\simeq 1.70\) meV and \(\Gamma^*_{\rm diag}\simeq 1.75\) meV at \(a=0\), while fitting the experimental \(\Gamma=1.46\) meV gives \(a\simeq 0.05\) from perturbation theory and \(a\simeq 0.06\) from exact diagonalization [2107.06576]. The concrete implication is that modest correlated hopping can materially reduce antiferromagnetic exchange.

In mean-field treatments of the extended Falicov–Kimball model, the simplest Hartree–Fock decoupling makes correlated hopping a bandwidth-renormalization channel:
\[
t_d \longrightarrow t_d^* = t_d + 2\,t'_d\,n_f.
\]
This feedback alters both the effective \(d\)-band parity and bandwidth and thereby changes the stability of ferroelectric, antiferroelectric, and charge-density-wave phases [1510.06562]. Positive \(t'_d\) enlarges the excitonic ferroelectric region at negative \(E_f\), whereas sufficiently negative \(t'_d\) suppresses it and can stabilize an antiferroelectric phase with \(\Delta=0\), \(\Delta_P\neq 0\) [1510.06562].

In narrow-band flat-band settings, three distinct correlated-hopping amplitudes \(t_1,t_2,t_3\) govern different physical processes: double–single, single–empty, and double–empty or opposite-spin single–single motion. The effective hopping at uniform filling \(n\) is
\[
t_{\rm eff}
= t_2(1-n)^2 + 2\,t_3\,n(1-n) + t_1 n^2,
\]
so the flat-band condition is \(t_{\rm eff}=0\), reducing at half filling to
\[
t_2 + 2\,t_3 + t_1 = 0.
\]
These amplitudes have qualitatively different roles in the formation of half-metallic ferromagnetic states, with \(t_2\) favoring ferromagnetism via band narrowing and \(t_3\) favoring antiferromagnetic tendencies via kinetic superexchange \(\sim -t_3^2/U\) [2202.12757].

Correlated hopping can also generate exact symmetries. In the one-dimensional Floquet-engineered fermionic model, the Hamiltonian retains spin-SU(2) symmetry for all \(X\), and at the integrable point \(X=1\) it acquires charge-SU(2) symmetry generated by \(\eta^\pm\) and \(\eta^z\), with \([H_{\rm eff}(X=1),\eta^\pm]=0\) and \([H_{\rm eff}(X=1),\eta^z]=0\) [1310.7959]. This opens a direct route to \(\eta\)-pairing physics in a correlated-hopping setting.

## 4. Phases and dynamical phenomena induced by correlated hopping

In bosonic lattice systems, correlated hopping reshapes the phase diagram beyond the standard Mott-insulator/superfluid dichotomy. In the one-dimensional extended correlated-hopping Bose–Hubbard model, the effective bridge-dependent amplitude is
\[
-\,t_{\rm eff}(n_{i+1}) = -\,t - t_c\,n_{i+1},
\]
so that the presence of a particle on the bridge site can suppress or enhance tunneling [2010.14459]. The reported zero-temperature phases include Mott insulator, conventional superfluid, paired-hole superfluid, paired-particle superfluid, droplet superfluid, and phase-separated superfluid [2010.14459]. At \(t_c=t\), a boson is completely blocked from hopping over an occupied site, and correlated hopping induces effective attraction among particles despite \(U>0\) [2010.14459]. A closely related two-species Bose–Hubbard study found a rich phase diagram including a pair superfluid and a superfluid quantum droplet phase, with the former arising from the interplay between single-particle and correlated hopping and the latter from large correlated hopping [1912.13359].

Correlated hopping can stabilize hidden nonlocal order. In the one-dimensional hole-superconductor model with next-nearest-neighbour correlated hopping,
\[
H  =  \sum_{i} [(-J + X\,n_{i+1})(c_i^\dagger c_{i+2} + c_{i+2}^\dagger c_i)] + V\sum_i n_i n_{i+1},
\]
the integrable point \(X=J, V=0\) yields exact and asymmetric spin-charge separation, a finite spin gap, gapless charge excitations, and a non-vanishing den Nijs–Rommelse type string correlator [1508.00404]. The spin gap is nonzero for all \(0<\rho<1\), and numerical studies away from the integrable point show persistence of both long-range string order and spin gap for all \(X>0\) at \(V=0\) [1508.00404].

In constrained bosonic dynamics, density-assisted hopping can produce many-body scars. The bosonic model \(H_1\) exhibits a bipartite Fock-space structure with an exponentially large number of exact zero-energy eigenstates at filling \(\nu=1\), weakly entangled scarred eigenstates at high energy densities, and periodic revivals after quenches from special product states such as \(|(210)^n\rangle\) [1910.09526]. The data report a dominant revival period \(T\approx \pi/4\) for \(H_1\), while a more constrained related model \(H_3\) gives exact revivals with \(F_3(t)=|\cos(2t)|^{2n}\) and period \(T_3=\pi/2\) [1910.09526]. These results establish that a “soft” density-dependent constraint is sufficient to generate anomalously slow thermalization.

Correlated hopping also supports topological mechanisms with no single-particle analogue. In the two-component fermionic mixture
\[
H = H_+ + H_-^0 - t' \sum_{i=1}^{L-1} n_{i,+}(c_{i,-}^\dagger c_{i+1,-}+\mathrm{H.c.}),
\]
the \(+\) species experiences a staggered potential \(\Delta_+\sum_i(-1)^i n_{i,+}\), and for \(\Delta_+>0\) forms a density wave on the even sublattice. Through \(t'\), the \(-\) species then sees an effective dimerization equivalent to an SSH model; for \(\Delta_+<0\), the dimerization is trivial [2503.00589]. The topological transition occurs at \(\Delta_+=0\), and MPS simulations show that \(G_{1,-}\) closes while \(G_{2,-}\) remains finite, with edge-state polarization, string order, and entanglement-spectrum degeneracy characterizing the topological regime [2503.00589]. In a Thouless pump protocol, the pumped charge of the \(-\) species is quantized, with \(Q=1\) when the loop encloses the topological point and \(Q=0\) otherwise [2503.00589]. This suggests an interaction-driven route to topological order mediated entirely by density-dependent kinetics.

## 5. Transport, thermoelectricity, and localization

In quantum dots, correlated hopping modifies both charge and heat transport and is especially visible in the non-linear regime. In the extended Anderson model, the term \(H_{CH}\) breaks the particle-hole symmetry of the standard \(x=0\) model, so conductance \(G(\delta)\) is no longer symmetric in \(\delta\equiv \varepsilon_d+U/2\) [2411.19850]. The lower Hubbard peak moves to more negative \(\delta\), while no compensating shift occurs at \(+U/2\) [2411.19850]. Differential conductance curves at \(eV\sim 4\Gamma\) and \(x=0.1\ldots 0.2\) show that the minimum between the two Hubbard peaks is shifted to \(\delta<0\), the left peak splits more strongly than the right, and the upper band is almost unchanged [2411.19850]. Differential thermopower similarly loses exact antisymmetry, develops higher, narrower extrema at negative \(\delta\), and can exhibit sign changes for moderate \(x\) in a range of \(\delta<0\) [2411.19850]. The proposed experimental signature is a systematic shift of the lower-energy peak to \(\delta<0\) without a corresponding shift of the upper-energy peak [2411.19850].

An exact transport symmetry can nevertheless survive. For the transport Green function
\[
G^r_\sigma(\omega)\equiv \langle\!\langle D_\sigma|D_\sigma^\dagger\rangle\!\rangle^r_\omega,
\]
the equation-of-motion treatment shows that the \(x\)-dependence enters through \(x(2-x)\) or \((1-x)^2\), producing an exact \(x\leftrightarrow 2-x\) symmetry of transport observables [2103.08985]. The average dot occupation also obeys this symmetry, whereas the spectral function calculated within an analogous decoupling scheme does not, because some lifetime-cutoff processes appear only at fourth order in the hybridization [2103.08985]. This distinction is a formal subtlety rather than a contradiction: transport and spectroscopy are controlled by different Green functions in the assisted-hopping Anderson model.

In the Falicov–Kimball model, dynamical mean-field theory shows that correlated hopping can strongly enhance thermoelectric response by reshaping the two-particle transport function more dramatically than the one-particle DOS. On the Bethe lattice, the transport coefficients follow from
\[
L_{lm}
=\frac{\sigma_0}{e^2}\int_{-\infty}^{\infty} d\omega
\Bigl(-\frac{\partial f}{\partial\omega}\Bigr) I(\omega)\,\omega^{l+m-2},
\]
with \(\sigma_{\rm dc}=e^2 L_{11}\), \(S=(1/eT)(L_{12}/L_{11})\), and \(\kappa_e=(1/T)(L_{22}-L_{12}^2/L_{11})\) [1609.07098]. Correlated hopping generates sharp resonant features in \(I(\omega)\) when a two-particle resonance condition is met, while the one-particle DOS remains comparatively smooth [1609.07098]. Bringing the Fermi level near the resonant frequency strongly increases electrical conductivity, thermal conductivity, and thermoelectric power [1609.07098]. In a related DMFT study on the hypercubic lattice, the most pronounced thermoelectric effects occur for \(-\,\tfrac12 t < X < 0\), and for \(U/t=2\), \(X/t=-0.3\), half filling of \(f\)-electrons, and \(n^c+n^f=1\), the Seebeck coefficient reaches \(S_{\max}\approx 2(k_B/e)\) at \(T/t\sim 0.05\), with power factor enhanced by more than an order of magnitude and \(ZT\sim 2\) at \(T/t\sim 0.1\) [1409.8495]. Doping to \(n^f=0.6\), \(n^c=0.4\) yields \(S\sim 3(k_B/e)\) and \(ZT>3\) [1409.8495].

When hopping between occupied sites is strongly reduced, correlated hopping can induce localization and an in-gap band. In the microdoped Mott-insulator phase of the correlated-hopping Falicov–Kimball model, taking \(t_2\to -0.5\) with \(t_3=0\) makes \(t^{++}=0\), so hopping is completely blocked on \(f\)-occupied sites [2003.02076]. For \(n_f>0.5\), this produces a third narrow band inside the Mott gap whose spectral weight is \(2n_f-1\) [2003.02076]. In the limit \(t^{++}=0\), the band shrinks to a \(\delta\)-peak carrying no transport weight; for slightly positive \(t^{++}\), it broadens into a narrow resonant mid-gap band, giving a giant low-temperature Seebeck response whose sign can be switched by microdoping [2003.02076].

Localization phenomena also arise in quasiperiodic systems with correlated off-diagonal structure. In a one-dimensional chain with AAH on-site modulation and Fibonacci or Bronze-Mean hopping sequences, the Hamiltonian
\[
H = \sum_{i=1}^{N-1} t_i(c_i^\dagger c_{i+1}+c_{i+1}^\dagger c_i)
+\sum_{i=1}^{N} (-1)^i\epsilon_i c_i^\dagger c_i
\]
uses \(t_i\in\{t_A,t_B\}\) arranged quasiperiodically [2506.02716]. The reported diagnostics—IPR, NPR, and fractal dimension—show a reentrant localization-delocalization-localization sequence: a fully localized regime beyond \(\lambda_{\rm c1}\), followed by reentrant delocalization windows \([1.55,2.10]\) for Fibonacci and \([0.70,1.30]\) for Bronze Mean, before a final localized phase at large \(\lambda\) [2506.02716]. The effect is attributed to the interplay of diagonal quasiperiodicity and correlated binary hopping with long-range order [2506.02716].

## 6. Methods, realizations, and recurring misconceptions

Correlated-hopping models are studied by a diverse set of methods, and the method is usually tightly matched to the physical regime. Strong-coupling and Schrieffer–Wolff projections are used when \(t\ll U\), as in the multi-orbital nanomagnet problem [2107.06576]. Cluster mean-field theory, exact diagonalization, and analytical strong-coupling expansion are combined for bosonic lattice models with extended correlated hopping [1912.13359] [2010.14459]. Matrix product state simulations diagnose equilibrium topology and quantized pumping in one-dimensional mixtures [2503.00589]. DMFT gives exact charge and heat transport in infinite-dimensional Falicov–Kimball models with correlated hopping [1409.8495] [1609.07098] [2003.02076]. Equation-of-motion Green-function methods address linear and nonlinear transport in Anderson-type quantum dots [2103.08985] [2411.19850].

Experimental implementations span several platforms. Ultracold fermions in optical lattices can realize correlated-hopping models by a modulated magnetic field near a Feshbach resonance; for \(^{40}K\) in a \(\lambda=1064\) nm lattice, the data give \(t/h\sim 200\ldots 500\) Hz, \(U/h\sim 1\) kHz, and \(\omega/(2\pi)\approx 500\ldots 600\) Hz, allowing \(X=1-J_0(K)\) to be tuned from the Hubbard limit to the \(\eta\)-symmetric point \(X=1\) [1310.7959]. In cavity QED, alkaline-earth atoms such as \(^{87}Sr\) and \(^{173}Yb\) with \(N\sim 10^5\) can reach \(\chi N\sim (2\pi)\times 10\ldots 100\) kHz while \(\gamma\sim (2\pi)\times 0.01\) kHz, with typical working parameters \(\Omega_A=\Omega_B\approx 0.05\,\chi N\), \(\Delta_A\approx -3\,\chi N\), and \(\Delta_B\approx (3\ldots 6)\chi N\) [2208.01896]. In atomic mixtures, density-dependent tunneling can be engineered via resonant Floquet modulation of inter-species interactions or Feshbach-induced occupation-dependent lattice shifts [2503.00589]. In mesoscopic transport, detection protocols emphasize bias- and gate-dependent asymmetries of differential conductance and thermopower [2411.19850].

Several recurrent misconceptions are corrected by the literature. First, correlated hopping is not merely a small perturbative correction to \(t\). In \(Cr_8\), \(a\approx 0.05\ldots 0.06\) suffices to reduce the calculated antiferromagnetic exchange from \(\approx 1.7\) meV to the experimental \(\approx 1.46\) meV [2107.06576]. In Falicov–Kimball models, even modest \(t'\approx 0.5\) can change the preferred ordered state from checkerboard to stripes or segregation and strongly alter finite-temperature critical behaviour [1007.2770]. Second, the main signatures are not limited to spectral functions. Several studies emphasize that two-particle transport functions or nonlinear response reveal stronger and cleaner correlated-hopping effects than one-particle DOS alone [1609.07098] [2411.19850]. Third, correlated hopping need not only renormalize pre-existing band topology or magnetism. The one-dimensional mixture with staggered \(\Delta_+\) and density-dependent \(t'\) shows that topological order can be induced through specific interaction couplings, without inheriting from a corresponding single-particle Hamiltonian [2503.00589].

Taken together, these results define correlated hopping as a kinetic interaction channel of broad scope rather than a model-specific embellishment. Across cold atoms, nanomagnets, quantum dots, Falicov–Kimball systems, and quasiperiodic lattices, the same formal ingredient—occupation-dependent tunneling—recurrently controls exchange renormalization, symmetry breaking, clustering, hidden order, anomalous transport, localization, and interaction-driven topology [1310.7959] [2107.06576] [2208.01896] [2503.00589].

Source: https://www.emergentmind.com/topics/correlated-hopping