Spinless Fermion Chains

A spinless fermion chain is the minimal lattice model in which Fermi statistics, coherent motion, interactions, filling, and one-dimensional collective behavior all matter. Each site carries one fermionic mode, so its local occupation is either zero or one. Nearest-neighbor hopping delocalizes particles, while a nearest-neighbor density interaction favors either alternating occupation or clustering.

The canonical interacting model is the $t$–$V$ chain. It is simple enough to solve exactly in important limits, yet rich enough to contain a free Fermi sea, a Luttinger liquid, a charge-density-wave phase, phase separation, boundary-twist responses, and the finite-size parity subtleties that appear when it is mapped to a periodic spin chain.

The Jordan–Wigner transformation relates this fermion model to the XXZ chain, but its boundary condition and correlation operators must also be transformed.

Helpful background. Tight-binding modes supplies the free band, while fermionic operators supplies the signs in the occupation basis.

Fermion occupations on a chain

At site $j$,

$$n_j = c_j^\dagger c_j, \qquad n_j^2 = n_j,$$

so the allowed eigenvalues are $n_j=0,1$. For an ordered set of $L$ sites, a convenient occupation basis is

$$\lvert n_1n_2\cdots n_L\rangle = (c_1^\dagger)^{n_1} (c_2^\dagger)^{n_2} \cdots (c_L^\dagger)^{n_L} \lvert\mathrm{vac}\rangle.$$

The ordering is part of the convention. Moving a fermionic operator through occupied earlier modes produces signs. Although each site has the same two-dimensional local vector space as a spin-$1/2$ degree of freedom, the fermionic operator algebra is graded; it is not an ordinary tensor-product spin algebra without additional parity strings.

The full Fock-space dimension is $2^L$. In a fixed-number sector with $N$ fermions,

$$\dim\mathcal H_N = \binom{L}{N}.$$

Number conservation therefore provides a substantial exact-diagonalization reduction. Additional commuting symmetries can reduce the problem further. Translation and inversion do not commute at generic momentum, and particle–hole symmetry relates different number sectors away from half filling.

The Centered t–V Hamiltonian

For an open chain, take $t>0$ and write

$$H_{tV}^{\mathrm{open}} = {} -t \sum_{j=1}^{L-1} \left( c_j^\dagger c_{j+1} + c_{j+1}^\dagger c_j \right) + V \sum_{j=1}^{L-1} \left(n_j-\frac12\right) \left(n_{j+1}-\frac12\right) - \mu \sum_{j=1}^{L} \left(n_j-\frac12\right).$$

The centered densities are not a different interaction. Expanding one bond gives

$$\left(n_j-\frac12\right) \left(n_{j+1}-\frac12\right) = n_j n_{j+1} - \frac12(n_j+n_{j+1}) + \frac14.$$

On a ring, each site touches two bonds, so centering shifts the uniform chemical potential by $V$ and adds $VL/4$. On an open chain it instead adds

$$-V\sum_{j=2}^{L-1}n_j-\frac V2(n_1+n_L)+\frac{V(L-1)}4.$$

The endpoint potentials differ from the bulk. A single uniform chemical-potential shift therefore does not make the centered and uncentered open models identical. Centering makes particle–hole symmetry visible at $\mu=0$ on a bipartite chain.

The terms have distinct roles:

• $t$ coherently transfers a fermion across a bond;
• $V>0$ penalizes equal neighboring density deviations and favors alternation;
• $V<0$ rewards equal neighboring density deviations and favors clustering;
• $\mu$ controls the filling in a grand-canonical description.

The word spinless means that no independent spin label is retained. It may describe genuinely polarized fermions, an effective single-component band, or a model obtained after projecting out other internal states. Pauli exclusion forbids two fermions in the same retained mode, but it does not remove interactions between different sites.

Boundary Conventions

For a ring, the bond sum includes $j=L$, and a twist $\phi$ may be imposed through

$$c_{L+1} = e^{i\phi}c_1.$$

Important special cases are

$$\phi=0 \quad\text{(periodic)}, \qquad \phi=\pi \quad\text{(antiperiodic)}.$$

Equivalently, one may distribute the twist uniformly over the bonds,

$$-t \sum_{j=1}^{L} \left( e^{i\phi/L}c_j^\dagger c_{j+1} + e^{-i\phi/L}c_{j+1}^\dagger c_j \right),$$

with strictly periodic operators. A site-dependent gauge transformation moves phase between the boundary and the bonds without changing the total phase accumulated around the ring.

Open and periodic chains are not interchangeable at finite $L$. The open chain has $L-1$ bonds and standing waves. The ring has $L$ bonds, translation sectors, and a flux-sensitive momentum grid. Their bulk energy densities usually agree as $L\to\infty$ away from singular limits, but their finite spectra and edge observables differ.

Symmetries and Conserved Quantities

For every $V$, $\mu$, and number-preserving boundary twist,

$$N = \sum_{j=1}^{L}n_j, \qquad [H,N] = 0.$$

This global $U(1)$ symmetry permits independent diagonalization in each fixed-$N$ sector.

Additional symmetries depend on geometry:

• a uniform ring has discrete translation symmetry;
• a uniform open chain has reflection symmetry about its midpoint;
• at zero twist, real couplings preserve spinless time-reversal symmetry by complex conjugation;
• at $\mu=0$, a real-hopping open chain of any length, or an even ring with zero or π twist, has the unitary particle–hole symmetry below;
• inversion sends a twist $\phi$ to $-\phi$ unless combined with another operation.

On such a bipartite chain, define

$$\mathcal C c_j\mathcal C^{-1} = (-1)^j c_j^\dagger.$$

Then

$$n_j-\frac12 \longmapsto -\left(n_j-\frac12\right).$$

Nearest-neighbor hopping and centered density interactions are invariant, whereas the chemical-potential term changes sign. Thus $\mu=0$ is the particle-hole-symmetric point. In a nondegenerate symmetric finite-volume state, it has

$$\langle N\rangle = \frac{L}{2}.$$

An odd ring is not bipartite. For an odd open chain, a symmetric Gibbs mixture has mean $L/2$, but no pure fixed-number state can have that half-integer number. A general flux is reversed by the unitary particle–hole map and therefore changes its invariance conditions.

Why This Is the Minimal Interacting Fermion Chain

The free hopping chain is diagonal in momentum space. Adding the nearest-neighbor density term produces quartic operators,

$$n_j n_{j+1} = \left(c_j^\dagger c_j\right) \left(c_{j+1}^\dagger c_{j+1}\right),$$

so generic momentum occupations cease to be individually conserved. Yet the interaction is short-ranged, translation invariant, and number conserving. The model therefore isolates the central competition:

$$\text{delocalization from }t \quad\text{versus}\quad \text{density organization from }V.$$

In one dimension this competition does not generically produce a conventional Landau Fermi liquid. In the intermediate regime it produces a Luttinger liquid whose elementary low-energy excitations are collective density waves.

Free Chain

Set $V=0$. The Hamiltonian is quadratic, so a one-particle diagonalization solves every fixed-number many-body sector.

Fourier Diagonalization on a Ring

Use

$$c_j = \frac1{\sqrt L} \sum_k e^{ikja}c_k,$$

where $a$ is the lattice spacing. The twisted boundary condition requires

$$e^{ikLa} = e^{i\phi},$$

and hence

$$k_m = \frac{2\pi m+\phi}{La}, \qquad m\in\mathbb Z \pmod L.$$

The free Hamiltonian becomes

$$H_0 = \sum_k \epsilon(k)c_k^\dagger c_k + \frac{\mu L}{2},$$

with

$$\epsilon(k) = -2t\cos(ka) - \mu.$$

The final constant follows from the centered chemical-potential convention and does not affect eigenvectors. A many-body eigenstate is a Slater determinant specified by occupied momenta,

$$\lvert k_1,\ldots,k_N\rangle = c_{k_1}^\dagger \cdots c_{k_N}^\dagger \lvert\mathrm{vac}\rangle,$$

with energy equal to the sum of occupied one-particle energies, plus the displayed constant.

Filling and Fermi Points

Let

$$\nu = \frac NL, \qquad 0\le\nu\le1.$$

In the thermodynamic ground state, occupied momenta fill a connected interval around $k=0$. For $0<\nu<1$,

$$k_F = \frac{\pi\nu}{a}.$$

At half filling,

$$\nu = \frac12, \qquad k_Fa = \frac\pi2.$$

The Fermi velocity is

$$v_F = \frac1\hbar \left. \frac{d\epsilon}{dk} \right|_{k_F} = \frac{2ta}{\hbar} \sin(\pi\nu).$$

Near the two Fermi points, the dispersion is approximately linear:

$$\epsilon(\pm k_F+q) - \epsilon_F \simeq \pm\hbar v_Fq.$$

This pair of right- and left-moving branches is the starting point for the low-energy field theory. The lattice remains essential because it fixes the bandwidth, commensurate fillings, and allowed scattering processes.

Ground-State Energy Density

For the free chain, the kinetic ground-state energy per site at fixed filling is

$$\begin{aligned} e_{\mathrm{kin}}(\nu) &= \frac{a}{2\pi} \int_{-k_F}^{k_F} dk\, [-2t\cos(ka)] \\ &= -\frac{2t}{\pi} \sin(\pi\nu). \end{aligned}$$

At half filling,

$$e_{\mathrm{kin}} = -\frac{2t}{\pi}.$$

The chemical potential obtained from this fixed-density energy is

$$\mu_{\mathrm{phys}} = \frac{\partial e_{\mathrm{kin}}}{\partial\nu} = -2t\cos(\pi\nu),$$

which is the band energy at $k_F$. It vanishes at half filling in the centered convention.

Open-Chain Standing Waves

For an open chain, define dimensionless wave numbers

$$q_m = \frac{\pi m}{L+1}, \qquad m=1,2,\ldots,L.$$

The normalized one-particle orbitals are

$$\varphi_m(j) = \sqrt{\frac{2}{L+1}} \sin(q_mj),$$

and the energies are

$$\epsilon_m = -2t\cos q_m - \mu.$$

At fixed $N$, the free ground state occupies $m=1,\ldots,N$. Translation symmetry is absent, but reflection parity and the node structure of standing waves remain useful.

Exact Free Correlations

At zero temperature in the infinite free chain,

$$C(r) \equiv \langle c_j^\dagger c_{j+r}\rangle = \frac{\sin(k_Fra)}{\pi r}, \qquad r\ne0,$$

while $C(0)=\nu$. The $1/r$ decay is algebraic: the free ground state is gapless whenever the band is partially filled.

Wick’s theorem gives the connected density correlation at distinct sites,

$$\begin{aligned} \langle n_j n_{j+r}\rangle_c &\equiv \langle n_j n_{j+r}\rangle - \langle n_j\rangle \langle n_{j+r}\rangle \\ &= -\lvert C(r)\rvert^2 \\ &= -\frac{\sin^2(k_Fra)}{\pi^2r^2}. \end{aligned}$$

The negative sign is the exchange hole: equal-component fermions suppress nearby joint occupation even without a dynamical repulsion.

At half filling, $C(r)$ vanishes for nonzero even $r$ and alternates in sign on odd separations. The density correlation contains a uniform $1/r^2$ part and an oscillatory component at wave number $2k_F=\pi/a$. This $2k_F$ structure survives interactions in the Luttinger liquid, but its exponent changes.

Nearest-Neighbor Interaction in Momentum Space

For a ring, the density interaction can be written schematically as

$$H_V = \frac1{2L} \sum_{k,k',q} \widetilde V(q) c_{k+q}^\dagger c_{k'-q}^\dagger c_{k'}c_k + \text{one-body and constant terms},$$

where the nearest-neighbor Fourier component is

$$\widetilde V(q) = 2V\cos(qa).$$

Total momentum is conserved modulo a reciprocal-lattice vector, but individual mode occupations $n_k$ are not. At commensurate filling, the lattice permits momentum transfer by a reciprocal vector. At half filling, $4k_F=2\pi/a$, so umklapp scattering can become important and drive the repulsive charge-ordering transition.

The exact phase boundary is nonperturbative. Weak-coupling scattering language explains which process is allowed, while the XXZ equivalence and Bethe ansatz fix the uniform model’s exact transition at $V=2t$.

Relation to the XXZ Chain

Use the Jordan–Wigner convention

$$s_j^z = n_j-\frac12,$$

with nonlocal strings in $s_j^\pm$. On an open chain, nearest-neighbor strings cancel inside the transverse exchange. After a staggered gauge transformation that reverses the hopping sign, the correspondence is

$$J = 2t, \qquad \Delta = \frac{V}{2t}, \qquad h = \mu.$$

Thus the fermion density deviation maps to spin magnetization,

$$N-\frac L2 = S_{\mathrm{tot}}^z.$$

The map gives immediate physical translations:

Fermion language	Spin language
hopping	transverse $xy$ exchange
density interaction	longitudinal $z$ exchange
chemical potential	longitudinal field
half filling	zero total magnetization
charge-density wave	Néel order along $z$
phase separation	ferromagnetic sector after the staggered convention
particle current	spin current

The equivalence does not mean that fermion and spin correlation functions are always identical local objects. Density is local under the map, but a single-fermion operator corresponds to a spin flip multiplied by a nonlocal string. Consequently a simple spin correlator can encode a string correlator in fermion variables, and vice versa.

Use XXZ Spin Chain for the canonical spin Hamiltonian, exact phase structure, Bethe-ansatz orientation, and spin observables.

Half-Filled Phase Diagram

Assume a uniform chain, $t>0$, $\mu=0$, half filling, and the thermodynamic limit.

Coupling	Regime	Gap and long-distance behavior
$V<-2t$	phase separated	particles and holes form macroscopic domains at fixed half filling
$V=-2t$	phase-separation boundary	singular endpoint with enhanced degeneracy and quadratic low-energy structure
$-2t<V<2t$	Luttinger liquid	gapless, compressible, algebraic correlations, central charge $c=1$
$V=2t$	BKT transition	gapless endpoint with logarithmic corrections; $K=1/2$
$V>2t$	charge-density wave	gapped bulk with two translation-related ordered patterns

At half filling, attraction with $V<-2t$ favors macroscopic domains, while repulsion with $V>2t$ favors alternating charge order. The displayed occupations illustrate their strong-coupling limits; finite symmetric eigenstates can superpose translated patterns. Between the endpoints the uniform model is a gapless Luttinger liquid.

These are thermodynamic phases, not labels that can be read from one small spectrum. A finite translation-invariant ring does not choose one charge pattern by itself. Near the Berezinskii–Kosterlitz–Thouless transition, the correlation length can be exponentially large and finite-size convergence can be slow.

Luttinger-Liquid Regime

The Luttinger-liquid treatment develops the common low-energy Hamiltonian. Here its parameters can be obtained exactly from the half-filled $t$–$V$ model.

Write

$$\Delta = \frac{V}{2t} = \cos\gamma, \qquad 0<\gamma<\pi.$$

For the half-filled uniform integrable chain with $-2t<V<2t$, the exact Luttinger parameter and mode velocity are

$$K = \frac{\pi}{2(\pi-\gamma)},$$

and

$$v = \frac{\pi ta}{\hbar} \frac{\sin\gamma}{\gamma}.$$

Checks at the free point are immediate. For $V=0$, $\gamma=\pi/2$, so

$$K=1, \qquad v=\frac{2ta}{\hbar}=v_F.$$

Repulsion lowers $K$ from one toward $1/2$ at the charge-ordering transition. Attraction raises $K$ and sends it to infinity as the phase-separation endpoint is approached from above.

The long-distance density correlation has the universal structure

$$\langle n_j n_{j+r}\rangle_c \sim -\frac{K}{2\pi^2r^2} + A \frac{\cos(2k_Fra)}{r^{2K}} + \cdots,$$

where $A$ is nonuniversal. The single-particle correlator obeys

$$\langle c_j^\dagger c_{j+r}\rangle \sim \frac{\text{oscillatory factor}} {r^{(K+K^{-1})/2}}.$$

At $K=1$, these powers reduce to the free-fermion results. Away from the free point, there is no quasiparticle jump in the momentum distribution. The low-energy excitations are collective modes even though the microscopic variables are fermions.

The displayed asymptotic formulas describe the leading universal powers. Lattice-scale amplitudes, subleading harmonics, finite temperature, boundaries, and logarithmic corrections at special points require additional care.

Repulsive Strong Coupling and Charge Order

At $t=0$ and $V>0$, each bond is minimized by opposite centered densities. At half filling on an even ring, the two classical ground patterns are

$$\lvert1010\cdots\rangle, \qquad \lvert0101\cdots\rangle.$$

They are related by translation by one site. In the bulk, a local hop creates neighboring $11$ and $00$ defects, equivalently a pair of domain walls. In the strict $t=0$ limit, the interaction cost of this local defect pair is $V$.

For finite $t$ with $V>2t$, quantum fluctuations dress these patterns but do not remove the thermodynamic charge-density-wave order. Order Parameters gives the general operator construction, while Long-Range Order gives the correlation-limit and finite-size scaling tests. For this model, a useful order parameter is

$$m_{\mathrm{CDW}} = \lim_{L\to\infty} \frac1L \sum_{j=1}^{L} (-1)^j \left\langle n_j-\frac12\right\rangle_{\mathrm{sb}},$$

where the subscript indicates a symmetry-broken thermodynamic state or an explicit order-of-limits prescription.

In a finite symmetric ring, $m_{\mathrm{CDW}}$ can vanish. The order is then detected through the structure factor

$$S(q) = \frac1L \sum_{j,\ell} e^{iq(j-\ell)a} \left\langle \left(n_j-\nu\right) \left(n_\ell-\nu\right) \right\rangle,$$

whose peak at $q=\pi/a$ grows extensively in the ordered phase.

Attractive Strong Coupling and Phase Separation

For $V<0$, equal neighboring occupations lower the interaction energy. At fixed intermediate filling and $\lvert V\rvert\gg t$, particles gather into one dense domain and holes into another. Only the interfaces are costly, so a macroscopic cluster wins over a homogeneous arrangement.

Phase separation is not a paired superfluid. Strong attraction here reorganizes the density macroscopically. The one-mode local space forbids an onsite pair, though spinless fermions can have intersite pair correlations in other regimes or models; number conservation alone does not rule out such correlations.

Finite translation-invariant rings have momentum eigenstates rather than a cluster pinned to a particular location. A localized domain appears after translation symmetry is weakly broken, through measurement conditioning, or in suitable linear combinations of nearly degenerate states.

Density, Current, and Twist Response

The local continuity equation follows from the Heisenberg equation. For the number density,

$$\frac{dn_j}{dt} = \mathcal J_{j-1,j} - \mathcal J_{j,j+1},$$

where the oriented particle current is

$$\mathcal J_{j,j+1} = \frac{it}{\hbar} \left( c_{j+1}^\dagger c_j - c_j^\dagger c_{j+1} \right).$$

The density interaction commutes with every $n_j$ and therefore does not add a separate local transfer term. It changes current expectation values and dynamics through the interacting state.

A boundary twist probes transport around a ring. If $E_0(\phi)$ is the ground-state energy, the persistent response is obtained by differentiating with respect to the applied flux or twist. A common stiffness diagnostic, with energy units, is proportional to

$$L \left. \frac{\partial^2E_0(\phi)}{\partial\phi^2} \right|_{\phi=0},$$

provided the ground-state branch is twice differentiable there. If a level crossing makes a cusp at zero twist, use a smooth branch or the appropriate energy-minimizing twist and state that choice. The precise prefactor depends on charge and convention. A gapless clean Luttinger liquid has nonzero zero-temperature stiffness, while the thermodynamic charge-density-wave insulator does not.

Use Density Operators and Current Operators for the canonical continuity-equation derivation and current conventions.

Boundary Parity from Jordan–Wigner

A fermionic ring defined directly can be assigned periodic, antiperiodic, or twisted boundary conditions as part of the model. The situation changes when the fermions are introduced by transforming a periodic spin chain.

With the convention

$$s_j^+ = c_j^\dagger \exp\!\left( i\pi\sum_{\ell<j}n_\ell \right),$$

define total fermion parity

$$P = (-1)^N.$$

In a fixed-parity sector, a periodic spin boundary is equivalent to

$$c_{L+1} = -P c_1.$$

Thus, in this convention,

Fermion number parity	Effective fermion boundary
$P=+1$ (even $N$)	antiperiodic
$P=-1$ (odd $N$)	periodic

A staggered gauge transformation may shift signs or twists, but the invariant lesson is that the fermionic boundary sector is tied to parity. Choosing a momentum grid before fixing parity can therefore shift finite-size energies, degeneracies, and apparent gaps.

This parity constraint belongs to the spin-to-fermion map. It should not be imposed automatically on a microscopic fermion ring whose boundary condition was specified independently. The boundary sign and matching parity projection are derived in Jordan–Wigner Transformation.

Finite-Size Shell Effects

At finite $L$, the location of allowed momenta relative to $\pm k_F$ matters. A level can lie exactly at the Fermi energy for one twist but not another. This changes ground-state degeneracy without changing the thermodynamic phase.

For example, a periodic four-site free ring has momenta

$$ka \in \left\{ 0,\frac\pi2,\pi,\frac{3\pi}{2} \right\},$$

with energies

$$-2t,\ 0,\ 2t,\ 0.$$

At $N=2$, one fermion occupies $k=0$ and the other can occupy either zero-energy mode. Antiperiodic boundary conditions instead give $ka=\pm\pi/4,\pm3\pi/4$; the two negative-energy orbitals are both occupied, and the free ground-state energy is

$$E_0 = -2\sqrt2\,t.$$

The two choices approach the same bulk energy density. Their small-system spectra differ because they sample the cosine band differently.

Entanglement as a Phase Diagnostic

For a periodic critical ground state described by a conformal field theory, the interval entanglement entropy scales as

$$S(\ell) = \frac c3 \ln\!\left[ \frac{L}{\pi} \sin\!\left(\frac{\pi\ell}{L}\right) \right] + s_1 + \cdots.$$

Here $L$ and $\ell$ count sites; the physical lengths are $La$ and $\ell a$, so the lattice cutoff cancels inside the logarithm. Entropy is dimensionless, $-\operatorname{Tr}\rho\ln\rho$. The $t$–$V$ Luttinger liquid has $c=1$. In a gapped phase, the entropy instead saturates with interval size once $\ell$ exceeds the correlation length, apart from finite-size and symmetry-sector effects.

Near $V=2t$, logarithmic and BKT crossover corrections can make a naive central-charge fit misleading. Entanglement should be combined with gap scaling, density correlations, stiffness, and structure factors. Use Entanglement Entropy for the canonical definitions and scaling framework.

What Is Exactly Solvable

Several different statements are sometimes compressed into “the chain is solvable”:

1. At $V=0$, Fourier or standing-wave modes diagonalize the Hamiltonian completely.
2. For uniform nearest-neighbor $t$ and $V$, the XXZ correspondence makes the model Bethe-ansatz integrable.
3. Exact integrability determines thermodynamic and excitation properties, but extracting a desired finite-size correlator may still require substantial work.
4. Generic next-nearest-neighbor hopping, longer-range interactions, quasiperiodicity, or disorder usually destroy Bethe-ansatz integrability.
5. Jordan–Wigner changes variables; it diagonalizes only those mapped models that become quadratic after the transformation.

Integrable does not mean noninteracting. For $V\ne0$, two-body scattering changes momentum quantization and the elementary low-energy excitations are collective.

Common Variants

The minimal model can be extended in controlled directions:

$$H' = -t_2\sum_j \left(c_j^\dagger c_{j+2}+\text{h.c.}\right) + V_2\sum_j n_j n_{j+2} + \sum_jw_jn_j.$$

Here $t_2$ frustrates the simple cosine band and generally breaks integrability, $V_2$ introduces competing density order, and $w_j$ can represent a trap, superlattice, quasiperiodic potential, or disorder.

A pairing term such as

$$\sum_j \left(\Delta_p c_jc_{j+1}+\Delta_p^*c_{j+1}^\dagger c_j^\dagger\right)$$

breaks $U(1)$ number conservation to fermion parity and leads toward the Kitaev-chain setting. That is a different canonical model, not merely another parameter value of the number-conserving $t$–$V$ chain.

Exercises

Parity and boundary conditions

Under the Jordan–Wigner convention used above, determine the fermion boundary condition induced by a periodic spin chain in sectors with $N=6$ and $N=7$. State why this rule need not apply to a directly defined fermion ring.

Solution

For $N=6$,

$$P = (-1)^6 = +1,$$

and $c_{L+1}=-Pc_1=-c_1$, so the fermions are antiperiodic.

For $N=7$,

$$P = (-1)^7 = -1,$$

and $c_{L+1}=-Pc_1=c_1$, so the fermions are periodic.

The rule arises because a nonlocal Jordan–Wigner string crosses the spin-chain boundary. A microscopic fermion model does not inherit that string automatically; its boundary condition is an independent part of its definition.

Current from continuity

Derive the bond current for the hopping Hamiltonian and show that the density interaction contributes no explicit source term.

Solution

Only the two hopping bonds adjacent to site $j$ fail to commute with $n_j$. Using

$$\frac{dn_j}{dt} = \frac{i}{\hbar}[H,n_j],$$

the right bond gives an outward term and the left bond an inward term. Collecting them yields

$$\frac{dn_j}{dt} = \mathcal J_{j-1,j} - \mathcal J_{j,j+1},$$

with

$$\mathcal J_{j,j+1} = \frac{it}{\hbar} \left( c_{j+1}^\dagger c_j - c_j^\dagger c_{j+1} \right).$$

Every density operator commutes with every other density operator, so

$$\left[ \sum_\ell \left(n_\ell-\frac12\right) \left(n_{\ell+1}-\frac12\right), n_j \right] = 0.$$

The interaction affects current dynamics through the state and through $d\mathcal J/dt$, but it creates no local violation of number conservation.

Centering changes endpoint energies

At $t=0$, compare the interaction energy of one fermion at an endpoint and one fermion at an interior site of a four-site open chain. Use both the centered interaction and $V\sum_jn_jn_{j+1}$.

Solution

For the uncentered interaction both configurations have energy zero: there are no occupied neighboring pairs. For the centered interaction, $1000$ gives the bond sum $(-1+1+1)V/4=V/4$, whereas $0100$ gives $(-1-1+1)V/4=-V/4$. The difference is $V/2$. A uniform chemical potential shifts both one-particle states equally and cannot remove this endpoint effect.

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