Hubbard Models: Symmetries and Controlled Limits

For the repulsive one-band Hubbard model, the generic internal symmetry is charge $U(1)$ times spin $SU(2)$. Real hopping only between opposite sublattices adds particle–hole symmetry at $\mu=U/2$ and an $SU(2)$ charge pseudospin, giving $SO(4)$. The exact $U=0$ and $t=0$ anchors expose band and local-charge descriptions, while a separated doublon sector permits a controlled expansion in $t/U$ whose low-energy variables are spins and projected carriers. These facts constrain every later calculation, but none of them alone proves that a state is a Mott insulator.

The hopping and interaction on this page are declared low-energy inputs. The effective-Hamiltonian construction explains how they arise and how orbital, screening, range, and double-counting choices limit their empirical adequacy.

Required background. Second-quantized fermions supplies the operator algebra; Dyson equations supply the spectral and self-energy language; and emergent effective degrees of freedom explain why electrons can reduce to spins or projected carriers. Helpful background. Tensor-network ansätze and lattice error budgets govern numerical claims.

The one-band Hubbard model card

Take a finite graph with $L$ sites, specified boundary conditions, and one spinful orbital on each site. Its local basis is $|0\rangle$, $|\uparrow\rangle$, $|\downarrow\rangle$, and $|\uparrow\downarrow\rangle$. With $n_{i\sigma}=c_{i\sigma}^\dagger c_{i\sigma}$, $n_i=n_{i\uparrow}+n_{i\downarrow}$, and $N=\sum_i n_i$, the grand-canonical Hamiltonian is

$$H=-\sum_{ij\sigma}t_{ij}c_{i\sigma}^\dagger c_{j\sigma} +U\sum_i n_{i\uparrow}n_{i\downarrow}-\mu N, \qquad U>0.$$

The kinetic sum is over ordered pairs, so both directions of a bond are included and Hermiticity requires $t_{ji}=t_{ij}^*$. A uniform diagonal one-body energy is absorbed into $\mu$, and we set $t_{ii}=0$. The canonical Hamiltonian used for fixed-$N$ energies is the same expression without $-\mu N$.

This minimal competition between itinerancy and local repulsion was introduced in Hubbard 1963, pp. 238–257. Its economy is also its limitation: every application must justify what was omitted when reducing the underlying system to one static orbital per site.

The density is $n=N/L$, with $0\leq n\leq2$ and half filling at $n=1$. For a translationally invariant model, let $\epsilon_{\mathbf k}$ be the noninteracting dispersion and

$$W=\max_{\mathbf k}\epsilon_{\mathbf k} -\min_{\mathbf k}\epsilon_{\mathbf k}$$

its bandwidth. A typical hopping $t$ and $U/W$ are useful regime parameters, but neither is an observable or a phase label.

Model-card item	Declaration
Degrees of freedom	One spin-$\tfrac12$ fermionic orbital per site; local occupancies $0,1,2$
Defining data	Graph or lattice, unit cell, boundary conditions, $t_{ij}$, $U$, $\mu$ or $N$, and temperature
Principal observables	Density, double occupancy, addition energies, charge response, spectral function, charge transport, and spin correlations
Exact anchors	$U=0$ bands and $t_{ij}=0$ independent sites
Controlled reductions	Weak coupling only above any infrared instability; large-$U$ projection only while the eliminated charge sector remains separated
Scope	Static repulsive one-band model; multiorbital, nonlocal, retarded, and disordered extensions require additional terms and checks

This declaration is part of the physics. Changing the unit cell can fold a band; changing the hopping graph can remove particle–hole symmetry; and changing the ensemble or order of limits can change what a finite calculation appears to show.

Exact symmetries and their hypotheses

Charge and spin. Number-conserving hopping gives charge $U(1)$ generated by $N$. Spin-independent hopping and the density interaction give spin $SU(2)$ generated by

$$\mathbf S=\frac12\sum_i c_{i\alpha}^\dagger \boldsymbol\sigma_{\alpha\beta}c_{i\beta}.$$

These are internal symmetries of the stated model. A lattice transformation is a symmetry only if it preserves $t_{ij}$ and the boundary conditions. Time reversal is present when the hopping can be put in a time-reversal-invariant gauge, for example a real gauge with no time-reversal-breaking loop flux.

Particle–hole symmetry. Up to an additive constant, the onsite terms are

$$U\sum_i\left(n_{i\uparrow}-\frac12\right) \left(n_{i\downarrow}-\frac12\right) -\widetilde\mu\sum_i(n_i-1), \qquad \widetilde\mu=\mu-\frac U2.$$

Suppose the graph is bipartite, every nonzero hopping joins opposite sublattices, and the hopping is real. Under

$$c_{i\sigma}\longmapsto\varepsilon_i c_{i\sigma}^\dagger, \qquad \varepsilon_i= \begin{cases} +1,&i\in A,\\ -1,&i\in B, \end{cases}$$

the kinetic and shifted interaction terms are invariant, while $n_i-1$ changes sign. The symmetric point is therefore $\widetilde\mu=0$, or $\mu=U/2$ in the unshifted convention. Particle–hole symmetry pins $n=1$ there; it does not say whether the state is metallic or insulating.

Charge pseudospin and $SO(4)$. At the same bipartite point, define

$$\eta^+=\sum_i\varepsilon_i c_{i\uparrow}^\dagger c_{i\downarrow}^\dagger, \qquad \eta^z=\frac12(N-L), \qquad \eta^-=(\eta^+)^\dagger.$$

The pseudospin rotates the empty and doubly occupied charge states into one another, whereas ordinary spin rotates the singly occupied states. Its algebra is

$$[\eta^+,\eta^-]=2\eta^z, \qquad [\eta^z,\eta^\pm]=\pm\eta^\pm, \qquad [S^a,\eta^b]=0,$$

and the unshifted Hamiltonian obeys

$$[H,\eta^\pm]=\pm(U-2\mu)\eta^\pm.$$

Thus $\mu=U/2$ has the connected continuous symmetry

$$SO(4)\simeq \frac{SU(2)_{\mathrm{spin}}\times SU(2)_\eta}{\mathbb Z_2},$$

where the common central action is fermion parity Yang and Zhang 1990, pp. 759–766. Because $\eta^\pm$ change $N$ by two, only the number-preserving part of pseudospin acts within one fixed-$N$ sector.

Hopping signs and phases. A site-dependent change of orbital phase, $c_i\mapsto e^{i\phi_i}c_i$, changes individual hopping phases but not the spectrum. A uniform nearest-neighbor sign can therefore be reversed on a bipartite graph by choosing opposite phases on its two sublattices. In contrast, the phase

$$\Phi_C=\arg\prod_{(ij)\in C}t_{ij}$$

around a closed oriented loop is gauge invariant. Odd loops, same-sublattice hopping, and magnetic flux can make the sign or phase physical and can remove the simple particle–hole transformation above.

The noninteracting anchor and weak-coupling ceiling

For translationally invariant hopping $t_{ij}=t(\mathbf R_i-\mathbf R_j)$, the Fourier transform gives

$$H_0=\sum_{\mathbf k\sigma} (\epsilon_{\mathbf k}-\mu)c_{\mathbf k\sigma}^\dagger c_{\mathbf k\sigma}, \qquad \epsilon_{\mathbf k}=-\sum_{\mathbf r}t(\mathbf r)e^{i\mathbf k\cdot\mathbf r}.$$

For real nearest-neighbor hopping on a $d$-dimensional hypercubic lattice of spacing $a$,

$$\epsilon_{\mathbf k}=-2t\sum_{\ell=1}^{d}\cos(k_{\ell}a), \qquad W=4d\lvert t\rvert.$$

This one-site-unit-cell example has a half-filled Fermi surface. A noninteracting band insulator requires additional one-body structure, such as multiple bands, a sublattice potential, or complete filling. The exact $U=0$ solution is therefore an anchor, not a claim that every Hubbard lattice is metallic.

Small $U/W$ permits perturbation theory at a fixed energy scale only while enhanced infrared channels remain small. Nesting means that a wavevector $\mathbf Q$ maps an extended part of the Fermi surface onto another part with opposite dispersion. Repeated particle–hole or Cooper scattering can then generate large logarithms, so the limit $U/W\to0$ need not commute with energy or temperature tending to zero.

The nearest-neighbor square lattice makes the distinction concrete. At half filling,

$$\epsilon_{\mathbf k+\mathbf Q}=-\epsilon_{\mathbf k}, \qquad \mathbf Q=(\pi,\pi).$$

At the symmetric chemical potential, a staggered Hartree–Fock ansatz sets

$$m=\frac12(-1)^i \langle n_{i\uparrow}-n_{i\downarrow}\rangle, \qquad \Delta=Um.$$

For spin label $\sigma=\pm1$, the reduced-zone one-particle block is

$$h_{\mathbf k\sigma}= \begin{pmatrix} \epsilon_{\mathbf k}&-\sigma\Delta\\ -\sigma\Delta&-\epsilon_{\mathbf k} \end{pmatrix}, \qquad E_{\mathbf k,\pm}=\pm\sqrt{\epsilon_{\mathbf k}^2+\Delta^2}.$$

The gap comes from broken spin and translation symmetry: setting $m=0$ closes it. It is therefore a Slater mechanism, not evidence for a symmetry-preserving Mott gap Slater 1951, pp. 538–541. For the standard half-filled square-lattice model, antiferromagnetic correlations drive an insulating ground state, while the finite-temperature pseudogap and crossover structure require non-mean-field treatment Schäfer et al. 2021, Abstract and § I.

The atomic limit and charge plateau

At $t_{ij}=0$, each site is independent. The empty, singly occupied, and doubly occupied grand-canonical energies are

$$\mathcal E_0=0, \qquad \mathcal E_1=-\mu, \qquad \mathcal E_2=U-2\mu.$$

For $0<\mu<U$, the two singly occupied states have the lowest energy. At zero temperature the density is pinned to one electron per site: removing an electron costs $\mu$, and adding one costs $U-\mu$. The full energy to create a separated empty site, or holon, and a doubly occupied site, or doublon, is $U$.

The finite-temperature one-site calculation shows exactly how the plateau is rounded. With $\beta=1/T$,

$$Z_{\mathrm{site}}=1+2e^{\beta\mu}+e^{\beta(2\mu-U)}, \qquad n(\mu,T)= \frac{2e^{\beta\mu}+2e^{\beta(2\mu-U)}}{Z_{\mathrm{site}}}.$$

At the particle–hole point, $n(U/2,T)=1$ at every temperature, but the density response is

$$\chi_c(T)= \left.\frac{\partial n}{\partial\mu}\right|_{\mu=U/2} =\frac{\beta}{1+e^{\beta U/2}}.$$

It is positive at every $T>0$ and becomes exponentially small only as $T\to0$. Depending on convention, the compressibility differs from $\chi_c$ by a density normalization.

For a finite system, let $E_L(N)$ be the canonical ground-state energy. The addition and removal chemical potentials and their difference are

$$\begin{aligned} \mu_+(L,N)&=E_L(N+1)-E_L(N),\\ \mu_-(L,N)&=E_L(N)-E_L(N-1),\\ \Delta_c(L,N)&=E_L(N+1)+E_L(N-1)-2E_L(N). \end{aligned}$$

At atomic half filling, $N=L$, one finds $\mu_+=U$, $\mu_-=0$, and $\Delta_c=U$. The spin sector is completely different: all $2^L$ singly occupied spin configurations remain degenerate.

Every finite system has discrete addition energies and a staircase $n(\mu)$, so $\Delta_c(L)>0$ or a finite-size plateau is not yet an insulating phase. The thermodynamic charge gap is

$$\Delta_c= \lim_{\substack{L\to\infty\\N/L=1}} \Delta_c(L,N),$$

with a stated size sequence and boundary condition. For a grand-canonical response, take the thermodynamic limit before interpreting $T\to0$; otherwise a shell gap can masquerade as incompressibility.

Large-U degrees of freedom and their control

Let $P$ project onto states without doublons. When the eliminated charge sector is separated from $P$ by an energy of order $U$ and hopping is small compared with that separation, virtual hopping produces, to second order, the bond Hamiltonian

$$H_J= \sum_{\langle ij\rangle} \frac{4\lvert t_{ij}\rvert^2}{U} \left(\mathbf S_i\cdot\mathbf S_j-\frac14n_in_j\right),$$

acting inside $P$. Each bond appears once. The positive exchange lowers a singlet relative to a triplet; the density term makes the correction vanish on an empty bond and becomes a constant at exactly one particle per site.

The exact two-site spectrum supplies a compact calibration. For one real bond at two-electron filling,

$$E_t-E_s= \frac{\sqrt{U^2+16t^2}-U}{2} =\frac{4t^2}{U}-\frac{16t^4}{U^3} +O(t^6/U^5).$$

This checks the antiferromagnetic sign and factor of four. The bond-singlet derivation displays the exact matrix, and the full superexchange and t–J construction performs the canonical transformation MacDonald, Girvin, and Yoshioka 1988, pp. 9754–9755. That page also explains why observables must be transformed along with the Hamiltonian.

On a generic lattice, the omitted effective terms can begin at $O(t^3/U^2)$ through triangular loops; for real nearest-neighbor hopping on a bipartite lattice, the next exchange and ring corrections begin at $O(t^4/U^3)$. With holes, projected hopping of order $t$ becomes active, while exchange and three-site motion are both of order $t^2/U$. Three-site terms therefore cannot be discarded by power counting alone.

Leaving the $n=1$ plateau removes that commensurate gap from the low-energy carrier sector; absent a different ordering, commensurability, or localization mechanism, the doped homogeneous state is compressible. It need not lose the high-energy separation to states with an additional doublon. The $t/U$ reduction can therefore remain controlled even though the low-energy theory now contains mobile holes. Control is lost when the eliminated sector is no longer energetically separated, not merely when $n\ne1$.

The chapter reduction map keeps this projected low-energy theory distinct from the charge spectrum of the full Hubbard model. It is the right visual summary; a second leaf-page diagram would duplicate that relationship.

What a Mott diagnosis must establish

The word insulator and the word Mott answer different questions. The first concerns charge response; the second attributes the insulating behavior to interactions rather than to band filling, conventional order, charge transfer, or localization.

Establish the insulating statement. At a declared commensurate filling, extrapolate the canonical addition gap, density response, and charge transport or stiffness with system size and temperature. A positive thermodynamic $\Delta_c$ measures charged addition and removal. A single-particle spectral gap measures electron insertion and removal. An optical onset is a neutral fixed-$N$ excitation selected by current matrix elements and can differ because of excitons or selection rules. Agreement is informative, but the three gaps are not definitions of one another.

Then discriminate the mechanism. Test whether the gap survives restoration of conventional broken symmetries; compare adiabatic paths to band, Slater, charge-transfer, and localized limits; and examine interaction dependence, double occupancy, local moments, and spectral redistribution. Interaction-scale transfer of weight is strong supporting evidence in a strongly correlated regime, but it is not a universal definition: the weak-coupling one-dimensional Mott gap is exponentially small.

Three examples show why dimension and order of limits belong in the claim:

• For the uniform nearest-neighbor chain with $t>0$ and lattice spacing set to one, $\epsilon_k=-2t\cos k$. At $n=1$, $T=0$, and in the thermodynamic limit, the exact solution has a positive charge gap for every $U>0$ Lieb and Wu 1968, pp. 1447–1448, with erratum. In the normalization above,

  $$\Delta_c(U)=\frac{16t^2}{U} \int_1^\infty \frac{\sqrt{y^2-1}\,\mathrm dy}{\sinh(2\pi t y/U)} \sim\frac{8}{\pi}\sqrt{Ut}\,e^{-2\pi t/U}$$

  as $U/t\to0^+$. Thus $U_c=0$ is compatible with a gap that is extraordinarily hard to resolve numerically. The spin sector remains gapless, and its long-distance correlations decay rather than approach a nonzero Néel order parameter Schulz 1990, p. 2831 and discussion.

• In one and two dimensions with short-range hopping and continuous spin symmetry, magnetic long-range order is forbidden at every $T>0$ in the thermodynamic limit Ghosh 1971, pp. 1584–1586. This does not forbid two-dimensional order at $T=0$, a rapidly growing correlation length, or a finite-temperature pseudogap caused by strong short-range antiferromagnetic fluctuations. A finite-$T$ ordered Hartree–Fock solution in two dimensions is therefore not a literal phase of the exact model.

• Paramagnetic single-site DMFT, controlled as coordination tends to infinity with scaled hopping, can have metallic and insulating solutions between two spinodals and a first-order line ending at a finite-temperature critical point Georges et al. 1996, § VII.D.1, pp. 65–66. This is a symmetry-constrained infinite-coordination result, not a theorem for every lattice; allowing antiferromagnetism can preempt or conceal the paramagnetic transition.

There is consequently no dimension-independent critical value of $U/W$. Before attaching a mechanism label, use the chapter’s correlated-electron validity table, then the dedicated spectral-weight analysis and insulator diagnostic matrix. The broader evidence hierarchy is reviewed by Imada, Fujimori, and Tokura 1998, §§ II–III, pp. 1047–1106.

Common pitfalls

A large interaction is not a phase diagnosis. The ratio $U/W$ organizes approximations, but it does not measure a gap, incompressibility, or localization. State the observable and the competing mechanisms.

Half filling is not always $\mu=U/2$. That equality follows from the stated bipartite particle–hole transformation. Same-sublattice hopping, a physical loop phase, site inequivalence, or extra orbitals generally shifts the chemical potential at which $n=1$.

A finite cluster always has a level spacing. Extrapolate addition energies, response, and transport across a controlled size sequence. A finite-$L$ plateau or vanishing derivative between steps is not by itself a thermodynamic gap.

A charge gap is not a spin gap. The atomic limit has $\Delta_c=U$ and exactly degenerate spins. Exchange, dimension, and fluctuations determine the spin sector at much lower energies.

A sign convention is not a flux. A sublattice gauge transformation can reverse a bipartite nearest-neighbor hopping sign, but it cannot remove a gauge-invariant loop phase. Report the loop structure before attributing physics to the sign of $t$.

Exercises

For a one-dimensional nearest-neighbor chain, Fourier-transform the hopping and find its bandwidth.

Solution

With $c_{j\sigma}=L^{-1/2}\sum_k e^{ikja}c_{k\sigma}$ and hopping $-t\sum_j(c_{j\sigma}^\dagger c_{j+1,\sigma}+\mathrm{h.c.})$, orthogonality of the Fourier modes gives

$$\epsilon_k=-t(e^{ika}+e^{-ika})=-2t\cos(ka).$$

Its maximum and minimum are $2\lvert t\rvert$ and $-2\lvert t\rvert$, so $W=4\lvert t\rvert$.

Derive the atomic partition function and the density response at $\mu=U/2$. Why does particle–hole pinning of $n$ not imply zero finite-temperature response?

Solution

The empty state contributes $1$, the two singly occupied states contribute $2e^{\beta\mu}$, and the doublon contributes $e^{\beta(2\mu-U)}$. Hence

$$Z_{\mathrm{site}}=1+2e^{\beta\mu}+e^{\beta(2\mu-U)}.$$

Differentiating $\ln Z_{\mathrm{site}}$ gives the displayed $n(\mu,T)$. At $\mu=U/2$, particle–hole symmetry makes empty and doubly occupied probabilities equal, so $n=1$. Their probabilities nevertheless change under an infinitesimal shift of $\mu$; direct differentiation gives

$$\chi_c(T)=\frac{\beta}{1+e^{\beta U/2}}>0$$

for every finite $T$. Symmetry fixes the value at the central point, not the slope through it.

Expand the exact Hubbard-dimer singlet–triplet splitting for large $U/t$ and identify the first correction to $J=4t^2/U$.

Solution

Using $\sqrt{1+x}=1+x/2-x^2/8+O(x^3)$ with $x=16t^2/U^2$,

$$\begin{aligned} E_t-E_s &=\frac U2\left(\sqrt{1+\frac{16t^2}{U^2}}-1\right)\\ &=\frac{4t^2}{U}-\frac{16t^4}{U^3}+O(t^6/U^5). \end{aligned}$$

The positive leading term is antiferromagnetic. The correction is relative order $(t/U)^2$ on the isolated bond.

Apply a site-dependent gauge transformation to a real hopping pattern. Show that the product of hopping phases around a closed loop is invariant, and explain why a uniform hopping sign is removable on a bipartite graph but not on a triangle.

Solution

Under $c_i\mapsto e^{i\phi_i}c_i$, the hopping changes as $t_{ij}\mapsto t_{ij}e^{i(\phi_j-\phi_i)}$. Multiplying around a loop makes every site phase cancel, so $\arg\prod_C t_{ij}$ is invariant. On a bipartite graph, choosing a phase difference $\pi$ between the two sublattices reverses every nearest-neighbor sign. Around a triangle, reversing all three real bonds changes the loop product by a minus sign; no site-phase choice can do that.

A half-filled calculation finds a single-particle gap only when antiferromagnetic order is allowed; the gap closes in a symmetry-restored calculation at the same parameters. What is established, and what remains open?

Solution

The comparison supports an order-induced, Slater-like contribution to the gap and does not establish a paramagnetic Mott insulator. It does not show that interactions are irrelevant: $U$ generates the ordered mean field and may also cause self-energy renormalization or spectral redistribution. One must still control size, temperature, symmetry restoration, charge response, transport, and competing mechanisms.

What you can now do

You can state the one-band Hubbard model without hiding its lattice, filling, gauge, or ensemble assumptions; derive its generic and bipartite symmetries; solve its band and atomic anchors; identify where weak-coupling perturbation theory fails in the infrared; and test the leading large-$U$ exchange scale without confusing a projection with the parent model’s charge spectrum.

Continue by function: use effective lattice Hamiltonians to justify $t_{ij}$ and $U$; superexchange and the t–J projection for the full canonical transformation and transformed observables; spectral-weight transfer for charge-sector dynamics; insulator distinctions for mechanism attribution; and DMFT mapping for a controlled local-solver route. Throughout, $U/W$ is a regime parameter, not a phase observable.

References

• Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg, “Dynamical Mean-Field Theory of Strongly Correlated Fermion Systems and the Limit of Infinite Dimensions,” Reviews of Modern Physics 68 (1996) 13–125, doi:10.1103/RevModPhys.68.13.
• Dipan K. Ghosh, “Nonexistence of Magnetic Ordering in the One- and Two-Dimensional Hubbard Model,” Physical Review Letters 27 (1971) 1584–1586, doi:10.1103/PhysRevLett.27.1584.
• John Hubbard, “Electron Correlations in Narrow Energy Bands,” Proceedings of the Royal Society A 276 (1963) 238–257, doi:10.1098/rspa.1963.0204.
• Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura, “Metal–Insulator Transitions,” Reviews of Modern Physics 70 (1998) 1039–1263, doi:10.1103/RevModPhys.70.1039.
• Elliott H. Lieb and F. Y. Wu, “Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension,” Physical Review Letters 20 (1968) 1445–1448, with erratum Physical Review Letters 21 (1968) 192, doi:10.1103/PhysRevLett.20.1445, erratum: doi:10.1103/PhysRevLett.21.192.2.
• Allan H. MacDonald, Steven M. Girvin, and Daijiro Yoshioka, “$t/U$ Expansion for the Hubbard Model,” Physical Review B 37 (1988) 9753–9756, doi:10.1103/PhysRevB.37.9753.
• Thomas Schäfer et al., “Tracking the Footprints of Spin Fluctuations: A MultiMethod, MultiMessenger Study of the Two-Dimensional Hubbard Model,” Physical Review X 11 (2021) 011058, doi:10.1103/PhysRevX.11.011058.
• H. J. Schulz, “Correlation Exponents and the Metal–Insulator Transition in the One-Dimensional Hubbard Model,” Physical Review Letters 64 (1990) 2831–2834, doi:10.1103/PhysRevLett.64.2831.
• John C. Slater, “Magnetic Effects and the Hartree-Fock Equation,” Physical Review 82 (1951) 538–541, doi:10.1103/PhysRev.82.538.
• Chen Ning Yang and Shou-Cheng Zhang, “SO4 Symmetry in a Hubbard Model,” Modern Physics Letters B 4 (1990) 759–766, doi:10.1142/S0217984990000933.