Lattice Models Overview

Quantum lattice models replace unrestricted spatial motion by a specified set of local degrees of freedom and couplings. Their Hamiltonians are often short enough to fit on one line, yet they describe band motion, magnetism, Mott localization, superfluidity, quantum criticality, topological phases, impurity screening, and controlled numerical benchmarks.

Their power comes from disciplined simplification. A lattice model declares which local states are retained, which sites or links interact, which symmetries are exact, and which energy scales have been integrated out. Those choices can represent a physical crystal, an optical-lattice experiment, a numerical regulator, a network of qubits, or an effective low-energy theory. The same algebraic formula can mean different physics unless this modeling data is stated.

Helpful background. Occupation representation explains mode states and signs; real-space representation distinguishes physical orbital projection from a numerical grid.

Local states and couplings define the model

Consider two models on the same chain. A spin-$1/2$ chain has two states at each site and exchange couplings between neighboring spins. A spinful Hubbard chain has four states per site—empty, up, down and doubly occupied—and hopping changes the local charge. Drawing the same line of sites does not make their Hilbert spaces or dynamics equivalent.

To read a Hamiltonian, first identify the local states and any constraints, then the operators and coefficient conventions. Specify the graph, boundary conditions, filling or spin sector, and the observables being studied. If the model approximates a microscopic system, the retained energy window and omitted processes matter too.

For example, “repulsive Hubbard chain at half filling” still needs a chain length, hopping and interaction scales, spin content, and boundary choice before it defines a finite spectrum. The chapter guide connects the individual model families; this page explains their shared ingredients.

Lattice, Graph, and Mode Language

A regular lattice carries geometric information such as spacing, dimension, primitive vectors, and point-group symmetry. A graph retains vertices and edges without requiring an embedding in Euclidean space. A mode model may use labels that are localized orbitals, circuit nodes, cavity modes, or synthetic dimensions.

The same graph Hamiltonian can be embedded in different physical geometries. Conversely, two models on the same geometric lattice can have different local Hilbert spaces or constraints and therefore describe different physics.

When distance matters, define a metric such as shortest-path distance

$$d(i,j) = \text{minimum number of edges from }i\text{ to }j.$$

Statements about finite range, decay, and locality require such a notion of separation. The complete interaction graph has graph distance one between any two sites; an all-to-all coupling is long-ranged relative to an underlying spatial metric, while its growing coordination prevents the usual bounded-degree estimates even in the graph metric.

Sites, Links, and Plaquettes

Degrees of freedom need not live only on sites.

• Site variables include spins, orbitals, bosonic modes, fermionic modes, and qudits.
• Link variables include hopping phases, quantum rotors, gauge fields, and dimer occupations.
• Plaquette or cell operators measure flux, ring exchange, or multi-site constraints.
• Global modes can couple to all sites, as in cavity-mediated or collective-spin models.

An interaction term can also be supported on a link or plaquette even when the elementary degrees of freedom live on sites.

A lattice model is assembled from local data at several supports. A site carries $\mathcal H_i$, a link can carry hopping $t_{ij}$ or exchange $J_{ij}$, and a plaquette can carry a multi-site term $K_\square$.

Local Hilbert Spaces

For distinguishable site degrees of freedom with local spaces $\mathcal H_i$, the global space is

$$\mathcal H = \bigotimes_{i=1}^{L} \mathcal H_i.$$

If every site has dimension $q$, then

$$\dim\mathcal H = q^L.$$

This exponential growth is present even when the Hamiltonian contains only order-$L$ local terms. Scaling of Hilbert Space develops the counting and computational consequences.

The tensor-product form needs qualifications for identical particles. Bosonic and fermionic lattice modes live naturally in Fock space. After choosing an ordering, finite fermionic Fock space is isomorphic as a vector space to a product of local occupation spaces, but the operator algebra retains fermion-parity signs.

Common local spaces

Local degree of freedom	States	Dimension
Spin $s$	$m=-s,-s+1,\ldots,s$	$2s+1$
One spinless fermion mode	Empty, occupied	$2$
One spinful fermion orbital	Empty, up, down, doubly occupied	$4$
One boson mode	$n=0,1,2,\ldots$	Infinite
Numerically truncated boson	$0\le n\le n_{\max}$	$n_{\max}+1$
Hard-core boson	$n=0,1$	$2$

Spin operators here are dimensionless: for spin one half, $\mathbf s=\boldsymbol\sigma/2$. A qudit is simply a finite local space of dimension $q$; it need not represent a literal angular momentum.

A numerical boson cutoff changes the commutator: $[b,b^\dagger]=I-(n_{\max}+1)|n_{\max}\rangle\langle n_{\max}|$. The weighted top-state probability and the target observables must converge as the cutoff increases. Hard-core occupation is instead a stated physical or effective constraint. Fermionic operators also require a consistent global mode ordering; equal local dimensions do not erase their signs.

Constrained Local Spaces

Some models retain only a subspace of an underlying local Fock space. Examples include:

• no-double-occupancy states in a $t$–$J$ description;
• one dimer touching each vertex in a quantum dimer model;
• fixed electric-flux or Gauss-law sectors in lattice gauge models;
• Rydberg blockade constraints;
• fixed total spin in a projected atomic multiplet.

If $P$ projects onto the allowed space, a projected operator is

$$\widetilde O = POP.$$

In general,

$$\widetilde{AB} \ne \widetilde A\,\widetilde B,$$

because the intermediate state in $AB$ may leave the retained subspace. Constraints therefore alter operator algebra as well as state counting. A constraint connecting several sites, such as a dimer covering or Gauss law, also means that the allowed global space need not factor into independent local spaces.

Operator Support

An operator $O_X$ is supported on a set of sites $X$ if it acts as the identity outside $X$:

$$O_X = O_X^{(X)} \otimes I_{X^{\mathrm c}}.$$

For fermionic systems, strictly local observable algebras are most naturally built from parity-even operators. Odd fermion operators on disjoint regions anticommute rather than commute, even though physical even observables can satisfy the expected locality structure.

Support identifies where an operation acts. Range identifies the diameter of that support in the graph metric.

Local Hamiltonians

A general lattice Hamiltonian can be organized as

$$H = \sum_{X\subseteq\Lambda} h_X,$$

where $h_X$ is supported on a finite subset $X$ of the lattice $\Lambda$.

For a finite-range Hamiltonian, there is a range $R$ such that

$$h_X=0 \qquad \text{when} \qquad \operatorname{diam}(X)>R.$$

For a short-range Hamiltonian, terms may extend farther but their norms decay sufficiently rapidly with diameter or separation.

Local does not mean noninteracting, weak, classical, or exactly solvable. It is a statement about operator support and geometry.

Geometric Locality Versus k-Locality

In quantum information, a Hamiltonian is called $k$-local when every term acts on at most $k$ subsystems. A two-site term connecting opposite ends of a large graph is $2$-local but not geometrically short-ranged.

Thus

$$|X|\le k$$

does not imply

$$\operatorname{diam}(X)\le R.$$

The distinction matters for Lieb–Robinson bounds, tensor networks, finite-size effects, and physical realizability.

Extensivity and Stability

For bounded local terms on a graph of bounded coordination, a finite-range Hamiltonian typically has order-$L$ terms. If

$$\lVert h_X\rVert \le J$$

and each site participates in at most a bounded number of terms, then

$$\lVert H\rVert \le \sum_X \lVert h_X\rVert = O(L).$$

This upper bound supports an extensive energy scale. It does not prove a particular thermodynamic phase or guarantee that every state has energy proportional to $L$.

These operator-norm bounds do not apply directly to unrestricted bosonic onsite terms. For such systems one needs suitable domain, stability and particle-density conditions, rather than assuming a finite norm. Long-range interactions require more care. For pair couplings $J_{ij}$, a useful stability condition is a uniform bound of the form

$$\sup_i \sum_{j\ne i} |J_{ij}| < \infty$$

along the chosen sequence of larger systems. All-to-all models often need size-dependent normalization to retain a finite energy per site.

Propagation under local interactions

For bounded spin interactions of finite range on a suitable bounded-coordination lattice, a Lieb–Robinson bound limits the commutator of initially separated local observables outside an effective propagation cone. The commutator is small, not generally exactly zero: this is not relativistic microcausality. Long-range interactions and unbounded bosonic modes need different assumptions and can change the shape of the bound. Lieb and Robinson’s original theorem concerns quantum spin systems.

Onsite Terms

An onsite Hamiltonian has the form

$$H_{\mathrm{site}} = \sum_i h_i.$$

Examples include:

$$\sum_i\epsilon_i n_i, \qquad -\sum_i\mathbf h_i\boldsymbol\cdot\mathbf s_i, \qquad \frac U2\sum_i n_i(n_i-1).$$

Onsite terms can set local energy splittings, chemical potentials, fields, anisotropies, constraints, or interactions between different modes at one site.

Calling a term onsite does not make it one-body. The bosonic operator $n_i(n_i-1)$ and the fermionic operator $n_{i\uparrow}n_{i\downarrow}$ are two-body interactions localized at one site.

Hopping Terms

A number-conserving hopping Hamiltonian is

$$H_t = -\sum_{i,j} \sum_{\alpha,\beta} t_{i\alpha,j\beta} c_{i\alpha}^\dagger c_{j\beta}.$$

Hermiticity requires

$$t_{i\alpha,j\beta} = t_{j\beta,i\alpha}^*.$$

The minus sign is conventional and can be absorbed into $t$ if definitions are adjusted consistently. On a graph, nonzero $t_{ij}$ defines directed or undirected transfer edges. Complex phases can encode magnetic flux or synthetic gauge fields.

Hopping changes local occupations while preserving total number:

$$\widehat N = \sum_{i,\alpha} n_{i\alpha}, \qquad [H_t,\widehat N]=0.$$

Quadratic Does Not Mean Single Particle

For ordinary canonical bosonic or fermionic modes without additional constraints, a number-conserving quadratic hopping Hamiltonian acts on every particle-number sector, and its spectrum is assembled from independent one-particle energies. It can describe arbitrarily many particles. The same-looking bilinear in constrained hard-core or projected operators need not be a free-particle problem: those operators no longer obey the canonical algebra.

Diagonalizing the one-body matrix gives

$$H_t = \sum_\lambda \varepsilon_\lambda d_\lambda^\dagger d_\lambda.$$

Particle statistics then control how these one-particle energies are occupied. The dedicated tight-binding page develops translation-invariant dispersions, Fourier diagonalization, and the connection to bands.

Density Interactions

A general density interaction is

$$H_V = \frac12 \sum_{i,j} V_{ij} {:}n_i n_j{:}_0.$$

For separated sites, a common nearest-neighbor form is

$$H_V = V \sum_{\langle i,j\rangle} n_i n_j.$$

Density interactions are diagonal in the site-occupation basis. They do not directly move particles, although they change the energies and dynamics of configurations through their competition with hopping.

Long-range $V_{ij}$ can represent Coulomb, dipolar, cavity-mediated, or phenomenological interactions. Boundary and summation conventions must accompany slowly decaying kernels.

Exchange Interactions

A general bilinear spin interaction is

$$H_J = \sum_{\langle i,j\rangle} \sum_{a,b} J_{ij}^{ab} s_i^a s_j^b.$$

Special choices include:

• Ising exchange with only $J^{zz}$;
• XXZ exchange with $J^{xx}=J^{yy}\ne J^{zz}$;
• isotropic Heisenberg exchange with $J^{ab}=J\delta_{ab}$;
• antisymmetric Dzyaloshinskii–Moriya exchange;
• bond-dependent compass or Kitaev-type exchange.

The sign convention must be declared. In the common Heisenberg convention $H=J\sum\mathbf s_i\cdot\mathbf s_j$, positive $J$ favors smaller total spin on a bond and negative $J$ favors larger total spin. On a lattice these local preferences interact with geometry and quantum fluctuations.

Pairing and Conversion Terms

Not every lattice Hamiltonian conserves particle number. Pairing terms have the form

$$H_\Delta = \sum_{i,j} \left( \Delta_{ij}c_i^\dagger c_j^\dagger + \Delta_{ij}^*c_j c_i \right).$$

For ordinary fermions, only the antisymmetric part of $\Delta_{ij}$ contributes; a convention with a full antisymmetric coefficient often includes an explicit factor $1/2$. Every displayed term changes particle number by two and preserves fermion parity exactly. Pair hopping, by contrast, annihilates a pair elsewhere and conserves total number. Bosonic models can also contain linear sources or species-conversion terms whose conserved weighted charges must be checked.

The conserved quantities must be derived from the full Hamiltonian rather than inferred from a familiar model name.

Multi-Site and Ring Terms

Effective Hamiltonians can contain terms supported on three or more sites. A schematic four-site ring exchange is

$$H_K = K \sum_\square \left( P_\square + P_\square^{-1} \right),$$

where $P_\square$ cyclically permutes states around a plaquette.

Such terms can arise from higher-order virtual processes. They are local if each plaquette has bounded size, even though they are not pairwise.

Discarding multi-site terms is an approximation, not a consequence of locality.

Spin and Particle Models Are Related but Distinct

A spin-$1/2$ site and a hard-core boson site both have two local states. One can identify

$$b_i^\dagger \longleftrightarrow s_i^+, \qquad n_i \longleftrightarrow s_i^z+\frac12.$$

For hard-core bosons on different sites, these local operator algebras match a spin model. The mapping translates hopping into XY exchange and density interactions into Ising exchange plus lower-rank terms.

Spinless fermions also have two local states, but mapping them to spins in one dimension requires the parity strings and boundary-sector care derived in Jordan–Wigner Transformation. Equal local dimensions do not imply identical exchange statistics.

Effective Spin Models from Particle Models

At strong coupling, virtual particle motion can generate spin exchange within a low-energy subspace. For the repulsive half-filled Hubbard model,

$$J \sim \frac{4t^2}{U}$$

at leading order under standard assumptions.

This is a controlled effective relation in the regime $t/U\ll1$, not an operator identity between the full Hubbard and Heisenberg Hilbert spaces. Charge fluctuations are projected out, and higher-order terms remain.

The derivation and its corrections are in Effective Hamiltonians in Many-Body Systems.

Symmetries of Lattice Models

Symmetries can act on internal states, spatial labels, or both. Common examples include:

• global particle-number U(1);
• fermion parity;
• spin rotations or discrete spin flips;
• lattice translations and point-group operations;
• time reversal;
• particle-hole or sublattice transformations;
• Gauss-law constraints and, separately, physical global symmetries in gauge models;
• subsystem symmetries acting on lines or planes.

A physical symmetry must preserve the Hamiltonian, boundaries and constraints. Gauge transformations generated by Gauss-law constraints are redundancies on physical states, so they should not be counted as additional physical degeneracies. A symmetry must preserve the graph and coefficients, possibly together with an allowed gauge transformation. A uniform bulk formula can lose translation or point-group symmetry on a finite open sample.

Onsite and Spatial Symmetries

A unitary onsite internal symmetry factors as

$$U(g) = \bigotimes_i u_i(g).$$

A spatial symmetry instead permutes sites and may also rotate internal orbitals:

$$U_g c_{i\alpha}U_g^\dagger = \sum_\beta [D_g]_{\beta\alpha} c_{g(i),\beta}.$$

Nonsymmorphic and magnetic space-group structures require additional translations, phases, or antiunitary actions. This overview uses only the distinction needed to read generic lattice Hamiltonians.

Boundary Conditions Are Model Data

Open boundaries omit couplings beyond an edge. Periodic boundaries identify opposite edges and add wraparound bonds. Twisted boundaries insert a net phase around a periodic direction.

These choices affect:

• exact spatial symmetries;
• allowed momentum sectors;
• edge and surface states;
• finite-size gaps and degeneracies;
• fermion-parity sectors under nonlocal mappings;
• topological flux sectors.

Boundary effects can vanish in some bulk thermodynamic observables, but that is a conclusion to establish, not permission to omit the finite-system definition.

Simple Formulas, Noncommuting Terms

Many canonical models have the structure

$$H = H_A+H_B,$$

where each part is separately simple but

$$[H_A,H_B] \ne 0.$$

In the Hubbard model, hopping is simple in momentum space and onsite repulsion is simple in real space. In the transverse-field Ising model, Ising exchange is diagonal in the $z$ basis while the transverse field is diagonal in the $x$ basis.

Noncommuting terms need not share all eigenvectors, and their competition can generate entanglement and collective behavior. Noncommutation alone does not prove frustration or a phase transition: two noncommuting positive operators can still share a zero-energy ground state. The exercise below gives an explicit example.

Representative Model Families

Model family	Local degrees of freedom	Characteristic terms	Central competition
tight binding	bosonic or fermionic orbitals	hopping, onsite energy	geometry versus delocalization
transverse-field Ising	spin $1/2$	Ising exchange, transverse field	order versus quantum fluctuations
Heisenberg and XXZ	fixed spins	exchange, anisotropy, field	alignment, frustration, dimensionality
Hubbard	spinful fermions	hopping, onsite interaction	itinerancy versus local correlation
Bose–Hubbard	bosons	hopping, onsite pair repulsion	phase coherence versus number localization
spinless fermion chain	one fermion mode per site	hopping, density interaction	transport versus charge order
t–J	constrained spinful carriers	projected hopping, exchange	doped motion versus spin background
Kondo family	local spins plus itinerant fermions	exchange, hopping	screening versus magnetic correlations
Anderson impurity	localized and bath orbitals	hybridization, onsite interaction	charge fluctuation versus local moment

The table classifies ingredients, not full phase diagrams. Dimension, filling, coupling sign, range, and symmetry can change the physics within each family.

Why Lattice Models Can Be Universal

A model is a universal low-energy description when long-distance observables become insensitive to many microscopic details after relevant parameters and symmetries are matched.

The logic is not that every microscopic difference is unimportant. Rather, coarse graining can make some operators irrelevant while retaining:

• dimensionality and locality;
• symmetry and its possible anomalies;
• conserved quantities;
• low-energy degrees of freedom;
• topology and filling constraints;
• relevant and marginal couplings.

Different materials or platforms can then share critical exponents, scaling functions, collective modes, or effective Hamiltonians without sharing atomic details.

Projection and Scale Separation

Suppose the microscopic Hilbert space splits into a low-energy subspace $P$ and its complement $Q=I-P$. A schematic effective Hamiltonian is

$$H_{\mathrm{eff}} = PHP + O\!\left( \frac{V^2}{\Delta} \right),$$

where $V$ mixes the sectors and $\Delta$ is a separation scale.

Controlled use requires more than writing the ratio. One must identify the retained states, estimate omitted terms, transform observables, and state the energy or time window of validity.

Eliminating virtual excursions into the discarded subspace can generate longer-range and multi-site terms. Plain projection $PHP$ omits those corrections; it does not generate them by itself.

Renormalization-Group View

Renormalization group methods describe coarse graining and the resulting flow of couplings. For a lattice Hamiltonian, coarse graining generally generates every operator allowed by the retained symmetries and degrees of freedom. A simple effective model is stable only when omitted operators are irrelevant, parametrically small, forbidden, or included when estimating the accuracy of the approximation.

Near a continuous transition, the correlation length can satisfy

$$\xi \gg a,$$

where $a$ is the microscopic lattice spacing. Long-distance behavior can then approach a continuum field theory even though the microscopic Hamiltonian is discrete.

Emergent rotational, Lorentz-like, or internal symmetry is possible, but it must be demonstrated in the low-energy limit. The lattice Hamiltonian does not possess that larger symmetry microscopically.

Lattice as Physical Model or Regulator

A physical lattice model can represent localized orbitals in a material or optical potential. Its spacing, coordination, and retained bands are part of the physics.

A regulator lattice approximates a continuum theory and is intended to approach

$$a\to0$$

with parameters matched appropriately. The Real-Space Representation gives this normalization and coupling dictionary.

These roles can overlap, but they should not be conflated. A fixed-spacing Hubbard model is not automatically a discretization destined for a continuum limit.

Filling and Sector Data

For particle models on $L$ sites, filling is commonly

$$\nu = \frac NL$$

or number per unit cell when several orbitals are present. Spin-resolved fillings may also be required.

The same Hamiltonian at different filling can realize qualitatively different regimes. Half filling, integer boson filling, dilute density, and a single impurity are distinct model specifications.

Canonical calculations fix conserved particle numbers. Grand-canonical calculations include

$$K = H-\mu N.$$

The chemical potential convention must be included when comparing phase boundaries or local energies.

Geometry and Frustration

Geometry affects more than coordination number. Odd cycles, non-bipartite graphs, and competing couplings can prevent every local interaction from being simultaneously minimized.

For antiferromagnetic Ising variables on a triangle, each bond favors opposite signs, but no configuration satisfies all three bonds. This is geometric frustration.

Quantum frustration also arises from noncommuting local terms even on simple geometries. The word should be tied to a specified inability to minimize compatible local tendencies, not used as a synonym for difficulty.

From finite systems to bulk phases

A finite lattice with finite local dimension has a discrete spectrum; an unbounded bosonic local space needs additional confinement and stability assumptions. Symmetry eigenstates need not have a nonzero order parameter. Useful ways to distinguish phases across a sequence of larger systems include:

• correlation functions approaching long-range order;
• gaps that remain open or close with size;
• quasi-degenerate low-energy sectors;
• scaling collapse near a critical point;
• stiffness or topological response;
• entanglement and boundary signatures.

The Thermodynamic Limit explains why finite-size crossover and true bulk singularity must be distinguished.

Symmetry Blocks and Basis Ordering

Before diagonalization, identify commuting charges such as

$$[H,Q_a]=0.$$

Simultaneous sectors require mutually commuting charges. A conserved crystal momentum and reflection, for example, cannot generally both be fixed away from momenta unchanged by reflection. Suitable sectors reduce memory and label the spectrum. Common sectors include total particle number, spin projection, crystal momentum, reflection parity, and fermion parity.

For fermions, declare the global mode ordering used to construct occupation basis states. A change of ordering is a unitary convention change when implemented consistently, but partial sign changes produce a different matrix.

Exercises

Extensivity bound for a bond Hamiltonian

Let a graph have $L$ sites and maximum degree $z$. Suppose

$$H = \sum_{(i,j)\in E} h_{ij}, \qquad \lVert h_{ij}\rVert\le J.$$

Show that $\lVert H\rVert\le JzL/2$.

Solution

The triangle inequality gives

$$\lVert H\rVert \le \sum_{(i,j)\in E} \lVert h_{ij}\rVert \le J|E|.$$

The handshaking identity is

$$2|E| = \sum_i z_i.$$

Since every degree obeys $z_i\le z$,

$$|E| \le \frac{zL}{2}.$$

Combining the bounds yields

$$\lVert H\rVert \le \frac{JzL}{2}.$$

Bounded coordination and bounded bond strength therefore produce an operator-norm upper bound linear in system size.

Hard-core bosons and spin one-half

On each site identify

$$\lvert0\rangle \longleftrightarrow \lvert\downarrow\rangle, \qquad \lvert1\rangle \longleftrightarrow \lvert\uparrow\rangle.$$

Show that $b_i^\dagger\leftrightarrow s_i^+$ and $n_i\leftrightarrow s_i^z+1/2$. Translate nearest-neighbor boson hopping into spin operators.

Solution

On the two-state local space,

$$b_i^\dagger\lvert0\rangle_i = \lvert1\rangle_i, \qquad b_i^\dagger\lvert1\rangle_i = 0,$$

which is the action of $s_i^+$ under the stated identification. Also,

$$s_i^z+\frac12$$

has eigenvalues $0$ on $\lvert\downarrow\rangle$ and $1$ on $\lvert\uparrow\rangle$, so it equals $n_i$.

The hopping term becomes

$$\begin{aligned} -t\sum_{\langle i,j\rangle} (b_i^\dagger b_j+b_j^\dagger b_i) &= -t\sum_{\langle i,j\rangle} (s_i^+s_j^-+s_j^+s_i^-) \\ &= -2t\sum_{\langle i,j\rangle} (s_i^x s_j^x+s_i^y s_j^y). \end{aligned}$$

This local mapping applies to hard-core bosons. Mapping spinless fermions requires additional parity strings in one dimension.

Competing terms in a two-spin Ising dimer

For

$$H = -J\sigma_1^z\sigma_2^z - h(\sigma_1^x+\sigma_2^x),$$

identify the ground states in the limits $h=0$ with $J>0$, and $J=0$ with $h>0$. Explain why the two terms compete.

Solution

At $h=0$ and $J>0$, the exchange energy is minimized by aligned $z$ spins:

$$\lvert\uparrow\uparrow\rangle, \qquad \lvert\downarrow\downarrow\rangle.$$

They are degenerate with energy $-J$.

At $J=0$ and $h>0$, each site minimizes $-h\sigma^x$ in the $+1$ eigenstate of $\sigma^x$. The ground state is

$$\lvert+\rangle_x \otimes \lvert+\rangle_x.$$

The operators $\sigma^z$ and $\sigma^x$ do not commute. A state sharp in the exchange-preferred $z$ basis is not sharp in the field-preferred $x$ basis, so the two terms cannot generally be minimized simultaneously.

Noncommuting terms with a common minimum

In the three-state space $|0\rangle,|1\rangle,|2\rangle$, set $A=|1\rangle\langle1|$ and $B=|v\rangle\langle v|$, where $|v\rangle=(|1\rangle+|2\rangle)/\sqrt2$. Are $A$ and $B$ commuting? Can one state minimize both?

Solution

Their commutator is $[A,B]=( |1\rangle\langle2|-|2\rangle\langle1|)/2$, which is nonzero. Yet both are positive projectors and both annihilate $|0\rangle$. This state minimizes each term and their sum. Frustration concerns the impossibility of attaining all local minima together; noncommutation by itself is insufficient.

References

• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press (2010).
• A. Auerbach, Interacting Electrons and Quantum Magnetism, Springer (1994).
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015).
• P. Fazekas, Lecture Notes on Electron Correlation and Magnetism, World Scientific (1999).
• E. Fradkin, Field Theories of Condensed Matter Physics, 2nd ed., Cambridge University Press (2013).
• T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press (2004).
• M. Lewenstein, A. Sanpera, and V. Ahufinger, Ultracold Atoms in Optical Lattices, Oxford University Press (2012).
• E. H. Lieb and D. W. Robinson, “The finite group velocity of quantum spin systems,” Communications in Mathematical Physics 28, 251–257 (1972), doi:10.1007/BF01645779.
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press (2011).
• X.-G. Wen, Quantum Field Theory of Many-Body Systems, Oxford University Press (2004).