Common Spin Hamiltonians

A spin Hamiltonian is specified by more than its coupling symbols. Operator normalization, spin length, bond orientation, graph, boundary conditions, and the sign placed in front of each term all affect numerical spectra and sometimes even the verbal meaning of “ferromagnetic.”

Use this sheet to compare exchange, anisotropy, magnetic fields and bond-dependent interactions. A coefficient multiplying dimensionless spins has energy units; a magnetic field in tesla must first be multiplied by the appropriate magnetic moment.

Required background. Spin introduces the local algebra, and lattice models explains sites and bonds. For choosing a spin description of a material, see Magnetic Moments.

Local Algebra and Normalization

This sheet uses dimensionless spin operators $\mathbf s_i$ unless stated otherwise:

$$\begin{aligned} [s_i^a,s_j^b] &= i\delta_{ij}\epsilon_{abc}s_i^c, \\ \mathbf s_i^2 &= s_i(s_i+1), \\ \dim\mathcal H_i &= 2s_i+1. \end{aligned}$$

Physical angular momentum is

$$\mathbf S_i = \hbar\mathbf s_i.$$

For spin $1/2$,

$$\mathbf s_i = \frac12\boldsymbol\sigma_i,$$

so

$$\begin{aligned} \mathbf s_i\cdot\mathbf s_j &= \frac14 \boldsymbol\sigma_i\cdot\boldsymbol\sigma_j, \\ s_i^a s_j^a &= \frac14\sigma_i^a\sigma_j^a, \\ s_i^a &= \frac12\sigma_i^a. \end{aligned}$$

Consequently,

$$J_s\, \mathbf s_i\cdot\mathbf s_j = J_\sigma\, \boldsymbol\sigma_i\cdot\boldsymbol\sigma_j$$

describes the same operator only when

$$J_\sigma = \frac{J_s}{4}.$$

Likewise, $-h_s s_i^a=-h_\sigma\sigma_i^a$ requires $h_\sigma=h_s/2$. Ratios such as $h/J$ therefore depend on whether both coefficients have been translated consistently.

Bond Counting

For an undirected graph with edge set $E$,

$$\sum_{\langle i,j\rangle}$$

should mean that each edge $\{i,j\}\in E$ appears once. The same interaction can be written as

$$\sum_{\langle i,j\rangle} H_{ij} = \frac12 \sum_{i\ne j} H_{ij}$$

only when the ordered-pair form satisfies $H_{ij}=H_{ji}$ and includes both orientations.

For a chain of $L$ sites:

• an open nearest-neighbor sum has $L-1$ bonds;
• a standard periodic sum has $L$ bonds for generic $L$;
• very small rings require an explicit edge list to avoid accidental duplicate bonds.

The safest numerical implementation constructs the bond list first and assembles the Hamiltonian second.

General Bilinear Spin Hamiltonian

A broad Hermitian family is

$$\begin{aligned} H &= H_{\mathrm{ex}} + H_h + H_{\mathrm{on}} + H_{\mathrm{multi}}, \\ H_{\mathrm{ex}} &= \frac12 \sum_{i\ne j} \sum_{a,b} J_{ij}^{ab} s_i^a s_j^b, \\ H_h &= -\sum_i \mathbf h_i\cdot\mathbf s_i, \\ H_{\mathrm{on}} &= \frac12 \sum_i\sum_{a,b} A_i^{ab} \{s_i^a,s_i^b\}. \end{aligned}$$

Every distinct-site product $s_i^a s_j^b$ is Hermitian, so any real coefficients give a Hermitian operator. To avoid specifying the same bond twice independently, choose the nonredundant convention

$$J_{ji}^{ba} = J_{ij}^{ab}.$$

The part antisymmetric under the combined interchange $(i,a)\leftrightarrow(j,b)$ cancels from the ordered sum; this is a redundancy, not a necessary Hermiticity condition on the original real coefficients. The onsite matrix $A_i$ may be taken real and symmetric. $H_{\mathrm{multi}}$ collects genuine three-spin, four-spin, ring-exchange, or other non-bilinear terms.

For one oriented bond, decompose the exchange tensor as

$$\mathbf J_{ij} = J_{ij}^{\mathrm{iso}}\mathbf 1 + \boldsymbol\Gamma_{ij} + \mathbf A_{ij},$$

where

$$\begin{aligned} J_{ij}^{\mathrm{iso}} &= \frac13\operatorname{tr}\mathbf J_{ij}, \\ \boldsymbol\Gamma_{ij} &= \frac12 (\mathbf J_{ij}+\mathbf J_{ij}^{\mathsf T}) - J_{ij}^{\mathrm{iso}}\mathbf 1, \\ \mathbf A_{ij} &= \frac12 (\mathbf J_{ij}-\mathbf J_{ij}^{\mathsf T}). \end{aligned}$$

The three pieces are isotropic exchange, symmetric traceless anisotropy, and antisymmetric exchange. Writing

$$A_{ij}^{ab} = \sum_c \epsilon_{cab}D_{ij}^c$$

gives

$$\sum_{a,b} A_{ij}^{ab}s_i^a s_j^b = \mathbf D_{ij}\cdot (\mathbf s_i\times\mathbf s_j).$$

Reversing the declared bond orientation sends $\mathbf D_{ij}\to-\mathbf D_{ij}$. A Dzyaloshinskii–Moriya vector therefore has no meaning without an orientation convention.

Quick Family Map

Family	Representative form	Immediate diagnostic
Heisenberg	$\sum_{\langle i,j\rangle}J_{ij}\mathbf s_i\cdot\mathbf s_j$	isotropic spin rotations
XXZ	$J\sum_j(s_j^xs_{j+1}^x+s_j^ys_{j+1}^y+\Delta s_j^zs_{j+1}^z)$	conserves total $s^z$
XYZ	$\sum_{j,a}J_a s_j^a s_{j+1}^a$	generic discrete spin rotations only
Ising	$-\sum_{\langle i,j\rangle}K_{ij}s_i^zs_j^z$	diagonal in the $s^z$ basis
transverse-field Ising	$-J\sum_j\sigma_j^z\sigma_{j+1}^z-h\sum_j\sigma_j^x$	global $\mathbb Z_2$ parity
XY	$\sum_j(J_xs_j^xs_{j+1}^x+J_ys_j^ys_{j+1}^y)-h\sum_js_j^z$	$J_x=J_y$ restores $U(1)$
Dzyaloshinskii–Moriya	$\sum_{\langle i,j\rangle}\mathbf D_{ij}\cdot(\mathbf s_i\times\mathbf s_j)$	oriented antisymmetric exchange
single-ion anisotropy	$D\sum_i(s_i^z)^2$	nontrivial zero-field splitting only for $s\ge1$
$J_1$–$J_2$	$J_1\sum_j\mathbf s_j\cdot\mathbf s_{j+1}+J_2\sum_j\mathbf s_j\cdot\mathbf s_{j+2}$	competing bond lengths
Kitaev or compass	$-\sum_{\langle i,j\rangle_a}K_a s_i^a s_j^a$	spin component tied to bond type
long range	$\sum_{i<j}J_{ij}\mathcal O_i\mathcal O_j$	extensivity may require normalization

The table identifies operator structure, not a phase diagram. Dimension, spin length, lattice, signs, fields, and coupling ratios remain essential model data.

Isotropic Heisenberg Exchange

The graph-level Heisenberg Hamiltonian is

$$H_H = \sum_{\langle i,j\rangle} J_{ij} \mathbf s_i\cdot\mathbf s_j - \sum_i \mathbf h_i\cdot\mathbf s_i.$$

With the displayed plus-sign convention:

• $J_{ij}>0$ favors the smallest allowed bond total spin and is called antiferromagnetic;
• $J_{ij}<0$ favors the largest allowed bond total spin and is called ferromagnetic.

Some authors instead write $-J\mathbf s_i\cdot\mathbf s_j$ and attach the word “ferromagnetic” to $J>0$. Compare the entire signed term, not the symbol $J$ alone.

At zero field, isotropic exchange commutes with every component of

$$\mathbf S_{\mathrm{tot}} = \sum_i\mathbf s_i.$$

Uniform fields preserve the component parallel to the field but break full spin-rotation symmetry. Spatially varying fields generally remove even that global conservation law.

The Heisenberg Model explains dimensional dependence, exact limits, ordered phases, magnons, and quantum-fluctuation physics.

XXZ and XYZ Exchange

A standard spin-chain convention is

$$H_{\mathrm{XXZ}} = J\sum_j \big( s_j^x s_{j+1}^x + s_j^y s_{j+1}^y + \Delta s_j^z s_{j+1}^z \big) - h\sum_j s_j^z.$$

The limits include:

$$\begin{array}{c|c} \Delta=1 & \text{isotropic Heisenberg} \\ \Delta=0 & \text{XX model} \\ |\Delta|\gg1 & \text{Ising-dominated exchange} \end{array}$$

The Hamiltonian conserves

$$S_{\mathrm{tot}}^z = \sum_j s_j^z.$$

The more general diagonal exchange tensor gives the XYZ chain,

$$\begin{aligned} H_{\mathrm{XYZ}} &= H_x+H_y+H_z, \\ H_a &= J_a\sum_j s_j^a s_{j+1}^a, \qquad a=x,y,z. \end{aligned}$$

Generic unequal $J_x,J_y,J_z$ leave discrete $\pi$ rotations but no continuously conserved total-spin component. The XXZ Spin Chain explains its phase structure, Bethe-ansatz regime, and low-energy interpretation.

Ising and Transverse-Field Ising Forms

A longitudinal Ising Hamiltonian is

$$H_I = -\sum_{\langle i,j\rangle} K_{ij}s_i^z s_j^z - \sum_i h_i^z s_i^z.$$

Every displayed term commutes with the others, so equilibrium state counting is a classical configuration sum in the product $s^z$ basis. Dynamics of a coherent superposition can still be quantum: relative phases change, and transverse observables need not be constant.

Adding a transverse field gives noncommuting quantum dynamics. A widely used Pauli convention is

$$H_{\mathrm{TFIM}} = -J\sum_j \sigma_j^z\sigma_{j+1}^z - h\sum_j\sigma_j^x.$$

The global spin-flip operator

$$U_x = \prod_j\sigma_j^x$$

satisfies

$$[H_{\mathrm{TFIM}},U_x] = 0.$$

A longitudinal term $-h_z\sum_j\sigma_j^z$ breaks this $\mathbb Z_2$ symmetry. The one-dimensional model without that longitudinal field is exactly solvable; generic perturbations need not be. Use the Transverse-Field Ising Model for phases, critical normalization, fermionization, and finite-size sectors.

XY Exchange and Ladder Operators

Define

$$s_i^\pm = s_i^x \pm i s_i^y.$$

Then

$$\mathbf s_i\cdot\mathbf s_j = s_i^z s_j^z + \frac12 \left( s_i^+s_j^- + s_i^-s_j^+ \right).$$

For anisotropic planar exchange,

$$\begin{aligned} \mathcal J_\pm &\equiv \frac{J_x\pm J_y}{4}, \\ H_{ij}^{\mathrm{XY}} &\equiv J_xs_i^xs_j^x + J_ys_i^ys_j^y, \\ H_{ij}^{\mathrm{XY}} &= \mathcal J_+ \left( s_i^+s_j^- + s_i^-s_j^+ \right) + \mathcal J_- \left( s_i^+s_j^+ + s_i^-s_j^- \right). \end{aligned}$$

The term proportional to $\mathcal J_+$ exchanges one unit of $s^z$ between sites and conserves total $S_{\mathrm{tot}}^z$. The term proportional to $\mathcal J_-$ changes total $s^z$ by two units. Thus the XY model has a continuous $U(1)$ spin-rotation symmetry about $z$ only when $J_x=J_y$; for generic anisotropy, a $\mathbb Z_2$ parity remains.

In one dimension, the uniform nearest-neighbor spin-$1/2$ XY chain maps to a quadratic fermion problem. Boundary and parity sectors are part of that statement; see the Jordan–Wigner Transformation and XY Model card.

Dzyaloshinskii–Moriya Exchange

Antisymmetric exchange has the form

$$H_{\mathrm{DM}} = \sum_{\langle i,j\rangle} \mathbf D_{ij}\cdot (\mathbf s_i\times\mathbf s_j).$$

It is bilinear and Hermitian for real $\mathbf D_{ij}$. It can favor canting or handed spin textures and can shift excitation minima. Its allowed direction is constrained by the spatial symmetries of the bond environment.

Important convention data are:

• an orientation for each bond;
• $\mathbf D_{ji}=-\mathbf D_{ij}$ under orientation reversal;
• the coordinate frame used for spin components;
• whether a site-dependent spin rotation has moved the term into twisted exchange or boundary conditions.

The bilinear DM term is even under time reversal because both spins reverse. An inversion center at the midpoint of that bond forces its DM vector to vanish for ordinary axial spins. Global inversion may instead relate two different bonds with nonzero DM vectors; distinguish the symmetry of one bond from that of the full lattice.

Single-Ion Anisotropy

For spins $s\ge1$, a common onsite term is

$$\begin{aligned} H_{\mathrm{SIA}} &= H_D+H_E, \\ H_D &= D\sum_i(s_i^z)^2, \\ H_E &= E\sum_i \left[ (s_i^x)^2-(s_i^y)^2 \right]. \end{aligned}$$

For the axial term as written:

• $D<0$ favors large $|m_z|$ and is often called easy-axis;
• $D>0$ favors small $|m_z|$ and is often called easy-plane.

For spin $1/2$,

$$(s_i^x)^2 = (s_i^y)^2 = (s_i^z)^2 = \frac14\mathbf 1.$$

The axial term is therefore a constant and the rhombic difference vanishes. A claimed spin-$1/2$ zero-field splitting from this quadratic onsite form signals either a convention error or additional degrees of freedom hidden in an effective pseudospin.

For an easy-axis term $-K(s^z)^2$ with $K>0$, lowering a saturated site from $m=s$ to $m=s-1$ costs $K(2s-1)$. The often used $2Ks$ is its leading large-$s$ limit; the exact cost vanishes for spin $1/2$.

Magnetic Anisotropy connects this operator dictionary to crystal symmetry, magnetocrystalline and shape energies, anisotropy fields, and texture length scales.

Bond-Directional Kitaev and Compass Exchange

In a bond-directional model, the coupled spin component depends on the bond label. A representative form is

$$H_K = -\sum_{a=x,y,z} K_a \sum_{\langle i,j\rangle_a} s_i^a s_j^a.$$

Here $\langle i,j\rangle_a$ denotes bonds of type $a$, not a sum over all spin components on every bond. The honeycomb spin-$1/2$ Kitaev model uses three bond types and is exactly solvable by a Majorana representation with conserved flux sectors. Generic compass models, extra Heisenberg exchange, fields, or longer-range couplings need not retain that solution.

Do not confuse this two-dimensional spin model with the one-dimensional Kitaev fermion chain. They share historical and conceptual connections but have different Hilbert spaces and Hamiltonians.

Frustrated Chains and Ladders

The $J_1$–$J_2$ Heisenberg chain is

$$\begin{aligned} H_{J_1J_2} &= H_1+H_2, \\ H_r &= J_r\sum_j \mathbf s_j\cdot\mathbf s_{j+r}, \qquad r=1,2. \end{aligned}$$

For antiferromagnetic $J_1,J_2>0$, the two bond lengths compete. The ratio $J_2/J_1$, boundary conditions, and chain length are indispensable data.

A two-leg Heisenberg ladder is

$$H_{\mathrm{lad}} = J_{\parallel} \sum_n\sum_{a=1}^{2} \mathbf s_{a,n}\cdot\mathbf s_{a,n+1} + J_{\perp} \sum_n \mathbf s_{1,n}\cdot\mathbf s_{2,n}.$$

$J_{\parallel}$ couples neighboring spins along each leg, while $J_{\perp}$ couples the two legs across each rung. A ladder is not merely one chain with a larger nearest-neighbor coupling: its graph, elementary loops, and strong-rung limit are different.

Long-Range and Dipolar Couplings

A power-law Heisenberg family can be written

$$H_{\mathrm{LR}} = \frac{J}{\mathcal N_L} \sum_{i<j} \frac{ \mathbf s_i\cdot\mathbf s_j }{ (r_{ij}/a)^\alpha }.$$

For sufficiently slowly decaying interactions, the unnormalized pair sum grows faster than system size. One useful Kac factor is

$$\mathcal N_L = \frac1L \sum_{i\ne j} \frac{1}{(r_{ij}/a)^\alpha}.$$

Whether this normalization is included changes the thermodynamic limit and must be stated. A finite experimental array with physical long-range forces and a Kac-normalized theoretical model are not automatically the same limit.

Magnetic dipoles have the tensor interaction

$$\begin{aligned} H_{\mathrm{dd}} &= \frac{\mu_0}{4\pi} \sum_{i<j}V_{ij}, \\ V_{ij} &= \frac{ \boldsymbol\mu_i\cdot\boldsymbol\mu_j -3\mu_i^\parallel\mu_j^\parallel }{r_{ij}^3}, \\ \mu_i^\parallel &= \boldsymbol\mu_i\cdot \widehat{\mathbf r}_{ij}. \end{aligned}$$

Its anisotropy is fixed by the real-space bond direction. High-field secular Hamiltonians are approximations to this interaction and require their own rotating-frame and energy-scale assumptions.

Fields and Zeeman Conventions

The fundamental magnetic coupling is

$$H_Z = -\sum_i \boldsymbol\mu_i\cdot\mathbf B_i.$$

If $\boldsymbol\mu_i=\gamma_i\mathbf S_i$, then

$$H_Z = -\sum_i \gamma_i\hbar \mathbf B_i\cdot\mathbf s_i.$$

For an electronic spin, one commonly has $\boldsymbol\mu=-g\mu_{\mathrm B}\mathbf s$, so the sign relative to $\mathbf B$ differs from a model convention that simply writes $-h\sum_i s_i^z$ with $h>0$. In spin models, $\mathbf h_i$ usually denotes an energy vector that has already absorbed the magnetic moment and sign.

Uniform and staggered fields are distinguished by

$$H_h = -h_u\sum_i s_i^z - h_s\sum_i\eta_i s_i^z,$$

where $\eta_i=\pm1$ labels sublattices. A staggered field can preserve no ordinary one-site translation even when the underlying exchange is uniform.

Multi-Spin and Chiral Terms

Not every effective spin interaction is bilinear. A scalar-chirality term is

$$H_\chi = \kappa \sum_{(i,j,k)} \mathbf s_i\cdot (\mathbf s_j\times\mathbf s_k),$$

where each oriented triangle $(i,j,k)$ must be declared. Reversing its orientation changes the sign. Unlike the bilinear DM term, scalar chirality is odd under time reversal because it contains three spin operators.

Ring exchange is often written using cyclic permutation operators,

$$H_{\mathrm{ring}} = K\sum_{\square} \left( P_{1234}+P_{1234}^{-1} \right).$$

Expanding a permutation into spin operators depends on the local representation. It should not be replaced by an arbitrary product of four spin components without deriving the coefficient and constant terms.

Conserved spin components

For any proposed continuous generator $Q$, test

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

Useful default expectations are:

Hamiltonian	Generic internal symmetry or conserved quantity
zero-field isotropic Heisenberg	global spin rotations; $\mathbf S_{\mathrm{tot}}^2$ and every component
XXZ with a $z$ field	rotations about $z$; $S_{\mathrm{tot}}^z$
generic XYZ	discrete $\pi$ spin rotations; no total component
longitudinal Ising	every local $s_i^z$ commutes with $H$
transverse-field Ising	global $\mathbb Z_2$ spin flip
isotropic XY or XX	$S_{\mathrm{tot}}^z$
anisotropic XY	$S_{\mathrm{tot}}^z$ parity, not $S_{\mathrm{tot}}^z$ itself
uniform field	only rotations about the field axis, if exchange permits

Spatial symmetries require a separate check of the graph and couplings. Uniform algebraic coefficients do not guarantee translation, inversion, or point-group symmetry on an irregular graph or at an open boundary.

Under time reversal,

$$\mathsf T\mathbf s_i\mathsf T^{-1} = -\mathbf s_i.$$

Real bilinear exchange is time-reversal even. Zeeman fields and scalar spin chirality are time-reversal odd. Complex coefficients require an explicit antiunitary test rather than visual inspection.

Exact Two-Spin Checks

For two spins,

$$\mathbf s_i\cdot\mathbf s_j = \frac12 \left[ (\mathbf s_i+\mathbf s_j)^2 - \mathbf s_i^2 - \mathbf s_j^2 \right].$$

For two spin-$1/2$ sites, the bond eigenvalues are

$$\begin{array}{c|c|c} \text{sector} & s_{\mathrm{tot}} & \mathbf s_i\cdot\mathbf s_j \\ \hline \text{singlet} & 0 & -3/4 \\ \text{triplet} & 1 & 1/4 \end{array}$$

The projectors are

$$\begin{aligned} P_{ij}^{(0)} &= \frac14 - \mathbf s_i\cdot\mathbf s_j, \\ P_{ij}^{(1)} &= \frac34 + \mathbf s_i\cdot\mathbf s_j. \end{aligned}$$

The spin-swap operator is

$$P_{ij} = 2\mathbf s_i\cdot\mathbf s_j + \frac12.$$

These identities expose normalization and sign errors immediately. For $H=J\mathbf s_i\cdot\mathbf s_j$, the signed splitting $E_t-E_s$ is $J$, while the positive excitation gap is $|J|$.

Boundary and Twist Conventions

For a periodic spin chain,

$$\mathbf s_{L+1} = \mathbf s_1.$$

A $U(1)$ twist may instead be represented as

$$\begin{aligned} s_{L+1}^+ &= e^{i\phi}s_1^+, \\ s_{L+1}^- &= e^{-i\phi}s_1^-, \\ s_{L+1}^z &= s_1^z. \end{aligned}$$

A site-dependent rotation can move phases between bulk exchange and the boundary. The spectrum is preserved only when the transformed boundary condition and observables are carried along. See Boundary Conditions on Lattices for the full finite-size treatment.

Natural Scales and Extensivity

Choose an energy scale $J_\ast>0$, for example the largest exchange magnitude or root-mean-square bond strength. Dimensionless controls then include

$$\frac{h}{J_\ast}, \qquad \frac{D}{J_\ast}, \qquad \frac{J_2}{J_1}, \qquad \frac{k_{\mathrm B}T}{J_\ast}.$$

For a bounded-degree graph with bounded local spin, fields and couplings, the operator norm grows at most extensively:

$$\|H\| = O(L).$$

All-to-all or slowly decaying interactions can violate this scaling unless couplings decrease with system size. State whether the goal is a physical finite system, a conventional thermodynamic limit, or a mean-field/Kac-scaled limit.

Solvability Boundaries

Common exact statements are narrow:

• the longitudinal Ising Hamiltonian is diagonal in a product basis;
• the uniform one-dimensional spin-$1/2$ XY and transverse-field Ising chains become quadratic fermions after sector-aware Jordan–Wigner mapping;
• selected one-dimensional Heisenberg and XXZ chains are Bethe-ansatz integrable;
• the honeycomb spin-$1/2$ Kitaev model has an exact Majorana and flux-sector solution;
• an isolated dimer or other very small cluster can be diagonalized exactly.

Generic fields, disorder, cross-component exchange, longer-range couplings, or multi-spin terms can destroy an exact solution. “Related to an integrable model” is not itself a solvability theorem. The Exact Solutions Preview explains method boundaries and integrability caveats.

Spin correlations and excitations

Typical spin-model outputs include:

• energy gaps and symmetry-resolved spectra;
• uniform or staggered magnetization;
• equal-time and dynamical spin correlations;
• static and dynamical structure factors;
• spin stiffness, susceptibilities, and transport coefficients;
• entanglement entropy and finite-size scaling;
• magnon or domain-wall dispersions where quasiparticles are controlled.

Use Structure Factors for Fourier and normalization conventions, Quantum Phase Transitions for scaling claims, and Magnons for controlled spin-wave excitations. The corresponding measurements depend on operator normalization and momentum conventions as well as the Hamiltonian.

Exercises

1. Translate spin and Pauli couplings

For two spin-$1/2$ sites, translate

$$H = J\mathbf s_1\cdot\mathbf s_2 - h(s_1^z+s_2^z)$$

into Pauli matrices. Then give the zero-field singlet and triplet energies.

Solution

Using $\mathbf s_i=\boldsymbol\sigma_i/2$,

$$H = \frac J4 \boldsymbol\sigma_1\cdot\boldsymbol\sigma_2 - \frac h2 (\sigma_1^z+\sigma_2^z).$$

At $h=0$,

$$\mathbf s_1\cdot\mathbf s_2 = \begin{cases} -3/4, & s_{\mathrm{tot}}=0,\\ 1/4, & s_{\mathrm{tot}}=1. \end{cases}$$

Therefore

$$E_{\mathrm s} = -\frac{3J}{4}, \qquad E_{\mathrm t} = \frac J4.$$

The signed splitting $E_{\mathrm t}-E_{\mathrm s}=J$ is a quick normalization check; the excitation gap is $|J|$.

2. Extract the DM vector

Let the antisymmetric part of a bond exchange tensor satisfy

$$A^{ab} = \sum_c\epsilon_{cab}D^c.$$

Show that its bond energy is $\mathbf D\cdot(\mathbf s_i\times\mathbf s_j)$ and determine what happens when the bond orientation is reversed.

Solution

Substitution gives

$$\begin{aligned} \sum_{a,b}A^{ab}s_i^a s_j^b &= \sum_{a,b,c} D^c\epsilon_{cab}s_i^a s_j^b \\ &= \sum_cD^c (\mathbf s_i\times\mathbf s_j)^c \\ &= \mathbf D\cdot (\mathbf s_i\times\mathbf s_j). \end{aligned}$$

Reversing the sites gives

$$\mathbf s_j\times\mathbf s_i = -\mathbf s_i\times\mathbf s_j.$$

The same physical bond energy therefore requires $\mathbf D_{ji}=-\mathbf D_{ij}$.

3. Find the XY symmetry condition

For $J_xs_i^xs_j^x+J_ys_i^ys_j^y$, identify the terms that violate conservation of total $s^z$ and state when they vanish.

Solution

The ladder-operator form is

$$\begin{aligned} \mathcal J_\pm &= \frac{J_x\pm J_y}{4}, \\ H_{ij}^{\mathrm{XY}} &= \mathcal J_+ (s_i^+s_j^-+s_i^-s_j^+) + \mathcal J_- (s_i^+s_j^++s_i^-s_j^-). \end{aligned}$$

The term proportional to $\mathcal J_-$ changes total $s^z$ by $+2$ or $-2$. It vanishes exactly when

$$J_x = J_y.$$

At that isotropic planar point, total $S_{\mathrm{tot}}^z$ is conserved. Away from it, only its parity is generically conserved.

4. Diagnose spin-half single-ion anisotropy

Show that

$$D(s_i^z)^2 + E\left[(s_i^x)^2-(s_i^y)^2\right]$$

cannot split an isolated true spin-$1/2$ doublet.

Solution

For spin $1/2$, $s_i^a=\sigma_i^a/2$ and $(\sigma_i^a)^2=\mathbf1$. Therefore

$$(s_i^a)^2 = \frac14\mathbf1$$

for every axis. The onsite term reduces to

$$D(s_i^z)^2 + E\left[(s_i^x)^2-(s_i^y)^2\right] = \frac D4\mathbf1.$$

It shifts both states equally and produces no splitting. Nontrivial zero-field splitting requires $s\ge1$ or additional microscopic structure beyond an isolated spin-$1/2$ degree of freedom.

References

• A. Auerbach, Interacting Electrons and Quantum Magnetism, Springer (1994) — exchange models, spin waves, antiferromagnets, and continuum limits.
• E. Fradkin, Field Theories of Condensed Matter Physics, 2nd ed., Cambridge University Press (2013) — spin chains, duality, fermionization, and field-theory limits.
• T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press (2004) — XXZ chains, spin–particle mappings, and low-energy one-dimensional physics.
• A. Kitaev, “Anyons in an Exactly Solved Model and Beyond,” Annals of Physics 321, 2–111 (2006) — bond-directional honeycomb model and exact Majorana solution.
• E. Lieb, T. Schultz, and D. Mattis, “Two Soluble Models of an Antiferromagnetic Chain,” Annals of Physics 16, 407–466 (1961) — exact one-dimensional XY and related chain methods.
• T. Moriya, “Anisotropic Superexchange Interaction and Weak Ferromagnetism,” Physical Review 120, 91–98 (1960) — microscopic symmetry analysis of antisymmetric exchange.
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press (2011) — Ising models, quantum criticality, and order-parameter dynamics.