Spin-1/2 Chain

A spin-$1/2$ chain is a one-dimensional tensor product of two-level local spin spaces coupled by specified onsite and finite-range operators; it is a model family whose symmetry, integrability, phases, and numerics depend decisively on normalization, couplings, fields, graph, and boundary conditions.

The XYZ family below provides a common starting point for comparing Ising, Heisenberg, and XXZ chains. The two-site example shows why a missed factor of two in spin normalization changes a physical energy gap.

Helpful background. Spin-1/2 states supplies Pauli matrices; lattice models explains tensor products of local spaces.

The XYZ Spin-1/2 Hamiltonian

The reference family is a uniform nearest-neighbor XYZ chain in a uniform field.

Field	Reference choice
sites	$L\geq3$ sites labeled $j=1,\ldots,L$
local space	$\mathcal H_j\simeq\mathbb C^2$
local operators	dimensionless $s_j^\alpha=\sigma_j^\alpha/2$
geometry	one-dimensional chain
exchange	real diagonal couplings $J_x,J_y,J_z$
field	real energy vector $\mathbf h=(h_x,h_y,h_z)$
boundaries	open or periodic, stated explicitly
energy offset	none
thermodynamic limit	$L\to\infty$ at fixed local couplings
disorder and longer range	absent unless declared

For periodic boundaries, the reference Hamiltonian is

$$\begin{aligned} H &= H_{\mathrm{ex}} + H_h, \\ H_{\mathrm{ex}} &= \sum_{\alpha=x,y,z} J_\alpha \sum_{j=1}^{L} s_j^\alpha s_{j+1}^\alpha, \\ H_h &= - \sum_{\alpha=x,y,z} h_\alpha \sum_{j=1}^{L} s_j^\alpha. \end{aligned}$$

with

$$s_{L+1}^\alpha = s_1^\alpha.$$

This reference family contains several canonical chains as parameter restrictions, but an arbitrary point with fields is not generically integrable. Cross-component exchange, Dzyaloshinskii–Moriya terms, bond alternation, disorder, and multi-spin interactions are variants, not implicit parts of this Hamiltonian.

Degrees of Freedom and Basis Convention

The local basis is chosen as

$$s^z\lvert\uparrow\rangle = \frac12\lvert\uparrow\rangle, \qquad s^z\lvert\downarrow\rangle = -\frac12\lvert\downarrow\rangle.$$

Equivalently,

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

The many-body Hilbert space is

$$\mathcal H_L = \bigotimes_{j=1}^{L} \mathbb C^2,$$

with dimension

$$\dim\mathcal H_L = 2^L.$$

The site operator is embedded as

$$s_j^\alpha = I^{\otimes(j-1)} \otimes \frac{\sigma^\alpha}{2} \otimes I^{\otimes(L-j)}.$$

Operators on distinct sites commute:

$$[s_i^\alpha,s_j^\beta] = 0 \qquad (i\ne j),$$

whereas the onsite algebra is

$$[s_j^\alpha,s_j^\beta] = i\epsilon_{\alpha\beta\gamma} s_j^\gamma.$$

For reproducible bit-basis examples, we order kets as

$$\lvert b_1b_2\cdots b_L\rangle,$$

and map them to integers by

$$x = \sum_{j=1}^{L} b_j2^{L-j}.$$

Thus site $1$ is the most significant bit. Another ordering is equally valid if all operators, basis labels, and reported vectors use it consistently.

Spin Versus Pauli Normalization

We use dimensionless spin operators

$$s^\alpha = \frac{\sigma^\alpha}{2},$$

whose eigenvalues along any axis are $\pm1/2$. Physical angular momentum is

$$S^\alpha = \hbar s^\alpha.$$

A bond and field transform as

$$\sigma_i^\alpha\sigma_j^\alpha = 4s_i^\alpha s_j^\alpha, \qquad \sigma_i^\alpha = 2s_i^\alpha.$$

Therefore a Pauli-form Hamiltonian

$$H = \sum_{\langle i,j\rangle,\alpha} K_\alpha \sigma_i^\alpha\sigma_j^\alpha - \sum_{i,\alpha} g_\alpha\sigma_i^\alpha$$

uses

$$J_\alpha = 4K_\alpha, \qquad h_\alpha = 2g_\alpha$$

in the present spin convention. Spectra, critical fields, and trace moments can differ by factors of two or four when this conversion is missed.

Local Operator Decomposition

Define

$$s_j^\pm = s_j^x \pm is_j^y.$$

The anisotropic transverse bond is

$$J_xs_j^xs_{j+1}^x + J_ys_j^ys_{j+1}^y ={} \frac{J_x+J_y}{4} \left( s_j^+s_{j+1}^- + s_j^-s_{j+1}^+ \right) + \frac{J_x-J_y}{4} \left( s_j^+s_{j+1}^+ + s_j^-s_{j+1}^- \right).$$

The first line of processes exchanges an up and a down spin and preserves total $z$ magnetization. The second creates or removes two down spins relative to the all-up reference and changes total $z$ magnetization by two units.

This operator decomposition immediately diagnoses a central symmetry condition:

$$[H,S_{\mathrm{tot}}^z] = 0, \qquad S_{\mathrm{tot}}^z = \sum_j s_j^z,$$

when

$$J_x = J_y, \qquad h_x = h_y = 0.$$

In that case, a sector with $N_\downarrow$ down spins has

$$S_{\mathrm{tot}}^z = \frac L2-N_\downarrow$$

and dimension

$$\dim\mathcal H_{N_\downarrow} = \binom{L}{N_\downarrow}.$$

Family Map

The parameter restrictions below identify standard nearest-neighbor descendants in the present spin convention.

Family	Parameter restriction	Characteristic symmetry	Standard solution status
longitudinal Ising	$J_x=J_y=0$, $h_x=h_y=0$	all $s_j^z$ commute with $H$	product-basis exact
transverse-field Ising	one Ising exchange plus a perpendicular field	global $\mathbb Z_2$	quadratic fermions in the standard chain
XY	$J_z=0$, field normal to the exchange plane	usually $\mathbb Z_2$	quadratic after Jordan–Wigner
XX	$J_x=J_y$, $J_z=0$, longitudinal field	$U(1)$	free spinless fermions
XXZ	$J_x=J_y$, $J_z$ independent, longitudinal field	$U(1)$	Bethe-ansatz integrable in the uniform chain
Heisenberg	$J_x=J_y=J_z$, zero field	$SU(2)$	Bethe-ansatz integrable for the uniform spin-$1/2$ chain
XYZ	unequal $J_x,J_y,J_z$, zero field	discrete spin rotations	integrable in the uniform zero-field chain via the eight-vertex structure
generic field XYZ	unequal exchange and arbitrary field	often only lattice symmetries	generally nonintegrable

Axis labels are conventional. A global spin rotation can turn a $zz$ Ising exchange with an $x$ field into an $xx$ exchange with a $z$ field. The transformed Hamiltonian is physically equivalent only when every operator, observable, boundary term, and state is rotated consistently.

Symmetries and Conserved Quantities

For uniform periodic couplings, the Hamiltonian commutes with one-site translation. It also has spatial reflection because the exchange is symmetric under interchanging the two sites of a bond.

At zero field, real bilinear exchange is time-reversal invariant. For an odd number of spin-$1/2$ sites,

$$\mathcal T^2 = -1,$$

so every finite-size energy level is at least Kramers degenerate. For even $L$, $\mathcal T^2=+1$ on the many-spin space and time reversal alone does not enforce pairwise degeneracy.

Internal symmetry depends on parameters:

• $J_x=J_y$ with a longitudinal field has axial $U(1)$ symmetry generated by $S_{\mathrm{tot}}^z$.
• $J_x=J_y=J_z$ and $\mathbf h=0$ has global $SU(2)$ spin-rotation symmetry.
• A field at the isotropic point reduces $SU(2)$ to rotations about the field axis.
• At zero field, diagonal XYZ exchange is invariant under global $\pi$ rotations about each principal axis.
• The transverse-field Ising restriction has a global spin-flip $\mathbb Z_2$ symmetry about the field axis.

Translation, reflection, spin flip, magnetization, and total spin are distinct labels. They should not be assumed simultaneously. A numerical block decomposition is valid only after checking

$$[H,Q] = 0$$

for each claimed sector operator $Q$. To use several eigenvalues as simultaneous labels, the symmetry operators must also be mutually compatible. For example, reflection $R$ and translation $T$ satisfy $RTR^{-1}=T^{-1}$. Reflection maps momentum $q$ to $-q$, so a state of definite generic momentum cannot also be assigned reflection parity. Momentum $0$ and, when present, $\pi$ are the exceptions. Noncommuting symmetries can instead organize multidimensional representation blocks.

Boundary Conditions

For open boundaries,

$$\begin{aligned} H_{\mathrm O} ={}& \sum_{j=1}^{L-1} h_{j,j+1} + \sum_{j=1}^{L} h_j. \end{aligned}$$

There are $L-1$ nearest-neighbor bonds. For periodic boundaries,

$$H_{\mathrm P} = H_{\mathrm O} + h_{L,1},$$

so a ring with $L\geq3$ has $L$ undirected bonds.

For $L=2$, modulo indexing is ambiguous: the formal terms $j=1$ and $j=2$ may describe the same undirected pair twice. A two-site benchmark must state whether it uses one bond, two parallel bonds, or a coupling rescaled to compensate. The minimal example below is an open one-bond dimer.

Open boundaries break translation symmetry but may preserve reflection. Periodic boundaries remove physical ends and support crystal-momentum sectors. A boundary spin rotation can define a modified Hamiltonian generally. Interpreting a twist as a flux for a conserved spin component, and deriving its associated stiffness, requires that continuous symmetry. In an axial chain one may replace the wrapping flip-flop terms by

$$s_L^+s_1^- \longmapsto e^{i\phi} s_L^+s_1^-,$$

and its Hermitian conjugate by the opposite phase.

After a Jordan–Wigner transformation, periodic spin boundaries do not become one universal fermion boundary condition. The fermionic boundary sign depends on parity. Boundary Conditions on Lattices derives that parity dependence.

Natural Scales and Control Parameters

A convenient local energy scale is

$$J_* = \max \left( |J_x|, |J_y|, |J_z|, |h_x|, |h_y|, |h_z| \right).$$

When $J_*>0$, useful dimensionless controls include

$$\frac{J_y}{J_*}, \quad \frac{J_z}{J_*}, \quad \frac{h_\alpha}{J_*}, \quad \frac{k_{\mathrm B}T}{J_*}, \quad L.$$

For the XXZ restriction, one usually writes

$$J_x = J_y = J, \qquad J_z = J\Delta,$$

so $\Delta$ is the exchange anisotropy. For a transverse-field Ising restriction, a field-to-exchange ratio organizes the competition, but its numerical critical value depends on whether Pauli or spin operators define those coefficients.

Finite-size control requires comparing several $L$ values, boundary choices, and symmetry sectors. A generic XYZ field produces virtual pair flips even when large. Special factorizing parameter choices can also give exact product states, so exact polarization or separability should be checked by applying the Hamiltonian rather than inferred solely from a large field.

A small gap can represent a true bulk gap, a critical $1/L$ level spacing, tunneling between finite-size symmetry partners, or a crossing between different conserved sectors.

Solvable Limits and General Chains

Different restrictions of the XYZ family admit different solution methods.

Quantity or subfamily	Status	Caveat
finite-$L$ matrix	exactly defined once basis, graph, and coefficients are fixed	dimension grows as $2^L$
full exact diagonalization	numerically exact up to arithmetic and eigensolver tolerances	feasible only for limited $L$
commuting longitudinal Ising limit	exact product-basis spectrum	a transverse field removes commutativity
standard XY and transverse-field Ising chains	reducible to quadratic fermions	boundaries and parity sectors remain essential
uniform spin-$1/2$ Heisenberg and XXZ chains	Bethe-ansatz integrable	finite roots, thermodynamics, and correlators are separate tasks
uniform zero-field XYZ chain	integrable through the eight-vertex correspondence	generic fields destroy that structure
generic finite-range chain	no general analytic solution	tensor networks or other controlled numerics may still be accurate
real-time generic dynamics	unitary and exactly defined	thermalization, transport, and hydrodynamics are model-dependent
thermodynamic phases	require a controlled $L\to\infty$ limit	finite-chain Gibbs thermodynamics is analytic at positive temperature; zero-temperature level crossings can still occur

Integrability is not synonymous with triviality. An exact spectral construction may still leave difficult correlation functions, quench overlaps, or finite-temperature limits. Conversely, a nonintegrable finite chain is still an exact finite-dimensional quantum problem even when no closed-form thermodynamic solution exists.

Trace Moments and High-Temperature Limits

With no added identity offset, every term in the reference Hamiltonian is a nonidentity Pauli string, so

$$\operatorname{Tr}H = 0.$$

For a uniform periodic chain with $L\geq3$, Pauli-string orthogonality gives

$$\frac{1}{2^L} \operatorname{Tr}H^2 ={} L \left[ \frac{ J_x^2+J_y^2+J_z^2 }{16} \qquad + \frac{ h_x^2+h_y^2+h_z^2 }{4} \right].$$

This identity is independent of integrability and is a strong assembly test. It changes if bonds are counted differently, an identity shift is added, or Pauli matrices replace spin operators without rescaling.

At infinite temperature,

$$\frac{1}{2^L} \operatorname{Tr} (s_i^\alpha s_j^\beta) = \frac14 \delta_{ij}\delta_{\alpha\beta}.$$

The entropy is

$$S_\infty = Lk_{\mathrm B}\ln2.$$

When $\operatorname{Tr}H=0$, the high-temperature energy begins as

$$U = -\beta \frac{\operatorname{Tr}H^2}{2^L} +O(\beta^2),$$

and therefore

$$C_V = k_{\mathrm B}\beta^2 \frac{\operatorname{Tr}H^2}{2^L} +O(\beta^3).$$

These are normalization checks, not phase diagnostics. They do not determine low-energy order, criticality, or transport.

Important Limits

Limit	Exact structure	What it does not imply
all $J_\alpha=0$	independent spins in a uniform field	no exchange correlations
only $J_z,h_z$ nonzero	commuting classical Ising energy in the $z$ basis	no transverse quantum dynamics
strong field	a polarized reference with small corrections in generic XYZ models	axial chains can be exactly polarized above a finite saturation field; generic pair-flip terms dress that state
$J_x=J_y$, longitudinal field	fixed-$S_{\mathrm{tot}}^z$ sectors	not necessarily free or integrable
$J_x=J_y=J_z$, zero field	$SU(2)$ multiplets	not classical alignment for antiferromagnetic exchange
$L$ finite	discrete spectrum; analytic positive-temperature Gibbs averages	eigenvalue crossings can make the ground energy nonanalytic without producing a thermodynamic phase transition
$L\to\infty$	possible phases and critical scaling	boundary and parity sequences must still be controlled
spin $s\to\infty$	semiclassical limit of a different family	a fixed spin-$1/2$ chain has no tunable large-spin parameter

Changing the sign of a coupling is not always removable. On a bipartite open chain, some signs can be changed by staggered spin rotations. On an odd periodic ring, the same transformation can leave a frustrated wrapping bond. Geometry and boundary conditions are part of every sign-convention claim.

Typical Observables

The uniform magnetization density is

$$m^\alpha = \frac1L \sum_j \langle s_j^\alpha\rangle.$$

A translation-averaged connected correlator is

$$C^{\alpha\beta}(r) ={} \frac1L \sum_j \left[ \langle s_j^\alpha s_{j+r}^\beta \rangle \qquad - \langle s_j^\alpha\rangle \langle s_{j+r}^\beta\rangle \right].$$

For the periodic chain, with dimensionless crystal momentum $q$ in units of inverse site spacing, the static structure factor is

$$S^{\alpha\beta}(q) = \frac1L \sum_{j,k} e^{-iq(j-k)} \langle s_j^\alpha s_k^\beta \rangle.$$

Other standard observables include:

• symmetry-resolved excitation gaps;
• uniform and staggered susceptibilities;
• dynamical structure factors;
• spin currents when a component is conserved;
• domain-wall, chirality, or string observables in suitable variants;
• bipartite entanglement entropy and entanglement spectra;
• fidelity and response to boundary twists.

The phrase “the gap” is incomplete. A calculation should identify the reference state, target symmetry sector, momentum, parity, and boundary condition. Equal-Time Correlations and Structure Factors explain the normalization conventions.

Minimal Worked Example

Consider one open XYZ bond with no field:

$$H_2 = J_xs_1^xs_2^x + J_ys_1^ys_2^y + J_zs_1^zs_2^z.$$

The Bell basis is

$$\begin{aligned} \lvert\Phi_\pm\rangle &= \frac{ \lvert\uparrow\uparrow\rangle \pm \lvert\downarrow\downarrow\rangle }{\sqrt2}, \\ \lvert\Psi_\pm\rangle &= \frac{ \lvert\uparrow\downarrow\rangle \pm \lvert\downarrow\uparrow\rangle }{\sqrt2}. \end{aligned}$$

These states diagonalize all three commuting two-site products

$$\sigma_1^x\sigma_2^x, \qquad \sigma_1^y\sigma_2^y, \qquad \sigma_1^z\sigma_2^z.$$

Their energies are

$$\begin{aligned} E_{\Phi_+} &= \frac{ J_x-J_y+J_z }{4}, \\ E_{\Phi_-} &= \frac{ -J_x+J_y+J_z }{4}, \\ E_{\Psi_+} &= \frac{ J_x+J_y-J_z }{4}, \\ E_{\Psi_-} &= \frac{ -J_x-J_y-J_z }{4}. \end{aligned}$$

The energies sum to zero, as required by $\operatorname{Tr}H_2=0$. At the isotropic point,

$$J_x = J_y = J_z = J,$$

the first three states form the triplet with energy $J/4$, while $\lvert\Psi_-\rangle$ is the singlet with energy $-3J/4$. This dimer fixes the spin normalization but does not represent a two-site periodic ring unless its bond convention is stated separately.

Numerical Benchmark

Use the periodic $L=4$ reference chain with

$$J_x = 1, \qquad J_y = 2, \qquad J_z = 3,$$

and

$$h_x = 1, \qquad h_y = h_z = 0.$$

The four undirected bonds are

$$(1,2), \quad (2,3), \quad (3,4), \quad (4,1).$$

The Hilbert-space dimension is

$$2^4 = 16.$$

The basis-independent trace targets are

$$\operatorname{Tr}H = 0,$$

and

$$\frac1{16} \operatorname{Tr}H^2 = \frac92.$$

Equivalently,

$$\operatorname{Tr}H^2 = 72.$$

Although $S_{\mathrm{tot}}^z$ is not conserved, the global $\pi$ rotation about $x$ is:

$$P_x = \prod_{j=1}^{4} \sigma_j^x, \qquad [H,P_x] = 0.$$

A valid implementation should:

1. reproduce dimension $16$, Hermiticity, and the four-bond graph;
2. verify the trace and second moment without using eigenvectors;
3. check $P_x^2=I$ and $[H,P_x]=0$;
4. confirm that $[H,S_{\mathrm{tot}}^z]\ne0$ for these parameters;
5. reconstruct the full spectrum from both $P_x$ sectors if block diagonalization is used;
6. compare the one-bond dimer spectrum before trusting the ring assembly;
7. test specializations against the stable benchmark suite.

Two useful specializations connect to worked benchmarks. For the Pauli-form transverse-field Ising Hamiltonian

$$H_{\mathrm{TFIM}} = -J_{\mathrm I} \sum_j \sigma_j^z\sigma_{j+1}^z - h_{\mathrm I} \sum_j \sigma_j^x,$$

use

$$\begin{aligned} J_z &= -4J_{\mathrm I}, & h_x &= 2h_{\mathrm I}, \\ J_x &= J_y = 0. \end{aligned}$$

Then apply MB-B001. For the spin-form Heisenberg ring, set

$$J_x = J_y = J_z = J, \qquad \mathbf h = 0,$$

and apply MB-B002.

The trace test detects many normalization and bond-assembly mistakes, but is not a proof that a matrix represents the intended model: different Hamiltonians can share these moments. Combine it with local matrix elements, symmetry commutators, and known special limits before using the result for phase or dynamics calculations.

Finite-Size Spectra and Symmetry Sectors

For a finite chain:

• the spectrum is discrete and analytic in generic parameters away from exact crossings;
• a symmetry-preserving eigenstate has zero expectation value for an odd order parameter;
• nearly degenerate symmetry partners can precede bulk symmetry breaking;
• momenta depend on boundary conditions and, after nonlocal mappings, parity sectors;
• even and odd rings can have different frustration and Kramers structure;
• the lowest excitation in the full Hilbert space need not be the gap relevant to a chosen response operator.

If $S_{\mathrm{tot}}^z$ is conserved, fixed-magnetization blocks must reconstruct the full dimension:

$$\sum_{N_\downarrow=0}^{L} \binom{L}{N_\downarrow} = 2^L.$$

If translation sectors are used, basis states fall into orbits whose lengths divide $L$. Momentum-block dimensions are therefore not generally equal. Symmetry Sectors in Many-Body Numerics explains how the blocks reconstruct the full spectrum.

To compare finite-chain results, specify:

• $L$ and its parity;
• open, periodic, or twisted boundaries;
• operator normalization;
• exact symmetry sector;
• bond list;
• energy offset;
• convergence or extrapolation sequence.

Additional Couplings and Evolution

Cross-component exchange

A general bilinear bond uses a real exchange tensor:

$$H_{ij} = \sum_{\alpha,\beta} J_{ij}^{\alpha\beta} s_i^\alpha s_j^\beta.$$

Its antisymmetric part is equivalent to a Dzyaloshinskii–Moriya vector. Such terms can break inversion, change conserved spin components, and shift spiral correlations.

Longer range and frustration

Next-nearest-neighbor exchange introduces competing paths. The $J_1$–$J_2$ chain can dimerize and frustrate simple antiferromagnetic order. The ratio $J_2/J_1$ becomes new model data.

Disorder and quasiperiodicity

Random fields or bonds break translation symmetry and require sample ensembles. Localization and rare-region claims need disorder-size and realization convergence, not one spectrum.

Open-system evolution

Adding measurement, noise, or dissipation requires a master equation or quantum channel. A non-Hermitian effective Hamiltonian alone does not specify the unconditional dynamics.

Mappings to particles

Hard-core bosons share the local two-state algebra. Spinless fermions in one dimension require Jordan–Wigner strings, and periodic boundaries require parity-sector care. Equal local dimensions do not erase exchange statistics or boundary data.

Exercises

Exercise 1: Count a magnetization sector

For $L$ spin-$1/2$ sites, show that the sector with $N_\downarrow$ down spins has dimension $\binom{L}{N_\downarrow}$ and verify that all sectors reconstruct the full Hilbert space.

Solution

A computational-basis state in the sector is fixed by choosing which $N_\downarrow$ of the $L$ sites carry bit $1$. Therefore

$$\dim\mathcal H_{N_\downarrow} = \binom{L}{N_\downarrow}.$$

Each bit string belongs to exactly one such sector. Summing over all allowed down-spin counts gives the binomial theorem:

$$\sum_{N_\downarrow=0}^{L} \binom{L}{N_\downarrow} = (1+1)^L = 2^L.$$

The magnetization eigenvalue in this sector is

$$S_{\mathrm{tot}}^z = \frac12(L-N_\downarrow) - \frac12N_\downarrow = \frac L2-N_\downarrow.$$

Exercise 2: Derive the magnetization condition

Use the raising-and-lowering decomposition to determine when the reference Hamiltonian commutes with $S_{\mathrm{tot}}^z$.

Solution

The flip-flop operators

$$s_j^+s_{j+1}^-, \qquad s_j^-s_{j+1}^+$$

raise one site and lower the other, so their net change in total $z$ magnetization is zero.

The pair operators

$$s_j^+s_{j+1}^+, \qquad s_j^-s_{j+1}^-$$

change total magnetization by $+2$ and $-2$. Their coefficient is $(J_x-J_y)/4$, so they vanish precisely when $J_x=J_y$.

A longitudinal field $h_z$ commutes with $S_{\mathrm{tot}}^z$, while $h_xs_j^x+h_ys_j^y$ contains single-spin raising and lowering terms. Thus

$$[H,S_{\mathrm{tot}}^z] = 0$$

for the reference family when

$$J_x = J_y, \qquad h_x = h_y = 0.$$

Exercise 3: Diagonalize the XYZ dimer

Derive the four Bell-basis energies of the one-bond XYZ dimer and recover the singlet–triplet splitting at the isotropic point.

Solution

The Bell states are simultaneous eigenstates of the three Pauli-pair operators. Their eigenvalue triples for

$$(\sigma^x\sigma^x,\sigma^y\sigma^y,\sigma^z\sigma^z)$$

are

$$\begin{array}{c|ccc} & xx & yy & zz \\ \hline \Phi_+ & +1 & -1 & +1 \\ \Phi_- & -1 & +1 & +1 \\ \Psi_+ & +1 & +1 & -1 \\ \Psi_- & -1 & -1 & -1 \end{array}.$$

Since $s^\alpha s^\alpha=\sigma^\alpha\sigma^\alpha/4$, multiplying by the couplings gives

$$\begin{aligned} E_{\Phi_+} &= \frac{J_x-J_y+J_z}{4}, \\ E_{\Phi_-} &= \frac{-J_x+J_y+J_z}{4}, \\ E_{\Psi_+} &= \frac{J_x+J_y-J_z}{4}, \\ E_{\Psi_-} &= \frac{-J_x-J_y-J_z}{4}. \end{aligned}$$

At $J_x=J_y=J_z=J$, the first three energies equal $J/4$, while

$$E_{\Psi_-} = -\frac{3J}{4}.$$

The Bell state $\Psi_-$ is the spin singlet; the other three form the triplet.

Exercise 4: Prove the trace-moment formula

Use Pauli-string orthogonality to derive the normalized second moment of the uniform periodic reference chain for $L\geq3$.

Solution

Distinct Pauli strings are orthogonal under the Hilbert–Schmidt inner product:

$$\frac1{2^L} \operatorname{Tr}(P_aP_b) = \delta_{ab}.$$

A spin bond is a Pauli string divided by four:

$$s_j^\alpha s_{j+1}^\alpha = \frac14 \sigma_j^\alpha\sigma_{j+1}^\alpha.$$

Its normalized squared trace is therefore $1/16$. A field operator is a Pauli string divided by two, so its normalized squared trace is $1/4$.

For $L\geq3$, all bond and onsite strings in the stated uniform sum are distinct. Cross terms vanish, and there are $L$ copies of each component. Hence

$$\frac{1}{2^L} \operatorname{Tr}H^2 ={} L \left[ \frac{J_x^2+J_y^2+J_z^2}{16} \qquad + \frac{h_x^2+h_y^2+h_z^2}{4} \right].$$

For the numerical target, this is

$$4 \left( \frac{1+4+9}{16} + \frac14 \right) = \frac92.$$

Multiplying by $2^4=16$ gives $\operatorname{Tr}H^2=72$.

Exercise 5: Translate benchmark conventions

Convert the Pauli-form transverse-field Ising chain to the dimensionless-spin convention, and explain why the Heisenberg specialization needs no further rescaling.

Solution

Using $\sigma^\alpha=2s^\alpha$,

$$-J_{\mathrm I} \sigma_j^z\sigma_{j+1}^z = -4J_{\mathrm I} s_j^zs_{j+1}^z,$$

and

$$-h_{\mathrm I}\sigma_j^x = -2h_{\mathrm I}s_j^x.$$

Comparing with

$$J_zs_j^zs_{j+1}^z - h_xs_j^x$$

gives

$$J_z = -4J_{\mathrm I}, \qquad h_x = 2h_{\mathrm I}.$$

The Heisenberg benchmark is already written as

$$J \sum_j \mathbf s_j\cdot\mathbf s_{j+1}$$

with $\mathbf s=\boldsymbol\sigma/2$. Therefore its specialization is simply

$$J_x = J_y = J_z = J, \qquad \mathbf h = 0.$$

The remaining issue is graph data: the $L=4$ ring has four undirected bonds, whereas a one-bond $L=2$ dimer must not be generated by an ambiguous modulo sum.

References

• I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, “Rigorous results on valence-bond ground states in antiferromagnets”, Physical Review Letters 59, 799–802 (1987).
• R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press (1982).
• H. Bethe, “Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette”, Zeitschrift für Physik 71, 205–226 (1931).
• 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).
• E. Lieb, T. Schultz, and D. Mattis, “Two soluble models of an antiferromagnetic chain”, Annals of Physics 16, 407–466 (1961).
• D. C. Mattis, The Theory of Magnetism Made Simple, World Scientific (2006).
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press (2011).
• A. W. Sandvik, “Computational studies of quantum spin systems”, AIP Conference Proceedings 1297, 135–338 (2010).
• S. R. White, “Density matrix formulation for quantum renormalization groups”, Physical Review Letters 69, 2863–2866 (1992).