Transverse-Field Ising Model

One-Sentence Description

The one-dimensional transverse-field Ising model is a chain of spin-$1/2$ degrees of freedom in which ferromagnetic Ising exchange competes with a noncommuting transverse field, yielding an exactly solvable quantum phase transition between an ordered phase and a quantum paramagnet.

This dossier fixes the model record, conventions, solution-status claims, observables, limiting cases, and validation targets. The Transverse-Field Ising Model teaching article owns the detailed physical development and quasiparticle results. Jordan–Wigner Transformation owns the nonlocal spin-to-fermion derivation, and Quantum Phase Transitions owns the general scaling framework.

Baseline Model Record

Unless a variant is named explicitly, this dossier uses the following model.

Field	Baseline choice
geometry	a one-dimensional chain of $L$ sites
local degree of freedom	one spin-$1/2$ space $\mathbb C^2$ per site
operators	dimensionless Pauli matrices $\sigma_j^\alpha$ with eigenvalues $\pm1$
interaction	nearest-neighbor Ising exchange along $z$
field	uniform transverse field along $x$
couplings	$J>0$ and $h\ge0$
dimensionless control	$g=h/J$ when $J\ne0$
boundaries	open or periodic, stated explicitly
energy offset	none
longitudinal field	absent
disorder and long-range terms	absent
equilibrium limit	$L\to\infty$ after the finite-chain problem is defined

For bulk formulas, the periodic chain with $L\ge3$ is the default unless stated otherwise. Small periodic chains require extra care: for $L=2$, writing both $j=1$ and $j=2$ terms in a nearest-neighbor sum counts the same undirected bond twice. The numerical contracts below therefore specify their bond lists rather than relying on an ambiguous shorthand.

The sign of $h$ is inessential for this isolated model because a global $\pi$ rotation about $z$ sends $\sigma_j^x\mapsto-\sigma_j^x$ while leaving every Ising bond unchanged. The restriction $h\ge0$ removes this unitary redundancy. The sign of $J$ has more substantial boundary consequences and is discussed under variants.

Degrees of Freedom and Basis

The local and many-body Hilbert spaces are

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

so

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

The computational basis is the simultaneous eigenbasis of all $\sigma_j^z$:

$$\begin{aligned} \sigma^z\lvert0\rangle &= +\lvert0\rangle, \\ \sigma^z\lvert1\rangle &= -\lvert1\rangle. \end{aligned}$$

A basis vector can therefore be labeled by a bit string

$$\lvert s_1s_2\cdots s_L\rangle, \qquad s_j\in\{0,1\}.$$

The exchange term is diagonal in this basis. The transverse-field operator $\sigma_j^x$ flips bit $j$, so the full Hamiltonian connects bit strings that differ at one site. This gives an immediate sparse-matrix construction with at most $L+1$ nonzero entries per row before duplicate contributions are combined.

The Spin-$1/2$ Chain dossier owns the generic tensor-product ordering, local-operator embedding, and family-wide matrix checks. Here they are specialized to one Ising coupling and one transverse field.

Hamiltonian and Conventions

For open boundary conditions,

$$\begin{aligned} H_{\mathrm O} ={}& -J \sum_{j=1}^{L-1} \sigma_j^z\sigma_{j+1}^z \\ &- h \sum_{j=1}^{L} \sigma_j^x. \end{aligned}$$

For periodic boundary conditions,

$$\begin{aligned} H_{\mathrm P} ={}& -J \sum_{j=1}^{L} \sigma_j^z\sigma_{j+1}^z \\ &- h \sum_{j=1}^{L} \sigma_j^x, \end{aligned}$$

with

$$\sigma_{L+1}^{\alpha} = \sigma_1^{\alpha}.$$

The open chain has $L-1$ exchange bonds; an ordinary ring has $L$. Both $J$ and $h$ have units of energy. For $J\ne0$, the overall scale and dimensionless competition are separated by

$$H = J\widehat H(g), \qquad g = \frac{h}{J}.$$

Pauli and spin-operator translations

This dossier never uses $\sigma^\alpha$ and a spin operator interchangeably. With dimensionless spin operators

$$s_j^\alpha = \frac12\sigma_j^\alpha,$$

the same Hamiltonian becomes

$$H = -4J \sum_{\langle j,k\rangle} s_j^z s_k^z - 2h \sum_j s_j^x.$$

If dimensionful angular-momentum operators $S_j^\alpha=\hbar s_j^\alpha$ are used, the coupling constants acquire the corresponding powers of $\hbar^{-1}$. A critical ratio quoted without the operator normalization is not a complete statement.

No hidden constant

Some authors write the exchange in terms of domain-wall projectors,

$$n_{j,j+1}^{\mathrm{DW}} = \frac{1-\sigma_j^z\sigma_{j+1}^z}{2}.$$

Then

$$-J\sigma_j^z\sigma_{j+1}^z = -J + 2J n_{j,j+1}^{\mathrm{DW}}.$$

The projector form contains an explicit constant $-J$ per bond. The baseline Hamiltonian above retains that constant; no later recentering is implied.

Symmetries

Global Ising parity

Define the global spin-flip operator

$$P = \prod_{j=1}^{L} \sigma_j^x.$$

Because operators on different sites commute,

$$P^2 = \mathbb I,$$

and conjugation gives

$$\begin{aligned} P\sigma_j^zP &= -\sigma_j^z, \\ P\sigma_j^xP &= \sigma_j^x. \end{aligned}$$

Every Ising bond contains two $\sigma^z$ factors, so

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

The Hilbert space splits into parity sectors $P=\pm1$. For $L\ge1$, each sector has dimension

$$\dim\mathcal H_{+} = \dim\mathcal H_{-} = 2^{L-1}.$$

The longitudinal magnetization is odd under $P$, while the energy, transverse magnetization, and even-$z$ correlators are parity even.

Spatial symmetries

The uniform periodic chain is invariant under one-site translations and reflections. The uniform open chain retains reflection about its midpoint but not translation symmetry. Disorder, nonuniform fields, boundary pinning, or a defect bond can remove these symmetries without necessarily removing Ising parity.

Reality versus physical time reversal

In the computational basis, $H$ is a real symmetric matrix, so complex conjugation $K$ is an antiunitary symmetry of that matrix representation. This should not be confused with physical spin time reversal. Physical time reversal sends every spin component to its negative; therefore a nonzero transverse field changes sign under it. The exchange-only model is time-reversal invariant, whereas the field selects an oriented external magnetic field.

Nonconserved magnetization

Neither total $z$ magnetization nor total $x$ magnetization commutes with the generic Hamiltonian:

$$\left[ H, \sum_j\sigma_j^z \right] \ne 0, \qquad \left[ H, \sum_j\sigma_j^x \right] \ne 0$$

when $Jh\ne0$. Ising parity, rather than a continuous spin component, is the model’s internal conserved quantum number.

Important Limits and Deformations

Regime	Ground-state or low-energy picture	Exact statement
$h=0$, $J>0$	two aligned $z$ configurations	classical commuting Ising limit
$J=0$, $h>0$	unique product state polarized along $+x$	independent-site limit
$0<g<1$	ordered thermodynamic phase	two symmetry-related pure phases; finite chains retain parity eigenstates
$g=1$	scale-invariant critical point	gap closes linearly; Ising universality class
$g>1$	quantum paramagnet	unique bulk phase adiabatically connected to the $+x$ product state
$h<0$	reversed transverse polarization	unitarily equivalent to $h>0$
$J<0$	antiferromagnetic Ising exchange	equivalent to $J>0$ on a bipartite chain, subject to boundary frustration
longitudinal field nonzero	explicit Ising-symmetry breaking	generic lattice model is no longer quadratic-fermion integrable

Ordered endpoint

At $h=0$, the ferromagnetic ground configurations are

$$\lvert\uparrow\uparrow\cdots\uparrow\rangle \quad\text{and}\quad \lvert\downarrow\downarrow\cdots\downarrow\rangle.$$

For a periodic chain their common energy is

$$E_0 = -JL,$$

and for an open chain it is $-J(L-1)$. A flipped domain costs $2J$ at each domain wall. Periodic boundary conditions force the total number of domain walls to be even.

Paramagnetic endpoint

At $J=0$ and $h>0$, the unique ground state is

$$\lvert\rightarrow\rightarrow\cdots\rightarrow\rangle,$$

where

$$\sigma^x\lvert\rightarrow\rangle = +\lvert\rightarrow\rangle.$$

Its energy is

$$E_0 = -hL.$$

The first excited manifold contains one site polarized along $-x$ and lies $2h$ above the ground state.

Antiferromagnetic exchange

Conjugating every even site by $\sigma^x$ reverses $\sigma_j^z$ on that sublattice while leaving $\sigma_j^x$ unchanged. It therefore maps $J<0$ to $J>0$ on an open chain and on an even periodic ring. An odd periodic ring is frustrated and cannot be mapped to an unfrustrated ferromagnetic ring without leaving a twisted bond.

Longitudinal field

Adding

$$V_z = -\lambda \sum_j\sigma_j^z$$

breaks $P$ explicitly because $PV_zP=-V_z$. It also destroys the generic free-fermion closure of the lattice Hamiltonian. The resulting model may still have controlled limits, continuum descriptions, or special integrable field-theory regimes, but those do not make the generic finite lattice model exactly solvable.

Exact Solution Status

The uniform one-dimensional model with nearest-neighbor exchange and no longitudinal field is exactly reducible to independent fermionic quasiparticles. The standard route is

$$\text{spins} \xrightarrow{\text{Jordan–Wigner}} \text{quadratic fermions} \xrightarrow{\text{Fourier}} \text{momentum pairs} \xrightarrow{\text{Bogoliubov}} \text{normal modes}.$$

The phrase “exactly solvable” is quantity dependent.

Quantity	Status in the baseline chain
finite-size spectrum	exact after boundary and parity sectors are fixed
bulk quasiparticle dispersion	elementary closed form
thermodynamic ground-state energy	one-dimensional integral
gap and critical point	exact
spontaneous order parameter	exact in the thermodynamic ordered phase
partition function	exact from independent quasiparticles
local spin correlations	exact determinant or Pfaffian representations; asymptotics require further analysis
entanglement entropy	exact correlation-matrix construction for fermionic Gaussian states
generic real-time Gaussian observables	reducible to mode evolution and Wick contractions
model with longitudinal field	not generically a quadratic-fermion problem
disordered or long-range variants	solvability must be reassessed model by model

Exact diagonalization of a finite $2^L$ matrix is not, by itself, an analytic integrability claim. Conversely, exact quasiparticle energies do not make every nonlocal spin correlation a one-line calculation.

Exact Bulk Fingerprints

For the baseline convention and the thermodynamic chain, the positive quasiparticle dispersion is

$$\varepsilon(q) = 2J \sqrt{ 1+g^2-2g\cos q }, \qquad q\in[-\pi,\pi).$$

Its minimum gives the bulk gap

$$\Delta_{\mathrm{bulk}} = 2J\lvert1-g\rvert.$$

The gap closes at

$$g_c = 1,$$

or $h=J$ in this Pauli convention. At criticality,

$$\varepsilon(q) = 4J \left| \sin\frac q2 \right|.$$

For small physical wave number $k=q/a$,

$$\varepsilon \sim 2Ja\lvert k\rvert,$$

so the emergent velocity is

$$v = \frac{2Ja}{\hbar}.$$

The thermodynamic ground-state energy density is

$$e_0(g) = -\frac{J}{\pi} \int_0^\pi \sqrt{ 1+g^2-2g\cos q } \, \mathrm dq.$$

Useful exact critical data are

$$z=1, \qquad \nu=1, \qquad \beta=\frac18, \qquad \eta=\frac14, \qquad c=\frac12.$$

Here $\beta$ is the order-parameter exponent, not inverse temperature, and $c$ is the central charge of the critical continuum theory. The spontaneous longitudinal magnetization in a selected thermodynamic pure phase is

$$m_z = \left(1-g^2\right)^{1/8}, \qquad 0\le g<1.$$

It vanishes for $g\ge1$. This nonzero value belongs to a symmetry-broken thermodynamic state. The exact parity eigenstate of any finite chain has $\langle M_z\rangle=0$.

Boundary and Parity Audit

The Jordan–Wigner string turns the spin boundary condition into a parity-dependent fermion boundary condition. For a periodic spin chain, the allowed momenta therefore depend on the eigenvalue of $P$. One must diagonalize the correct momentum grid in each parity sector and then impose the sector’s fermion-parity constraint.

Three finite-size energy differences are often called “the gap”:

1. the splitting between the lowest even- and odd-parity states;
2. the lowest excitation energy within a fixed parity sector;
3. the bulk quasiparticle gap inferred after taking $L\to\infty$.

They need not agree at finite $L$. In the ordered phase, the first can be exponentially small even while the bulk excitation gap remains nonzero. At criticality, the low-lying spacings scale as $L^{-1}$ and depend on boundary conditions and conformal sector.

At $h=0$ on a periodic ring, a local spin flip creates two domain walls and costs $4J$. The dispersion formula approaches a single-quasiparticle energy $2J$ as $g\to0$, but parity and boundary constraints decide which combinations are physical in a specified finite spin sector. This is a standard example of why the bulk dispersion alone is not a complete finite-size spectrum.

See Boundary Conditions on Lattices and Symmetry Sectors for the general bookkeeping.

Typical Observables

Energy and gaps

The ground-state energy, excitation energies, and sector-resolved gaps diagnose limiting states, criticality, and implementation errors. A complete finite-size report states both boundary conditions and the symmetry sectors being compared.

Longitudinal and transverse magnetization

Define the intensive operators

$$M_z = \frac1L \sum_{j=1}^{L} \sigma_j^z, \qquad M_x = \frac1L \sum_{j=1}^{L} \sigma_j^x.$$

$M_z$ is odd under $P$ and is the order parameter. $M_x$ is parity even and measures polarization along the applied field. In a finite parity eigenstate,

$$\langle M_z\rangle = 0,$$

so ordered behavior is instead exposed by $\langle M_z^2\rangle$, long-distance correlations, a small symmetry-breaking source, or the near-degeneracy of opposite-parity states.

Correlations and structure factor

Equal-time correlators include

$$C_{zz}(r) = \langle \sigma_j^z\sigma_{j+r}^z \rangle$$

and the connected function

$$C_{zz}^{\mathrm c}(r) = C_{zz}(r) - \langle\sigma_j^z\rangle \langle\sigma_{j+r}^z\rangle.$$

For a translation-invariant ring, one useful convention for the static structure factor is

$$S_{zz}(q) = \frac1L \sum_{j,k} e^{iq(j-k)} \langle \sigma_j^z\sigma_k^z \rangle.$$

Its normalization must be stated when comparing codes or references. Correlation Functions Overview and Structure Factors own the general conventions.

Domain-wall density

For a bond set $\mathcal E$,

$$N_{\mathrm{DW}} = \sum_{(j,k)\in\mathcal E} \frac{ 1-\sigma_j^z\sigma_k^z }{2}.$$

This operator counts antiparallel bonds in a computational-basis configuration and measures their quantum expectation in a superposition. It is directly tied to the exchange energy.

Entanglement

For a contiguous region $A$, the von Neumann entropy is

$$S_A = -\operatorname{Tr} \left( \rho_A\ln\rho_A \right).$$

Away from criticality it saturates with subsystem size in the ground state of the gapped chain. At the critical point it grows logarithmically, with a coefficient controlled by $c=1/2$ and by the boundary geometry. Entanglement and Criticality owns the general scaling laws.

Physical Phenomena

Quantum competition

Exchange and field favor incompatible local descriptions. The exchange term prefers definite $\sigma^z$ alignment, while the field term continually mixes those configurations through spin flips. Varying $g$ changes the ground state even at zero temperature because it changes quantum fluctuations rather than thermal populations.

Finite cats and thermodynamic order

For a finite chain with $h>0$, the exact ground state has definite parity. Deep in the ordered regime it approaches an even superposition of the two aligned configurations, while the lowest odd state becomes nearly degenerate. Spontaneous symmetry breaking requires an order of limits:

$$\lim_{\lambda\to0^+} \lim_{L\to\infty} \langle M_z\rangle_{\lambda,L} \ne 0,$$

where $\lambda$ is a longitudinal source. Reversing the limits leaves the finite-system parity symmetry intact. Spontaneous Symmetry Breaking develops this distinction.

Domain walls and quasiparticles

At small $g$, low-energy excitations are naturally described as fluctuating domain walls. The exact fermionic quasiparticles are the delocalized normal modes of those defects. Their statistics are a property of the nonlocal solution variables; the original Hilbert space remains a tensor product of spins.

Quantum criticality

At $g=1$, the correlation length and correlation time diverge, the gap closes, and long-distance observables lose sensitivity to microscopic details. The critical point belongs to the two-dimensional classical Ising universality class and is described by a $c=1/2$ conformal field theory in the scaling limit.

No finite-temperature ordered phase in one dimension

For the short-range one-dimensional chain, every nonzero temperature introduces a finite density of domain walls. The finite-temperature correlation length can be large at low temperature, but true long-range Ising order does not persist for $T>0$. This does not contradict the zero-temperature quantum critical point.

Minimal Worked Example: Two-Site Open Chain

For $L=2$ with one open bond,

$$H = -J\sigma_1^z\sigma_2^z - h \left( \sigma_1^x+\sigma_2^x \right).$$

Introduce parity-adapted states

$$\begin{aligned} \lvert F_{\pm}\rangle &= \frac{ \lvert00\rangle \pm \lvert11\rangle }{\sqrt2}, \\ \lvert A_{\pm}\rangle &= \frac{ \lvert01\rangle \pm \lvert10\rangle }{\sqrt2}. \end{aligned}$$

The $P=+1$ block in the basis $\{\lvert F_+\rangle,\lvert A_+\rangle\}$ is

$$H_+ = \begin{pmatrix} -J & -2h \\ -2h & J \end{pmatrix}.$$

The two negative-parity states do not mix:

$$\begin{aligned} H\lvert F_-\rangle &= -J\lvert F_-\rangle, \\ H\lvert A_-\rangle &= +J\lvert A_-\rangle. \end{aligned}$$

Therefore the complete spectrum is

$$\operatorname{spec}(H) = \left\{ -\sqrt{J^2+4h^2}, -J, +J, +\sqrt{J^2+4h^2} \right\}.$$

At $J=1$ and $h=0.7$, the sorted dimensionless energies are

$$\frac EJ \in \left\{ -1.720465053409, -1, +1, +1.720465053409 \right\}.$$

This example tests operator normalization, basis ordering, parity, Hermiticity, and the open-bond convention without requiring the many-mode exact solution.

Numerical Benchmarks

Finite-matrix contract

MB-B001 fixes the complete small-chain validation data. Its stronger assembly case uses $L=4$, periodic boundaries, $J=1$, and $h=0.7$. Required fingerprints include

$$\begin{aligned} \dim\mathcal H &= 16, \\ \frac{E_0}{J} &= -4.563856065203, \\ \operatorname{Tr}H &= 0, \\ \frac1{16} \operatorname{Tr}H^2 &= 4(J^2+h^2) = 5.96, \\ [H,P] &= 0. \end{aligned}$$

The benchmark page owns the complete level-and-degeneracy table. A code should pass the $L=2$ open-chain test before the periodic $L=4$ test, because a wrong two-site ring can accidentally conceal double-counted bonds.

The second moment follows without diagonalization. Distinct nonidentity Pauli strings are orthogonal under the Hilbert–Schmidt inner product, so for an ordinary $L$-site ring,

$$\frac1{2^L} \operatorname{Tr}H^2 = L(J^2+h^2).$$

This trace identity checks all matrix entries, not merely the lowest eigenvalue.

Critical-scaling contract

MB-B009 uses the critical open chain $h=J>0$. Its exact global gap is

$$\Delta_L = 4J \sin \left( \frac{\pi}{4L+2} \right).$$

For example,

$$\frac{\Delta_8}{J} = 0.369073437853208.$$

Writing $\ell=L+\tfrac12$ gives

$$\frac{ \ell\Delta_L }{ \pi J } = \frac{\sin x}{x}, \qquad x = \frac{\pi}{4\ell},$$

and hence

$$\frac{ \ell\Delta_L }{ \pi J } = 1 - \frac{\pi^2}{96\ell^2} + O(\ell^{-4}).$$

This contract tests the dynamical exponent $z=1$ and the leading boundary-aware correction. It is a global gap, not a gap restricted to one parity sector.

Reproducibility status

The artifact catalog reserves

notebooks/many-body-statistical/transverse_field_ising_ed.ipynb

for the spin-bit Hamiltonian and complete small-chain spectrum. The notebook is planned rather than silently treated as published. Until it passes the release gates on Reproducible Notebooks, the versioned benchmark contracts are the authoritative numerical targets.

Variants and Handoffs

Variant	What changes	Canonical route
classical Ising chain	remove the noncommuting transverse field and study thermal configurations	Ising Chain model card
antiferromagnetic chain	change the exchange sign and track frustration or sublattice rotation	Transverse-Field Ising Model
longitudinal-field chain	break Ising parity and generic free-fermion integrability	Quantum Phase Transitions
XY chain	add unequal $x$- and $y$-exchange couplings	Exact Solutions Preview
random Ising chain	make bonds or fields site dependent	separate disorder and strong-disorder methods are required
long-range Ising model	replace nearest-neighbor exchange by algebraically decaying couplings	universality and causal structure can change
higher-dimensional Ising model	change the bond graph	no generic Jordan–Wigner free-fermion solution
driven or quenched chain	specify an initial state and time-dependent protocol	nonequilibrium observables require their own contract

The Ising Chain Hamiltonian card is the compact formula locator. This dossier is the convention-complete record. The teaching article remains the place for derivation and interpretation.

Common Mistakes

• Calling the exchange-only diagonal Hamiltonian and the transverse-field quantum model the same problem without qualification.
• Replacing Pauli matrices by $s^\alpha=\sigma^\alpha/2$ without changing the couplings.
• Writing a periodic nearest-neighbor sum for $L=2$ and counting the same physical bond twice.
• Quoting $g_c=1$ without stating the Hamiltonian and operator normalization.
• Assuming a finite parity eigenstate has nonzero longitudinal magnetization.
• Confusing the exponentially small parity-partner splitting with the nonzero bulk quasiparticle gap in the ordered phase.
• Using one fermionic momentum grid for both parity sectors of a periodic spin chain.
• Interpreting exact quasiparticle energies as proof that every spin correlation has an elementary closed form.
• Treating a longitudinal-field perturbation as though the quadratic solution survived unchanged.
• Inferring a finite-temperature ordered phase from a large but finite low-temperature correlation length.

Exercises

1. Verify the Ising symmetry

Show directly that $P=\prod_j\sigma_j^x$ commutes with the open-chain Hamiltonian. Determine the parity of $M_z$ and $M_x$.

Solution

On one site,

$$\sigma^x\sigma^z\sigma^x = -\sigma^z, \qquad \sigma^x\sigma^x\sigma^x = \sigma^x.$$

Conjugation by $P$ therefore changes the sign of each $\sigma_j^z$. A bond contains two such factors:

$$P \left( \sigma_j^z\sigma_{j+1}^z \right) P = \sigma_j^z\sigma_{j+1}^z.$$

Each $\sigma_j^x$ is also invariant, so $PHP=H$ and $[H,P]=0$. The same conjugation gives

$$PM_zP = -M_z, \qquad PM_xP = M_x.$$

Thus $M_z$ is parity odd and $M_x$ is parity even.

2. Translate the normalization

An article writes

$$H' = -J_s \sum_j s_j^zs_{j+1}^z - h_s \sum_j s_j^x,$$

with $s^\alpha=\sigma^\alpha/2$. Express $J_s$ and $h_s$ in terms of this dossier’s $J$ and $h$. At what ratio $h_s/J_s$ does the critical point occur?

Solution

Using

$$\sigma_j^z\sigma_{j+1}^z = 4s_j^zs_{j+1}^z, \qquad \sigma_j^x = 2s_j^x,$$

gives

$$J_s = 4J, \qquad h_s = 2h.$$

The baseline critical point is $h=J$, so

$$\left. \frac{h_s}{J_s} \right|_{\mathrm c} = \frac{2J}{4J} = \frac12.$$

The apparent factor-of-two shift is entirely a normalization change.

3. Diagonalize the two-site chain

Use the parity-adapted basis to derive the complete $L=2$ open-chain spectrum. Which sector contains the ground state when $h>0$?

Solution

The positive-parity block is

$$H_+ = \begin{pmatrix} -J & -2h \\ -2h & J \end{pmatrix},$$

whose characteristic polynomial is

$$\det(E\mathbb I-H_+) = E^2-J^2-4h^2.$$

Its eigenvalues are $\pm\sqrt{J^2+4h^2}$. The two negative-parity states $\lvert F_-\rangle$ and $\lvert A_-\rangle$ have energies $-J$ and $+J$. Therefore

$$\operatorname{spec}(H) = \left\{ -\sqrt{J^2+4h^2}, -J, +J, +\sqrt{J^2+4h^2} \right\}.$$

For $h>0$,

$$\sqrt{J^2+4h^2} > \lvert J\rvert,$$

so the ground state is the negative eigenvalue of the $P=+1$ block.

4. Count domain walls on a ring

Prove that every computational-basis configuration on a periodic ring has an even number of domain walls. Why does this constrain the $h=0$ excitation energy?

Solution

Assign $z_j=\pm1$ to site $j$. A domain wall occurs when $z_jz_{j+1}=-1$. Around a ring,

$$\prod_{j=1}^{L} z_jz_{j+1} = \left( \prod_{j=1}^{L}z_j \right)^2 = 1.$$

If $N_{\mathrm{DW}}$ bonds have product $-1$, the same product is $(-1)^{N_{\mathrm{DW}}}$. Hence $N_{\mathrm{DW}}$ is even. Each wall changes one bond energy from $-J$ to $+J$ and costs $2J$, so the lowest domain-wall excitation of the periodic $h=0$ spin chain contains two walls and costs $4J$.

5. Extract the critical finite-size exponent

Starting from

$$\Delta_L = 4J \sin \left( \frac{\pi}{4L+2} \right),$$

derive its leading large-$L$ behavior and identify $z$.

Solution

Let $\ell=L+\tfrac12$. Then

$$\Delta_L = 4J \sin \left( \frac{\pi}{4\ell} \right).$$

Using $\sin x=x-x^3/6+O(x^5)$ gives

$$\begin{aligned} \Delta_L &= \frac{\pi J}{\ell} \left[ 1 - \frac{\pi^2}{96\ell^2} + O(\ell^{-4}) \right] \\ &= \frac{\pi J}{L} + O(L^{-2}). \end{aligned}$$

At a quantum critical point, a characteristic gap scales as $\Delta_L\propto L^{-z}$. Therefore $z=1$.

Cross-Links

• Model Encyclopedia for the shared dossier schema and model-family map.
• Spin-$1/2$ Chain dossier for the umbrella XYZ-family conventions and generic trace audit.
• Transverse-Field Ising Model for the full physical narrative, exact dispersion, phases, and entanglement.
• Jordan–Wigner Transformation for the operator mapping and boundary-sector derivation.
• Exact Solutions Preview for the wider hierarchy of free-fermion and Bethe-ansatz methods.
• Quantum Phase Transitions for scaling fields, finite-size scaling, and universality.
• Order Parameters for source fields and finite-system diagnostics.
• Benchmark Problems for the complete MB-B001 and MB-B009 contracts.
• Ising Chain model card and Hamiltonian card for compact lookup.

References

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2. P. Pfeuty, “The one-dimensional Ising model with a transverse field,” Annals of Physics 57, 79–90 (1970), doi:10.1016/0003-4916(70)90270-8.
3. E. Barouch and B. M. McCoy, “Statistical mechanics of the XY model. II. Spin-correlation functions,” Physical Review A 3, 786–804 (1971), doi:10.1103/PhysRevA.3.786.
4. J. B. Kogut, “An introduction to lattice gauge theory and spin systems,” Reviews of Modern Physics 51, 659–713 (1979), doi:10.1103/RevModPhys.51.659.
5. S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press (2011), doi:10.1017/CBO9780511973765.
6. B. K. Chakrabarti, A. Dutta, and P. Sen, Quantum Ising Phases and Transitions in Transverse Ising Models, Springer (1996), doi:10.1007/978-3-540-49865-0.
7. A. Dutta et al., Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information, Cambridge University Press (2015), doi:10.1017/CBO9781107706057.
8. G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, “Entanglement in quantum critical phenomena,” Physical Review Letters 90, 227902 (2003), doi:10.1103/PhysRevLett.90.227902.
9. P. Calabrese and J. Cardy, “Entanglement entropy and quantum field theory,” Journal of Statistical Mechanics P06002 (2004), doi:10.1088/1742-5468/2004/06/P06002.
10. S. Suzuki, J.-i. Inoue, and B. K. Chakrabarti, Quantum Ising Phases and Transitions in Transverse Ising Models, 2nd ed., Springer (2013), doi:10.1007/978-3-642-33039-1.