Quantum magnetism

Quantum magnetism studies collective phases and excitations of interacting quantum spins.

Geometric frustration occurs when the interaction graph prevents all local preferred configurations from being satisfied simultaneously. Antiferromagnetic spins can be antiparallel on every bond of a bipartite graph, but three equal spins on a triangle instead minimize their total exchange by summing to zero, giving a coplanar

120

-degree arrangement.

Hubbard model ( $H = H_{hop} + U_{H} \sum_{n} n_{n ↑} n_{n ↓}$ )

 1  0

A lattice fermion model combining hopping with local density interaction, conventionally written

U_{H} \sum_{n} n_{n ↑} n_{n ↓}

. Its interaction parameter must be matched when a spin-square representation is used:

- 2 U (S^{2} - 3 n /4)

equals

3 U n_{↑} n_{↓}

, so

U_{H} = 3 U

for that representation.

On-site spin-square normal-ordering identity ( $: S^{2} := S^{2} - 3 n /4 = - 3 n_{↑} n_{↓} /2$ )

 0  0

For a single spinful fermion orbital,

S^{2} = 3 (n - 2 n_{↑} n_{↓}) /4

. The contraction term is

3 n /4

, so

: S^{2} := S^{2} - 3 n /4 = - 3 n_{↑} n_{↓} /2

. Empty and doubly occupied states are spinless while singly occupied states have spin one half, which also proves the identity directly.

Itinerant ferromagnetism ( $ϵ_{kσ} = ϵ_{k} - σ M$ )

 0  0

Ferromagnetic spin polarization carried by mobile electrons rather than a separate fixed-spin lattice. A homogeneous spin field splits the electron bands and competes with a positive field cost. Mean-field instability is governed by the Stoner criterion; the ordered metal supports transverse spin waves as well as fermionic particle-hole excitations.

Mean-field itinerant ferromagnetic Goldstone mode ( $E_{q} \sim D q^{2}$ )

 0  0

An isotropic spin-polarized metal has a degenerate magnetization direction and a long-wavelength transverse magnon with quadratic dispersion. Nonzero spin density pairs the two transverse broken generators into one type-B Goldstone boson. Exchange-split Fermi surfaces also support fermionic low-energy excitations; the magnon is distinct from longitudinal amplitude motion and the spin-flip continuum.

Stoner criterion ( $U_{c} χ_{0} = 1, χ_{0} = \int ν (- n_{F}^{'}) d ϵ$ )

 1  0

In the homogeneous spin-vector mean-field scheme with field cost

M^{2} / (2 U)

and spin-summed density of states per site, the paramagnetic curvature is proportional to

U^{- 1} - \int ν (ϵ) [- n_{F}^{'} (ϵ)] d ϵ

. It changes sign at

U χ_{0} = 1

. For a smooth low-temperature density of states,

χ_{0} ≃ ν (μ)

, giving the familiar Fermi-level criterion. Interaction and spin-counting conventions must be specified before comparing coefficients.

Homogeneous spin-field effective action ( $S_{eff} = βN M^{2} / (2 U) - \sum_{kσ} ln [1 + e^{- β (ϵ_{k} - μ - σ M)}]$ )

 0  0

Integrating free fermions in a constant auxiliary spin field gives

S_{eff} = βN M^{2} / (2 U) - \sum_{kσ} ln [1 + e^{- β (ϵ_{k} - μ - σ M)}]

. Differentiation gives the self-consistent spin polarization. The extensive action is a mean-field restriction of the full auxiliary-field functional integral, not an exact omission of all spatial/time fluctuations.

Coordination number of a lattice ( $z$ )

 0  0

The coordination number

z

is the number of nearest neighbors of a lattice site. A Kac normalization divides a pair coupling by

z

so that the interaction energy per site remains finite as

z \to \infty

Néel state

 0  0

A Néel state is a two-sublattice product state with antiparallel spin polarization on the two sublattices. It is the classical ordered reference state of a bipartite antiferromagnet.

Néel order parameter

 0  0

The Néel order parameter is the slowly varying staggered magnetization of an antiferromagnet. On a bipartite lattice it changes sign between neighboring sublattices before coarse graining.

Quantum depletion of Néel order ( $M_{s} = S - N^{- 1} \sum_{k} v_{k}^{2}$ )

 0  0

For a linear spin-wave approximation, the ground state is empty of the diagonal quasiparticles but contains the original spin bosons. If

a_{k} = u_{k} α_{k} - v_{k} α_{- k}^{†}

, their occupation number is

v_{k}^{2}

. The staggered magnetization is reduced from

S

S - N^{- 1} \sum_{k} v_{k}^{2}

. For the one-dimensional nearest-neighbor Heisenberg antiferromagnet,

v_{k}^{2} = (∣ sin k ∣^{- 1} - 1) /2

and the integral diverges logarithmically. This invalidates finite ordered magnetization within the approximation; it is not itself a determination of the exact spectral gap.

Staggered magnetization

 0  0

The staggered magnetization weights the local magnetization with opposite signs on the two sublattices of a bipartite graph. It is nonzero in a Néel state even when the uniform magnetization vanishes.

Uniform magnetization

 0  0

The uniform magnetization is the nonalternating component of local spin polarization. In a nearly Néel-ordered antiferromagnet it describes the small canting shared by the two sublattices.

Long-range order

 0  0

Long-range order means that an appropriate connected or symmetry-resolved correlation remains nonzero at arbitrarily large separation. It cannot be created from a product state by a bounded-depth circuit of bounded-range gates.

Antiferromagnetic spin wave

 0  0

An antiferromagnetic spin wave is a long-wavelength fluctuation of staggered magnetic order. In an isotropic ordered antiferromagnet its low-energy dispersion is linear.

Hypercubic antiferromagnetic spin-wave dispersion ( $ω_{k} = 2 dJ S 1 - γ_{k}^{2}$ )

 0  0

For a nearest-neighbour Heisenberg antiferromagnet on a hypercubic lattice, the linear spin-wave approximation gives

γ_{k} = d^{- 1} \sum_{j} cos (k_{j} a)

and the displayed frequency. Near a gapless point its spin-wave velocity is

2 J S a d

. Its ground-state boson occupation is

[1/ 1 - γ_{k}^{2} - 1] /2

, producing quantum depletion of Néel order. Exact zero modes need an infrared regulator or separate global-rotation treatment.

Spin-wave velocity ( $c$ )

 0  0

The spin-wave velocity is the slope

c = lim_{k \to 0} ω (k) / k

of a linearly dispersing spin wave. For the leading square-lattice Heisenberg antiferromagnet continuum theory,

c = 22 J S a

k^{2}

Nearest-neighbour ferromagnetic magnon dispersion ( $E_{k} = 4 J S sin^{2} (k /2)$ )

 0  0

For

H = - J \sum_{n} S_{n} \cdot S_{n + 1}

and unit spacing, the one-magnon energy is

2 J S (1 - cos k) = 4 J S sin^{2} (k /2)

. The exact one-magnon hopping matrix has diagonal

2 J S

and neighbour entries

- J S

. The same result follows from the linear spin-wave approximation. Its long-wavelength stiffness is

J S

, with energy proportional to

k^{2}

Next-nearest-neighbour ferromagnetic magnon dispersion ( $E_{k} = 4 S [J_{1} sin^{2} (k /2) + J_{2} sin^{2} k]$ )

 0  0

Positive nearest- and next-nearest-neighbour exchanges give one-magnon energy

2 S [J_{1} (1 - cos k) + J_{2} (1 - cos 2 k)]

. At small momentum this is

S (J_{1} + 4 J_{2}) k^{2}

. The quadratic-boson result describes the harmonic many-magnon spectrum; higher-order spin-wave interactions remain in the full Hamiltonian.

Damped ferromagnetic spin wave

 0  0

A damped ferromagnetic spin wave has a complex pole

ω = \pm ω_{k} - i Γ_{k}

. Both poles of its retarded Green function must lie in the lower half of the complex frequency plane so that the response respects causality.

Spin coherent-state Berry phase

 0  0

The spin coherent-state path integral contains a Berry phase equal, modulo gauge choices, to the spin

S

times the solid angle swept on the Bloch sphere. Single-valued path-integral amplitudes require

2 S \in Z

Wess–Zumino term

 0  0

For a spin coherent-state history, a Wess--Zumino term records the oriented target-sphere area bounded by the spin path. Its variation is local even though a single gauge-patch expression for the area need not be global.

Antiferromagnetic Wess–Zumino term

 0  0

In a bipartite antiferromagnet, the smooth staggered contributions to the spin Wess--Zumino terms cancel between sublattices. Their term linear in uniform canting is proportional to

m \cdot (n \times \partial_{t} n)

CP1 spinor representation ( $C P^{1}$ spinor representation)

 0  0

The CP1 spinor representation writes a unit vector as

n_{i} = z^{†} σ_{i} z

with

z^{†} z = 1

. The phase redundancy

z \mapsto e^{i α} z

identifies normalized spinors that represent the same point of the two-sphere.

Ferromagnetic Wess–Zumino term

 0  0

The ferromagnetic Wess–Zumino term is the first-order kinetic term

iκ z^{†} \overset{z}{˙}

in the CP1 description. A local phase change of

z

changes it only by the total derivative

- κ \overset{α}{˙}

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Quantum magnetism is a field of study within condensed matter physics that explores the magnetic properties and behavior of materials at the quantum level. It primarily focuses on how quantum mechanical interactions among electrons, their spins, and lattice structures lead to a variety of magnetic phenomena. In classical terms, magnetism is commonly associated with the alignment of magnetic moments (small magnetic fields due to the spin and orbital motion of electrons).

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Quantum magnetism

Geometrical frustration

Hubbard model ( $H = H_{hop} + U_{H} \sum_{n} n_{n ↑} n_{n ↓}$ )

On-site spin-square normal-ordering identity ( $: S^{2} := S^{2} - 3 n /4 = - 3 n_{↑} n_{↓} /2$ )

Itinerant ferromagnetism ( $ϵ_{kσ} = ϵ_{k} - σ M$ )

Mean-field itinerant ferromagnetic Goldstone mode ( $E_{q} \sim D q^{2}$ )

Stoner criterion ( $U_{c} χ_{0} = 1, χ_{0} = \int ν (- n_{F}^{'}) d ϵ$ )

Homogeneous spin-field effective action ( $S_{eff} = βN M^{2} / (2 U) - \sum_{kσ} ln [1 + e^{- β (ϵ_{k} - μ - σ M)}]$ )

Coordination number of a lattice ( $z$ )

Néel state

Néel order parameter

Quantum depletion of Néel order ( $M_{s} = S - N^{- 1} \sum_{k} v_{k}^{2}$ )

Staggered magnetization

Uniform magnetization

Long-range order

Antiferromagnetic spin wave

Hypercubic antiferromagnetic spin-wave dispersion ( $ω_{k} = 2 dJ S 1 - γ_{k}^{2}$ )

Spin-wave velocity ( $c$ )

Zeeman splitting

Ferromagnetic spin chain

Easy-plane anisotropy

Ferromagnetic magnon

Nearest-neighbour ferromagnetic magnon dispersion ( $E_{k} = 4 J S sin^{2} (k /2)$ )

Next-nearest-neighbour ferromagnetic magnon dispersion ( $E_{k} = 4 S [J_{1} sin^{2} (k /2) + J_{2} sin^{2} k]$ )

Damped ferromagnetic spin wave

Spin coherent-state Berry phase

Wess–Zumino term

Antiferromagnetic Wess–Zumino term

CP1 spinor representation ( $C P^{1}$ spinor representation)

Ferromagnetic Wess–Zumino term

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