Spin gap, or Magnetic Interactions in Flatland

• The excitation spectrum of three-dimensional magnets does not have a gap. Spin gap is an essentially low-D phenomenon and has a purely quantum origin; its value is determined by the strength of exchange interactions between the spins.
• Spin gap is a phenomenon which, due to quantum fluctuations, destroys magnetically ordered ground state at low temperatures despite strong interactions between magnetic units

Direct exchange between the compass needles

Power drawn from the magnetosphere is an invisible source that compels the needle on this compass to point North.

Let us now fix this compass in a certain position and use it as a magnet to move other compass needles. An experience tells us that the needles arranhe parallel or antiparallel, depending on relative position. It means that magnetis interactions are directional, in contract to charge interactions.

Exchange interactions: the S = 1/2 Heisenberg Dimer

Two spins, S1 and S2, interact according to the Hamiltonian H = J S₁•S₂

H = J S_i•S_j

The spins can couple to form an S_tot = 0 (singlet state)
and an S_tot = 1 (triplet state) 

where S_tot is the total spin for the system.

The energies of the states depend on their mutual orientation and are calculated by taking an scalar product:

S₁•S₂ = ½ (S_tot•S_tot-S₁•S₁-S₂•S₂) = ½ (S_tot(S_tot+1)- ¾ – ¾)

Now we find a relationship between the spins in the excited and ground states.

N_ex/N_gr = exp (-J/T) /(1+3exp (-J/T)

Interactions of spins on a 2D plane

Create unpaired electrons. Let them interact.

Get two possibilities:

Align parallel
S = ½+ ½ = 1

Align antiparallel
S = ½- ½ = 0

Magnetic susceptibility

For singlet state S=0, and therefore magnnetic susceptibility equals to zero.

Let us increase the temperature.

Now we have some spins with S=1, and to calculate magnetic suscepribility, we must multiply the previous formula on transiiton probability:

Temperature dependence of magnetic susceptibility is characterized by exponentional decay at low temperatures and a broad maximum.

Dimensionless magnetic susceptibility for lowD systems

χ_JC = χ_molarJ/C, where C is the Curie Constant

χ_maxJ/C ∼ 0.58770511 at T_max/J = 0.6408510

The exchange integral can be directly calculated from the position of the maximum, and the spin concentration from the height at the maximum.

Very Important Note!

If the sample contains non-interacting paramagnetic spins, their contribution is 2 orders of magnitude larger than from interacting ones.

Examples of exchange coupled spin networks

Magnetic susceptibilities of exchange coupled spin networks

Theoretical models describing possible arrangement of S= ½ spins:

• The Heisenberg chain with isotropic antiferromagnetically coupled spins (Bonner-Fisher),
• Dimerised chain (Bleaney-Bowers),
• Alternating chain (Hatfield),
• Spin ladder (Troyer-Tsunetsugu-Würtz).

Why do we need spin gapped magnets?

• The physics of low-dimension quantum antiferromagnets (AF) is intriguing and surprising. AF spin chains or ladders display exotic behavior such as spin liquid, spin glass, and spin ice states; magnetic orders, spin Peierls state, etc. depending on the value of the spin, the dimensionality of the material, the anisotropy, the strengths and signs of the magnetic couplings.
• Quantum phase transitions have been extensively studied over the last decades both from a theoretical and an experimental point of view.
• It is well known that a magnetic system can show a crossover from a long range ordered state to quasi 1D magnetic behavior or even high-Tc superconductivity.

Quantum spin liquid (QSL)

• At sufficiently low temperatures, condensed-matter systems tend to develop order. An exception are quantum spin-liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures.
• Physicists started paying more attention to quantum spin liquids in 1987, when Nobel laureate Philip W. Anderson theorized that quantum spin liquid theory may relate to the phenomenon of high-temperature superconductivity,
• The QSL is a solid crystal, but its magnetic state is described as liquid: the magnetic orientations of the individual particles within it fluctuate constantly, resembling the constant motion of molecules within a true liquid.
• There is no static order to magnetic moments, but there is a strong interaction between them, and due to quantum effects, they don’t lock in place

Quantum spin liquids are exotic ground states of frustrated quantum magnets, in which local moments are highly correlated but still fluctuate strongly down to zero temperature.