The spin stiffness or spin rigidity is a constant which represents the change in the ground state energy of a spin system as a result of introducing a slow in-plane twist of the spins. The importance of this constant is in its use as an indicator of

quantum phase transitions In physics, a quantum phase transition (QPT) is a phase transition between different quantum phases (phases of matter at zero temperature). Contrary to classical phase transitions, quantum phase transitions can only be accessed by varying a physica ...

—specifically in models with metal-insulator transitions such as Mott insulators. It is also related to other

topological invariant In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...

s such as the

Berry phase In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the ...

and

Chern number In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches o ...

s as in the

Quantum Hall effect The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exhi ...

Mathematically

Mathematically it can be defined by the following equation: :

\rho_s = \cfrac\cfrac, _

where

E_0

is the ground state energy,

\theta

is the twisting angle, and N is the number of lattice sites.

Spin stiffness of the Heisenberg model

Start off with the simple Heisenberg spin Hamiltonian: :

H_\mathrm = -J \sum_ \left_i^z S_j^z + \cfrac(S_i^+ S_j^- + S_i^- S_j^+)\right /math>

Now we introduce a rotation in the system at site i by an angle θ

_i around the z-axis: :

S_i^+ \longrightarrow S_i^+e^

S_i^- \longrightarrow S_i^-e^

Plugging these back into the Heisenberg Hamiltonian: :

H(\theta_) = -J \sum_ \left_i^z S_j^z + \cfrac(S_i^+e^ S_j^-e^ + S_i^-e^ S_j^+e^)\right /math>

now let θ

_ij = θ_i - θ_j and expand around θ_ij = 0 via a MacLaurin expansion only keeping terms up to second order in θ_ij :

H \approx H_\mathrm - J \sum_ \left theta_J_^ - \cfrac\theta_^2 T_^\right /math>

where the first term is independent of θ and the second term is a

perturbation Perturbation or perturb may refer to: * Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly * Perturbation (geology), changes in the nature of alluvial deposits over time * Perturbati ...

for small θ. :

J_^ = \cfrac(S_i^+ S_j^- - S_i^- S_j^+)

is the z-component of the spin current operator :

T_ = \cfrac(S_i^+ S_j^- + S_i^- S_j^+)

is the "spin kinetic energy" Consider now the case of identical twists, θ_x only that exist along nearest neighbor bonds along the x-axis. Then since the spin stiffness is related to the difference in the ground state energy by :

E(\theta) - E(0) = N\rho_s\theta_x^2

then for small θ_x and with the help of second order perturbation theory we get: :

\rho_s = \cfrac \left \cfrac\langle T_x \rangle + \sum_ \cfrac\right /math>

References

* * * * * *{{cite journal , author=R. G. Melko, A. W. Sandvik, and D. J. Scalapino , title=Two-dimensional quantum XY model with ring exchange and external field , journal=

Physical Review B ''Physical Review B: Condensed Matter and Materials Physics'' (also known as PRB) is a peer-reviewed, scientific journal, published by the American Physical Society (APS). The Lead Editor of PRB is Stephen E. Nagler and the Chief Editor is Sarma ...

, volume=69 , issue=10 , pages=100408–100412 , year=2004 , doi=10.1103/PhysRevB.69.100408 , arxiv = cond-mat/0311080 , bibcode = 2004PhRvB..69j0408M , s2cid=119491422 Quantum mechanics Magnetism Statistical mechanics

Mathematically

Spin stiffness of the Heisenberg model

See also

References