Question

The Hamiltonian for the spin 1/2 ferromagnetic Heisenberg spin chain is H=−J∑iσ⃗ i⋅σ⃗ i+1

with J>0
and σ⃗ i
the Pauli matrices acting on ith lattice site.

The state with spin up at every lattice site is an eigenstate. This can be seen since clearly it is an eigenstate of the −Jσziσzi+1
term and the −J(σxiσxi+1+σyiσyi+1)
terms cancel acting on it. And of course there is nothing special about the z-axis. The state with all spin up belongs to the irreducible representation with net spin N/2
where N
is the number of lattice sites. Any state in this space should have the same energy by rotational symmetry.

Now my question is what happens in the thermodynamic limit at zero temperature? I thought that the Mermin-Wagner-Coleman theorem precludes spontaneous symmetry breaking even at zero temperature for a quantum system with one spatial dimension. Canonical examples include the 1+1 dimensional sigma models on a sphere which are closely related to the classical Heisenberg model in 2D.

But this model seems to have obvious spontaneous symmetry breaking for any finite N
. Is there a loophole here or is there something technical going on in taking the thermodynamic limit that restores symmetry?
Options:

A) There is a loophole in the Hamiltonian formalism that prevents spontaneous symmetry breaking.
B) The thermodynamic limit implies an infinite number of particles and states, which restores symmetry.
C) The symmetry breaking is an artifact of the finite size of the system and gets resolved in the thermodynamic limit.
D) Symmetry breaking occurs due to the nature of the Pauli matrices, and there's no restoration of symmetry in the thermodynamic limit.

Answer 1

The symmetry breaking observed in the Heisenberg spin chain model for finite systems is an artifact that gets resolved in the thermodynamic limit. In this limit, quantum fluctuations play a significant role and restore the symmetry, conforming to the predictions of the Mermin-Wagner-Coleman theorem.

Step-by-step explanation:

The question is about the behavior of the spin ½ ferromagnetic Heisenberg spin chain in the thermodynamic limit at zero temperature, specifically in the context of the Mermin-Wagner-Coleman theorem. The student states that the Hamiltonian for this system describes a spin chain where each spin interacts with its neighbor and the state with all spins up is an eigenstate. They correctly point out that by rotational symmetry, states with all spins aligned along any axis would have the same energy.

In the thermodynamic limit, the Mermin-Wagner-Coleman theorem suggests that in a one-dimensional quantum system at zero temperature, long-range order and spontaneous symmetry breaking shouldn't occur. However, the student observes that for a finite number of lattice sites N, the model seems to show spontaneous symmetry breaking. They are asking whether this breaking of symmetry persists in the thermodynamic limit.

The correct answer is C) The symmetry breaking is an artifact of the finite size of the system and gets resolved in the thermodynamic limit. In the thermodynamic limit, where the number of particles approaches infinity, fluctuations destroy long-range order, and quantum effects restore the symmetry that appeared to be broken in finite systems.