14
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Examples of Matrix Product States
You can think of an MPS as being built up by objects with three indices. How to easily represent such an object? We can think of this as a matrix where each entry is a vector (in particular the vector ...
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Is there Difference Between 1D and 2D in Spin model?
The resulting '1D' Ising system has long-range interactions (while the initial 2D system had only short-range interactions). Approximating the state of the system by a MPS works well for chains with ...
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Why can we choose spin-1/2 degrees of freedom to commute?
Qubits are neither fermionic nor bosonic, but you can use either fermionic or bosonic degrees of freedom to store qubits.
The notion of a qubit has nothing to do with exchange symmetry nor with the ...
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What new features does the Heisenberg Model have compared to the Ising Model?
As this is a list-like question, let me list a few things (without much discussion -- feel free to ask specific questions about individual points). Each item mentions what the Heisenberg model (HM) ...
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Bogoliubov-de-Gennes (BdG) formalism of Hamiltonians
The Bogoliubov-de-Gennes Hamiltonian is a mean-field Hamiltonian, that is, a one-body (quadratic) Hamiltonian: it is by no means equivalent to the original many-body Hamiltonian. The two-body forces ...
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Why do we disorder-average before/after taking the logarithm of the partition function for annealed/quenched disorder?
To fix the idea, let's consider a spin glass Hamiltonian $H(\sigma,J)$, where $\sigma$ are the spins and $J$ is a random variable with distribution $p(J)$ representing the couplings.
An example is ...
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10
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Goldstone's theorem, symmetry breaking and the Heisenberg model
The Heisenberg model is actually an example of an exception to the standard Goldstone theorem for relativistic QFT. In the standard case, we expect that each broken symmetry yields a gapless mode with ...
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The "replica trick" initial formula
I realize this thread is old, but I was also reading the "for Pedestrians" introduction and I had the same question. Although the previous response seems reasonable, it does not address the ...
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What is the difference between classical and quantum Ising model?
You are correct that for $h=0$ the quantum Ising model reduces to the classical model. Assuming a 2D square lattice this model has been solved exactly by Onsager. It undergoes a phase transition at a ...
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Number of Goldstone Modes in the Heisenberg Model?
Having a Goldstone mode at momentum $\boldsymbol k = (k_x,k_y,k_z)$ requires that the energy vanishes there, i.e. $\varepsilon(\boldsymbol k) = 0$. In the periodic Brillouin zone $\left[ -\frac{\pi}{a}...
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What do physicists mean by solving the Ising model?
Another commonly used notion of "solving a theory" is to find a procedure to compute, at least in principle, all local observables. Sometimes one might also add non-local observables to the ...
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Spin ice / spin liquid v.s. quantum spin ice / quantum spin liquid
When we talk about spin liquid, we typically mean quantum spin liquid. Thanks to @Stephen Powell's comment below, I learned that there is also a thing called "classical spin liquid". As far as I know, ...
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Antiferromagnetic and Ferromagnetic Ising Model on triangular lattice
The partition function of the Ising model on a triangular model has been computed by Plechko using Grasmann variables to decouple the spins. Some references are:
Plechko, V. N. « Anticommuting ...
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2D Ising model on curved surface
In fact, the two-dimensional Ising model was first solved by Lars Onsager on a cylinder/torus, which, while lacking intrinsic curvature, has a 'nontrivial' topology and moduli space (you can play with ...
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Time-reversal symmetry for spin Hamiltonian
In general the representation of the time-reversal operator depends on the system that you consider. Let us start with some basic remarks on time-reversal symmetry prior to the spinor question that ...
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What is the intuition behind the statement that non-equilibrium systems with static disorder are self-averaging?
I've seen this claim echoed in other references, e.g. in Jordan Rammer's book Quantum Field Theory of Non-Equilibrium States pg. 455 below equation 12.33, however I haven't found a satisfying ...
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What are $U(n)$ or $\mathbb{Z}_2$ quantum spin liquids?
They specify which gauge symmetry the Quantum Spin Liquids is subject to, when treated via a Lattice Gauge Theory. The gauge field (some sort of interaction) is defined over a discretised space(-time)....
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Is the fully connected Potts model exactly solvable?
You should not expect close form expressions for the finite-$N$ partition functions.
In fact, this is already the case when $q=2$. The latter is equivalent to the Curie-Weiss model,
in which the spins ...
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Why is the sequence of limits $\lim\limits_{V\to\infty}\lim\limits_{B\to 0}m(B,V)$ when reversed does not give the same result?
The physical meaning of the two different double limits is quite obvious:
$$\lim\limits_{V\to\infty}\lim\limits_{B\to 0}m(B,V)$$ corresponds to start with a finite volume $V$, making the external ...
6
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What is the minimal symmetry required for a spin Hamiltonian to describe a spin-liquid ground state?
The definition of a spin liquid as a spin system "with no spontaneously broken symmetries" is out of date and no longer used, partially for the reason you describe. If you perturb as spin-liquid ...
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What is the Kitaev Model and why has it become so popular?
The paper linked in the original post already answers some of the post's questions.
What is the Kitaev Model?
It's a lattice model where
nearest neighbor spin degrees of freedom interact via a ...
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Lagrange multiplier in spin liquid mean-field theory (Paper by X.G. Wen)
Lagrange multipliers in quantum systems are usually (always?) implemented on the level of expectation values - often specifically ground state expectation values. That's also the case for these two ...
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What do physicists mean by solving the Ising model?
I think most physicists mean computing the partition function. So, given a lattice $L$ with edges $E(L)$ and vertices $V(L)$, solving the corresponding Ising model would mean to compute
$$Z(\beta,h):=\...
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Partition function for 4 spins
(It seems that the notes in question can be found at https://arxiv.org/abs/0901.3492, in which this section is on pages 9-10.)
We can rewrite the integral (1.5) in an evaluable form by taking into ...
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What if we used "Schwinger Fermions" to study spin waves?
This is known is Abrikosov's pseudo-fermion representation, which is compactly written
$$ \mathbf{S}_j = \frac{1}{2} \sum_{\sigma,\sigma'} f_{j\sigma}^\dagger \vec{\tau}_{\sigma,\sigma'} f_{j\sigma'}, ...
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Why can we choose spin-1/2 degrees of freedom to commute?
Qubits in the quantum-circuit model are assumed to be distinguishable, so it makes no difference whether they're implemented using bosons or fermions.
"A qubit is a spin-1/2 system" is true ...
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5
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What is the momentum canonically conjugate to spin in QM?
First of all, there is a very elementary reason why $[\sigma_i,\Pi_j] = i \delta_{ij}$ is impossible for finite dimensional matrices. Because that identity would result in the following contradiction:
...
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What new features does the Heisenberg Model have compared to the Ising Model?
One of the main differences is that the Ising model lies on a discrete symmetry (the $Z_2$ symmetry) while the Heisenberg model lies on a continuous one (rotational symmetry). It will affect the phase ...
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Ground state degeneracy: Spin vs Fermionic language
Your claim is not correct: After a Jordan-Wigner transformation, the Ising model is mapped to a free fermion chain which has either periodic or anti-periodic boundary conditions, depending whether the ...
- 20.8k
5
votes
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Heisenberg ferromagnet in continuum limit
Remember that the Hamiltonian involves a sum over all pairs of neighbouring sites. Assume that the sites are located on a square lattice so that their positions are given by $\vec r=a(n_x\vec e_x+n_y\...
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