24
votes
Detailed derivation and explanation of the AKLT Hamiltonian
Let me try to answer my own questions to thank those who gave voice and support for undeleting this question. My main reference is the chapter 3 Basic quantum statistical mechanics
of spin systems of ...
- 669
17
votes
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Understanding Periodic and Anti-periodic boundary condition for Jordan-Wigner transformation
It is important to note that there are two different boundary conditions, the first is the boundary condition for real spin model, the second is for fermion model. In fact, for real spin model with a ...
- 1,652
16
votes
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Is there any relation between density matrix renormalization group (DMRG) and renormalization group (RG)?
The name DMRG is somewhat of a historical accident, and its modern day incarnations are not directly linked to the renormalization group or phase transitions. Instead, it is better understood as a ...
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13
votes
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What is meant exactly by "renormalization" in condensed matter physics, specifically in density matrix renormalization group (DMRG)?
It is instructive to come back first to NRG (Numerical Renormalization Group) proposed by Ken Wilson (Nobel prize laureate for his work on Renormalization Group in the context of critical phenomena). ...
- 3,708
12
votes
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Most general form of a spin rotation invariant Hamiltonian?
$\newcommand{\bm}[1]{\mathbf{#1}}$
You need to look at this in terms of spin quantum numbers (i.e., eigenvalues).
$(\bm S_1+\bm S_2)$ can take values $S_\mathrm{tot} = 0,\dots,2S$. Now if we restrict ...
- 21.4k
7
votes
Ground state energy of infinite Heisenberg XXX model with open or periodic boundary conditions equal?
The ground state energy for the one-dimensional spin-1/2 Heisenberg model can be obtained using Bethe ansatz methods. For periodic boundary conditions, this was done as early as 1938 by Lamek Hulthén, ...
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6
votes
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Uniqueness of AKLT Ground State vs. SU(2) symmetry and Lieb-Schultz-Mattis theorem
The Lieb-Schultz-Mattis (LSM) theorem states that for a rotationally invariant (SO(3) invariant) 1D spin chain, such as the Heisenberg chain, with half-integer spin, the system cannot be gapped with a ...
- 21.4k
6
votes
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Long-range correlations in transverse field Ising model
If you look at the paper by Hastings and Koma, https://arxiv.org/abs/math-ph/0507008, they claim to prove the following:
When two observables commute with each other at large distance, the connected ...
- 2,159
6
votes
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Ground state energy of infinite Heisenberg XXX model with open or periodic boundary conditions equal?
The ground state energy of finite-size quantum spin systems depends on the boundary conditions. For example, the energy of the ground state of a ferromagnetic one-dimensional XXX chain of $N$ spins $1/...
- 6,258
5
votes
What is the 'Drude Weight' and why is it important?
Thanks to the feedback of @guangcun I have changed parts of this answer.
As I understand it now, the Drude weight distinguishes between an insulator and a metal in a clean model system. This means, ...
- 311
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\...
- 3,708
5
votes
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Why do we use matrix product states?
Take a product state $|\psi\rangle=|+,+,\dots,+\rangle$, on $N$ spins. Then, to write it in the first form takes a tensor
$$
\psi_{i,j,\dots}
$$
with $2^N$ non-zero elements. For $N=100$, there is ...
- 21.4k
5
votes
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(Transverse) Ising Model Higher Than Four Dimensions
The criterion for determining whether mean-field theory is good or not is the Ginzburg criterion. You can estimate how accurate mean-field theory is by computing the leading corrections to it. In ...
- 2,159
5
votes
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Ground state magnetization of the Heisenberg XXZ chain
As you know from the reference you cite, the XXZ model is solvable using the algebraic Bethe Ansatz. It may be surprising, but although in principle one has an exact solution, actually extracting the ...
- 2,405
5
votes
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XXZ chain exact ground state energy
Such analytical expressions were derived for special cases by multiple authors. More general results were obtained by Yang and Yang in
C. N. Yang and C. P. Yang One-Dimensional Chain of Anisotropic ...
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5
votes
How to translate from a state/density matrix formalism to matrix product state representation?
The review does a good job of answering the question, but just wanted to present the material using less forbidding notation and to make the connection to density matrices, entanglement entropy, and ...
- 367
5
votes
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MPS canonical form
I'm not entirely sure what you mean "encode all Schmidt decompositions", as any MPS representation of a quantum state carries the full information about the state, and thus all of its ...
- 21.4k
4
votes
Goldstone mode as spin wave in 2D?
Usually, the argument relies on the fact that there is a band of spinons (i.e., a continuum of modes) above the ground states, so I don't see how your argument above works. One then argues that those ...
- 21.4k
4
votes
Localization length in Anderson localized systems
Localization in one dimension is often studied with techniques from random matrix theory. In particular, one can find the Lyapunov exponent corresponding to the random matrices that describe a given ...
4
votes
$\phi^4$ theory kinks as fermions?
Quantum fluctuations in the kink sectors of Sin-Gordon and the quartic interaction theory are described by reflectionless Pöschl-Teller-Operators, which form a SUSY-Chain with $N$ elements. The second ...
- 352
4
votes
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Does one-dimensional ferromagnetic chain have long range order at zero tempreture?
Yes and no. The quantum Ising model in 1+1 dimensions has a phase transition as you vary the coupling in the Hamiltonian. This model is directly related to the statistical mechanics of the classical ...
- 8,917
4
votes
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R-matrix for spin chains
This is essentially an answer to your questions R-matrix for spin chains, Elliptic R-matrix and Yang Baxter solution for XYZ model, $R$ matrix for XYZ spin chain, Algebraic Bethe Ansatz and $R$-...
- 607
4
votes
Average entropy of a subsystem
The formula you cite from Page is correct. Perhaps you are making a mistake when averaging over states? Have you checked that you averaged over enough states for the numerical average to have ...
- 9,311
4
votes
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How can I explicitly express the Ising Hamiltonian in matrix form?
For completeness I'll summarize the answer here. After a fun conversation in the comments, we saw that it will be more illuminating to write
$$H=-\sum_ {i=1}^{N-1}
\sigma_i^x \sigma_ {i+1} ^x + h \...
- 4,820
4
votes
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How can I simulate a ground state degenerate system numerically?
I'll go ahead and say that one can roughly classify systems into
Systems with a unique ground state.
Systems that have multiple (possibly infinitely many) degenerate ground states, but that will tend ...
- 2,749
4
votes
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Kagome Lattice: Spin-orbit coupling Hamiltonian in tight-binding models
I think the authors got it right.
The subtlety lies in the definition of $\mathbf{E}_{ij}$ and $\mathbf{R}_{ij}$. The authors consider $\mathbf{E}_{ij}$ as the electric field felt by the electron ...
- 499
4
votes
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How can you (computationally) calculate the halfchain entanglement entropy of a spinchain?
It all boils down to a two-partite system,
$$
\vert\psi\rangle = \sum c_{ij} \vert i,j\rangle\ ,
$$
where $i$ and $j$ are all indices in the left and right part, respectively. Then, the reduced ...
- 21.4k
4
votes
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What is the ground state wavefunction of $\hat{H}=-J\sum\limits_{\langle i,j\rangle}\hat{S}_i^z\hat{S}_j^z,~~ (J>0)$?
The system is a 1D lattice of spin-half particles. And the Hamiltonian considers nearest neighbour interaction with a coupling that favours alignment. Intuitively we can see that the ground state at ...
- 9,463
4
votes
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Why does a 1D hardcore bosonic chain have different ground state energy in bosonic and spin representations?
In your second approach, you correctly solve the hardcore boson problem (in essence, hardcore bosons are exactly two-level spin systems).
In your first approach, on the other hand, you diagonalize the ...
- 21.4k
4
votes
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Jordan-Wigner transformation for lattice models without $U(1)$ symmetry
The case of your model with the field along $\hat{x}$ is known as the transverse-field XXZ chain, see e.g. Dmitriev et al., One-dimensional anisotropic Heisenberg model in the transverse magnetic ...
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