Questions tagged [spin-chains]
One dimensional quantum systems which can either be multiple discrete spin particles or their continuum limit.
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Difference between boundary conditions in thermodynamic limit
Consider a model for a spin chain. I somehow am able to find a general formula for the expectation value of some observable in both periodic and open boundary conditions. ie.,
under PBC, I have
$\...
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Are strong correlations of boundary spins possible in the absence of long-range order in the bulk?
Question about one-dimensional models with short range interaction of quantum spins, such as transverse Ising and Heisenberg models. Are there any examples when, in the ground state of the system, the ...
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Example of an injective matrix product state (MPS)
I am struggling to understand what is an injective matrix product state (MPS). From the definition, it is said that an injective MPS $|M(A)\rangle$is one where the tensor $A$ has a projector $P(A)$ ...
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What is the Haldane gap?
The Haldane Phase is a topological phase of matter in which a Haldane gap opens due to the breaking of either time-reversal symmetry or inversion symmetry. Physically speaking, what is the "...
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Why is the excitation energy of antiferro 1/2 spin XXZ chain associated with the $\psi_{n+2}$ and $\psi_{n-2}$
I do not understand the equation on page 5 of nagaoso's book "quantum field theory in strongly correlated electronic systems".
The Hamiltonian is shown in this form
$H=\frac{J_\bot}{2}\sum_{...
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What is the appropriate 'Page Value' for entanglement entropy in a symmetry sector, say for example with a $U(1)$ symmetry?
Don Page derived the formula for the average entropy of a subsystem of a quantum system (assumed to be in a pure state), if the system is partitioned into two subsystems of dimensions $m$ and $n$, ...
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Transforming spin operators into fermionic operators and finding their anticommutation relations
The Jordan-Wigner transformation (JWT) is a method used in quantum mechanics to map spin operators, which are typically associated with spin-1/2 particles, to fermionic operators, which describe ...
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Gap of the XXZ model in fixed magnetisation sectors
I am wondering whether it is known, or whether it can easily seen from the Bethe ansatz solution, what the gap of the spin-1/2 XXZ model of finite size $N$ with periodic boundary conditions
($H=\sum\...
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Can the edge degeneracy in spin-$2$ AKLT go away on an arbitrarily small $SO(3)$-symmetric bulk perturbation?
I am learning about SPTs, or symmetry-protected-topological phases. There is a rich structure in antiferromagnetic spin chains. The Heisenberg point is gapless in half-integer-spin antiferromagnets ...
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Can a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry generated by $\prod_{i} \sigma^x_i$ and $\prod_{i} \sigma^z_i$ be broken in a spin-$1/2$ chain?
I am interested in understanding patterns of spontaneous symmetry breaking in spin chains. I want to understand what happens when I have "competing" orders, like symmetry breaking orders ...
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Jordan-Wigner transformation in transverse field Ising model
Jordan-Wigner transformation provides an exact solution for transverse field Ising model in both the ferromagnetic phase and the paramagnetic phase. Yet this seems to imply that in both phases, the ...
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What is the signal of a spin wave?
From what I understand, for example in the Ising model, we can probe the correlation function via neutron scattering, and the correlation function gives the magnetic susceptibility for the system. Is ...
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Why is $H = J \sum_i (S^x_i S^x_{i+1} + S^y_iS^y_{i+1})$ always gapless for any spin $S$?
In the following I have in mind antiferromagnetic spin chains in periodic boundary conditions on a chain of even length $L$.
Consider the spin-$S$ spin chain
$$H = J \sum_{i=1}^L (S^x_i S^x_{i+1} + S^...
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Relative Sign In XXZ Chain
This is a relatively simple question that I just want confirmation on. In literature, I have seen 2 ways of writing the Heisenberg XXZ Chain:
1.) $H = -J \sum_{n=1}^{N}\left(S_n^xS_{n+1}^x+ S_n^yS_{n+...
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Phase transitions in the XXZ model
Consider the one-dimensional quantum XXZ model defined by the Hamiltonian:
$$
H = J \sum_{i} \left (X_i X_{i+1} + Y_i Y_{i+1} + \Delta Z_i Z_{i+1} \right).
$$
First, let us focus at zero ...
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Parity of a 1d Ising model, and with higher order terms
I don't know if this should be asked here or in a math stack exchange, but I'll try here first.
Consider the classical 1d Ising model with periodic boundary condition:
\begin{equation}
H_2 (\vec{\...
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Can the hybridization of edge states in the 1D SSH model be observed numerically?
So I was reading the lecture notes by Asboth on topological insulators . In the first chapter the SSH model is discussed :
$H_{SSH} = \sum_{i = 1}^N v|i,A\rangle \langle i,B | + h.c. + \sum_{i = 1}^{N-...
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Is there a name for a Heisenberg-like model, but instead of the ZZ operator, we have one that favor only spin-up-spin-up configurations?
I understand that the Quantum Heisenberg XXZ model in 1D has the form:
$$\hat H = \frac{1}{2} \sum_{j=1}^{N} (J_x \sigma_j^x \sigma_{j+1}^x + J_y \sigma_j^y \sigma_{j+1}^y + J_z \sigma_j^z \sigma_{j+1}...
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A simple calculation in the XXX spin chain
I am currently studying the XXX Heisenberg spin chain using the Bethe ansatz. I am working in the string hypothesis and I am having troubles deriving a simple expression for Fourier transformation of ...
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Example of first-order quantum phase transitions between two gapped phases with unique ground state in local 1d spin chain without extra symmetry
I'm trying to better understand first-order phase transitions in local, 1d quantum systems, particularly spin chains. I realized that I don't have a strong understanding of what's possible and ...
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Generating Matrix Product States from a (random) vector
I try to decomposite an arbitrary quantum state into a matrix product state. For this i follow this paper by U. Schollwöck where especially section 4.1.3 is relevant.
So far I did the following:
...
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Mathematical meaning for Algebraic Bethe Ansatz
I'm a mathematician who's trying to understand the meaning of Algebraic Bethe Ansatz. What I understood is that when dealing with quantum integrable models (like XXZ Heisenberg spin chain), one is ...
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Limit of solving the 1D Heisenberg chain to find the dynamics numerically
I am trying to simulate the dynamics of a 1D Heisenberg chain using Python.
I am going step-by-step.
There is an external magnetic field along +Z direction.
At first we consider a single classical ...
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Time reversal symmetry and Bosonization
Bosonization of Spin 1/2s to fields $\phi(x)$, $\theta(x)$ is defined as (Ref: 'Quantum Physics in 1-D' by Giamarchi):
$S^z(x)=\frac{-1}{\pi}\nabla\phi(x)+\frac{(-1)^x}{\pi a}\cos 2\phi(x)$,
$S^x(x)=\...
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How non-local can the interactions be for Density Matrix Renormalization Group (DMRG) to still work?
I know that Density Matrix Renormalization Group (DMRG) / Tensor Networks (TN) work well for local Hamiltonians, where on each site I have a fermion or boson, which only have nearest-neighbor ...
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Coupling two Ising chains via an energy-energy interaction
Consider the transverse-field Ising model on a chain with periodic boundary conditions:
$$ H = -\sum_{i=1}^{L} \sigma_{i}^z \sigma_{i+1}^z + h \sigma_{i}^x$$
There's a phase transition at $h=1$, which ...
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Mapping a 1D quantum Ising chain to a 2-dimensional classical Ising system
Going through Ref. 1 (I'll stick with the book's equation numbering), I'm learning about the mapping of quantum systems into classical systems. First of all let me briefly recap notation and some ...
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Symmetry Protected Topology and Edge Modes
I have a spin 1/2 chain with open boundary conditions described by Hamiltonian $H=\sum_i \sigma_{2i}^z \sigma_{2i+1}^z$. From $H$ it's clear that boundary sites are decoupled from the rest of the ...
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Ground state energy of infinite Heisenberg XXX model with open or periodic boundary conditions equal?
I was wondering if there is anywhere a formal proof that shows that the ground state energy of a Heisenberg XXX model with periodic boundary conditions becomes equal to the ground state energy with ...
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Tunneling lowers the energy of a ground state superposition of spins up and down in the quantum Ising model
Considering an Ising model in the quantum scenario in quantum spatial dimension d=1 (that corresponds to classical D=2=d+1 dimension). Starting with the Ising model hamiltonian under the approximation ...
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Obtaining a Matrix Product State (MPS) using Schmidt Decomposition for a Tripartite State
I understand that one method to derive an MPS representation of a quantum state involves applying the Schmidt decomposition $ N−1$ times. While I'm familiar with the diagrammatic notation, I wanted to ...
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MPS canonical form
If I express a MPS in its (left, right or anything else) canonical form, does this representation encode all Schmidt decompositions between a subsystem and its complement,rather than only the Schmidt ...
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"Entropy" of a set of correlators in a quantum system
Please forgive the ill-posedness of this question; I am hoping someone can help me formulate what I am asking more clearly.
Consider the ground state of a one-dimensional quantum spin chain on $N$ ...
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Ground state of the Heisenberg XXX model with a coupling?
I have a one-dimensional Heisenberg chain with a Magnetic field with $N$ sites with $J>0$
\begin{equation}
\mathcal{H} = -J \sum_{i = 1}^{N-1} \vec{S_i}\cdot \vec{S_{i+1}}- \sum_{i = 1}^N \vec{H}\...
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Calculate partition function of 1D quantum Heisenberg models?
For the 1D Quantum Heisenberg Spin Model:
$\displaystyle {\hat H = -\frac{1}{2} \sum_{j=1}^{N} (J_x \sigma_j^x \sigma_{j+1}^x + J_y \sigma_j^y \sigma_{j+1}^y + J_z \sigma_j^z \sigma_{j+1}^z + h\...
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Emergent higher symmetry breaking without topological order?
In this paper prof. Wen states that (p.6)
a spontaneous higher symmetry broken state always corresponds to a topologically ordered state.
Are there examples of simple (or not) quantum spin models ...
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Exact ground state degeneracy for quantum spin system with non commuting terms and its quantum phase transition?
Let's say I have a 2D quantum spin model of N spin-1/2 particles, with two terms:
$$
H = -J \sum_N \prod_{i \in G} \sigma^x_i - h \sum_N \prod_{i \in G'} \sigma^z_i
$$
The first is a collection of ...
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Integrability of spin central model
I have a central model of this form $$H = \sum_{i=1}^{N} S^z_0\otimes S^z_i$$ where the $S^z_i$ acts on the $i$th element of the environment, i.e. the Hilbert space is of the following form $\mathcal{...
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Why one can say that the inner product in $\bigotimes\limits_{n=1}^{N}\mathbb{C}_n^2$ has the following form?
In the article "Quantum theory of measurement and
macroscopic observables" of Klaus Hepp it is said that for a lattice of $N$ spin $\frac{1}{2}$ systems each in $\mathbb{C}^2$, so that the ...
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Question about the 'reduced basis transformation'
I' ve been reading the review Ulrich Schollwöck: The density-matrix renormalization group in the age of
matrix product states (arXiv link)
and encountered with a question about the so called 'reduced ...
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Bethe ansatz and density of states for XXX spin chain
Consider the 1 dimensional Heisenberg antiferromagnet with Hamiltonian
$$ H = J\sum_{i=1}^L \vec S_i \cdot\vec S_{i+1}$$
and periodic boundary conditions.
I understand that this can be solved exactly ...
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1
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$G$-injective MPS and symmetry-broken phases
First, a little bit of motivation. I was reading the paper "Matrix Product States and Projected Entangled Pair States" to try to learn more about MPS representations of symmetry broken ...
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Is the dot product of spins the only way to create a scalar (Hamiltonian) invariant under spin rotation?
I wanted to generalize the result for the following question for four spins 1/2: Most general form of a spin rotation invariant Hamiltonian?.
Assume that we have a Hilbert space for four spins $(\vec{...
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Connection between diffusion and non-integrable 1D spin chains
My question concerns non-integrable (à la Bethe) 1D spin chains.
Consider, for example, the 1D non-integrable Ising model
\begin{equation}
H = \sum_{i \in \mathbb{Z}}\sigma_{i}^{z} \sigma_{i+1}^{z} + ...
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Most general form of a spin rotation invariant Hamiltonian?
I am told that the most general form of a spin rotation invariant Hamiltonian for two systems 1 and 2 both with spin $S$, i.e., the spin operators
\begin{align}
(\hat{S}_1^x)^2 +(\hat{S}_1^y)^2 + (\...
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0
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iTEBD real time evolution for 3-body time evolution operator
I am trying to implement the iTEBD algorithm for real-time evolution of the PXP model. Here, $P$ is the projector onto the ground state, and $X$ is the Pauli spin matrices.
I know for the 2-body case, ...
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CFT description of polynomially degenerate, critical spin-chain
For length $L$ spin chains described by conformal field theories, there's a nice a way to extract the central charge via fitting the following ansatz for the entanglement entropy of the ground state:
$...
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What is $v$ in conformal field theory?
In reading about conformal field theory applied to spin chains of length $N$, I've seen the following expression several times, describing how the central charge $c$ can be extracted from the ground ...
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Normalization in tensor networks [closed]
I am trying to implement the iTEBD algorithm for the $PXP$ model, i.e, the hamiltonian is
$$H = \sum_iP_{i-1}X_iP_{i+1}.$$
Here $P$ is the projector onto the ground state and $X$ is the usual pauli x ...
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How do you calculate the entanglement entropy of a tensor network?
I found that the entanglement entropy can be calculated using the Schmidt coefficients of the state, using
$S = -\sum_i|\alpha_i|^2\log(|\alpha_i|^2)$
In the case of tensor networks, does this simply ...
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