Questions tagged [spin-chains]

One dimensional quantum systems which can either be multiple discrete spin particles or their continuum limit.

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Is the magnon dispersion equivalent to a plot of energies vs state number?

I am using QuSpin to plot spin-chain energies against their state number. I get discrete plots that look remarkably like the magnon dispersions. Are they the same? I don't think they would be because $...
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Solving the quantum XX model using drone fermions

Suppose we wish to solve the XX model in 1D, which describes spin-$\frac{1}{2}$ particles interacting with their nearest neighbor. Assuming open boundary conditions for simplicity, the Hamiltonian ...
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How to stack two Haldane chains?

This questions is a follow up to a pervious question of mine: Inverse of Haldane phase? Now that I know that Haldane phase is it's own inverse, I am having trouble is visualizing how could we stack ...
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Inverse of Haldane phase?

Based on what I have learned so far, Haldane phases are a nontrivial SPT for 1D spin-1 chains. The trivial phase acts as an "identity" under the group of SPT phases ( with stacking as the ...
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How to take into account finite temperature in transverse Ising chain?

A similar question has already been asked here What I'm wondering is how to take into account finite temperature in the transverse Ising chain and see how that affects the magnetization. The reason ...
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Coding of Correlation function in spin chain

I have to code a correlation function for Spinoperators $S_x^i S_x^j$, $S_y^i S_y^j$ and $S_z^i S_z^j$ (basically $\vec{S}_i \vec{S}_j$) for two spins on two different sites of a spin chain. Can ...
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How do boundary conditions change during a spin transformation?

I am currently reading the following review paper: (1) Two Dimensional Model as a Soluble Problem for Many Fermions by Schultz et. al. Equation (3.2), which is reproduced below, introduces the Jordan-...
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Ground state of sums of commuting, translated projectors

I have in mind a spin chain of length $L$ with local Hilbert space dimension $d$ and projectors $\{ P_i \}$ that act on $r$ sites $i, i+1, ..., i+r-1$. The projectors are identical besides which sites ...
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Can bond dimension vary from bond to bond?

Consider a bipartite system composed of subsystems $A$ and $B$, with corresponding Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$, spanned by $\{\chi_1,...,\chi_n\}$ and $\{\phi_1,...,\phi_m\}$, ...
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Jordan-Wigner Transformations on fermionic system

I've been trying to use Jordan-Wigner Transformations on a given fermionic Hamiltonian. The given Hamiltonian is: $$ \hat{H}= -\sum_{m=1}^{N}(J_z \hat{S}_{m}^{z} \hat{S}_{m+1}^{z} + \frac{J_{\perp}}{2}...
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Long-range correlations in transverse field Ising model

The transverse field Ising model in 1+1d has two phases: a symmetric "disordered" phase and a symmetry-breaking "ordered" phase. Both of these phases have a finite excitation gap. ...
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What is the $XXX_s$ Hamiltonian in terms of $\vec{S}_i \cdot \vec{S}_{i+1}$?

Faddeev, Takhtajan, and others united and discovered many integrable models through the Algebraic Bethe Ansatz. For example, the integrable spin-1/2 Heisenberg model $$H_{1/2} = \sum_{i=1}^L \vec{S}_i ...
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How does the wave nature of a Spin Wave arise?

I have been learning about Goldstone's theorem recently and I am a bit confused about show Goldstone excitations arise. Consider spin waves in a magnet. The Hamiltonian is given by (see: https://en....
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$XXX_s$ spin chains as constrained $XXX_\frac{1}{2}$ spin chains

The eigenenergies of an $XXX_{s}$ spin chain are found by solving the Bethe Ansatz. For a closed chain of length $L$ and $N$ rapidities, $u_i$, encoding excitations from the ground state moving along ...
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How to add two Matrix Product States of different bond dimensions?

If I have the MPS representation of two quantum states, how do I add them? Note that the bond -dimensions need not be the same for the two MPSs.
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How to add anisotropic term in Hamiltonian for a Kagome antiferromagnet?

I have defined an antiferromagnet (AFM) on a Kagome lattice. It has three spins in one unit-cell and each spin is at $120^\circ$ angle from its neighbouring spin (for example see photo below). A ...
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Parity of XYZ model ground state

I am considering the XYZ Hamiltonian (with PBC) $$\widehat{H}_{\mathrm{XYZ}}=\sum_{i=1}^{N} \left(\hat{\sigma}_{i}^{x} \hat{\sigma}_{i+1}^{x}+J_{y}\hat{\sigma}_{i}^{y} \hat{\sigma}_{i+1}^{y}+J_{z}\hat{...
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Dispersion Relation in Spin Chains in Terms of Quantum Speed Limits

Going off the dispersion relation derived by He and Guo $$E_k = \sqrt{\left(\frac{J}{2}\right)^2 + h^2 + Jh\cos k}$$ Where $J$ is the nearest neighbour interaction strength and $h$ is the external ...
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Basis state of non-interacting fermions

I am trying to calculate the periodic dynamics of many-body systems (spin-$1/2$ $XY$) Hamiltonian, where, \begin{equation} H_1 = \sum_{i=1}^{N-1}(\sigma^{x}_{i}\sigma^{x}_{i+1}+\sigma^{y}_{i}\sigma^{...
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Relationship between Hartley entropy and local dimension

I am recently reading a paper about entanglement entropy. It mentions that if we consider a 1D spin chain and write a pure state in the matrix product state: \begin{align} |\psi\rangle = A^{\sigma_1}A^...
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Uniqueness of AKLT Ground State vs. SU(2) symmetry and Lieb-Schultz-Mattis theorem

I have a question in my mind regarding the uniqueness of AKLT ground state. Currently I am watching a video clip of MPS and I am curious why the AKLT ground state model is unique gapped ground state. ...
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Unphysical Solution of the Bethe Ansatz

I actually want to ask an elementary question regarding the algebraic Bethe-Ansatz. Say I have constructed the Bethe Ansatz Equations (BAE) in the algebraic framework with pseudovacuum $\phi$, $B(u)$ ...
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How to translate from a state/density matrix formalism to matrix product state representation?

From what I understand, MPS is just a simpler way to write out a state, compared to the density matrix. But how do I get those $A_i$ matrices? From all the examples I read, people just somehow "...
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Literature Request: Conformal Field Theory for 1D XXZ Chain

It is known that the Heisenberg XXZ chain in 1D \begin{equation} \hat{H} = - J \sum_{i=1}^N \left(S^x_jS^x_{j+1} + S^y_jS^y_{j+1} + \Delta S^z_jS^z_{j+1}\right) \end{equation} can be described by the ...
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Can we use Bosonization to study theories without $U(1)$ symmetry?

When studying lattice models using bosonization, we expect the total charge is conserved so that the elementary excitation is particle-hole-like bosonic degrees of freedom. How about models without $U(...
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Matrix product state representation for the "infinitely repulsive hardcore boson" state

Consider a one-dimensional spin-1/2 chain with $N$ spins, and let $|\psi\rangle$ be the equal weight superposition of all states with no adjacent spin-ups, e.g. for $N=3$ with open-boundary, $|\psi_{N=...
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Is it possible to diagonalize a Hamiltonian with both quadratic and linear terms in the fermi operators?

A quadratic Hamiltonian in the fermi operators is exactly diagonalizable. The most convenient way of describing these Hamiltonians is of the form: $$\mathcal{H}=\displaystyle \sum_{j,k}(\alpha_{jk}a_{...
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The dominant eigenvalue of the transfer matrix of a matrix product state

Consider a translation-invariant matrix product state \begin{equation} |\psi_L\rangle= \mathrm{Tr}[A(s_1)A(s_2)\ldots A(s_L)]|s_1 s_2\ldots s_L\rangle. \end{equation} I'm interested in the ...
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Truncation Problem of Density Matrix Renormalisation Group (DMRG)

I am wondering that is there any restrictions for the truncation in DMRG algorithm. Currently I am using DMRG to calculate ground state energy per site of a many-body system described by on-site ...
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Bogoliubov transforms for a Heisenberg antiferromagnet: inconsistencies with the excitation spectrum

Consider the standard 1D Heisenberg antiferromagnet (J>0) $$ \mathcal{H} = J \sum_i S_i \cdot S_{i+i} $$ We apply the standard Holstein-Primakoff expansion about the classical ground state, $S_i = \...
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1D transverse-field Ising model - what is the difference between its classical and quantum treatment?

The 1D transverse field Ising model: $$ H(\sigma)=-J\sum_{i\in Z} \sigma^x_i \sigma^x_{i+1} -h \sum_{i \in Z} \sigma^z_i$$ is usually solved in quantum way, but we can also solve it classically - ...
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Writing down a Hamiltonian that couples spin and phonons

I am studying spin dynamics and am trying to write down a Hamiltonian that couples the spins with the phonons. I have the following interacting spin Hamiltonian $$H_{s}=\sum h_{i}S_{i}+H_{\text{...
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Time evolution of spin with Anti-symmetric (Dzyaloshinkii-Moriya) interaction

I am trying to simulate the time evolution of a spin in spin chain interacting via Dzyaloshinkii-Moriya interaction. The Hamiltonian is of the form $$H_{A}=J_{A}\sum_{i}(S^{x}_{i}S_{i+1}^{y}-S^{x}_{i+...
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Heisenberg equation of motion and continuum limit

Given the quite simple Hamiltonian $$\hat{\mathcal{H}}=\sum_n\big(\hat{S}_n^+\hat{S}^-_{n+1}+\hat{S}_n^-\hat{S}^+_{n+1}\big)$$ on a 1D spin chain, it basically interchanges two spins lying next to ...
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Commutator of Hamiltonian and the spin sum

For a 1-D Heisenberg quantum spin chain the Hamiltonian is given by: $$H=-\sum_{j=0}^{N-1} J_{i,i+1}\boldsymbol{\sigma}_j^i \cdot\boldsymbol{\sigma}_{j+1}^i -\sum_{j=0}^{N}h_j\sigma_j^z$$ where $\...
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Efficient MPS Description of a given quantum state

If we know the amplitudes of a (pure) quantum state wrt some basis, is there an algorithmic procedure to ensure an efficient MPS description (one with the lowest bond dimension) of the state ?
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Magnetization ($z$-basis) of a 1D Transverse Ising Model

I'm trying to find the magnetization $\langle\sigma_{z} \rangle$ of a 1D transverse Ising chain and plot it as a function of the transverse field $\lambda$. More specifically, I want to plot this for ...
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3 votes
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Jordan-Wigner transformation for lattice models without $U(1)$ symmetry

The Jordan-Wigner transformation is a powerful approach to studying one-dimensional spin models. The following dictionary between spin operators and creation/annihilation operators for fermions allows ...
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Why does a 1D hardcore bosonic chain have different ground state energy in bosonic and spin representations?

Consider a simple periodic 1D chain with four sites with periodic boundary condition. The Hamiltonian reads $$ H = t c_1^\dagger c_2 + c_2^\dagger c_3 + t c_3^\dagger c_4 + c_4^\dagger c_1 + h.c. $$ ...
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Correlation Functions of the one dimensional spin-1/2 XY Model

I am currently working on studying how to diagonalize the spin-1/2 XY model using the method included in " Annals of Physics 16.3 (1961): 407-466" by Lieb et al. In fact, I'd like someone to ...
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Correlation Functions: How can I prove this simple equation [closed]

The correlation functions of the Transverse Ising Model is beautifully explained in "Quantum Ising Phases and Transitions in Transverse Ising Models" Quantum Ising Phases and Transitions in ...
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Do magnons exist in 1D?

I'm confused about the state of a system described by the quantum Heisenberg model $$ H = -J\sum_i \vec{S}_i \cdot \vec{S}_{i+1} $$ in 1 spacial dimension (1D spin chain). We can find the low energy ...
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$S^+$ acting on a spin chain raises the entropy by at most $\ln(2)$

Consider the operator $S^+ = \sum_{i=1}^L S^+_i$ acting on a spin-chain of spin-1/2 particles. Denote the half-chain Von Neumann entanglement entropy of a state $|\psi\rangle$ by $\mathbb{S}[|\psi\...
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Evaluating Spin 1 matrices $S_x*S_y$ in terms of $S_+$ and $S_-$ basis

I am a beginner to spin algebra. I am trying to represent $S_x*S_y$ in terms of $S_+$ and $S_-$ basis. What would this representation look like? I know that the Pauli properties for spin 1/2 does not ...
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Jordan-Wigner transformation on a circle and spin structures?

Is there an analog of the Jordan-Wigner transformation between fermion algebra on a circle and a Pauli algebra? For example, the continuum analog of bosonization of "compact boson $\...
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XXZ chain exact ground state energy

I would like to know the analytical expression of the ground state energy of the XXZ model, if such formula exists (probably from a Bethe Ansatz solution) and if it is valid in all parameter regimes.
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Why is $\sum_{i=0}^N S_i^z S_{i+1}^z |\uparrow ... \downarrow_n ... \uparrow \rangle = \frac{1}{4}(N-4)$?

I am following these (http://edu.itp.phys.ethz.ch/fs13/int/SpinChains.pdf) lecture notes and I can't understand why given the following XXX Heisenberg hamiltonian $$ \mathcal{H}=\frac{J N}{4}-J \sum_{...
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What does it mean for a system to be in its ground state?

From the measurement postulate, we are told that every observable quantity is represented by a Hermitian operator $\textbf{A}$, where the eigenvalues of said operator gives the possible measurement ...
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Antiferromagnetic chain from Altland/Simons book (p.81)

In Condensed Matter Field Theory (2nd edition) by Altland/Simons there considered antiferromagnetic chain with Hamiltonian: $$H = J\sum_{<n,m>} S_nS_m = J\sum_{<n,m>}[S^{z}_n S^{z}_m + \...
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Is there any study about using DMRG to simulate two spin chains coupled at only several sites on each chain?

Is there any study about the DMRG simulation of such kind of systems? or Each blue site is a spin, for example. Only one or several spins on each chain are coupled.
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