.. _sphx_glr_api_gallery_condensed_spin_chains:

Quantum spin chains
-------------------

Exact diagonalization of spin-1/2 chains in their symmetry sectors,
checked against the classic exact solutions: Bethe's ansatz for the
Heisenberg antiferromagnet, the free-fermion solution of the
transverse-field Ising chain, and the Calabrese-Cardy logarithmic
entanglement of a critical chain versus the area law of a gapped one.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Hans Bethe solved the spin-1/2 Heisenberg ring H = J\sum_i \mathbf{S}_i\cdot\mathbf{S}_{i+1} exactly in 1931 by writing its eigenstates as superpositions of interacting magnons. Here the ground-state energy from bethe_ansatz_xxx_ground_energy is compared with Lanczos exact diagonalization of xxz_hamiltonian in the S^z = 0 sector, and extrapolated to Hulthen&#x27;s thermodynamic limit E_0/N = J(1/4 - \ln 2).">

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  .. image:: /api/gallery/condensed/spin_chains/images/thumb/sphx_glr_plot_bethe_ansatz_heisenberg_chain_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/spin_chains/plot_bethe_ansatz_heisenberg_chain`

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      <div class="sphx-glr-thumbnail-title">Bethe's ansatz for the Heisenberg antiferromagnetic chain</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="In a gapped chain the entanglement entropy of a block saturates -- an &quot;area law&quot;, the boundary of a 1D block being two points. At a quantum critical point Calabrese and Cardy (2004) showed it instead grows as S(\ell) = \frac{c}{3}\ln\!\left[\frac{N}{\pi}\sin\frac{\pi\ell}{N}\right] + c&#x27;, with c the central charge of the conformal field theory. Exact ground states of the transverse-field Ising chain (~physicskit.condensed.spin_chains.tfim_hamiltonian) show both behaviors, and fit_central_charge recovers the Ising value c = 1/2.">

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  .. image:: /api/gallery/condensed/spin_chains/images/thumb/sphx_glr_plot_calabrese_cardy_entanglement_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/spin_chains/plot_calabrese_cardy_entanglement`

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      <div class="sphx-glr-thumbnail-title">Calabrese-Cardy: logarithmic entanglement at criticality vs. the area law</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Lieb, Schultz and Mattis (1961) and Pfeuty (1970) mapped the transverse-field Ising chain H = -J\sum\sigma^x_i\sigma^x_{i+1} - h\sum\sigma^z_i onto free fermions with dispersion \varepsilon_k = 2\sqrt{J^2 + h^2 - 2Jh\cos k}. This example checks the full many-body spectrum from tfim_free_fermion_spectrum against exact diagonalization of tfim_hamiltonian, and follows the gap as it closes at the quantum critical point h = J.">

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  .. image:: /api/gallery/condensed/spin_chains/images/thumb/sphx_glr_plot_transverse_field_ising_free_fermions_thumb.png
    :alt:

  :doc:`/api/gallery/condensed/spin_chains/plot_transverse_field_ising_free_fermions`

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      <div class="sphx-glr-thumbnail-title">The transverse-field Ising chain as free fermions</div>
    </div>


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    </div>


.. toctree::
   :hidden:

   /api/gallery/condensed/spin_chains/plot_bethe_ansatz_heisenberg_chain
   /api/gallery/condensed/spin_chains/plot_calabrese_cardy_entanglement
   /api/gallery/condensed/spin_chains/plot_transverse_field_ising_free_fermions