t-J model

In quantum physics, system's models are usually based on the Hamiltonian operator $\backslash hat\; H$, corresponding to the total energy of that system, including both kinetic energy and potential energy.

The t-J Hamiltonian can be derived from the $\backslash hat\; H$ of the Hubbard model using the Schrieffer–Wolff transformation, with the transformation generator depending on t/U and excluding the possibility for electrons to doubly occupy a lattice's site,This is done using a projector quantum operator $\backslash hat\; P$ that project $\backslash hat\; H\_t$ on the subspace where fermionic operators can not add an electron on a site already occupied (see next note) which results in:{{Cite book |last=Eckle |first=Hans-Peter |title=Models of Quantum Matter |publisher=Oxford University Press |year=2019 |isbn=9780199678839 |chapter=8.8.1 From the Hubbard to the t–J model: non-half filled band case |chapter-url=http://physics.gu.se/~tfkhj/HubbardEckle.pdf}}

:$\backslash hat\; H\; =\; -t\backslash sum\_\{\backslash langle\; ij\backslash rangle,\backslash sigma\}\; \backslash left(\; c\_\{i\backslash sigma\}^\{\backslash dagger\}\; c\_\{j\backslash sigma\}\; +\; \backslash mathrm\{h.c.\}\; \backslash right)$

+ J\sum_{\langle ij\rangle}\left(\mathbf{S}_{i}\cdot \mathbf{S}_{j}-\frac{n_in_j}{4}\right) + O(t^3/U^2)

where the term in t corresponds to the kinetic energy and is equal to the one in the Hubbard model. The second one is the potential energy approximated at the second order, because this is an approximation of the Hubbard model in the limit U >> t developed in power of t. Terms at higher order can be added.

The parameters are:

• {{underset|⟨ij⟩|Σ}} is the sum over nearest-neighbor sites i and j, for all sites, typically on a two-dimensional square lattice,
• c{{su|p=†|b=iσ}}, c{{su|b=iσ}} are the fermionic creation and annihilation operators at site i,
• σ is the spin polarization,
• t is the hopping integral,
• J is the antiferromagnetic exchange coupling, J = {{sfrac|4t²|U}},
• U is the on-site coulombic repulsion, that must satisfy the condition for U >> t,
• n_i = {{underset|σ|Σ}}c{{su|p=†|b=iσ}}c{{su|b=iσ}} is the particle number at site i and can be maximum 1, so that double occupancy is forbidden (in the Hubbard model is possible),
• S_i and S_j are the spins on sites i and j,
• h. c. stands for Hermitian conjugate,

If n_i = 1, that is when in the ground state, there is just one electron per lattice's site (half-filling), the model reduces to the Heisenberg model and the ground state reproduce a dielectric antiferromagnets (Mott insulator).{{Cite journal |last1=Izyumov |first1=Yu. A. |last2=Chashchin |first2=N. I. |year=1998 |title=tJ -model in terms of equations with variational derivatives |journal=Condensed Matter Physics |volume=1 |issue=1 |pages=41–56|doi=10.5488/CMP.1.1.41 |bibcode=1998CMPh....1...41I |s2cid=11254082 |doi-access=free }}

The model can be further extended considering also the next-nearest-neighbor sites and the chemical potential to set the ground state in function of the total number of particles:{{cite journal|doi=10.1103/PhysRevB.57.10913 |first=Naoum |last=Karchev |title=Generalized CP¹ model from the t₁-t₂-J model |journal=Phys. Rev. B |volume=57 |issue=17 |page=10913 |date=1998|arxiv=cond-mat/9706105 |bibcode=1998PhRvB..5710913K |s2cid=12865671 }}{{Cite journal |last=Yanagisawa |first=Takashi |date=2008-02-12 |title=Phase diagram of thet–U²Hamiltonian of the weak coupling Hubbard model |url=https://iopscience.iop.org/article/10.1088/1367-2630/10/2/023014 |journal=New Journal of Physics |language=en |volume=10 |issue=2 |pages=023014 |doi=10.1088/1367-2630/10/2/023014 |arxiv=0803.1739 |bibcode=2008NJPh...10b3014Y |s2cid=55405863 |issn=1367-2630}}

:$$

\mathcal{\hat H} = t_1 \sum\limits_{\langle i,j \rangle} \left( c_{i\sigma}^{\dagger} c_{j\sigma} + \mathrm{h.c.} \right) \ + \ t_2 \sum\limits_{\langle\langle i,j \rangle\rangle} \left( c_{i\sigma}^{\dagger} c_{j\sigma} + \mathrm{h.c.} \right) \ + \ J \sum\limits_{\langle i,j \rangle} \left( \mathbf{S}_{i} \cdot \mathbf{S}_{j} - \frac{ n_{i} n_{j} }{4}\right) - \ \mu\sum\limits_{i} n_{i} ,

where ⟨...⟩ and ⟨⟨...⟩⟩ denote the nearest and next-nearest neighbors, respectively, with two different values for the hopping integral (t₁ and t₂) and μ is the chemical potential.

{{Reflist}}

• {{Cite book |title=Lectures on Correlation and Magnetism |first=Patrik |last=Fazekas |publisher=World Scientific |year=1999 |isbn=978-981-4499-62-0 |series=Series in Modern Condensed Matter Physics: Volume 5 |volume=5 |page=199 |language=EN |doi=10.1142/2945}}
• {{cite journal|title=t-J model then and now: A personal perspective from the pioneering times|first=Józef|last=Spałek|arxiv=0706.4236|journal=Acta Phys. Pol. A|volume=111|issue=4|pages=409–424|date=2007|doi=10.12693/APhysPolA.111.409|bibcode=2007AcPPA.111..409S|s2cid=53117123}}
• {{Citation |last=Dr Mitchell |title=Electron interactions and the Hubbard model |url=https://www.youtube.com/watch?v=wtYQlELm3tk |language=en |access-date=2022-08-29}}

Category:Quantum lattice models