Magnetic topological insulator#Axion coupling

The $\backslash mathbb\{Z\}\_2$ classification of a 3D crystalline topological insulator can be understood in terms of the axion coupling $\backslash theta$. A scalar quantity that is determined from the ground state wavefunction{{cite journal |last1=Qi |first1=Xiao-Liang |last2=Hughes |first2=Taylor L. |last3=Zhang |first3=Shou-Cheng |title=Topological field theory of time-reversal invariant insulators |journal=Physical Review B |date=24 November 2008 |volume=78 |issue=19 |pages=195424 |doi=10.1103/PhysRevB.78.195424 |bibcode=2008PhRvB..78s5424Q |arxiv=0802.3537 |s2cid=117659977 }}

:$\backslash theta\; =\; -\backslash frac\{1\}\{4\backslash pi\}\backslash int\_\{\backslash rm\; BZ\}\; d^3k\; \backslash ,\; \backslash epsilon^\{\backslash alpha\; \backslash beta\; \backslash gamma\}\; \backslash text\{Tr\}\; \backslash Big[\; \backslash mathcal\{A\}\_\backslash alpha\; \backslash partial\_\backslash beta\; \backslash mathcal\{A\}\_\backslash gamma\; -i\backslash frac\{2\}\{3\}\; \backslash mathcal\{A\}\_\backslash alpha\; \backslash mathcal\{A\}\_\backslash beta\; \backslash mathcal\{A\}\_\backslash gamma\; \backslash Big]$ .

where $\backslash mathcal\{A\}\_\backslash alpha$ is a shorthand notation for the Berry connection matrix

:$\backslash mathcal\{A\}\_j^\{nm\}(\backslash mathbf\{k\})\; =\; \backslash langle\; u\_\{n\backslash mathbf\{k\}\}\; |\; i\backslash partial\_\{k\_j\}\; |\; u\_\{m\backslash mathbf\{k\}\}\; \backslash rangle$,

where $|\; u\_\{m\backslash mathbf\{k\}\}\; \backslash rangle$ is the cell-periodic part of the ground state Bloch wavefunction.

The topological nature of the axion coupling is evident if one considers gauge transformations. In this condensed matter setting a gauge transformation is a unitary transformation between states at the same $\backslash mathbf\{k\}$ point

:$|\backslash tilde\{\backslash psi\}\_\{n\backslash mathbf\{k\}\}\backslash rangle\; =\; U\_\{mn\}(\backslash mathbf\{k\})|\backslash psi\_\{n\backslash mathbf\{k\}\}\backslash rangle$.

Now a gauge transformation will cause $\backslash theta\; \backslash rightarrow\; \backslash theta\; +2\backslash pi\; n$ , $n\; \backslash in\; \backslash mathbb\{N\}$. Since a gauge choice is arbitrary, this property tells us that $\backslash theta$ is only well defined in an interval of length $2\backslash pi$ e.g. $\backslash theta\; \backslash in\; [-\backslash pi,\backslash pi]$.

The final ingredient we need to acquire a $\backslash mathbb\{Z\}\_2$ classification based on the axion coupling comes from observing how crystalline symmetries act on $\backslash theta$.

• Fractional lattice translations $\backslash tau\_q$, n-fold rotations $C\_n$: $\backslash theta\; \backslash rightarrow\; \backslash theta$.
• Time-reversal $T$, inversion $I$: $\backslash theta\; \backslash rightarrow\; -\backslash theta$.

The consequence is that if time-reversal or inversion are symmetries of the crystal we need to have $\backslash theta\; =\; -\backslash theta$

and that can only be true if $\backslash theta\; =\; 0$(trivial),$\backslash pi$(non-trivial) (note that $-\backslash pi$ and $\backslash pi$ are identified) giving us a $\backslash mathbb\{Z\}\_2$ classification. Furthermore, we can combine inversion or time-reversal with other symmetries that do not affect $\backslash theta$ to acquire new symmetries that quantize $\backslash theta$. For example, mirror symmetry can always be expressed as $m=I*C\_2$ giving rise to crystalline topological insulators,{{cite journal |last1=Fu |first1=Liang |title=Topological Crystalline Insulators |journal=Physical Review Letters |date=8 March 2011 |volume=106 |issue=10 |pages=106802 |doi=10.1103/PhysRevLett.106.106802 |pmid=21469822 |bibcode=2011PhRvL.106j6802F |arxiv=1010.1802 |s2cid=14426263 }} while the first intrinsic magnetic topological insulator MnBi$\_2$Te$\_4${{cite journal |last1=Gong |first1=Yan |display-authors=etal |title=Experimental realization of an intrinsic magnetic topological insulator |journal=Chinese Physics Letters |year=2019 |volume=36 |issue=7 |page=076801 |doi=10.1088/0256-307X/36/7/076801 |arxiv=1809.07926|bibcode=2019ChPhL..36g6801G |s2cid=54224157 }}{{cite journal |last1=Otrokov |first1=Mikhail M. |display-authors=etal |title=Prediction and observation of the first antiferromagnetic topological insulator |journal=Nature |year=2019 |volume=576 |issue=7787 |pages=416–422 |doi=10.1038/s41586-019-1840-9 |pmid=31853084 |arxiv=1809.07389|s2cid=54016736 }} has the quantizing symmetry $S=T*\backslash tau\_\{1/2\}$.

So far we have discussed the mathematical properties of the axion coupling. Physically, a non-trivial axion coupling ($\backslash theta\; =\; \backslash pi$) will result in a half-quantized surface anomalous Hall conductivity ($\backslash sigma^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}=e^2/2h$) if the surface states are gapped. To see this, note that in general $\backslash sigma^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}$ has two contribution. One comes from the axion coupling $\backslash theta$, a quantity that is determined from bulk considerations as we have seen, while the other is the Berry phase $\backslash phi$ of the surface states at the Fermi level and therefore depends on the surface. In summary for a given surface termination the perpendicular component of the surface anomalous Hall conductivity to the surface will be

:$\backslash sigma^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}\; =\; -\backslash frac\{e^2\}\{h\}\backslash frac\{\backslash theta-\backslash phi\}\{2\backslash pi\}\; \backslash \; \backslash text\{mod\}\; \backslash \; e^2/h$.

The expression for $\backslash sigma^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}$ is defined $\backslash text\{mod\}\; \backslash \; e^2/h$ because a surface property ($\backslash sigma^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}$) can be determined from a bulk property ($\backslash theta$) up to a quantum. To see this, consider a block of a material with some initial $\backslash theta$ which we wrap with a 2D quantum anomalous Hall insulator with Chern index $C=1$. As long as we do this without closing the surface gap, we are able to increase $\backslash sigma^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}$ by $e^2/h$ without altering the bulk, and therefore without altering the axion coupling $\backslash theta$.

One of the most dramatic effects occurs when $\backslash theta=\backslash pi$ and time-reversal symmetry is present, i.e. non-magnetic topological insulator. Since $\backslash boldsymbol\{\backslash sigma\}^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}$ is a pseudovector on the surface of the crystal, it must respect the surface symmetries, and $T$ is one of them, but $T\backslash boldsymbol\{\backslash sigma\}^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}\; =-\; \backslash boldsymbol\{\backslash sigma\}^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}$ resulting in $\backslash boldsymbol\{\backslash sigma\}^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}\; =\; 0$. This forces $\backslash phi\; =\; \backslash pi$ on every surface resulting in a Dirac cone (or more generally an odd number of Dirac cones) on every surface and therefore making the boundary of the material conducting.

On the other hand, if time-reversal symmetry is absent, other symmetries can quantize $\backslash theta=\backslash pi$ and but not force $\backslash boldsymbol\{\backslash sigma\}^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}$ to vanish. The most extreme case is the case of inversion symmetry (I). Inversion is never a surface symmetry and therefore a non-zero $\backslash boldsymbol\{\backslash sigma\}^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}$ is valid. In the case that a surface is gapped, we have $\backslash phi\; =\; 0$ which results in a half-quantized surface AHC $\backslash sigma^\{\backslash text\{surf\}\}\_\{\backslash text\{AHC\}\}\; =\; -\backslash frac\{e^2\}\{2h\}$.

A half quantized surface Hall conductivity and a related treatment is also valid to understand topological insulators in magnetic field {{Cite journal

| volume = 58

| issue = 18

| pages = 1799–1802

| last = Wilczek

| first = Frank

| title = Two applications of axion electrodynamics

| journal = Physical Review Letters

| date = 4 May 1987

| doi = 10.1103/PhysRevLett.58.1799

| pmid = 10034541

| bibcode=1987PhRvL..58.1799W

}} giving an effective axion description of the electrodynamics of these materials.{{Cite journal

| journal = Physical Review B

| volume = 78

| issue = 19

| page = 195424

| last1 = Qi | first1 = Xiao-Liang

| last2 = Hughes | first2 = Taylor L.

| last3 = Zhang | first3 = Shou-Cheng

| title = Topological field theory of time-reversal invariant insulators

| date = 24 November 2008

| doi = 10.1103/PhysRevB.78.195424

| bibcode = 2008PhRvB..78s5424Q |arxiv = 0802.3537

| s2cid = 117659977

}} This term leads to several interesting predictions including a quantized magnetoelectric effect.{{Cite journal

| doi = 10.1103/Physics.1.36

| volume = 1

| page = 36

| last = Franz | first = Marcel

| title = High-energy physics in a new guise

| journal = Physics

| date = 24 November 2008

| bibcode = 2008PhyOJ...1...36F

| doi-access = free

}} Evidence for this effect has recently been given in THz spectroscopy experiments performed at the Johns Hopkins University.{{Cite journal|last1=Wu|first1=Liang|last2=Salehi|first2=M.|last3=Koirala|first3=N.|last4=Moon|first4=J.|last5=Oh|first5=S.|last6=Armitage|first6=N. P.|date=2 December 2016|title=Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator|journal=Science|language=en|volume=354|issue=6316|pages=1124–1127|doi=10.1126/science.aaf5541|issn=0036-8075|pmid=27934759|arxiv=1603.04317|bibcode=2016Sci...354.1124W|s2cid=25311729}}

{{Expand section|date=February 2021}}

Magnetic topological insulators have proven difficult to create experimentally. In 2023 it was estimated that a magnetic topological insulator might be developed in 15 years' time.{{Cite web |last=Anirban |first=Ankita |title=15 years of topological insulators |url=https://wikipedialibrary.wmflabs.org/?next_url=/ezproxy/r/ezp.2aHR0cHM6Ly93d3cubmF0dXJlLmNvbS9hcnRpY2xlcy9zNDIyNTQtMDIzLTAwNTg3LXk- |access-date=2024-08-18 |website=Nature |language=en}}

A compound made from manganese, bismuth, and tellurium (MnBi₂Te₄) has been predicted to be a magnetic topological insulator. In 2024, scientists at the University of Chicago used MnBi2Te4 to develop a form of optical memory which is switched using lasers. This memory storage device could store data more quickly and efficiently, including in quantum computing.{{Cite web |date=2024-08-14 |title=MnBi2Te4 Unveiled: A Breakthrough in Quantum and Optical Memory Technology |url=https://scitechdaily.com/mnbi2te4-unveiled-a-breakthrough-in-quantum-and-optical-memory-technology/ |access-date=2024-08-18 |website=SciTechDaily |language=en-US}}

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Category:Condensed matter physics

Category:Magnetism