Operator K-Theory for Disordered Topological Insulators and Superconductors
Alexander Alldridge (Cologne)
Gauge Theory and Topological Quantum Matter, Bad Honnef, September 20, 2018
Based on joint work with Christopher Max (Cologne).Supported by DFG grants AL 698/3-1 and TRR 183.
The hospitality of the ITP Cologne and the UC Berkeley Maths Department are gratefully acknowledged.
The importance of being real
The reality of Nambu space
Let \(\sh V\) be the Hilbert space of fermionic single particle creation operators \(c_i^\dagger\). The dual space \(\sh V^*\) consists of single particle annihilation operators \(c_i\). There is a fundamental anti-linear operation \[ c_i^\dagger\longmapsto c_i. \]
Formally, this is the Fréchet–Riesz isomorphism \[ R:\sh V\longrightarrow\sh V^*,\quad (R\psi_1)(\psi_2)\defi\langle\psi_1\mid\psi_2\rangle. \]
On Nambu space \(\sh W\defi\sh V\oplus\sh V^*\), this assembles to \[ \gamma\defi \begin{pmatrix} 0&R^{-1}\\ R&0 \end{pmatrix},\quad\gamma^*=\gamma,\quad\gamma^2=1. \]
Definition. An anti-unitary endomorphism \(\gamma\) of a Hilbert space \(\sh W\) is called a real structure if \(\gamma^2=1\). Such a pair \((\sh W,\gamma)\) is called a real Hilbert space.
Operators on Nambu space
Recall that a subalgebra \(A\subseteq\sh L(\sh H)\), where \(\sh H\) is some Hilbert space, is called a C\(^*\)-algebra if \(A\) is closed under the norm and taking the adjoint.
The real structure \(\gamma\) encodes the fermionic CAR relations: \[ \{\psi_1,\psi_2\}\defi\langle\gamma\psi_1\mid\psi_2\rangle\quad\Longrightarrow\quad\{c_i^\dagger,c_j\}=\delta_{ij}. \]
A fermionic Hamiltonian has to preserve the CAR relations (Fermi constraint) \[ \{H\psi_1,\psi_2\}+\{\psi_1,H\psi_2\}=0. \]
In terms of \(\gamma\), this can be equivalently expressed as \[ \gamma H^*\gamma=-H. \] (Here, \(H^*\) is a mathematician’s notation for the adjoint operator \(H^\dagger\).)
Definition. Let \(A\) be a C\(^*\)-algebra. An anti-linear involutive \(*\)-automorphism \(x\longmapsto\overline x\) is called a real structure. The pair \((A,\overline\cdot)\) is a real C\(^*\)-algebra. We say \(x\) is real/imaginary if \(\overline x=\pm x\).
The algebra \(\sh L(\sh W)\) of bounded operators on \(\sh W\) carries the real structure \[ \overline x\defi \gamma x\gamma. \] Thus, the Fermi constraint on \(H\) is equivalent to \(H\) being imaginary (and Hermitian).
Time reversal changes reality
Time reversal is an anti-unitary endomorphism of \(\sh V\) such that \[ T^2=-1. \]
Definition. An anti-unitary endomorphism \(T\) of a Hilbert space \(\sh V\) is called a quaternionic structure if \(T^2=-1\). The pair \((\sh V,T)\) is a quaternionic Hilbert space.
Given a quaternionic structure \(T\) on \(\sh V\), the bounded operators obtain a real structure: \[ \overline x\defi T^*xT=-TxT. \]
Example. On \(\def\cplxs{\mathbb C}V=\cplxs^2\), we may consider the quaternionic structure \[ \def\ger{\mathfrak} \ger c\defi \begin{pmatrix} 0&\mathsf c\\-\mathsf c&0 \end{pmatrix}, \] where \(\mathsf c\) is complex conjugation. The real elements in \(M_2(\cplxs)=\sh L(\cplxs^2)\) for \(\ger c\) are the quaternions \[ \overline x=x\quad\Longleftrightarrow\quad x= \begin{pmatrix} \alpha&\beta\\ -\overline\beta&\overline\alpha \end{pmatrix}. \] We denote the corresponding real C\(^*\)-algebra by \(\mathbb H_\cplxs\).
Ground states
The algebra of tight-binding observables
We work on a square lattice \(\Lambda=\ints^d\) with on-site degrees of freedom encoded in the finite-dimensional Hilbert space \(V\). Thus \[ \sh V=\ell^2(\Lambda)\otimes V,\quad\sh W=\sh V\oplus\sh V^*=\ell^2(\Lambda)\otimes W,\quad W\defi V\oplus V^*. \]
An operator \(O\in\sh L(\sh W)\) has finite hopping (or is controlled) if \[ \exists R>0\colon\quad O=\textstyle\sum_{\norm{x-y}<R}O_{xy}\ket x\bra y,\quad O_{xy}\in\endo W. \] (Here, the convergence is in the weak operator topology.)
Definition/Proposition (Roe 2003). Let \(C^*_u(\Lambda,W)\subseteq\sh L(\sh W)\) be the norm closure of the set of controlled operators. Then \(C^*_u(\Lambda,W)\) is a real C\(^*\)-algebra.
If we replace \(W\) by \(\cplxs\), the resulting C\(^*\)-algebra is denoted \(C^*_u(\Lambda)\) and called the uniform Roe algebra. In general, we might call \(C^*_u(\Lambda,W)\) the algebra of tight-binding observables.
Proposition (Roe 2003). The algebra of translation-invariant tight-binding observables is \[ C^*_u(\Lambda,W)^{\ints^d}\cong\sh C(\reals^d/\Lambda)\otimes\endo W. \] The isomorphism is given by \(O\longmapsto\Parens{k\longmapsto O(k)}\).
Enter disorder
Disorder breaks translational invariance microscopically; but at a macroscopic level, it is preserved. How to model this fact?
An argument of (Bellissard 1986; Bellissard 1992) goes roughly as follows:
- Observables are operator-valued functions \(\omega\longmapsto O_\omega\) on the space of disorder configurations \(\Omega\).
- Such configurations are given by crystalline defects, so the translations \(\ints^d\) act on \(\Omega\).
- After compactification, \(\Omega\) is a compact Hausdorff space equipped with an action of \(\ints^d\).
- Shifting \(\omega\) by \(x\) corresponds to transforming \(O_\omega\) by the translation action \(U_x\) on \(\sh W\): \[ O_{\omega\cdot x}=U_x^*O_\omega U_x,\quad\forall\omega\in\Omega,x\in\ints^d. \]
Definition. The algebra of macroscopically homogeneous tight-binding observables is \[ \mathbb A\defi\Set{O\in\sh C\Parens{\Omega,C^*_u(\Lambda,W)}}{\forall\omega,x\colon O_{\omega\cdot x}=U_x^*O_\omega U_x}, \] the real C\(^*\)-algebra of equivariant maps from disorder configurations to tight-binding operators.
Ground states as operators
Let \(H\) be a gapped Hamiltonian on \(\sh W\). Using the spectral decomposition \[ H=\int_{\reals\setminus(-\eps,\eps)}\lambda\,dP(\lambda), \] we define \[ J\defi i\int_\eps^\infty dP(\lambda)-i\int_{-\infty}^{-\eps}dP(\lambda). \]
Then we have \[ J^2=-1,\quad J=-J^*, \] and the Fermi constraint \(\overline H=-H\) translates to \[ \overline J=J. \] We call \(J\) an invariant quasi-particle vacuum (or IQPV) if \(H\) is translation invariant, cf. (Kennedy and Zirnbauer 2016).
Definition. A real unitary \(J\in\mathrm U(\mathbb A)\) is called a disordered IQPV if \(J^2=-1\).
Symmetries, in a class of their own
Physical symmetries
| \(s\) | class | symmetries | comments |
|---|---|---|---|
| \(0\) | \(D\) | none | – |
| \(1\) | \(D\mathrm{I\!I\!I}\) | \(T\) | time reversal |
| \(2\) | \(A\mathrm{I\!I}\) | \(T,Q\) | charge |
| \(3\) | \(C\mathrm{I\!I}\) | \(T,Q,C\) | particle-hole symmetry |
| \(4\) | \(C\) | \(S_1,S_2,S_3\) | spin rotations |
| \(5\) | \(C\mathrm I\) | \(S_1,S_2,S_3,T\) | – |
| \(6\) | \(A\mathrm I\) | \(S_1,S_2,S_3,T,Q\) | – |
| \(7\) | \(BD\mathrm I\) | \(S_1,S_2,S_3,T,Q,C\) | – |
Time-reversal symmetry \(T\) is a quaternionic structure on \(V\). We extend it to \(W\) by \[ T\equiv \begin{pmatrix} T&0\\ 0&RTR^{-1} \end{pmatrix}. \]
Physical spin rotations \(S_\mu\in\mathrm U(V)\), \(\mu=1,2,3\), satisfy \[ S_1S_2=-S_2S_1=S_3,\quad S_\mu^2=1,\quad TS_\mu=-S_\mu T, \] and are extended to \(W\) by \[ S_\mu\equiv \begin{pmatrix} S_\mu&0\\0&-RS_\mu R^{-1} \end{pmatrix}. \]
Charge \(Q\) assigns \(\pm1\) to particle creation/annihilation operators, so \[ Q= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}\in\endo{V\oplus V^*}=\endo W. \]
Particle-hole symmetry is \(C\defi\gamma S\) where \(S\in\mathrm U(V)\) satisfies \[ S^2=1,\quad[S,T]=[S,S_\mu]=0, \] and is extended to \(W\) by \[ S\equiv \begin{pmatrix} S&0\\0&RSR^{-1} \end{pmatrix}. \]
| \(s\) | class | symmetries | comments |
|---|---|---|---|
| \(0\) | \(A\) | \(Q\) | charge |
| \(1\) | \(A\mathrm{I\!I\!I}\) | \(Q,C\) | particle-hole conjugation |
Note: If \([H,Q]=0\) and \(\gamma H\gamma=-H\), then \[ H\equiv \begin{pmatrix} H&0\\ 0&-RHR^{-1} \end{pmatrix} \] so in this case, \(H\) is determined by its restriction to \(\sh V\).
Symmetries, under another name
How should the symmetries of a Hamiltonian be reflected on the ground state?
Idea (Kennedy and Zirnbauer 2016): assign pseudo-symmetries to symmetries \(g\longmapsto J_\alpha\) such that \[ \forall g\in G_s\colon gH=Hg\quad\Longleftrightarrow\quad \forall \alpha=1,\dotsc,s\colon J_\alpha J=-JJ_\alpha \] where \[ \overline{J_\alpha}=J_\alpha\in\mathrm U(W),\quad J_\alpha J_\beta+J_\beta J_\alpha=-2\delta_{\alpha\beta}. \]
For \(s=0,1,2,3\), this can be achieved by setting \[ J_1\defi J_T\defi \gamma T,\quad J_2\defi J_Q\defi i\gamma TQ,\quad J_3\defi J_C\defi i\gamma CQ=iSQ. \]
For \(s\sge 4\), one doubles \(W\) to \(W\otimes\cplxs^2\) and sets \[ J_\mu\defi iS_\mu\otimes\sigma_z,\quad J_4\defi {\id}_W\otimes i\sigma_y,\quad J_{5/6/7}\defi J_{T/Q/C}\otimes\sigma_x. \]
Definition. A disordered IQPV \(J\) is said to be in symmetry class \(s\) if (with \(J\equiv J\otimes\sigma_x\) in case \(s\sge4\)) \[ JJ_\alpha+J_\alpha J=0,\quad\forall\alpha=1,\dotsc,s. \]
Mathematical interlude ;)
Clifford algebras
Definition. Let \(\Ccl_{r,s}\) be the C\(^*\)-algebra defined by unitary generators \(k_1,\dotsc,k_r,j_1,\dotsc,j_s\) and relations \[ \left. \begin{aligned} &k_ak_b+k_bk_a=2\delta_{ab}\\ &j_\alpha j_\beta+j_\beta j_\alpha=-2\delta_{\alpha\beta}\\ &k_aj_\alpha+j_\alpha k_a=0 \end{aligned}\right\}\quad\forall a,b=1,\dotsc,r,\alpha,\beta=1,\dotsc,s. \] This is the Clifford algebra with \(r\) positive and \(s\) negative generators.
Universal property. Let \(B\) be a unital C\(^*\)-algebra and \(K_1,\dotsc,K_r,J_1,\dotsc,J_s\in\mathrm U(B)\) be such that \[ \left. \begin{aligned} &K_aK_b+K_bK_a=2\delta_{ab}\\ &J_\alpha J_\beta+J_\beta J_\alpha=-2\delta_{\alpha\beta}\\ &K_aJ_\alpha+J_\alpha K_a=0 \end{aligned}\right\}\quad\forall a,b=1,\dotsc,r,\alpha,\beta=1,\dotsc,s. \] Then there is a unique \(*\)-morphism \[ \Ccl_{r,s}\longrightarrow B,\quad \left. \begin{aligned} &k_a\\ &j_\alpha \end{aligned}\right\}\longmapsto\left\{ \begin{aligned} &K_a\\ &J_\alpha \end{aligned}\right.. \]
In particular, we may define a real structure on \(\Ccl_{r,s}\) by \[ \overline{k_a}\defi k_a,\quad\overline{j_\alpha}\defi j_\alpha. \] Equipped with this real structure, we denote \(\Ccl_{r,s}\) by \(\Cl_{r,s}\).
Real universal property. Let \(B\) be a unital real C\(^*\)-algebra and \(K_1,\dotsc,J_s\in\mathrm U(B)\) be real such that \[ \left. \begin{aligned} &K_aK_b+K_bK_a=2\delta_{ab}\\ &J_\alpha J_\beta+J_\beta J_\alpha=-2\delta_{\alpha\beta}\\ &K_aJ_\alpha+J_\alpha K_a=0 \end{aligned}\right\}\quad\forall a,b=1,\dotsc,r,\alpha,\beta=1,\dotsc,s. \] Then there is a unique real \(*\)-morphism \[ \Ccl_{r,s}\longrightarrow B,\quad \left. \begin{aligned} &k_a\\ &j_\alpha \end{aligned}\right\}\longmapsto\left\{ \begin{aligned} &K_a\\ &J_\alpha \end{aligned}\right.. \]
In particular, \(J_1,\dotsc,J_s\in\mathrm U(W)\) define a real \(*\)-morphism \[ \Cl_{0,s}\longrightarrow\endo W. \]
Periodicity
There is an isomorphism of real C\(^*\)-algebras \(\Cl_{1,1}\longrightarrow M_2(\cplxs)\), defined by \[ k_1\longmapsto\sigma_x,\quad j_1\longmapsto i\sigma_y. \]
This induces the (1,1) periodicity isomorphisms \[ \Cl_{r+1,s+1}\cong \Cl_{r,s}\mathrel{\hat\otimes} M_2(\cplxs). \] Here, the tensor product is graded on the RHS, which means that \[ (I\otimes\sigma)\cdot(I'\otimes\sigma')\defi-(II')\otimes(\sigma\sigma'),\quad\forall I,I'\in\Braces{k_a,j_\alpha},\sigma,\sigma'\in\Braces{\sigma_x,i\sigma_y}. \]
Proposition (Atiyah, Bott, and Shapiro 1964). There are isomorphisms of real C\(^*\)-algebras \[ \Cl_{0,r+2}\cong \Cl_{0,2}\otimes \Cl_{r,0},\quad \Cl_{r+2,0}\cong \Cl_{2,0}\otimes \Cl_{0,r} \]
Together with \[ \Cl_{0,2}\cong\mathbb H_\cplxs,\quad \Cl_{2,0}\cong M_2(\cplxs),\quad\mathbb H_\cplxs\otimes\mathbb H_\cplxs\cong M_4(\cplxs), \] this implies eightfold periodicity \[ \Cl_{0,8}\cong\Cl_{8,0}\cong \Cl_{0,2}\otimes \Cl_{0,2}\otimes \Cl_{2,0}\otimes \Cl_{2,0}\cong \mathbb H_\cplxs\otimes\mathbb H_\cplxs\otimes M_4(\cplxs)\cong M_{16}(\cplxs). \]
Real K-theory
Let \(B\subseteq\sh L(\sh H)\) be a real C\(^*\)-algebra such that \(B\sh H=\sh H\). The multiplier algebra \(\mathrm M(B)\) is the idealiser \[ \mathrm M(B)\defi\Set{x\in\sh L(\sh H)}{xB\cup Bx\subseteq B}. \] Let \(\knums\) be the real C\(^*\)-algebra of compact operators on some (separable) real Hilbert space. The stable multiplier algebra and Calkin algebra of \(B\) are, respectively, \[ \mathrm M^s(B)\defi\mathrm M(B\otimes\knums),\quad\mathrm Q^s(B)\defi\mathrm M^s(B)/(B\otimes\knums). \]
Definition (Kasparov 1975). A \(KR^{-p,q}\) cycle is a pair \((\phi,\hat J)\), where \(\phi:\Cl_{q+1,p}\longrightarrow\mathrm M^s(B):k_a,j_\alpha\longmapsto K_a,J_\alpha\) is a real \(*\)-morphism and \(\hat J\in\mathrm M^s(B)\) satisfies \[ \hat J^2\equiv-1\pmod{B\otimes\knums},\quad\hat J=-\hat J^*=\overline{\hat J},\quad \forall a,\alpha\colon\quad\hat JK_a+K_a\hat J=\hat JJ_\alpha+J_\alpha\hat J=0. \] A cycle is called degenerate if \(\hat J^2=-1\). The real \(K\)-theory of \(B\) is \[ KR^{-p,q}(B)\defi\Braces{\text{$KR^{-p,q}$ cycles}}\bigm/\text{homotopy}. \] This set has a natural Abelian group structure; degenerate cycles represent zero (Skandalis 1984).
We have an isomorphism \(KR^{-(p+1),q+1}(B)\cong KR^{-p,q}(B)\), and may define \[ KR^{-p+q}(B)\defi KR^{-p,q}(B). \] Then we have \(KR^{-p+8}(B)\cong KR^{-p}(B)\).
Invariants unperturbed by disorder
Bulk-to-boundary sequence
We introduce a boundary \(\Lambda_\partial\defi\ints^{d-1}\times\{0\}\). To connect bulk and boundary, we consider the half-space \[ \hat\Lambda\defi\ints^{d-1}\times\nats,\quad\hat{\sh W}\defi\ell^2(\hat\Lambda)\otimes W. \]
As a closed \(*\)-subalgebra of \(\sh C(\Omega,\sh L(\sh W))\), \(\mathbb A\) is generated by \[ \begin{gathered} \textstyle\sum_{x\in\Lambda}f(\omega\cdot x)\ket x\bra x,\ f\in\sh C(\Omega,\endo W),\\ U_1\defi U_{(1,0,\dotsc,0)},U_2\defi U_{(0,1,0,\dotsc,0)},\dotsc,U_d\defi U_{(0,\dotsc,0,1)}. \end{gathered} \]
Define the half-space algebra \(\hat{\mathbb A}\) to be the closed \(*\)-subalgebra of \(\sh C(\Omega,\sh L(\hat{\sh W}))\) generated by \[ \textstyle\sum_{x\in\Lambda}f(\omega\cdot x)\ket x\bra x,\ f\in\sh C(\Omega,\endo W),\ U_1,\dotsc,U_{d-1},\ \hat U_d, \] where \(\hat U_d\) is the unilateral shift operator \[ \hat U_d\ket{x_1,\dotsc,x_d}\otimes w\defi\ket{x_1,\dotsc,x_{d-1},x_d+1}\otimes w,\quad x_d\sge0. \]
The boundary algebra \(\mathbb A_\partial\) is defined just as \(\mathbb A\), but with \(\Lambda_\partial\cong\ints^{d-1}\) in place of \(\Lambda=\ints^d\).
Proposition (Schulz-Baldes, Kellendonk, and Richter 2000). The assignment \(\hat U_d\longmapsto U_d\) induces a short exact sequence \[ \begin{CD} 0 @>>> {\mathbb A_\partial\otimes\knums(\ell^2(\nats))} @>>> {\hat{\mathbb A}} @>>> {\mathbb A} @>>> 0. \end{CD} \]
Manufacturing invariants
Let \(J\in\mathbb A\) be a disordered IQPV in symmetry class \(s\). Let \[ q:\sh W\longrightarrow\hat{\sh W} \] be the orthogonal projection and define \[ \hat J\defi qJq\in\sh C\Parens{\Omega,\sh L(\hat{\sh W})}. \]
Then we have \[ \hat J^2\equiv-1\pmod{\mathbb A_\partial\otimes\knums},\quad\hat J=-\hat J^*=\overline{\hat J},\quad \forall \alpha=1,\dotsc,s\colon\quad\hat JJ_\alpha+J_\alpha\hat J=0. \] Doubling \(W\) to \(W\otimes\cplxs^2\) and using \((1,1)\) periodicity, we obtain \[ K_1\defi 1\otimes\sigma_x,\quad \hat J\equiv\hat J\otimes\sigma_x,\ J_1\equiv J_1\otimes\sigma_x,\dotsc,J_s\equiv J_s\equiv J_s\otimes\sigma_x,\ J_{s+1}\defi1\otimes i\sigma_y. \]
Because \(\hat J\) idealises \(\mathbb A_\partial\otimes\knums(\ell^2(\nats))\), we have \(\hat J\in M^s(\mathbb A_\partial\otimes\knums)\), so that \(\hat J\) represents a \(KR\) class \[ [J]_\partial\in KR^{-(s+1)}(\mathbb A_\partial). \]
Definition. We call \([J]_\partial\) the boundary class of the disordered IQPV \(J\).
Effects of disorder (or their absence)
Assume that \(\Omega\) is a Bernoulli shift, i.e. an infinite product of on-site disorder configurations: \[ \Omega=\textstyle\prod_\Lambda\Omega_0. \] Fix some disorder configuration \(\omega_0\in\Omega\). There are real \(*\)-morphisms \[ \iota:C^*_u(\Lambda,W)^{\ints^d}\longrightarrow\mathbb A:O\longmapsto(\omega\longmapsto O),\quad\eps_{\omega_0}:\mathbb A\longrightarrow C^*_u(\Lambda,W)^{\ints^d}:O\longmapsto O_{\omega_0}. \]
Then \(\eps_{\omega_0}\circ\iota={\id}\) and if \(\Omega_0\) is contractible, then \(\iota\circ\eps_{\omega_0}\simeq{\id}\).
Proposition. If \(\Omega\) is the Bernoulli shift of a contractible space, then \[ \begin{gathered} KR^{-s}(\mathbb A)\cong KR^{-s}(\reals^d/\Lambda)=KR^{-s}(\mathbb T^d),\\ KR^{-(s+1)}(\mathbb A_\partial)=KR^{-(s+1)}(\mathbb T^{d-1}). \end{gathered} \] So the \(KR\)-groups are not sensitive to sufficiently ‘small’ disorder.
In particular, we obtain in \(d=0\): \[ KR^{-(s+2)}(\mathbb A)=KR^{-(s+2)}(\mathrm{pt}). \]
| \(s\) | class | \(KR^{-(s+2)}(\mathrm{pt})\) |
|---|---|---|
| \(0\) | \(D\) | \(\ints/2\) |
| \(1\) | \(D\mathrm{I\!I\!I}\) | \(0\) |
| \(2\) | \(A\mathrm{I\!I}\) | \(\ints\) |
| \(3\) | \(C\mathrm{I\!I}\) | \(0\) |
| \(4\) | \(C\) | \(0\) |
| \(5\) | \(C\mathrm I\) | \(0\) |
| \(6\) | \(A\mathrm I\) | \(\ints\) |
| \(7\) | \(BD\mathrm I\) | \(\ints/2\) |
Bulk-boundary correspondence
There is a way to assign, to every disordered IQPV \(J\) in symmetry class \(s\), a bulk class \[ [J]\in KR^{-(s+2)}(\mathbb A). \] This uses a different picture of \(KR\)-theory (Van Daele 1988).
The bulk-to-boundary sequence induces a long exact sequence in \(KR\) theory, \[ \begin{CD} \dotsm @>>> KR^{-(s+2)}(\mathbb A_\partial) @>>> KR^{-(s+2)}(\hat{\mathbb A}) @>>> KR^{-(s+2)}(\mathbb A)\\ @>{\partial}>> KR^{-(s+1)}(\mathbb A_\partial) @>>> KR^{-(s+1)}(\hat{\mathbb A}) @>>> KR^{-(s+1)}(\mathbb A) @>>> \dotsm \end{CD} \]
Theorem (A–C. Max). Under the boundary map in the long exact \(KR\) theory sequence, bulk and boundary classes correspond: \[ \partial[J]=[J]_\partial. \]
As a corollary, if the bulk phase \([J]\neq0\), then \(\hat J\) is gapless.
References
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