# Definition Time-reversal symmetric topological insulators (TRS-TIs) are a class of [[Topological Insulator|topological insulators]] protected by [[Time-Reversal Operator|time-reversal symmetry]]. In 2-dimensions, they are known as [[quantum spin-hall insulator|quantum spin-hall insulators]] and are characterized by a single $Z_2$ ([[Cyclic Group of Order 2]]) [[topological invariant]]. In 3-dimensions, there are 4 such invariants. They are characterized by gapless states on the boundary (i.e. the edge in 2D and the surface in 3D). In contrast to [[Antiferromagnetic Topological Insulator|antiferromagnetic topological insulators]], these surface states exist on all surfaces in contact with a [[Trivial Insulator]], such as vacuum. These metallic, gapless surface states cannot change as long as the bulk gap is not closed. # Classification TRS-TIs are classified into [[Weak Topological Insulator|weak topological insulators]] and [[Strong Topological Insulator|strong topological insulators]]. # Examples The first experimentally demonstrated 3D TRS-TI is Bi$_{x}$Sb$_{1-x}$. The second generation materials are include Sb$_2$Te$_3$ ([[Sb2Te3]]), Bi$_2$Se$_3$ ([[Bi2Se3]]) and Bi$_2$Te$_3$ ([[Bi2Te3]]). All of these systems are [[Strong Topological Insulator|strong topological insulators]]. The $\beta$ phase of Bi$_4$I$_4$ ([[Bi4I4]]) is a [[Weak Topological Insulator]] # Mathematical Description The $Z_2$ [[Z2 Topological Invariant|invariants]] can be computed using the [[Fu-Kane Formula]].