PHYSICAL REVIEW B 81, 134509 ͑2010͒ Effects of interactions on the topological classification of free fermion systems Lukasz Fidkowski and Alexei Kitaev California Institute of Technology, Pasadena, California 91125, USA ͑Received 4 January 2010; published 7 April 2010͒ We describe in detail a counterexample to the topological classification of free fermion systems. We deal with a one-dimensional chain of Majorana fermions with an unusual T symmetry. The topological invariant for the free fermion classification lies in Z, but with the introduction of interactions the Z is broken to Z8.We illustrate this in the microscopic model of the Majorana chain by constructing an explicit path between two distinct phases whose topological invariants are equal modulo 8, along which the system remains gapped. The path goes through a strongly interacting region. We also find the field-theory interpretation of this phenomenon. There is a second-order phase transition between the two phases in the free theory, which can be avoided by going through the strongly interacting region. We show that this transition is in the two-dimensional Ising universality class, where a first-order phase transition line, terminating at a second-order transition, can be avoided by going through the analog of a high-temperature paramagnetic phase. In fact, we construct the full phase diagram of the system as a function of the thermal operator ͑i.e., the mass term that tunes between the two phases in the free theory͒ and two quartic operators, obtaining a first-order Peierls transition region, a second-order transition region, and a region with no transition. DOI: 10.1103/PhysRevB.81.134509 PACS number͑s͒: 71.10.Pm, 64.60.ae I. INTRODUCTION In the free fermion setting, this symmetry is described by one positive Clifford generator, hence p=−1, q=p+2=1, and for 1–4 ෈␲ ͑ ͒ The discovery of the quantum spin Hall effect and of d=1 we get a topological invariant k 0 Rq−d =Z. the strong three-dimensional ͑3D͒ topological insulator,5–8 We can get some intuition for this integer by thinking both of which are novel band insulators, has prompted re- about boundary states. We start by comparing it to the usual newed interest in the study of topological phases of free fer- Z2 classification of one-dimensional systems without symme- mion systems. Indeed, a full classification of all possible try. The Z2 classification of systems without symmetry is topological phases in such systems has been put forward in ␣ reflected in the fact that for a pair of Majorana chains cˆ j , Ref. 9, where it is related to the enumeration of symmetry ␣=1,2, we can gap out the dangling Majorana operators cˆ␣ 10 1 classes of matrices, and in Ref. 11, which uses the math- and cˆ␣ at the ends of the chain by introducing the terms ematical machinery of K theory. This classification is rather N icˆ1cˆ2 and icˆ1 cˆ2 ͑the i is necessary to make these terms successful, with physical representatives of the nontrivial to- 1 1 N N ͒ ˆ pological classes listed for dimensions one, two, and three, Hermitian . However, these terms are not T invariant: ˆ ͑ 1 2͒ ˆ −1 1 2 ˆ including the quantum spin Hall system HgTe and the 3D T icˆ j cˆ j T =−icˆ j cˆ j . Thus, in the T-symmetric case, we can- topological insulator BiSb. not gap out the dangling Majorana operators with quadratic The big open question now is how the presence of inter- interactions, for any number of chains—this is the origin of actions changes this classification. Specifically, it is possible the Z invariant, which just counts the number of boundary that phases that were distinct in the free classification can states in this setup. However, it turns out that we can use actually be adiabatically connected through a strongly inter- nontrivial quartic interactions to gap out the dangling Majo- acting region. Now, for certain systems the topological in- rana modes for the case of eight Majorana chains—this is variants can be defined in terms of physically measurable what we will focus on in this paper. quantities, and hence are stable to interactions. This occurs, Thus we study the setting of eight parallel Majorana for example, in the integer quantum Hall effect, where the chains, which has a phase transition characterized by k=8. integer Chern number is proportional to the Hall conductiv- We will see how the two phases separated by this transition ity, as well as in two-dimensional ͑2D͒ chiral superconduct- are actually adiabatically connected through an interacting ors and 2D topological insulators and superconductors. Also, phase. This means that the two phases are actually the same, the Z2 classification of the 3D topological insulator reflects and that the Z topological invariant is actually broken down the presence or absence of a ␲ theta term in the effective to Z8. Below we will demonstrate this fact by constructing an action for the electromagnetic field, extending the definition explicit path in Hamiltonian space connecting the two phases 12,13 of this invariant to include systems with interactions. of eight parallel Majorana chains. Adiabatic transformation In this paper we give an example where the free classifi- along this path connects the two phases through a strongly cation breaks down. The system is one dimensional, with an interacting, but everywhere gapped, region. We do the analy- ˆ unusual T symmetry: Tˆ 2 =1 instead of Tˆ 2 =͑−1͒N. For a con- sis first for the microscopic model in Sec. II, where we con- crete model, we consider the Majorana chain and its varia- struct a quartic Tˆ -invariant interaction that gaps out the eight ˆ ˆ ˆ −1 tions, where T acts on odd sites by Tcˆ jT =−cˆ j so that terms boundary Majoranas, and then for the continuum theory in like icˆ jcˆk are only allowed between sites of different parity. Sec. III. 1098-0121/2010/81͑13͒/134509͑9͒ 134509-1 ©2010 The American Physical Society LUKASZ FIDKOWSKI AND ALEXEI KITAEV PHYSICAL REVIEW B 81, 134509 ͑2010͒ 2n II. MICROSCOPIC MODEL i ␳͑ ͒ ͑ ͒ ␣ A = ͚ Ajkcˆ jcˆk. 3 We consider eight parallel Majorana chains cˆ j , where 4 j,k=1 ␣=1, ... ,8 is the chain index. The T symmetry still acts by ␣ ␣ The i in front is to make the matrix Hermitian, in order to Tˆ cˆ Tˆ −1=−cˆ . The Hamiltonian is the sum of the Hamilto- j j obtain a unitary representation. It is easy to check that nians for the individual chains ͓−i␳͑A͒,−i␳͑B͔͒=−i␳͓͑A,B͔͒, so that ␳ defines a map of Lie 8 algebras. The induced action on the cˆi Hˆ = ͚ Hˆ ␣ ͑1͒ → ͓␳͑ ͒ ͔͑͒ cˆl i A ,cl 4 ␣=1 Ј ͚ is just the standard cˆl = jAljcj. Before considering so͑8͒, we warm up by studying so͑4͒. n n−1 i Note that so͑4͒=so͑3͒ so͑3͒. Under ␳, the generators of ˆ ͩ ͚ ˆ␣ ˆ␣ ͚ ˆ␣ ˆ␣ ͪ ͑ ͒ H␣ = u c2l−1c2l + v c2lc2l+1 . 2 the two so͑3͒’s are 2 l=1 l=1 i i i We construct a path from a representative Hamiltonian of the ͫ ͑cˆ cˆ − cˆ cˆ ͒, ͑cˆ cˆ − cˆ cˆ ͒, ͑cˆ cˆ + cˆ cˆ ͒ͬ ͑5͒ 2 1 2 3 4 2 1 4 2 3 2 1 3 2 4 uϽv phase to one of the uϾv phase. While, in principle, we could start with any representative Hamiltonians for the two and phases, it will be especially convenient to choose so-called fully dimerized ones, i.e., u=1, =0 and u=0, =1. This i i i v v ͫ ͑cˆ cˆ + cˆ cˆ ͒, ͑cˆ cˆ + cˆ cˆ ͒, ͑− cˆ cˆ + cˆ cˆ ͒ͬ. ͑6͒ choice turns off the odd ͑or even͒ bond couplings and thus 2 1 2 3 4 2 1 4 2 3 2 1 3 2 4 breaks down the chains into easy to analyze independent ͉␺͘෈H finite-size systems. For example, for u=1, v=0, the finite- The condition for a state to be annihilated ͑ ͒ dimensional subsystems consist of the Majoranas by all generators of the first so 3 is equivalent to ͉␺͘ ͉␺͘ ͕ 1 1 8 8 ͖ cˆ1cˆ2cˆ3cˆ4 =− , and there are two states that satisfy this cˆ2l−1,cˆ2l ,...,cˆ2l−1,cˆ2l , and for u=0, v=1 they consist of ͕cˆ1 ,cˆ1 ,...,cˆ8 ,cˆ8 ͖. Both are 256 dimensional and we condition, forming a spin-1/2 representation of the other 2l 2l+1 2l 2l+1 ͑ ͒ ͉␺͘ ͉␺͘ ͉␺͘ will generically denote their Hilbert space H . so 3 . Similarly, cˆ1cˆ2cˆ3cˆ4 = for annihilated by the 0 ͑ ͒ ͑ ͒ ͑ ͒ H The key idea is now to work with these finite-dimensional second so 3 . Indeed, under so 3 so 3 , decomposes as systems, in order to make a fully analytic treatment possible. 1 1 Indeed, to connect the two phases, we start with one fully H = ͩ0, ͪ ͩ ,0ͪ. ͑7͒ dimerized Hamiltonian, say, u=1, v=0, and turn on an in- 2 2 teraction W which contains quartic terms, but only ones that From this analysis, we can already see that the above are products of Majoranas with the same site index ͑but dif- ␣ ␣ ␣ ␣ strategy for constructing the path through the space of ͒ 1 2 3 4 ferent chain indices , i.e., terms of the form cl cl cl cl . The gapped Hamiltonians connecting the two phases would fail virtue of such an interaction is precisely that it does not for the case of four Majorana chains.