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arXiv:2410.02059

An index for invertible phases of two-dimensional quantum spin systems

Nikita Sopenko

Published Oct 2, 2024• 0 citations
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Abstract

We define an index for invertible phases of two-dimensional fermionic and bosonic quantum spin systems without any additional symmetry. Conjecturally, it provides a microscopic definition of an invariant related to the chiral central charge of the boundary modes $c_- \bmod 24$ when the effective conformal field theory description is valid. Using this index, we prove that free fermionic systems with Chern number $ν\bmod 48 \neq 0$ are in a non-trivial invertible phase.

DOI
10.48550/arxiv.2410.02059

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We define an index for invertible phases of two-dimensional fermionic and bosonic quantum spin systems without any additional symmetry. Conjecturally, it provides a microscopic definition of an invariant related to the chiral central charge of the boundary modes c− mod 24c_- \bmod 24c−​mod24 when the effective conformal field theory description is valid. Using this index, we prove that free fermionic systems with Chern number ν mod 48≠0ν\bmod 48 \neq 0νmod48=0 are in a non-trivial invertible phase.
We analyze the validity of reflection positivity in the classification of invertible phases of quantum spin systems. We provide a mathematical model in which every 2d invertible state admits a reflection-positive representative. We prove that reflection positivity provides a canonical lift from the set of invertible phases to the set of invertible phases protected by a Z/N\mathbb{Z}/NZ/N-rotational symmetry. Using this, we define a refined version of the index recently introduced by the author. This refined version conjecturally provides a microscopic characterization of an invariant that coincides with the chiral central charge c−c_-c−​ when conformal field theory effectively describes the boundary modes.
We introduce and study (fermionic and bosonic) invertible quasi-local algebras over uniformly locally finite metric spaces XXX with infinite-dimensional local von Neumann algebras. We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and the group of stable equivalence classes of quantum cellular automata over X×ZX\times \mathbb{Z}X×Z. Using K-theory of the symmetric monoidal category of invertible quasi-local algebras and bounded spread isomorphisms, we propose a definition of an ΩΩΩ-spectrum of invertible phases as conjectured by Kitaev. We then show that the c=12c=\frac{1}{2}c=21​ chiral Majorana fermion net and the (E8)1(E_{8})_{1}(E8​)1​ conformal net provide Brauer non-trivial invertible quasi-local algebras, thus providing explicit constructions of non-trivial invertible states and quantum cellular automata on Z2\mathbb{Z}^{2}Z2. In addition, we show that the time-slice nets of rational diagonal conformal field theories admit lattice degrees of freedom, which implies the discretization of any holomorphic conformal net is invertible.
Effective field theory is a very useful technique for understanding quantum many body systems. We use this approach to study a certain class of critical quantum spin-1 chains and symmetry protected topological phases of matter in two spatial dimensions. In the first context we consider the SU(3) symmetric spin-1 chain with additional interactions. We use bosonization to demonstrate that the effective field theories describing the critical behaviour of these spin chains can be mapped to free compact boson conformal field theories (CFTs) with central charge c=2. We also describe how some predictions from field theory can be verified in numerical calculations using exact diagonalization (ED) and the density matrix renormalization group (DMRG) algorithm. In particular, the bosonization method gives a formula for the evolution of four Tomonaga-Luttinger liquid (TLL) parameters as a function of the lattice parameters. Using the analytic formulae for the scaling dimensions in terms of the TLL parameters and matching of lowest scaling dimensions, we numerically calculate these field-theoretic parameters and confirm that their evolution agrees with the prediction using bosonization. We also tackle aspects of the challenging problem of understanding interacting topological phases. In this context, we aim to understand the effects if interactions in certain classes of symmetry protected topological (SPT) phases of matter. We consider non-chiral SPT phases in two spatial dimensions protected by a discrete symmetry such as Z_K or Z_K x Z_K symmetry. We argue that modular invariance/noninvariance of the partition function of the one-dimensional edge theory can be used to diagnose whether, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons K-matrix theories and fermionic phases relevant to topological superconductors as examples, we demonstrate explicitly that when modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge. We also briefly discuss preliminary results of a numerical approach for simulating the 3 dimensional Landau Level problem. This includes a brief description of a potential application of variational Monte Carlo to spin-orbit coupled ab initio systems.
In this thesis, we study chiral topological phases of 2+1 dimensional quantum matter. Such phases are abstractly characterized by their non-vanishing chiral central charge ccc, a topological invariant which appears as the coefficient of a gravitational Chern-Simons (gCS) action in bulk, and of corresponding gravitational anomalies at boundaries. The chiral central charge is of particular importance in chiral superfluids and superconductors (CSF/Cs), where U(1)U(1)U(1) particle-number symmetry is broken, and ccc is, in some cases, the only topological invariant characterizing the system. However, as opposed to invariants which can be probed by gauge fields in place of gravity, the concrete physical implications of ccc in the context of condensed matter physics is quite subtle, and has been the subject of ongoing research and controversy. The first two parts of this thesis are devoted to the physical interpretation of the gCS action and gravitational anomalies in the context of CSF/Cs, where they are of particular importance, but have nevertheless remained poorly understood. We then turn to a seemingly unrelated aspect of chiral topological phases - their computational complexity. The infamous sign problemsign\ problemsign problem leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing an efficient classical simulation. The possibility of intrinsicintrinsicintrinsic sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. Here, we establish the existence of an intrinsic sign problem in a broad class of chiral topological phases, both bosonic and fermionic, defined by the requirement that e2πic/24e^{2πi c/24}e2πic/24 is notnotnot the topological spin of an anyon.
We study invertible phases protected by non-onsite symmetries carrying nontrivial lattice anomaly indices. While lattice anomalies are usually viewed as obstructions to symmetric short-range-entangled (SRE) states, we show that anomalous lattice symmetries can nevertheless admit invertible phases that are not SRE, and can moreover shift the set of symmetry-compatible invertible phases. In particular, we consider a fermionic Z4F\mathbb Z_4^FZ4F​ symmetry in (2+1) dimensions. For an onsite Z4F\mathbb Z_4^FZ4F​ symmetry, symmetric invertible phases have integer chiral central charge c−∈Zc_-\in\mathbb Zc−​∈Z, whereas a non-onsite symmetry with nontrivial lattice anomaly index obstructs all such phases despite having trivial continuum 't Hooft anomaly. We resolve this by constructing an exact exponentially quasi-local Z4F\mathbb Z_4^FZ4F​ symmetry of a p+ipp+ipp+ip superconductor with c−=1/2c_-=1/2c−​=1/2. We compute its lattice anomaly, and find that the symmetry indeed forbids c−∈Zc_-\in \mathbb{Z}c−​∈Z invertible states. The allowed invertible phases are consequently shifted to c−∈Z+1/2c_-\in\mathbb Z+1/2c−​∈Z+1/2. By gauging the fermion parity of the p+ipp+ipp+ip superconductor, one obtains an Ising topological order enriched by a Z4\mathbb Z_4Z4​ symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the p+ipp+ipp+ip state admits an enlarged non-onsite U(1)fU(1)^fU(1)f symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite U(1)fU(1)^fU(1)f symmetry again forbids Chern insulators carrying c−∈Zc_-\in\mathbb{Z}c−​∈Z, while being compatible with invertible states with c−∈Z+1/2c_-\in\mathbb{Z}+1/2c−​∈Z+1/2. This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.