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arXiv:2106.10897

Geometric and computational aspects of chiral topological quantum matter

Omri Golan

Published Jun 21, 2021• 2 citations
arXivHTMLPDF

Abstract

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 $c$, 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 $U(1)$ particle-number symmetry is broken, and $c$ 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 $c$ 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 $sign\ problem$ 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 $intrinsic$ 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 $e^{2πi c/24}$ is $not$ the topological spin of an anyon.

DOI
10.48550/arxiv.2106.10897

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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.
The infamous sign 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 efficient simulations on classical computers.Motivated by long-standing open problems in many-body physics, as well as fundamental questions in quantum complexity, the possibility of intrinsic 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 gapped, chiral, topological phases of matter.Within this class, we exclude the possibility of stoquastic Hamiltonians for bosons (or "qudits") and of sign-problem-free determinantal Monte Carlo algorithms for fermions.The intrinsically sign-problematic class of phases we identify is defined in terms of topological invariants with clear observable signatures: the chiral central charge and the topological spins of anyons.We obtain analogous results for phases that are spontaneously chiral, and present evidence for an extension of our results that applies to both chiral and nonchiral topological matter.
Topological phases of matter describe systems beyond the paradigm of spontaneous symmetry breaking, giving rise to unconventional phenomena with possible applications in topological quantum computing. Almost 40 years after their discovery, topological phases remain among the most active research fields of condensed matter physics. As evidenced by the recent discovery of higher order topological insulators (HOTIs), novel physics can emerge even in non-interacting systems. Theoretical investigations of strongly correlated systems are extremely difficult due to the exponential growth of the quantum-mechanical Hilbert space and frequently require the application of approximative methods such as model wave functions. Tensor network states (TNS) are a class of variational wave functions which allow an efficient encoding of relevant quantum many body states and which form the basis for very successful numerical algorithms. Projected entangled pair states (PEPS), a class of TNS in two and higher dimensions, permit the exact representation of many interacting topological phases with time reversal symmetry. However, the description of experimentally relevant chiral topological phases like the integer and fractional quantum Hall effects is much more subtle using the TNS framework. The first part of this thesis focuses on the description of chiral topological phases using PEPS. Firstly, we study a previously proposed chiral PEPS that was conjectured to possess anyonic excitations. By a careful analysis of its symmetries we are able to align important entanglement observables more closely with the expected universal behavior. This highlights that efficient PEPS can possess characteristic properties of chiral topological phases. Secondly, we focus on the issue that all known chiral PEPS have algebraic bulk correlation functions and therefore cannot be the ground states of gapped local Hamiltonians. Since this problem arises already for non-interacting chiral topological phases, we focus on two examples of such systems and show that their ground states can be represented exactly by efficient PEPS in a hybrid lattice with one momentum-space direction. After an inverse Fourier transform, the PEPS with only real-space coordinates requires an exponentially growing number of parameters for an exact representation. This provides a concrete illustration of the impossibility to encode gapped chiral phases exactly with efficient PEPS. Finally, we analyze a model wave function for a three-dimensional (3D) HOTI with strong intrinsic correlations using large-scale variational Monte Carlo simulations. This wave function is obtained by projection of two copies of a non-interacting HOTI with chiral hinge states. We characterize the gapless hinge states of the interacting system and show that they are of the same nature as the edge states of the 1/21/21/2 Laughlin state. Surprisingly, the gapped surfaces host a two-dimensional (2D) phase whose topological entanglement entropy is half of that of the 1/21/21/2 Laughlin state. Such a value cannot be obtained by any of the known 2D topological orders, showing a clear departure not only from the Laughlin 1/21/21/2 physics but from conventional 2D topological order. This demonstrates that 3D topological phases can host rich phenomena that are not yet fully understood.
We calculate the topological part of the electromagnetic response of bosonic integer quantum Hall (BIQH) phases in odd (space-time) dimensions, and bosonic topological insulator (BTI) and bosonic chiral semimetal (BCSM) phases in even dimensions. To do this, we use the nonlinear sigma model (NLSM) description of bosonic symmetry-protected topological (SPT) phases, and the method of gauged Wess-Zumino (WZ) actions. We find the surprising result that for BIQH states in dimension $2m\ensuremath{-}1$ $(m=1,2,\ensuremath{\cdots})$, the bulk response to an electromagnetic field ${A}_{\ensuremath{\mu}}$ is characterized by a Chern-Simons term for ${A}_{\ensuremath{\mu}}$ with a level quantized in integer multiples of m!m!m! (factorial). We also show that BTI states (which have an extra Z2{\mathbb{Z}}_{2}Z2​ symmetry) can exhibit a Z2{\mathbb{Z}}_{2}Z2​-breaking quantum Hall effect on their boundaries, with this boundary quantum Hall effect described by a Chern-Simons term at level m!2\frac{m!}{2}2m!​. We show that the factor of m!m!m! can be understood by requiring gauge invariance of the exponential of the Chern-Simons term on a general Euclidean manifold, and we also use this argument to characterize the electromagnetic and gravitational responses of fermionic SPT phases with U(1)\text{U}(1)U(1) symmetry in all odd dimensions. We then use our gauged boundary actions for the BIQH and BTI states to (i) construct a bosonic analog of a chiral semimetal (BCSM) in even dimensions, (ii) show that the boundary of the BTI state exhibits a bosonic analog of the parity anomaly of Dirac fermions in odd dimensions, and (iii) study anomaly inflow at domain walls on the boundary of BTI states. In a series of Appendixes we derive important formulas and additional results. In particular, in Appendix A we use the connection between equivariant cohomology and gauged WZ actions to give a mathematical interpretation of the actions for the BIQH and BTI boundaries constructed in this paper.
It is believed that not all quantum systems can be simulated efficiently using classical computational resources. This notion is supported by the fact that in quantum Monte Carlo (QMC) simulations for a large number of important problems it is not known how to express the partition function in a sign-free manner. The answer to the question --- whether there is an fundamental obstruction to such a sign-free representation in generic quantum systems --- remains unclear. Here, focussing on systems with bosonic degrees of freedom, we show that quantized gravitational responses appear as obstructions to local sign-free QMC. In condensed matter physics settings these responses, such as thermal Hall conductance, are associated with fractional quantum Hall effects. We show that similar arguments hold also in the case of spontaneously broken time-reversal (TR) symmetry such as in the chiral phase of a perturbed quantum Kagome antiferromagnet. The connection between quantized gravitational responses and the sign problem is also clearly manifested in certain vertex models, where TR symmetry is preserved.
It is believed that not all quantum systems can be simulated efficiently using classical computational resources. This notion is supported by the fact that it is not known how to express the partition function in a sign-free manner in quantum Monte Carlo (QMC) simulations for a large number of important problems. The answer to the question-whether there is a fundamental obstruction to such a sign-free representation in generic quantum systems-remains unclear. Focusing on systems with bosonic degrees of freedom, we show that quantized gravitational responses appear as obstructions to local sign-free QMC. In condensed matter physics settings, these responses, such as thermal Hall conductance, are associated with fractional quantum Hall effects. We show that similar arguments also hold in the case of spontaneously broken time-reversal (TR) symmetry such as in the chiral phase of a perturbed quantum Kagome antiferromagnet. The connection between quantized gravitational responses and the sign problem is also manifested in certain vertex models, where TR symmetry is preserved.
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.