arXiv subjects

Ville Lahtinen

Publications and source records attributed to Ville Lahtinen.

At least 19 recordsLinked to original sources

arXiv2021

Quantum criticality in many-body parafermion chains

We construct local generalizations of 3-state Potts models with exotic critical points. We analytically show that these are described by non-diagonal modular invariant partition functions of products of $Z_3$ parafermion or $u(1)_6$ conformal field theories (CFTs). These correspond either to non-trivial permutation invariants or block diagonal invariants, that one can understand in terms of anyon condensation. In terms of lattice parafermion operators, the constructed models correspond to parafermion chains with many-body terms. Our construction is based on how the partition function of a CFT depends on symmetry sectors and boundary conditions. This enables to write the partition function corresponding to one modular invariant as a linear combination of another over different sectors and boundary conditions, which translates to a general recipe how to write down a microscopic model, tuned to criticality. We show that the scheme can also be extended to construct critical generalizations of $k$-state clock type models.

cond-mat.str-el↗

arXiv2017

A Short Introduction to Topological Quantum Computation

This review presents an entry-level introduction to topological quantum computation -- quantum computing with anyons. We introduce anyons at the system-independent level of anyon models and discuss the key concepts of protected fusion spaces and statistical quantum evolutions for encoding and processing quantum information. Both the encoding and the processing are inherently resilient against errors due to their topological nature, thus promising to overcome one of the main obstacles for the realisation of quantum computers. We outline the general steps of topological quantum computation, as well as discuss various challenges faced by it. We also review the literature on condensed matter systems where anyons can emerge. Finally, the appearance of anyons and employing them for quantum computation is demonstrated in the context of a simple microscopic model -- the topological superconducting nanowire -- that describes the low-energy physics of several experimentally relevant settings. This model supports localised Majorana zero modes that are the simplest and the experimentally most tractable types of anyons that are needed to perform topological quantum computation.

cond-mat.mes-hall↗

arXiv2016

Anatomy of Fermionic Entanglement and Criticality in Kitaev Spin Liquids

We analyse in detail the effect of non-trivial band topology on the area law behaviour of the entanglement entropy in Kitaev's honeycomb model. By mapping the translationally invariant 2D spin model into 1D fermionic subsystems, we identify those subsystems responsible for universal entanglement contributions in the gapped phases and those responsible for critical entanglement scaling in the gapless phases. For the gapped phases we analytically show how the topological edge states contribute to the entanglement entropy and provide a universal lower bound for it. For the gapless semi-metallic phases and topological phase transitions the identification of the critical subsystems shows that they fall always into the Ising or the XY universality classes. As our study concerns the fermionic degrees of freedom in the honeycomb model, qualitatively similar results are expected to apply also to generic topological insulators and superconductors.

cond-mat.str-el↗

arXiv2016

Self-organized pseudo-graphene on grain boundaries in topological band insulators

Semi-metals are characterized by nodal band structures that give rise to exotic electronic properties. The stability of Dirac semi-metals, such as graphene in two spatial dimensions (2D), requires the presence of lattice symmetries, while akin to the surface states of topological band insulators, Weyl semi-metals in three spatial dimensions (3D) are protected by band topology. Here we show that in the bulk of topological band insulators, self-organized topologically protected semi-metals can emerge along a grain boundary, a ubiquitous extended lattice defect in any crystalline material. In addition to experimentally accessible electronic transport measurements, these states exhibit valley anomaly in 2D influencing edge spin transport, whereas in 3D they appear as graphene-like states that may exhibit an odd-integer quantum Hall effect. The general mechanism underlying these novel semi-metals -- the hybridization of spinon modes bound to the grain boundary -- suggests that topological semi-metals can emerge in any topological material where lattice dislocations bind localized topological modes.

cond-mat.mes-hall↗

arXiv2016

Diagnosing Topological Edge States via Entanglement Monogamy

Topological phases of matter possess intricate correlation patterns typically probed by entanglement entropies or entanglement spectra. In this work, we propose an alternative approach to assessing topologically induced edge states in free and interacting fermionic systems. We do so by focussing on the fermionic covariance matrix. This matrix is often tractable either analytically or numerically and it precisely captures the relevant correlations of the system. By invoking the concept of monogamy of entanglement we show that highly entangled states supported across a system bi-partition are largely disentangled from the rest of the system, thus appearing usually as gapless edge states. We then define an entanglement qualifier that identifies the presence of topological edge states based purely on correlations present in the ground states. We demonstrate the versatility of this qualifier by applying it to various free and interacting fermionic topological systems.

cond-mat.str-el↗

arXiv2015

Realizing all $so(N)_1$ quantum criticalities in symmetry protected cluster models

We show that all $so(N)_1$ universality class quantum criticalities emerge when one-dimensional generalized cluster models are perturbed with Ising or Zeeman terms. Each critical point is described by a low-energy theory of $N$ linearly dispersing fermions, whose spectrum we show to precisely match the prediction by $so(N)_1$ conformal field theory. Furthermore, by an explicit construction we show that all the cluster models are dual to non-locally coupled transverse field Ising chains, with the universality of the $so(N)_1$ criticality manifesting itself as $N$ of these chains becoming critical. This duality also reveals that the symmetry protection of cluster models arises from the underlying Ising symmetries and it enables the identification of local representations for the primary fields of the $so(N)_1$ conformal field theories. For the simplest and experimentally most realistic case that corresponds to the original one-dimensional cluster model with local three-spin interactions, our results show that the $su(2)_2 \simeq so(3)_1$ Wess-Zumino-Witten model can emerge in a local, translationally invariant and Jordan-Wigner solvable spin-1/2 model.

cond-mat.str-el↗

arXiv2014

Detection of Chern numbers and entanglement in topological two-species systems through subsystem winding numbers

Topological invariants, such as the Chern number, characterise topological phases of matter. Here we provide a method to detect Chern numbers in systems with two distinct species of fermion, such as spins, orbitals or several atomic states. We analytically show that the Chern number can be decomposed as a sum of component specific winding numbers, which are themselves physically observable. We apply this method to two systems, the quantum spin Hall insulator and a staggered topological superconductor, and show that (spin) Chern numbers are accurately reproduced. The measurements required for constructing the component winding numbers also enable one to probe the entanglement spectrum with respect to component partitions. Our method is particularly suited to experiments with cold atoms in optical lattices where time-of-flight images can give direct access to the relevant observables.

quant-ph↗

arXiv2014

Perturbed vortex lattices and the stability of nucleated topological phases

We study the stability of nucleated topological phases that can emerge when interacting non-Abelian anyons form a regular array. The studies are carried out in the context of Kitaev's honeycomb model, where we consider three distinct types of perturbations in the presence of a lattice of Majorana mode binding vortices -- spatial anisotropy of the vortices, dimerization of the vortex lattice and local random disorder. While all the nucleated phases are stable with respect to weak perturbations of each kind, strong perturbations are found to result in very different behavior. Anisotropy of the vortices stabilizes the strong-pairing like phases, while dimerization can recover the underlying non-Abelian phase. Local random disorder, on the other hand, can drive all the nucleated phases into a gapless thermal metal state. We show that all these distinct behavior can be captured by an effective staggered tight-binding model for the Majorana modes. By studying the pairwise interactions between the vortices, i.e. the amplitudes for the Majorana modes to tunnel between vortex cores, the locations of phase transitions and the nature of the resulting states can be predicted. We also find that due to oscillations in the Majorana tunneling amplitude, lattices of Majorana modes may exhibit a Peierls-like instability, where a dimerized configuration is favored over a uniform lattice. As the nature of the nucleated phases depends only on the Majorana tunneling, our results apply also to other system supporting localized Majorana mode arrays, such as Abrikosov lattices in p-wave superconductors, Wigner crystals in Moore-Read fractional quantum Hall states or arrays of topological nanowires.

cond-mat.str-el↗

arXiv2013

A hierarchy of exactly solvable spin-1/2 chains with so(N)_1 critical points

We construct a hierarchy of exactly solvable spin-1/2 chains with so(N)_1 critical points. Our construction is based on the framework of condensate-induced transitions between topological phases. We employ this framework to construct a Hamiltonian term that couples N transverse field Ising chains such that the resulting theory is critical and described by the so(N)_1 conformal field theory. By employing spin duality transformations, we then cast these spin chains for arbitrary N into translationally invariant forms that all allow exact solution by the means of a Jordan-Wigner transformation. For odd N our models generalize the phase diagram of the transverse field Ising chain, the simplest model in our hierarchy. For even N the models can be viewed as longer ranger generalizations of the XY chain, the next model in the hierarchy. We also demonstrate that our method of constructing spin chains with given critical points goes beyond exactly solvable models. Applying the same strategy to the Blume-Capel model, a spin-1 generalization of the Ising chain in a generic magnetic field, we construct another critical spin-1 chain with the predicted CFT describing the criticality.

cond-mat.str-el↗

arXiv2013

Kitaev spin models from topological nanowire networks

We show that networks of topological nanowires can realize the physics of exactly solvable Kitaev spin models with two-body interactions. This connection arises from the description of the low-energy theory of both systems in terms of a tight-binding model of Majorana modes. In Kitaev spin models the Majorana description provides a convenient representation to solve the model, whereas in an array of topological nanowires it arises, because the physical Majorana modes localized at wire ends permit tunnelling between wire ends and across different Josephson junctions. We explicitly show that an array of junctions of three wires -- a setup relevant to topological quantum computing with nanowires -- can realize the Yao-Kivelson model, a variant of Kitaev spin models on a decorated honeycomb lattice. Translating the results from the latter, we show that the network can be constructed to give rise to collective states characterized by Chern numbers ν= 0, +/-1 and +/-2, and that defects in an array can be associated with vortex-like quasi-particle excitations. Finally, we analyze the stability of the collective states as well as that of the network as a quantum information processor. We show that decoherence inducing instabilities, be them due to disorder or phase fluctuations, can be understood in terms of proliferation of the vortex-like quasi-particles.

cond-mat.mes-hall↗

arXiv2013

Seeing Majorana fermions in time-of-flight images of spinless fermions coupled by s-wave pairing

The Chern number, nu, as a topological invariant that identifies the winding of the ground state in the particle-hole space, is a definitive theoretical signature that determines whether a given superconducting system can support Majorana zero modes. Here we show that such a winding can be faithfully identified for any superconducting system (p-wave or s-wave with spin-orbit coupling) through a set of time-of-flight measurements, making it a diagnostic tool also in actual cold atom experiments. As an application, we specialize the measurement scheme for a chiral topological model of spinless fermions. The proposed model only requires the experimentally accessible s-wave pairing and staggered tunnelling that mimics spin-orbit coupling. By adiabatically connecting this model to Kitaev's honeycomb lattice model, we show that it gives rise to nu = \pm 1 phases, where vortices bind Majorana fermions, and nu=\pm 2 phases that emerge as the unique collective state of such vortices. Hence, the preparation of these phases and the detection of their Chern numbers provide an unambiguous signature for the presence of Majorana modes. Finally, we demonstrate that our detection procedure is resilient against most inaccuracies in experimental control parameters as well as finite temperature.

cond-mat.supr-con↗

arXiv2013

Condensate-induced transitions and critical spin chains

We show that condensate-induced transitions between two-dimensional topological phases provide a general framework to relate one-dimensional spin models at their critical points. We demonstrate this using two examples. First, we show that two well-known spin chains, namely the XY chain and the transverse field Ising chain with only next-nearest-neighbor interactions, differ at their critical points only by a non-local boundary term and can be related via an exact mapping. The boundary term constrains the set of possible boundary conditions of the transverse field Ising chain, reducing the number of primary fields in the conformal field theory that describes its critical behavior. We argue that the reduction of the field content is equivalent to the confinement of a set of primary fields, in precise analogy to the confinement of quasiparticles resulting from a condensation of a boson in a topological phase. As the second example we show that when a similar confining boundary term is applied to the XY chain with only next-nearest-neighbor interactions, the resulting system can be mapped to a local spin chain with the u(1)_2 x u(1)_2 critical behavior predicted by the condensation framework.

cond-mat.str-el↗

arXiv2012

Topological liquid nucleation induced by vortex-vortex interactions in Kitaev's honeycomb model

We provide a comprehensive microscopic understanding of the nucleation of topological quantum liquids, a general mechanism where interactions between non-Abelian anyons cause a transition to another topological phase, which we study in the context of Kitaev's honeycomb lattice model. For non-Abelian vortex excitations arranged on superlattices, we observe the nucleation of several distinct Abelian topological phases whose character is found to depend on microscopic parameters such as the superlattice spacing or the spin exchange couplings. By reformulating the interacting vortex superlattice in terms of an effective model of Majorana fermion zero modes, we show that the nature of the collective many-anyon state can be fully traced back to the microscopic pairwise vortex interactions. Due to RKKY-type sign oscillations in the interactions, we find that longer-range interactions beyond nearest neighbor can influence the collective state and thus need to be included for a comprehensive picture. The omnipresence of such interactions implies that corresponding results should hold for vortices forming an Abrikosov lattice in a p-wave superconductor, quasiholes forming a Wigner crystal in non-Abelian quantum Hall states or topological nanowires arranged in regular arrays.

cond-mat.mes-hall↗

arXiv2011

Interacting non-Abelian anyons as Majorana fermions in the honeycomb lattice model

We study the collective states of interacting non-Abelian anyons that emerge in Kitaev's honeycomb lattice model. Vortex-vortex interactions are shown to lead to the lifting of the topological degeneracy and the energy is discovered to exhibit oscillations that are consistent with Majorana fermions being localized at vortex cores. We show how to construct states corresponding to the fusion channel degrees of freedom and obtain the energy gaps characterizing the stability of the topological low energy spectrum. To study the collective behavior of many vortices, we introduce an effective lattice model of Majorana fermions. We find necessary conditions for it to approximate the spectrum of the honeycomb lattice model and show that bi-partite interactions are responsible for the degeneracy lifting also in many vortex systems.

cond-mat.mes-hall↗

arXiv2011

Engineering complex topological memories from simple Abelian models

In three spatial dimensions, particles are limited to either bosonic or fermionic statistics. Two-dimensional systems, on the other hand, can support anyonic quasiparticles exhibiting richer statistical behaviours. An exciting proposal for quantum computation is to employ anyonic statistics to manipulate information. Since such statistical evolutions depend only on topological characteristics, the resulting computation is intrinsically resilient to errors. So-called non-Abelian anyons are most promising for quantum computation, but their physical realization may prove to be complex. Abelian anyons, however, are easier to understand theoretically and realize experimentally. Here we show that complex topological memories inspired by non-Abelian anyons can be engineered in Abelian models. We explicitly demonstrate the control procedures for the encoding and manipulation of quantum information in specific lattice models that can be implemented in the laboratory. This bridges the gap between requirements for anyonic quantum computation and the potential of state-of-the-art technology.

quant-ph↗

arXiv2010

Universal quantum computation with a non-Abelian topological memory

An explicit lattice realization of a non-Abelian topological memory is presented. The correspondence between logical and physical states is seen directly by use of the stabilizer formalism. The resilience of the encoded states against errors is studied and compared to that of other memories. A set of non-topological operations are proposed to manipulate the encoded states, resulting in universal quantum computation. This work provides insight into the non-local encoding non-Abelian anyons provide at the microscopical level, with an operational characterization of the memories they provide.

quant-ph↗

arXiv2010

Topological phase transitions driven by gauge fields in an exactly solvable model

We demonstrate the existence of a new topologically ordered phase in Kitaev's honeycomb lattice model. This new phase appears due to the presence of a vortex lattice and it supports chiral Abelian anyons. We characterize the phase by its low-energy behavior that is described by a distinct number of Dirac fermions. We identify two physically distinct types of topological phase transitions and obtain analytically the critical behavior of the extended phase space. The Fermi surface evolution associated with the transitions is shown to be due to the Dirac fermions coupling to chiral gauge fields. Finally, we describe how the new phase can be understood in terms of interactions between the anyonic vortices.

cond-mat.str-el↗

arXiv2009

Non-Abelian statistics as a Berry phase in exactly solvable models

We demonstrate how to directly study non-Abelian statistics for a wide class of exactly solvable many-body quantum systems. By employing exact eigenstates to simulate the adiabatic transport of a model's quasiparticles, the resulting Berry phase provides a direct demonstration of their non-Abelian statistics. We apply this technique to Kitaev's honeycomb lattice model and explicitly demonstrate the existence of non-Abelian Ising anyons confirming the previous conjectures. Finally, we present the manipulations needed to transport and detect the statistics of these quasiparticles in the laboratory. Various physically realistic system sizes are considered and exact predictions for such experiments are provided.

cond-mat.mes-hall↗