Majorana zero modes are well studied in the gapped phases of topological systems. We investigate Majorana zero modes at the topological quantum criticality in one dimensional topological superconducting model with longer range interaction. We identify stable localized Majorana zero modes appearing at criticality under certain conditions. Topological invariant number for these non-trivial criticalities is obtained from zeros of a complex function associated with the Hamiltonian. Behavior of parametric curve at criticalities validate the invariant obtained and account for the appearance of Majorana zero modes at criticality. Trivial and non-trivial topological nature of criticality due to the presence of multicritical point cause an unusual topological transition along the critical line. We observe and investigate this unique transition in terms of eigenvalue spectrum. Appearance of MZMs at criticality demands integer value of topological invariant number in order to validate the concept of bulk-boundary correspondence. Hence we propose a scheme to separate the invariant number into fractional and integer contribution to establish bulk-boundary correspondence at criticality.
Time reversal invariant (TRI) topological superfluids (TSFs) and topological superconductors (TSCs) are robust symmetry protected gapped topological states. In this article, we study the evolution of these topological states in the presence of time reversal symmetry breaking (TRB) fields and/or sufficiently large TRI fields. Physically, one of the realizations of TRB fields can be internal spin exchange fields due to background magnetic ordering. We find that the fully gapped TSFs and TSCs are generically separated from other nodal states by various zero temperature quantum critical points tha…
Vortices in chiral topological superconductors are known for trapping Majorana zero modes as a signature response to their bulk-state topologies. In this work, we establish a similar bulk-vortex correspondence for two-dimensional $C_n$-protected class D higher-order topological superconductors, in which a quantum vortex will universally trap a pair of $C_n$-protected Majorana bound states with $\hat{z}$-directional angular momenta $J_z=0$ and $J_z=n/2$, respectively. This intriguing one-to-one mapping between anomalous vortex modes and higher-order topology can be systematically derived throug…
An Attempt is made to study the non-Hermitian effect on the topological quantum criticality and also in the physics of Majroana zero mode (MZMs). In this work, the effects and modifications done by the non-Hermitian factor $γ$ on the topological phases, criticality and also in the MZMs is studied. We use the zero mode solutions (ZMS) to construct the phase diagram. We find a correspondence between Hermitian and non-Hermitian model Hamiltonian. The MZMs appear at criticality and has a stability dependence on the new passage created because of the introduction of non-Hermiticity. The multicritic…
Disorder-enhanced topological protection and universal quantum criticality in a spin-3/2 topological superconductor
Texas Center for Superconductivity and Department of Physics, University of Houston（休斯顿大学物理系超导电性中心）；Department of Physics and Astronomy, Rice University（莱斯大学物理与天文学系）；Rice Center for Quantum Materials, Rice University（莱斯大学量子材料中心）
There has been much recent progress on exotic surface critical behavior, yet the classical-quantum correspondence of special and extraordinary-log criticality remains largely unclear. Employing worm Monte Carlo simulations, we explore the surface criticality at an emergent superfluid-Mott insulator critical point in the Villain representation, which is believed to connect classical and quantum O(2) critical systems. We observe a special transition with the thermal and magnetic renormalization exponents $y_t=0.58(1)$ and $y_h=1.690(1)$ respectively, which are close to recent estimates from mode…
The Kitaev model on the honeycomb lattice is a paradigmatic system known to host a wealth of nontrivial topological phases and Majorana edge modes. In the static case, the Majorana edge modes are nondispersive. When the system is periodically driven in time, such edge modes can disperse and become chiral. We obtain the full phase diagram of the driven model as a function of the coupling and the driving period. We characterize the quantum criticality of the different topological phase transitions in both the static and driven model via the notions of Majorana-Wannier state correlation functions…
Majorana fermions are a fascinating medium for discovering new phases of matter. However, the standard analytical tools are very limited in probing the non-perturbative aspects of interacting Majoranas in more than one dimensions. Here, we employ the holographic correspondence to determine the specific heat of a two-dimensional interacting gapless Majorana system. To perform our analysis we first describe the interactions in terms of a pseudo-scalar torsion field. We then allow fluctuations in the background curvature thus identifying our model with a $(2+1)$-dimensional Anti-de Sitter (AdS) g…
In this work, we investigate the conductance and current correlations properties of a quantum topological inteferometer consisting of a QD coupled to two Majorana Bound States (MBSs) confined at both ends of a 1D topological superconductor ring nanowire. We analyze the ring in its topological non trivial and trivial phases to show that the tunneling conductance, shot noise and fano factor present unique characteristics to distinguish the hallmark of MBSs. We reinforce our findings by taking advantage of the correspondence between the quantum topological interferometer and a dot effectively cou…
We propose the platform of a Yang-Lee anyon system which is constructed from Majorana bound states in topological superconductors. Yang-Lee anyons, described by the non-unitary conformal field theory with the central charge $c=-22/5$, are non-unitary counterparts of Fibonacci anyons, obeying the same fusion rule, but exhibiting non-unitary non-Abelian braiding statistics. We consider a topological superconductor junction system coupled with dissipative electron baths, which realizes a non-Hermitian interacting Majorana system. Numerically estimating the central charge, we examine the condition…
In this paper, we formulate and quantitatively examine the effect of dissipation on topological systems. We use a specific model of Kitaev quantum wire with an onsite Ohmic dissipation, and perform a numerically exact quantum Monte Carlo simulation to investigate this interacting open quantum system with a strong system-bath (SB) coupling beyond the scope of Born-Markovian approximation. We concentrate on the effect of dissipation on the topological features of the system (e.g. the Majorana edge mode) at zero temperature, and find that even though the topological phase is robust against weak S…
We discuss a scheme for performing Jordan-Wigner transformation for various lattice fermion systems in two and three dimensions which keeps internal and spatial symmetries manifest. The correspondence between fermionic and bosonic operators is established with the help of auxiliary Majorana fermions. The current construction is applicable to general lattices with even coordination number and an arbitrary number of fermion flavors. The approach is demonstrated on the single-orbital square, triangular and cubic lattices for spin-1/2 fermions. We also discuss the relation to some quantum spin liq…
We propose a simple solvable variant of the Sachdev-Ye-Kitaev (SYK) model which displays a quantum phase transition from a fast-scrambling non-Fermi liquid to disordered Fermi liquid. Like the canonical SYK model, our variant involves a single species of Majorana fermions connected by all-to-all random four-fermion interactions. The phase transition is driven by a random two-fermion term added to the Hamiltonian whose structure is inspired by proposed solid-state realizations of the SYK model. Analytic expressions for the saddle point solutions at large number $N$ of fermions are obtained and …
We study quantum criticality of spinless fermions on the quasi one dimensional $π$-flux square lattice in cylinder geometry, by using the infinite density matrix renormalization group and abelian bosonization. For a series of the cylinder circumferences $L_y=4n+2=2, 6, \cdots$ with the periodic boundary condition, there are quantum phase transitions from gapped Dirac fermion states to charge density wave (CDW) states. We find that the quantum phase transitions for such circumferences are continuous and belong to the (1+1)-dimensional Ising universality class. On the other hand, when $L_y=4n=4,…
In contrast to elementary Majorana particles, emergent Majorana fermions (MFs) in condensed-matter systems may have electromagnetic multipoles. We developed a general theory of magnetic multipoles for surface helical MFs on time-reversal-invariant superconductors. The results show that the multipole response is governed by crystal symmetry, and that a one-to-one correspondence exists between the symmetry of Cooper pairs and the representation of magnetic multipoles under crystal symmetry. The latter property provides a way to identify nonconventional pairing symmetry via the magnetic response …
The issue on the effect of interactions in topological states concerns not only interacting topological phases but also novel symmetry-breaking phases and phase transitions. Here we study the interaction effect on Majorana zero modes (MZMs) bound to a square vortex lattice in two-dimensional (2D) topological superconductors. Under the neutrality condition, where single-body hybridization between MZMs is prohibited by an emergent symmetry, a minimal square-lattice model for MZMs can be faithfully mapped to a quantum spin model, which has no sign problem in the world-line quantum Monte Carlo sim…
We study a link between the ground-state topology and the topology of the lattice via the presence of anomalous states at disclinations -- topological lattice defects that violate a rotation symmetry only locally. We first show the existence of anomalous disclination states, such as Majorana zero-modes or helical electronic states, in second-order topological phases by means of Volterra processes. Using the framework of topological crystals to construct $d$-dimensional crystalline topological phases with rotation and translation symmetry, we then identify all contributions to $(d-2)$-dimension…
It is often thought that emergent phenomena in topological phases of matter are destroyed when tuning to a critical point. In particular, topologically protected edge states supposedly delocalize when the bulk correlation length diverges. We show that this is not true in general. Edge states of topological insulators or superconductors remain exponentially localized---despite a vanishing band gap---if the transition increases the topological index. This applies to all classes where the topological classification is larger than $Z_2$, notably including Chern insulators. Moreover, these edge sta…
Bipartite entanglement and surface criticality: The extra contribution of non-ordinary edge in entanglement
Fudan University（复旦大学）；Westlake University（西湖大学）；Westlake Institute for Advanced Study（西湖高等研究院）；Beihang University（北京航空航天大学）；Tianmushan Laboratory（天目山实验室）
The Sachdev-Ye-Kitaev (SYK) model is an all-to-all interacting Majorana fermion model for many-body quantum chaos and the holographic correspondence. Here we construct fermionic all-to-all Floquet quantum circuits of random four-body gates designed to capture key features of SYK dynamics. Our circuits can be built using local ingredients in Majorana devices, namely charging-mediated interactions and braiding Majorana zero modes. This offers an analog-digital route to SYK quantum simulations that reconciles all-to-all interactions with the topological protection of Majorana zero modes, a key fe…
Low dimensional structures in the non-trivial topological phase can host the in-gap Majorana bound states, identified experimentally as zero-bias peaks in differential conductance. Theoretical methods for studying Majorana modes are mostly based on the bulk-boundary correspondence or exact diagonalization of finite systems via, e.g., Bogoliubov-de Gennes formalism. In this paper, we develop an efficient method for identifying the Majorana in-gap (edge) states via looking for extreme eigenvalues of symmetric matrices. The presented approach is based on the Krylov method and allows for study the…
Models for non-unitary quantum dynamics, such as quantum circuits that include projective measurements, have been shown to exhibit rich quantum critical behavior. There are many complementary perspectives on this behavior. For example, there is a known correspondence between d-dimensional local non-unitary quantum circuits and tensor networks on a D=(d+1)-dimensional lattice. Here, we show that in the case of systems of non-interacting fermions, there is furthermore a full correspondence between non-unitary circuits in d spatial dimensions and unitary non-interacting fermion problems with stat…
Separation of the Kibble-Zurek Mechanism from Quantum Criticality
Institute for Advanced Studies in Basic Sciences (IASBS)（基础科学高等研究院）；Institute for Research in Fundamental Sciences (IPM)（基础科学研究院）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学与应用研究所）；Max Planck Institute for the Chemical Physics of Solids（马克斯·普朗克固体化学物理研究所）
We establish an important duality correspondence between topological order in quantum many body systems and criticality in ferromagnetic classical spin systems. We show how such a correspondence leads to a classical and simple procedure for characterization of topological order in an important set of quantum entangled states, namely the Calderbank-Shor-Steane (CSS) states. To this end, we introduce a particular quantum Hamiltonian which allows us to consider the existence of a topological phase transition from quantum CSS states to a magnetized state. We study the ground state fidelity in orde…
We present a Green's function formalism to investigate the topological properties of weakly interacting one-dimensional topological insulators, including the bulk-edge correspondence and the quantum criticality near topological phase transitions, and using interacting Su-Schrieffer-Heeger model as an example. From the many-body spectral function, we find that closing of the bulk gap remains a defining feature even if the topological phase transition is driven by interactions. The existence of edge state in the presence of interactions can be captured by means of a T-matrix formalism combined w…
Topological crystalline superconductors have attracted rapidly rising attention due to the possibility of higher-order phases, which support Majorana modes on boundaries in $d-2$ or lower dimensions. However, although the classification and bulk topological invariants in such systems have been well studied, it is generally difficult to faithfully predict the boundary Majoranas from the band-structure information due to the lack of well-established bulk-boundary correspondence. Here we propose a protocol for deriving symmetry indicators that depend on a minimal set of necessary symmetry data of…
We study the exactly solvable 1D model: the dimerized $XY$ chain with uniform and staggered transverse fields, equivalent upon fermionization to the noninteracting dimerized Kitaev-Majorana chain with modulation. The model has three known gapped phases with local and nonlocal (string) orders, along with the gapless incommensurate (IC) phase in the $U(1)$ limit. The criticality is controlled by the properties of zeros of model's partition function, analytically continued onto the complex wave numbers. In the ground state they become complex zeros of the spectrum of the Hamiltonian. The analysis…
Topological crystalline superconductors are known to have possible higher-order topology, which results in Majorana modes on $d-2$ or lower-dimensional boundaries. Given the rich possibilities of boundary signatures, it is desirable to have topological invariants that can predict the type of Majorana modes from band structures. Although symmetry indicators, a type of invariant that depends only on the band data at high-symmetry points, have been proposed for certain crystalline superconductors, there exist symmetry classes in which symmetry indicators fail to distinguish superconductors with d…
Department of Physics and Anthony J. Leggett Institute for Condensed Matter Theory, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校物理系和安东尼·J·莱格特凝聚态理论研究所）
Quantum fluctuating loops in 2+1 dimensions give gapless many-body states that are beyond current field theory techniques. Microscopically, these loops can be domain walls between up and down spins, or chains of flipped spins similar to those in the toric code. The key feature of their dynamics is that the reconnection of a pair of strands is forbidden. This happens at previously-studied multi-critical points between topologically nontrivial phases. We show that this topologically constrained dynamics leads to universality classes with unusual scaling properties. For example, scaling operators…
The exactly-solvable Kitaev model of two-dimensional honeycomb magnet leads to a quantum spin liquid (QSL) characterized by Majorana fermions, relevant for fault-tolerant topological quantum computations. In the high-field paramagnetic state of $α$-RuCl3, half-integer quantization of thermal Hall conductivity has been reported as a signature of edge current, but the bulk nature of this state remains elusive. Here, from high-resolution heat capacity measurements under in-plane field rotation, we find strongly angle-dependent low-energy excitations in bulk $α$-RuCl3. The excitation gap has a sex…
We study classification of interacting fermionic symmetry-protected topological (SPT) phases with both rotation symmetry and Abelian internal symmetries in one, two, and three dimensions. By working out this classification, on the one hand, we demonstrate the recently proposed correspondence principle between crystalline topological phases and those with internal symmetries through explicit block-state constructions. We find that for the precise correspondence to hold it is necessary to change the central extension structure of the symmetry group by the $\mathbb{Z}_2$ fermion parity. On the ot…
Monolayer WTe$_2$, a centrosymmetric transition metal dichacogenide, has recently been established as a quantum spin Hall insulator and found superconducting upon gating. Here we study the pairing symmetry and topological nature of superconducting WTe$_2$ with a microscopic model at mean-field level. Surprisingly, we find that the spin-triplet phases in our phase diagram all host Majorana modes localized on two opposite corners. Even when the conventional pairing is favored, we find that an intermediate in-plane magnetic field exceeding the Pauli limit stabilizes an unconventional equal-spin p…
