Spin-mixing-tunneling network model for Anderson transitions in two-dimensional disordered spinful electrons
We consider Anderson transitions in two-dimensional spinful electron gases subject to random scalar potentials with time-reversal-symmetric spin-mixing tunneling (SMT) and spin-preserving tunneling (SPT) at potential saddle points (PSPs). A symplectic quantum network model, named as SMT-QNM, is constructed in which SMT and SPT have the same status and contribute independent tunneling channels rather than sharing a total-probability-fixed one. Two-dimensional continuous Dirac Hamiltonian is then extracted out from this discrete network model as the generator of certain unitary transformation. With the help of high-accuracy numerics based on transfer matrix technique, finite-size analysis on two-terminal conductance and normalized localization length provides a phase diagram drawn in the SMT-SPT plane. As a manifestation of symplectic ensembles, a normal-metal (NM) phase emerges between the quantum spin Hall (QSH) and normal-insulator (NI) phases when SMT appears. We systematically analyze the quantum phases on the boundary and in the interior of the phase space. Particularly, the phase diagram is closely related to that of disordered three-dimensional weak topological insulators by appropriate parameter mapping. At last, if time-reversal symmetry in electron trajectories between PSPs is destroyed, the system falls into unitary class with no more NM phase. A direct SMT-driven transition from QSH to NI phases exists and can be explained by spin-flip backscattering between the degenerate doublets at the same sample edge.
pacs:71.30.+h, 72.15.Rn, 73.20.Fz, 73.43.Nq
I I. Introduction
Anderson transitions (ATs), i.e., transitions between localized and delocalized quantum phases in disordered electronic systems, have attracted intense and continuous attention since its proposalAnderson_1958 () due to its fundamental significance in condensed matter physicsLee_1985 (); Kramer_1993 (); Huckestein_1995 (); Evers_Mirlin_2008 (). In 1970s and 1980s, scaling-theory and field-theory approaches revealed the connections between Anderson transition and conventional second-order phase transitionsLee_1985 (); Kramer_1993 (); Huckestein_1995 (). In 1990s, the symmetry classification of disordered systems was achieved based on its relation to the classical symmetric spacesZirnbauer_1996 (); Zirnbauer_1997_prb (); Caselle_2004 (). Later, the completeness of this classification is proved in 2005Heinzner_2005 (). Now we know there are totally ten symmetry classes according to how many discrete symmetries are obeyed by the underlying physical system. When a system only has symmetries translationally invariant in energy, such as the time-reversal symmetry (TRS) and spin-rotation symmetry (SRS), it falls into one of the three traditional Wigner-Dyson classes (unitary, orthogonal and symplectic)Wigner_1951 (); Dyson_1962 (). However, if we focus on some particular value of energy, extra discrete symmetries could arise and lead to novel symmetry classes. In condensed matter systems described by tight-binding models on a bipartite lattice with randomness only residing in hopping terms, three chiral classes are identifiedZirnbauer_1996 (). The remaining four were discovered in superconducting systems and known as the Bogoliubov-de Gennes classesZirnbauer_1997_prb (). In the past decades, ATs in these ten classes have been investigated intensively and considerable progress has been made in various directions, such as their scaling-theory and field-theory descriptionsLee_1985 (); Kramer_1993 (); Huckestein_1995 (), multifractality in critical wave functionsJanssen_1994 (); Mudry_1996 (); Evers_Mirlin_2000 (); Evers_2001 (); Obuse_2004_prb (); Obuse_2007_prl (); Obuse_2010_prb_1 () and level statistics at criticalityMirlin_2000 (); XiongGang_2001_PRL (); XiongGang_2006_JPCM (); Obuse_2005_prb (); GarciaGarcia_2007 (), etc.
Recently, the spin-orbit-induced topological materials, named as topological insulators (TIs), have received intensive attentionRMP_2010_Hasan_Kane (); RMP_2011_XLQi_SCZhang (); RMP_2015_Beenakker (); RPP_2016_ZHQiao_QNiu (); RMP_2016_Bansil (); RMP_2016_Witten (); RMP_2016_Ryu (). In TIs, the interplay between topology and symmetry greatly enriches our knowledge of quantum statesRMP_2016_Ryu (); Fu_2012_prl (); Fulga_2012_prb (); Ryu_2007_prl (); Ryu_2010_njp (). First, the TRS is crucial for their realization and stabilization. Second, the spin-orbit coupling (SOC) destroys the SRS, thus makes TIs belong to the Wigner-Dyson symplectic class. In two dimensions (2D), they are the well-known quantum spin Hall (QSH) ensembles. In disordered QSH systems, ATs can be extended from traditional metal-insulator transitions to a broader sense which includes transition between topologically trivial and nontrivial phasesEvers_Mirlin_2008 (). In the past decade, great efforts have been devoted into this issueRyu_2010_njp (); Obuse_2007_prb (); Obuse_2008_prb (); Obuse_2010_prb_2 (); Obuse_2014_prb (); Slevin_2012 (); Ryu_2012_prb (); Slevin_2014_njp (); XRWang_2015_PRL (); CWang_2017_PRB (). The widely-used framework is to construct a quantum network model which consists of two copies of Chalker-Coddington random network model (CC-RNM) describing up and down spins, as well as certain coupling describing spin-flip process. If spin flip occurs in electron trajectories between potential saddle points (PSPs), it is the well-known spin-orbit coupling (SOC). While if it takes place at the PSPs, it is the spin-mixing tunneling (SMT) which is the main focus in this work. Recently, a quantum network model (-QNM) is proposedRyu_2010_njp (); Obuse_2007_prb (); Obuse_2008_prb (); Obuse_2010_prb_2 (); Obuse_2014_prb (); Slevin_2012 () in which SMT at PSPs are considered. It belongs to the Wigner-Dyson symplectic class and a series of work declare that it provides a good description of ATs in 2D disordered spinful electron gases(2D-DSEGs). In -QNM, at PSPs the total tunneling probability are fixed, which means SMT takes away part of the probability from the spin-preserving tunneling (SPT) process. However, from the basic principles of quantum tunneling SMT provides an additional channel and should not affect the existing SPT. In this work, we treat the SMT as an independent quantum tunneling channel and build a new network model, namely the “SMT-QNM”, to provide an alternative perspective to understand ATs in 2D-DSEGs.
This paper is organized as follows. In Sec. II the SMT-QNM is systematically built up based on probability conservation and TRS at PSPs. Then the 2D continuous Dirac Hamiltonian with “valley” degree of freedom is extracted out. In Sec. III numerical algorithms using transfer matrix technique for finite-size analysis on two-terminal conductance and normalized localization length are reviewed. Based on them, in Sec. IV the quantum phases of SMT-QNM are investigated and a phase diagram is then obtained. We discuss its close connection with that of the disordered 3D weak TIs. In Sec. V, we consider the case when TRS in electron trajectories between PSPs is destroyed. The system then falls into unitary class. We briefly summarize the quantum phases and phase transitions therein. Finally, the concluding remarks are provided in the last section.
Ii II. The SMT-QNM
ii.1 II.A Brief review of CC-RNM
Under a strong magnetic field , the motion of an electron in a smooth enough 2D random scalar potential can be decomposed into a rapid cyclotron gyration and a slow drift of the guiding center along an equipotential contour which is generally composed of numerous loops around potential valleys or peaksChalker_Coddington_1988 (); Kramer_2005 (). The drifting direction of electrons in each loop is uni-directional (chiral): . At PSPs, electrons’ reflecting along equipotential lines and their mutual tunneling are the essential physical ingredients for constructing a network model describing quantum criticality in disordered 2D systems. For modelization, the PSPs are arranged to form a 2D square lattice with the interconnected links representing electron flows along equipotential lines. The potential peaks and valleys distribute alternatively in the square plaquettes enclosed by the links. This endues definite propagating direction of electron flows on the links and then divides the PSPs into two subgroups: the S- and S’-types (see Fig. 1a and 1b). At each PSP, two incoming and two outgoing electron flows intersect hence lead to a scattering matrix. Quantum tunneling only occurs at PSPs and in the simplest case can be assumed identical. At last, disorder is introduced by random phases along links. This is the basic framework of CC-RNM. In all illustration figures in this paper, we adopt the following sketch rules: if , the reflecting (tunneling) routes are depicted by solid (dash) curves and vice versa.
For a S-type PSP at , its scattering matrix is,
where is the outgoing (incoming) electron flow amplitude at link , is a diagonal matrix, with being the dynamical phase an electron acquires when propagating on link between the observation point and the PSP at . The kernel matrix has the general form,
where () measuring the reflecting (tunneling) amplitude at a PSP, and is related to the Fermi level of the systemKramer_2005 (). are undetermined coefficients. In steady states, probability conservation at any PSP requires and . Clearly,
which means TRS is broken thus the CC-RNM belongs to the unitary class. Throughout this work, which is also the choice in most literatures.
ii.2 II.B Scatter matrices of SMT-QNM
To describe ATs in 2D-DSEGs, the CC-RNM should be generalized to include spins, providing the following hypotheses. First, the potential profile is identical for any spin orientation. Second, the absence of external magnetic fields makes TRS possible which turns the original uni-directed electron flow on each link to a Kramers doublet. Opposite spin components then “feel” opposite effective magnetic fields, forming two copies of CC-RNM with opposite chirality. Third, appropriate coupling should be introduced between the two copies of CC-RNM to describe spin-flip process. Generally, spin flip can occur anywhere. In real modelization, two strategies are most common: (a) it only occurs on the links between PSPs; (b) it only occurs at the PSPs. The first strategy reflects the SOC while the second one is the SMT.
The -QNM proposed in Refs.Ryu_2010_njp (); Obuse_2007_prb (); Obuse_2008_prb (); Obuse_2010_prb_2 (); Obuse_2014_prb () follows the second strategy, however views SMT and SPT as two competing processes sharing a fixed probability “”. In this work, the SPT channel remains unperturbed. Meantime we treat SMT as an independent quantum tunneling channel and construct the SMT-QNM to understand ATs in 2D-DSEGs. For S-type PSPs (See Fig. 1c), the scattering matrix at position reads,
where is the outgoing (incoming) electron flow amplitude at link with spin , with representing the phase an electron acquires when propagating on link between the observation point and the PSP at . We have neglected the spin index since the Kramers pair of electron flows have the same accumulated phase on the same link. To mimic the randomness in PSP distribution, these phases are distributed uniformly and independently in the region . If we focus on the very point where a PSP locates, then becomes unity. The kernel matrix describes the reflecting and tunneling at a general S-type PSP and has the following structure,
where “” means matrix complex conjugate. For this scattering matrix, several points need to be clarified. First, it is hermitian due to TRS. Second, since SMT is an additional tunneling channel hence takes probability away from reflecting rather than SPT process. For simplicity, can be defined as (thus is real), with describing the strength of SMT. Third, probability conservation in steady states at any PSP requires the scattering matrix to be unitary,
where is the unit matrix. Fourth, TRS requires
where are the Pauli matrices. This gives,
By writing as
Eq. (7) turns to
in which is the 3D Levi-Civita symbol. In addition Eq. (9) gives
Summarizing these two conditions, a reasonable solution to is
leading to a physical realization of as
Obviously and are the phase shifts associated with SPT and SMT processes, respectively. At last, by rotating S-type PSPs 90 degrees clockwise, we get S’-type PSPs and their scattering matrix can be easily obtained from Eq. (4).
To summarize, in our SMT-QNM at any PSP (S- and S’-type), for an incoming electron flow with some certain spin orientation and probability 1, it tunnels into an outgoing flow with the same spin orientation via SPT process with probability “” and also into an outgoing flow with opposite spin orientation via SMT process with probability “”, leaving a probability “” residing in the original equipotential line.
ii.3 II.C 2D Dirac Hamiltonian from SMT-QNM
The mapping from CC-RNM to 2D Dirac Hamiltonian was accomplished in 1996Ho_Chalker_1996 (), and the connection between the -QNM and 2D Dirac Hamiltonian was established in 2010Ryu_2010_njp (). The main strategy of both works is to view the unitary (due to probability conservation) scattering matrices as a unitary time-evolution operation whose infinitesimal generator is the required Hamiltonian, as we all know that a unitary matrix is the exponential of a Hermitian one. In this subsection, we follow this strategy and succeed in extracting the 2D Dirac Hamiltonian from our SMT-QNM and recognizing the roles of phase shifts in SMT and SPT at PSPs. Also, this part of work lays the foundation for understanding the close connection between the phase diagram of our SMT-QNM and that of disordered 3D weak TIs (see Sec. IV.D).
ii.3.1 II.C.1 Preparations
We arrange the S-type and S’-type PSPs alternatively in a 2D Cartesian plane to form a bipartite square lattice, as shown in Fig. 2. Then following the sketch rules in Fig. 1c and 1d, a series of closed square plaquettes are obtained, with each edge bearing two opposite-directed links. For , the centers of these closed plaquettes are the potential valleys, while the potential peaks reside in the blanks outside. For , the situation is just reversed. Quantum tunnelings (SPT and SMT) occur at the plaquette corners, which are the PSPs. We take the case as the framework for our discussion, which does not affect the generality of our results. If one of these plaquettes is assigned with coordinate , then the position of anyone in this set is
They form a square lattice and is our main concern. The eight directed links on the edges of a plaquette are labeled by with and or .
For the plaquette with coordinate , the scattering event at the S-type PSP on its upper-right corner (urc) is as follows
in which TRS has be invoked in writing and . While that of the S’-type PSP on the lower-right corner (lrc) reads
Next the displacement operators are introduced as
where is an arbitrary function defined at . By definition, they are commutative and
By rearranging the amplitudes in the order of “2,4,1,3”, we rewrite Eq. (16) into the form
in which “d (od)” means diagonal (off-diagonal). Similarly, Eq. (18) is rewritten as
By defining the total amplitude vector composed of all eight links along the edges of plaquette at as
with the superscript “” indicating matrix transpose and introducing , the elementary imaginary discrete-time evolution of is,
To acquire decoupled equations, the “two-step” time evolution,
is more convenient since it is diagonal. We will focus on in the rest of this work. Also we make the transformation
to raise the reference point of the total phase flux of each plaquette by [see Eqs. (22) and (25)], which is crucial for the extraction of 2D Dirac Hamiltonian. It can be easily checked that is unitary, thus provide a Hamiltonian as its infinitesimal generator,
We then demonstrate that in the close vicinity of the CC-RNM critical point
how is mapped to 2D Dirac Hamiltonian by expanding to the leading-order powers of
ii.3.2 II.C.2 2D Dirac Hamiltonian around
(a) displacement operators act on smooth enough functions thus
(b) the phases and are small enough hence
(c) in the close vicinity of CC-RNM critical point one has
acting as a scalar/vector potential, respectively.
Then the system is driven away slightly from the critical point (32) along the -line. Hence and , and to the leading order of one has
Correspondingly, the SMT Hamiltonian turns to
After performing a unitary transformation
we get the final Hamiltonian
Obviously describes a pair of Dirac fermions (with mass ) subject to the same random scalar potential and respective random vector potential , meantime bearing a mutual coupling . By introducing a “valley” space distinguishing these two Dirac fermions (different locations of Dirac cones in Brillouin zone), the final Hamiltonian can be rewritten as
where and are identity and Pauli matrices in valley space. Therefore our SMT-QNM belongs to the symplectic class and should be an effective model for ATs in QSH ensembles. Also, the above analytics shows that the phase shifts in SPT and SMT processes at PSPs have different roles during the extraction of 2D Dirac Hamiltonian. The former () enters the vector potentials thus could have impacts on geometric phase accumulated along the plaquette edges. While the latter () resides in the coupling matrix between and then manifests itself in the unitary transformation that changes to , hence acts as a gauge field describing the spin-flip interaction.
Iii III. Algorithms for finite-size analysis
iii.1 III.A Two-terminal conductance
For numerical convenience, by rotating Fig.2 45 degrees clockwise, we obtain a 2D PSP lattice composed of principal layers (PLs), as shown in Fig. 3. Each PL consists of S-type and S’-type PSPs. At a S-type PSP, the “left-ro-right” transfer matrix is obtained from its scattering matrix [see Eq. (4)] as
While at a S’-type PSP, the counterpart is
Then the transfer matrix for the th PL is
where the boundary nodes are selected to be S’-type PSPs as an example (see Fig. 3). is the transfer matrix of the sub-layer composed merely by S-type PSPs with the following form
is the transfer matrix of the S’-type sub-layer
where are matrices and determined by the choice of boundary condition in transverse direction. When we focus on edge modes, the reflecting boundary condition (RBC) is imposed. The Kramers pair is totally reflected without any spin flip at boundary nodes, thus
If bulk behaviors are the main concern, the periodic boundary condition (PBC) is adopted, which means
At last, and are diagonal matrices,
describing the left-to-right intra- and inter-PL random phases in links connecting S-type and S’-type PSPs in adjacent sub-layers. Note that TRS ensures in any link, spin-up electron flowing in a certain direction acquires the same dynamical phase with that of a spin-down electron in the opposite direction. Thus one has the “phase pairing rule”
In practice, for certain the phases are independently and uniformly distributed in .
Multiplying sequentially, the total transfer matrix , which relates the electron flows on the left of the network and those on the right , is then obtained
By introducing a unitary matrix with