Wide-range tunable Dirac-cone band structure in a chiral-time symmetric non-Hermitian system

Wide-range tunable Dirac-cone band structure in a chiral-time symmetric non-Hermitian system

S. Lin and Z. Song songtc@nankai.edu.cn School of Physics, Nankai University, Tianjin 300071, China
Abstract

We establish a connection between an arbitrary Hermitian tight-binding model with chiral () symmetry and its non-Hermitian counterpart with chiral-time () symmetry. We show that such a kind of non-Hermitian Hamiltonian is pseudo-Hermitian. The eigenvalues and eigenvectors of the non-Hermitian Hamiltonian can be easily obtained from those of its parent Hermitian Hamiltonian. It provides a way to generate a class of non-Hermitian models with a tunable full real band structure by means of additional imaginary potentials. We also present an illustrative example that could achieve a cone structure from the energy band of a two-layer Hermitian square lattice model.

pacs:
11.30.Er, 03.75.Ss, 11.30.Rd

I Introduction

Extra imaginary potentials induce many unusual features even in certain simple or trivial systems, which include quantum phase transition occurred in a finite system Znojil1 (); Znojil2 (); Bendix (); LonghiPRL (); LonghiPRB1 (); Jin1 (); Znojil3 (); LonghiPRB2 (); LonghiPRB3 (); Jin2 (); Joglekar1 (); Znojil4 (); Znojil5 (); Zhong (); Drissi (); Joglekar2 (); Scott1 (); Joglekar3 (); Scott2 (); Tony (), unidirectional propagation and anomalous transport LonghiPRL (); Kulishov (); LonghiOL (); Lin (); Regensburger (); Eichelkraut (); Feng (); Peng (); Chang (), invisible defects LonghiPRA2010 (); Della (); ZXZ (), coherent absorption Sun () and self sustained emission MostafazadehPRL (); LonghiSUS (); ZXZSUS (); Longhi2015 (); LXQ (), loss-induced revival of lasing PengScience (), as well as laser-mode selection FengScience (); Hodaei (). Most of these phenomena are related to the critical behaviours near exceptional or spectral singularity points. It opens a way for exploring novel quantum states. The basis of such approaches is to seek various non-Hermitian systems with exact solutions. Recently, the graphene-like materials with Dirac cones at the Fermi energy and a number of unique mechanical, electrical, and optical properties, have attracted much attention RMP (). Its linear-Dirac dispersion makes it an active topic in various research fields. However, for materials in nature, it is very hard to realize experimentally with tuneable parameters. An artificial system, such as photonic simulator, would provide a platform to simulate some aspects in various band structures. Previous efforts mainly focus on the Hermitian systems. A natural question would emerge that whether one can find some artificial materials which have a cone band structure.

In this paper, we consider a method of constructing a variety of non-Hermitian systems which have full real spectra. We focus on the connection between an arbitrary Hermitian tight-binding model with chiral () symmetry and its non-Hermitian counterpart with chiral-time () symmetry. We show that such a kind of non-Hermitian Hamiltonian is pseudo-Hermitian. The obtained result indicates that the eigenvalues and eigenvectors of the non-Hermitian Hamiltonian can be easily obtained from those of its parent Hermitian Hamiltonian and the reality of the spectrum is robust to the disorder. It also provides a way to generate a class of non-Hermitian models with a tunable full real band structure by means of additional imaginary potentials. We present an illustrative example, which is a two-layer square lattice model. By adding staggered imaginary potentials, exact result shows that a cone band structure can be achieved.

The remainder of this paper is organized as follows. In Sec. II, we present a general formalism for the solution of an arbitrary non-Hermitian -symmetric system. Sec. III is devoted to present an illustrative example of a two-layer square lattice model. Finally, we present a summary and discussion in Sec. IV.

Ii Model and formalism

The main interest of this work is focused on the relation between an arbitrary Hermitian tight-binding model with symmetry and a non-Hermitian model which is constructed based on the former by adding additional imaginary potentials. The latter is a non-Hermitian counterpart of the former in the context of this work.

Consider the Hamiltonian of a non-Hermitian tight-binding model

(1)

with

(2)
(3)

on a bipartite lattice which can be decomposed into two sublattices and . Here we only consider the case with identical sublattice numbers  for simplicity. A schematic illustration of the model is presented in Fig. 1(a).

Figure 1: (color online). Schematics for the system with to illustrate the connection between the systems of Eqs. (1) and (15). (a) A bipartite lattice consists of sublattices A and B, which are connected by bond which is across the th site in sublattice A and the th site in sublattice B. In the absence of imaginary potentials, i.e., , it has symmetry, which ensures that the system has the spectrum (). In the presence of imaginary potentials, it has symmetry and becomes a pseudo-Hermitian system. (b) An ensemble of non-interacting half spins in a complex external magnetic field. It is an equivalent system of (a) when the local magnetic field for spin has the form .

The Hamiltonian  has both and time-reversal () symmetries, i.e.,

(4)

where the operators  and  are defined as

(5)
(6)

The Hamiltonian  has  symmetry, i.e.,

(7)
(8)

We note that Hamiltonian has symmetry, which is broken in its non-Hermitian counterpart in the presence of imaginary potentials . The situation here is a little different from the case associated with parity-time () symmetry, where the combined operator commutes with the Hamiltonian. In quantum mechanics, we say that a Hamiltonian  has a symmetry represented by a operator  if . The word “symmetry” is also used in a different sense in condensed matter physics. We say that a system with Hamiltonian  has chiral symmetry, if . The physics of  depends on the model discussed Asboth2016lnp (); Malzard2015prl (); Guo2015prb (); ChenS2015prb (); Peng2016np (); Lee2016prl (). Here we emphasize “chiral symmetry” due to its anticommutation relation with its Hamiltonians. Specifically, the anticommutation relation between operators and results in the equations

(9)
(10)

However, the symmetry is like anti symmetry Peng2016np (). Actually, an anti--symmetric Hamiltonian can be simply constructed from a conventional -symmetric Hamiltonian by multiplying . Here we present two tables demonstrate the difference and connection between  and symmetry.

symmetry

(a)

Table 1: The difference and connection between and  symmetry.

symmetry

(b)

Since the relation  cannot guarantee operators  and  possess a common complete set of eigensates, it is difficult to define the  symmetry of a state . In order to define the  symmetry of a state, we consider the operator  which obeys the relation . Then  and  can have a common complete set of eigensates. The  symmetry of a state  is defined as usual, . Accordingly, in the exact -symmetric region, all the eigenstate obeys  and  has fully real spectrum. For the concerned model, the eigenenergy of  is either real or pure imaginary. When all the eigenstates break the  symmetry, i.e., , the Hamiltonian has fully real spectrum, and  and  have the opposite real eigenenergies.

Now we investigate the Hamiltonian in a pseudo spin representation. We will show that is a pseudo-Hermitian Hamiltonian and there is a simple relation between the spectra of and . Due to the symmetry, the Hamiltonian  can be diagonalized as the form

(11)

where is the positive energy spectrum with , and

(12)

are eigenstates with eigenenergies . Here states  and are single-particle states with particle probability only distributed on sublattices A and B, respectively. Due to the symmetry of , it is easy to check that . One can express the Hamiltonian in the representation of pseudo spins

(13)

where

(14)

is the -component of the Pauli matrix. Accordingly, we could rewrite the Hamiltonian  as the form

(15)

which describes an ensemble of non-interacting half spins in a complex external magnetic field. Here the field and the Pauli matrices are

(16)
(17)
(18)

Based on this analysis, the eigenstates and eigenenergies of Hamiltonian  are

(19)
(20)

where and the Dirac normalized coefficients are .

This result has many implications. (i) Non-Hermitian Hamiltonian  is pseudo-Hermitian, since it has either a real spectrum or else its complex eigenvalues always occur in complex conjugate pairs MostafazadehJPA (). (ii) It explicitly connects the complete set  to . Only an extra phase is added in  from , which indicates that the two states have the same Dirac probability distribution when  is real. (iii) The exceptional points occur at , which correspond to the symmetry breaking of states . It allows a variety of non-Hermitian models with a wide range of disorder parameters to have a full real spectrum and the modulation of band structure is due to the non-Hermiticity. In the next section, we will show its application in an example.

To demonstrate these features, we consider an example model, a generalized non-Hermitian Rice-Mele model, which has been investigated in Ref. HWHPRA (); LSPRA (). The corresponding Hermitian Hamiltonian has the form

(21)

with the periodic boundary condition . The hopping amplitude between two sublattices is where is the dimerization factor. The generalized non-Hermitian Rice-Mele Hamiltonian has been completely solved and the obtained result can be recovered by the present method. In this case, we have

(22)
(23)

with , . In the absence of , the energy gap is , which determines the exceptional point occurring at  for the non-Hermitian Rice-Mele Hamiltonian. In other words, the energy gap of protects the symmetry of the eigenstates of . This is still true in the presence of noise in . In contrast to a symmetric system, the reality of the spectrum for a symmetric one is more robust to the disorder of coupling constants.

Figure 2: (color online). (a) Schematic illustration of a bilayer square lattice with staggered imaginary potentials. The two sublattices are denoted by A (red) and B (green), respectively. The intra and interlayer hopping strengths are and , respectively. For , the system has both and symmetries, while nonzero  breaks the symmetry but maintains the symmetry. The additional staggered imaginary potentials make the simple lattice have a tunable cone band structure.
Figure 3: (color online). 3D plots of band structures of bilayer square lattice with periodic boundary condition. Staggered imaginary potentials are applied throughout the lattice and here we set . The parameters are (a) (b) (c) Here the key difference between the cases for (a) and (c) is that the dispersion relation in the bottom of band is quadratic for (a) but linear (cone) for (c). The band structure in case (a) is trivial in the context that Dirac cone in lattice system has been shown to exhibit some novel features. Although a Hermitian system can support Dirac dispersion (e.g., honeycomb lattice), the speed of electron (slope of the cone) is not tunable.

Iii Cone structure

The connection between and can be employed to modulate the band structure of , which has some intriguing properties induced by the non-Hermitian term . In traditional condensed matter theories, the energy band structure plays a crucial role in the theory of electron conductivity in the solid state and explains why materials can be classified as insulators, conductors and semiconductors. Moreover, much attention has been paid to the honeycomb lattice RMP (), which is relevant to high electron mobility and topological phase, as exemplified by the graphene.

In the Hermitian realm, the band structures of most kinds of systems have been well studied. Nevertheless, non-Hermitian parameters may induce an unusual band structure which is difficult to achieve in a Hermitian system. As an example, Eq. (20) provides a way to accomplish this task that imaginary potentials can deform the shape of a given band structure without altering its topology except the situation when the system contains the exceptional points. In the following, we will present an example which realizes a cone structure on a square lattice.

We consider a bilayer square lattice model which is shown in Fig. 2. The corresponding Hermitian Hamiltonian has the form

(24)
(25)
(26)

where or is the index that respectively labels the position in the top or bottom layers, and is the in-plane site index. Parameters and of this model are intra and interlayer hopping strengths. In this paper, we only consider the case of . And the distribution of imaginary potentials is given as the form

(27)

The Hamiltonian  can be easily diagonalized via Fourier transformation. Let us consider an individual rung, i.e. two sites with the same in-plane site index on the opposite layers. An occupied rung has two possible states that are bond and antibond states. The bond (antibond) state of a rung can only be transited to the bond (antibond) state next to it. Therefore it can be decomposed into two independent single layer square lattices with on-site potentials  and , respectively. The spectra and eigenvectors are

(28)

and

(29)

where  denotes the two independent single layers, and , with , . States  and  are the position states of sublattices A and B with the layer labels and . This band structure is trivial, but it would be a good parent to construct a cone structure by adding staggered imaginary potentials. Now we consider the corresponding non-Hermitian Hamiltonian . According to the above result, the spectra and eigenvectors of  are

(30)

and

(31)

where

(32)

is real when the symmetry is not broken. In the exact -symmetric region, there are local maxima (minima) on the valence (conduction) band at points ( odd). The energy band gap is and the exceptional points occur at . In the vicinity of and considering the case , we have an approximate relation

(33)

with and , which indicates that the band structure is a hyperboloid of two sheets. For , it reduces to a Dirac cone. Note that the difference between the cases for  and  is that the dispersion relation in the bottom of band is quadratic for  but linear (cone) for . Although a Hermitian system can support Dirac dispersion (e.g., honeycomb lattice), the speed of electron (slope of the cone) is less tunable. In contrast, the group velocity at the linear region for our model is

(34)

which indicates that  strongly depends on the ratio of and (), while it only depends on the hopping strength in a honeycomb lattice. In this sense, imaginary extension may make something easier to achieve than that in a Hermitian system. Furthermore, it seems that it has a similar band structure with that of graphene near the zero-energy plane. The difference between them is that the vertices of the cone of graphene are degenerate points, while the ones in the present model are exceptional points. For , the energy gap and the group velocity are tunable by , , and . The cone band structures for different  are plotted in Fig. 3.

Figure 4: (color online). DOS per unit cell as a function of energy (in units of ) computed from the energy dispersion Eq. (30) with several typical values of (blue), (red), (black). And here we set . Also shown is a zoom-in of the densities of states close to the zero-energy point, which can be approximated by . The approximate expression in Eq. (36) is also plotted (solid green) as comparison.

We introduce density of states (DOS) to characterize the band structure. DOS is essentially the number of different states at a particular energy level that electrons are allowed to occupy, i.e., the number of electron states per unit volume per unit energy. DOS calculations allow one to capture various electronic properties, such as specific heat, paramagnetic susceptibility, and other transport phenomena of conductive solids. The DOS of energy bands for a square lattice can be expressed as follows

(35)

which describes the number of states per unit energy per unit cell and therefore the function is properly normalized to . Due to the symmetry of spectrum, we have . Here the densities of states for different are plotted in Fig. 4. In the vicinity of and considering the case , Eq. (33) allows us to derive an approximate expression for the density of states

(36)

which is a linear function of energy. We plot this expression in Fig. 4 as comparison. It indicates that  shows a semimetallic behavior as that in graphene.

Iv Summary

In conclusion, we have studied the connection between an arbitrary Hermitian tight-binding model with symmetry and its non-Hermitian counterpart with symmetry. It has been shown that such a kind of non-Hermitian Hamiltonian is pseudo-Hermitian, providing a way to generate a class of non-Hermitian models with a tunable full real band structure by adding additional imaginary potentials. Based on the exact results, it is found that, the eigenvalues and eigenvectors of the non-Hermitian Hamiltonian can be easily obtained from those of its parent Hermitian Hamiltonian. The reality of the spectrum is robust to the disorder due to the protection of energy gap. Furthermore, as an illustrative example, we investigate the band structure of a two-layer square lattice model with staggered imaginary potentials. We find that a tunable cone band structure can be achieved. It should have wide applications in non-Hermitian synthetic graphene-like materials.

Acknowledgements.
We acknowledge the support of the National Basic Research Program (973 Program) of China under Grant No. 2012CB921900 and CNSF (Grant No. 11374163).

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