Discrete Wigner - Weyl calculus for the finite lattice
Abstract
Abstract
We develop the approach of Felix Buot to construction of Wigner - Weyl calculus for the lattice models. We apply this approach to the tight - binding models with finite number of lattice cells. For simplicity we restrict ourselves to the case of rectangular lattice. We start from the original Buot definition of the symbol of operator. This definition is corrected in order to maintain self - consistency of the algebraic constructions. It appears, however, that the Buot symbol for simple operators does not have a regular limit when the lattice size tends to infinity. Therefore, using a more dense auxiliary lattice we modify the Buot symbol of operator in order to build our new discrete Weyl symbol. The latter obeys several useful identities inherited from the continuum theory. Besides, the limit of infinitely large lattice becomes regular. We formulate Keldysh technique for the lattice models using the proposed Weyl symbols of operators. Within this technique the simple expression for the electric conductivity of a two dimensional non - equilibrium and non - homogeneous system is derived. This expression smoothly approaches the topological one in the limit of thermal equilibrium at small temperature and large system area.
I Introduction
The development of original Wigner-Weyl calculus began from the works of H. Groenewold [1] and J. Moyal [2]. This calculus replaces the conventional operator formulation of quantum mechanics by the formulation in terms of the Weyl symbols of operators. The Weyl symbol of operator is a function on phase space. This calculus was based on the ideas of H. Weyl [3] and E. Wigner [4]. Within the Wigner - Weyl calculus for quantum mechanics Wigner distribution is used instead of the wave function, while Weyl symbols are used instead of the operators of physical observables. Moyal product of the two functions defined in phase space replaces the product of two operators [5, 6]. Applications of this calculus to several problems in quantum mechanics have been proposed [7, 8]. In the context of field theory certain modifications of Wigner-Weyl formalism were built. Such constructions were applied to the high energy physics theory and to the condensed matter physics [9, 10, 11, 12, 13, 14, 15]. In particular, the notion Wigner distribution has been used in QCD [16, 17]. In the framework of quantum kinetic theory the Wigner distribution has been used widely [18, 19]. The applications to the noncommutative field theories have been proposed as well [20, 21].
Attempts to build lattice formulation of Wigner-Weyl calculus encountered certain difficulties. Some work in this direction has been undertaken already in the works by Schwinger [22]. Below we will discuss in more details the work by Buot [23, 15, 24]. Here we would like to mention the physical applications discussed by Wooters [25], and Leonhardt [26], as well as the mathematical constructions reported by Kasperowitz [27], and Ligabó [28]. It is worth mentioning that the deformational quantization is related intimately to Wigner - Weyl calculus [29, 30, 31, 32].
The so - called approximate version of the lattice Wigner-Weyl calculus (more details see below) has been proposed in the works with the participation of the present author [33]. The application of this calculus is limited to the systems with weak inhomogeneity and slowly varying external fields. Using this formalism it is possible to express through the topological invariants the response of various nondissipative currents to external field strength [34, 35, 36, 37, 38, 39]. This formalism has also been applied to the investigation of scale magnetic effect [40, 35].
Motivation of the present study originates from the idea to describe rigorously the topological properties of the non - homogeneous systems using Wigner - Weyl calculus. The methodology developed in the series of our previous publications [33, 41, 42, 43], and in the present paper, may, in principle, be applied to the study of various non - dissipative transport phenomena. The corresponding conductivities are typically expressed as the topological invariants of various kinds. For definiteness we concentrate here at one of the most remarkable non - dissipative transport phenomena, which is the Quantum Hall Effect (QHE). The topological description of the QHE has been proposed originally for the uniform systems in the presence of constant external magnetic field. The conductivity of such an idealized system is expressed through the TKNN invariant [44]. This invariant can also be used for the description of the intrinsic anomalous quantum Hall effect (AQHE) in topological insulators. This version of the QHE exists without external magnetic field. The TKNN topological invariant is expressed through the integral of Berry curvature over the occupied energy levels. If the one - particle Hamiltonian of the system is modified smoothly, then the TKNN invariant (i.e. the Hall conductivity) is not changed [45, 46, 47, 48, 49, 50]. Although expression for the QHE conductivity through the TKNN invariant is the most popular topological expression for the QHE, its applicability is limited to unphysically idealized systems, which are uniform (except for the presence of external magnetic field) and non - interacting. This excludes, in particular, consideration of strong Coulomb interactions between electrons, the role of disorder, and boundary. The alternative topological description of the QHE may be given in terms of the Green functions. The development of the corresponding methodology has a long history. It began from the consideration of intrinsic AQHE in homogenous topological insulators without interactions [51, 52, 53]. For the tight - binding model defined on the infinite lattice the corresponding expression is given by
where
| (1) |
Here is the two - point Green function depending on momentum.
The more simple topological invariant expressed through the two - point Green function is the one responsible for the stability of the Fermi surface [53] . Here , while is the contour that windes in momentum space around the Fermi surface. The more complicated topological invariant is responsible for the stability of Fermi points [54, 53]. In fact, algebraically its expression is identical to the one of Eq. (1). The similar constructions are also used in the other branches of condensed matter physics [55, 56, 57, 58, 59]. In particular, such topological invariants protect gapless fermions existing along the boundaries of topological insulators [60, 61] and in Dirac/Weyl semimetals [53, 59], in the bulk. The similar constructions are also discussed for the 3He-superfluid [62]. Various incarnations of these constructions may be found also in the high energy physics [63, 64, 65, 51, 66, 67, 53, 68, 69, 70].
As it was mentioned above, the TKNN invariant has been obtained for the uniform system without interactions. The same refers also to expression of Eq. (1). However, later it has been proven that Eq. (1) remains valid in the presence of interactions when the two-point Green function is taken with the interaction corrections [71, 72, 73]. Notice, that the original TKNN invariant cannot be extended in a similar way to the interacting systems as the matter of principle. However, the role of interaction corrections to QHE conductivity was considered even before the mentioned above proof has been given [74, 75, 76, 77].
Although Eq. (1) solves the problem with interaction corrections, by construction it is defined for the homogeneous systems only. Direct extension of this expression to the non - homogeneous systems has been given in [33]. Namely, for the tight - binding model of a two - dimensional system defined on infinite lattice the Hall conductivity averaged over the system area is given by with
| (2) |
Here is temperature, is the area of the system, is Wigner transformation of the two-point Green function . Correspondingly, is lattice Weyl symbol of operator ( is inverse to the Green function itself). Symbol means the Moyal product. Notice that the Weyl symbol of an operator is defined for any real values of its arguments, and not only for the discrete coordinate lattice points . Eq. (2) has been derived originally for the non - interacting systems with weak inhomogeneity. This assumes, in particular, that the external magnetic fields remain not large (although are admitted to vary in space). Namely, the magnitude of magnetic field is much smaller than several thousands Tesla, while the wavelengths of external electromagnetic fields are much larger than Angstrom. Except for the artificial lattices, the bound on the magnetic field strength is satisfied for any real system. In [78, 42] it has been proven that in the presence of interactions the Hall conductivity is still given by the expression of Eq. (2), when the Green function is replaced by the complete interacting two-point Green’s function. The version of the Wigner-Weyl calculus for the lattice models with the Weyl symbol of operator denoted above as was called approximate because some of the basic identities of continuum Wigner - Weyl calculus are satisfied by the given Weyl symbols only approximately [79]. The approximate nature of the formalism results in the requirement of the weakness of inhomogeneity. As a result Eq. (2) remains valid only approximately, and is to be modified for artificial lattices or in case of strong inhomogeneities. Notice, that these are the systems, where the so - called Hofstadter butterfly appears.
Extension of Eq. (2) to the case of arbitrarily strong inhomogeneity has been given in [41]. There the tight - binding model on the infinite rectangular lattice has been considered. The version of lattice Wigner - Weyl calculus has been proposed that was called ”precise” because the corresponding Weyl symbol of an operator satisfies the basic identities of continuous Wigner - Weyl calculus precisely. Using the developed formalism it was derived that Eq. (2) is to be replaced by
| (3) |
Here , is the point in space, is imaginary time that varies between and , but Wigner transformation of the Green function and Weyl symbol of its inverse do not depend on . is the area of the two-dimensional lattice cell while is the overall area of the system. By we denote the refined lattice, in which the extra lattice sites are added with the half - integer coordinates (the coordinates of the original lattice are implied to be integer).
The silent feature of the construction proposed in [41] is that Eq. (3) contains the sum over the lattice of large but finite area . Such a finite lattice approximates the infinite lattice. The limit of infinite is to be taken at the end of calculation. At the same time the very definition of the Weyl symbol (entering Eq. (3)) as well as the derivation of its basic properties are valid for the infinite lattice with continuous compact Brillouin zone. Intuitively it is clear that Eq. (3) remains valid under these conditions. However, it is better to build the rigorous infrared regularization of the theory with the version of Wigner - Weyl calculus that obeys all necessary axioms inherited from the continuous theory, and that is reduced smoothly to the ”precise” Wigner - Weyl calculus of [41]. In the present paper we report the corresponding construction. Moreover, we extend the considerations of the Wigner - Weyl field theory to the kinetic domain, and build the version of Keldysh technique for the models defined on finite lattices, written on the language of Weyl symbols of operators instead of the conventional operator formalism. Here we follow the methodology developed ealier in [80, 81, 82, 83] for continuous models and extended in [43] to the lattice models with weak inhomogeneity using the mentioned above ”approximate” Wigner - Weyl calculus.
The first attempt to build the version of Wigner - Weyl calculus adopted for the models defined on finite lattices was performed in the works by Felix Buot [23, 15, 24]. He defined the Weyl symbol of an operator as a function on discrete phase space composed of the original coordinate lattice and discrete momentum space with the same number of points. Unfortunately, the further constructions of [23, 15, 24] contain certain algebraic inconsistencies. As a result, in particular, the so - called star property (see below Eq. (10)) is valid only approximately under the condition that external fields vary slowly (as well as in the mentioned above ”approximate” Wigner - Weyl calculus of [33]). Nevertheless, certain results remain valid precisely under the assumption that the proper lattice Weyl symbol is constructed and replaces the original definition by Buot. The very proposition to use lattice version of Wigner - Weyl calculus adopted for the finite lattice seems to us extremely important. Moreover, the ideas developed in the works of F.Buot are very interesting and fruitful. We pay tribute to the physical intuition of Felix Buot, to his constructions and physical results that remain correct in spite of certain technical mistakes. We consider our present study as a logical prolongation of his work.
We perform the correction of the original constructions by F.Buot in three steps. At the first step we build the symbol of operator that is called by us the Buot symbol of operator. This construction is maximally close to the original definition of F.Buot (see below Eq. (100)). In principle, it is possible to build the self - contained version of lattice field theory completely in the language of the Buot symbol of operators. However, it appears that the limit of infinite lattice volume is not regular being written in terms of this type of the symbol of operator. The same refers to the attempt to approach continuum limit. More specifically, the Buot symbol of physically relevant operators (including the unit operator) is the fast oscillating function of both coordinates and momenta. As a result we proceed constructing the two subsequent modifications of the Buot symbol. The first modification results from the consideration of the model on the auxiliary more dense momentum lattice. The second modification is the mirror construction of the first modification based on the consideration of the auxiliary coordinate lattice that contains times more lattice points than the original lattice. The doubly modified Buot symbol still contains the fast oscillating factors depending on space coordinates and momenta. However, these factors are common for all considered operators. Our final construction appears when we simply omit these factors in the doubly modified Buot symbol. We call the resulting symbol of operator the Weyl symbol, and prove its basic properties. It appears, that this Weyl symbol obeys the basic axioms inherited from the continuous Wigner - Weyl calculus. Moreover, the limit of infinitely large lattice as well as the continuum limit are regular, i.e. the Weyl symbol as a function on phase space has the well - defined limit when the number of lattice points tends to infinity. As expected, this continuum limit is the Weyl symbol of ”precise” Wigner - Weyl calculus proposed in [41]. Written in terms of our new Weyl symbol the Hall conductivity of the two - dimensional system is given by with
| (4) |
Here the is the Weyl symbol of the Matsubara Green function while is Weyl symbol of its inverse. Momentum space is Euclidean one, its points are denoted by . In particular, is the Matsubara frequency. The original coordinate space lattice contains lattice sites, while the Brillouin zone is discrete as well, and also contains points. By we denote the twice more dense spatial lattice that contains points. Correspondingly, is the twice more dense Brillouin zone containing points. Eq. (4) represents the rigorous infrared regularization of Eq. (3).
The paper is organized as follows. In Sect. II we start from the formulation of the main results. In Sect. III we define the Buot symbol of operator and derive its basic properties. In Sect. IV we consider the modification of the notion of Buot symbol and build the final Weyl symbol of operator defined on finite lattice. In Sect. V we describe the dynamics of quantum kinetic theory written in terms of Weyl symbols, and derive the desired expression for the QHE conductivity. In Sect. VI we end with the conclusions.
II Statement of the main results
II.1 Definition of Weyl symbol and its properties
We are considering the rectangular lattice
with the number of points and periodic boundary conditions. Momentum space is
Hilbert space of one - particle states is spanned on ket vectors
Another set of basis vectors is
We assume here and . Besides, we define the refined lattice
with the number of points that is times larger. We also define the refined momentum space
and extended refined spaces
and
We illustrate these constructions by Fig. 1, where lattices are represented schematically. Besides, we represent in this figure the douply refined lattice , which will be defined and used in the following sections. Lattices and will also be used. Their definition, obviously, contains application of both extention and the transition to the refined lattice. The similar constructions refer also to various modifications of momentum space .
For we define Weyl symbol as follows
| (5) | |||||
or, equivalently,
| (6) | |||||
Using periodicity of these expressions () they are naturally extended to . This definition may be extended further to any continuous values of and as follows
| (7) |
The matrix elements of operator are expressed through Weyl symbol of this operator as follows:
| (8) |
and
| (9) |
We formulate the following properties of the Weyl symbol:
-
1.
Trace property.
-
2.
Second trace identity.
-
3.
Periodicity.
with .
-
4.
Star property
(10) -
5.
Weyl symbol of unity
-
6.
Weyl symbol of translation to one lattice spacing is given by
(11) One can see that the limit of this expression exists and gives
(12) -
7.
In the limit of infinitely large lattice the Weyl symbol defined here smoothly approaches the Weyl symbol of [41] given by
Here vectors are normalized as
These vectors are defined as
II.2 Quantum Hall effect in condensed matter system defined on finite rectangular lattice
We consider inhomogeneous system defined on the - dimensional lattice . Time remains continuous. The fermionic field is defined on lattice sites. We assume that the system remains non - interacting. Keldysh Green function is defined as the following matrix
| (15) |
Here Heisenberg fermionic field operator depends on time and spatial coordinates . By we denote the time ordering while is anti - time ordering. means average with respect to the initial (in general, non - equilibrium) statistical ensemble.
In the following the dimensional vectors (with space and time components) are denoted by large Latin letters. For an operator we denote its matrix elements by . Since we deal with the lattice models the space components of - vectors are discrete while the time components are continuous. We then define Weyl symbol of an operator as the mixture of lattice Weyl symbol (with respect to discrete space components) and Wigner transformation with respect to the time component:
momentum is denoted by , and . Here is spatial momentum with components. Keldysh Green function is an operator inverse to . Weyl symbol of Keldysh Green function is denoted by , while Weyl symbol of Keldysh is .
We obtain the following results for the dynamics of lattice system written in terms of Weyl symbols of operators.
-
1.
Weyl symbols and obey Groenewold equation
(17) Here the Moyal product is defined as
(18) It is worth mentioning that for the complete description of the system we need the values of Weyl symbols defined on discrete spatial phase space . For such values of spatial momenta and coordinates the Weyl symbol of unity is equal to .
-
2.
We express DC conductivity (averaged over the lattice area) of the two - dimensional non - interacting systems as
(19) through the lesser component of expression that contains Wigner transformed Keldysh Green function and its inverse that obey Groenewold equation . Using representation of Keldysh Green function of Eq. (15) the lesser component in the above representation may be rewritten explicitly as
(20) where trace is taken over Keldysh components as well as over the internal indices, while
Here means anti-symmetrization.
-
3.
We show that the above expression for the conductivity (averaged over the system area) is reduced to the following expression in case of the equilibrium system at zero temperature
where
(21) Here the is the Weyl symbol of the Matsubara Green function while is Weyl symbol of its inverse. Momentum space is Euclidean one, its points are denoted by . is Matsubara frequency. This expression represents the infrared regularization of the one obtained in [41].
-
4.
One can check that Eq. (21) is a topological invariant in the limit of infinitely large lattice when the sum over momenta is replaced by an integral. For that we need that the system was in thermal equilibrium originally, and that the thermal equilibrium corresponds to zero temperature. Moreover, we need that the Hamiltonian does not depend on time, which means that the system remains in thermal equilibrium during the whole process. The value of the average conductivity is then robust to smooth variations of the system. (This does not refer, however, to local values of conductivity that may depend on space coordinates.) Notice, that the sum over is important for the topological invariance of this quantity.
III Buot symbol and its properties
III.1 Definitions
Here we consider the D - dimensional rectangular lattice with lattice sites. We enumerate lattice sites by variable with , and denote the lattice by :
Periodic boundary conditions are chosen. Therefore, the set of lattice sites may be extended to the set of integer numbers with equivalence . In turn, conjugate momenta are with with . The set of momenta is denoted by :
The values of may be taken from the set of all integer numbers if equivalence relation is imposed. In addition, we introduce the extended lattice
And we represent it as
We also define the refined momentum space
It may be represented as
Thus we define two operations on the lattices: enlarges the lattice by adding points between any pair of the adjacent lattice sites. Operation defines the lattice with the lattice sites of that complement the points of . For the further convenience we also define here the operation . :
It doubles the lattice in all dimensions.
Hilbert space of one - particle states is spanned on ket vectors
Another set of basis vectors is
We assume normalization of states
which implies completeness relations
Any operator acting in may be represented as follows
| (22) |
The next step is an introduction of new variables and :
Here is assumed to be taken modulo while is modulo . Since , and , in order to reproduce all possible values of these variables, we have to require that
while
At the same time extra factor is to be added in order to account for the overcounting of degrees of freedom. As a result we rewrite the above expression for as follows:
| (23) | |||||
Factor is introduced in order to provide that both and are either integer of half - integer. Similar, factor provides that .
III.2 Buot symbol of operator
We obtain the following representation for :
| (24) |
where
| (25) |
while
| (26) |
The above defined function represents the lattice version of Weyl symbol of operator. For the case of the finite lattice it will be called below the Buot symbol of operator . Notice, that this definition differs somehow from the original definition by F.Buot. However, we feel this appropriate to call this symbol in his name.
In order to understand better the properties of this symbol let us calculate the Buot symbol of unity operator:
| (27) | |||||
For even we have
| (28) |
For odd we have
| (29) |
For the operator of translation to the lattice spacing the Buot symbol is
| (30) | |||||
Let us express the Buot symbol of operator through the matrix elements in momentum space
| (31) |
Here in the transition to the second line we change variables:
Factor accounts for the overcounting of degrees of freedom during this transformation. In the similar way we obtain
| (32) |
Notice, that in the original paper by Buot it was not specified that the sum in Eq. (31) is over . Correspondingly, constraint resulted in function was not imposed. This resulted in mistakes in the further expressions. By the present paper we correct these mistakes and introduce the proper definition of the lattice Weyl symbol. Strictly speaking our definition of function differs from the original definition of Buot. We feel, however, appropriate to use this notation because we evaluate the very idea of such a formalism proposed for the first time by F.Buot.
We can express the matrix elements of an operator through the Buot symbol as follows:
| (34) |
Using property of the Buot symbol (for integer - valued vector ) we obtain:
| (35) |
As an example let us calculate the last row of this expression for and even :
| (36) | |||||
In the similar way we can derive
| (38) |
The above expressions give the following two representations of the trace of an operator:
| (39) |
Notice that
| (40) | |||||
and
| (41) |
As a result we can write
| (42) |
Besides, we have
| (43) | |||||
and
| (44) |
Therefore, for even we have also
| (45) |
while for odd :
| (46) |
III.3 Star product
Now let us consider the product of two operators and . Our aim is to express the Buot symbol of this product through Buot symbols and . We have
| (47) | |||||
and
| (48) | |||||
For the matrix elements of the product of two - operators we obtain:
Here several combinations of momenta lead to the non - vanishing matrix elements like :
The constraints on the values of appear:
| (49) |
with integer . We have solutions of these equations:
| (50) |
with . Here the values of are taken modulo .
Considering analytical continuation of functions and to real values of arguments, we come to
| (51) | |||||
We will consider the above expression for and .
However, we define function for any intermediate values of and :
| (52) |
One can easily check the following properties of the Buot symbol extended to real values of arguments ():
| (53) |
Namely, we have for :
| (54) | |||||
Therefore, for continuous values of and :
| (55) | |||||
In the similar way, for discrete and we have
| (56) | |||||
This equality may be extended also to the continuous values of and .
III.4 Alternative derivation of star identity
Let us give the alternative derivation of this equality. It is assumed that while . We start from the star product of Buot symbols and come back to the Buot symbol of the product as follows.
| (59) | |||||
One can see that for both and are integer. At the same time if is half integer, both and are half integer. In both cases is integer. As a result
| (60) | |||||
III.5 Trace of the product
Let us consider the trace of the product of two operators and . We have
| (61) | |||||
and
| (62) | |||||
For the matrix elements of the product of two - operators we obtain:
| (63) |
Here several combinations of momenta lead to the nonvanishing matrix elements like :
The constraints on the values of appear:
| (64) |
with integer . We have the following solutions of these equations:
| (65) |
with . Here all values of variables are taken modulo .
We are in the position to obtain the necessary expression for the trace of product:
| (66) | |||||
III.6 Summary of the properties of the Buot symbol
For we define
| (67) | |||||
This definition is extended further to any values of as follows:
| (68) |
The matrix elements of operator are expressed through the Buot symbol of this operator as follows:
| (69) |
and
| (70) |
We formulate the following properties of the Buot symbol:
-
1.
Trace property.
-
2.
Second trace identity.
-
3.
Periodicity.
with .
-
4.
Star property
(71) -
5.
Buot symbol of unity
IV Modifications of the Buot symbol of operator, and the definition of lattice Weyl symbol
IV.1 The construction that uses auxiliary lattice
First let us consider for simplicity the case of the one - dimensional system. In the following we will consider the even values of , and impose the following constraints on all operators.
for , while
for even and odd .
Now may be divided into the two pieces . Here
while
Notice that while . By we denote restriction of operator to space spanned on vectors with .
Hilbert space of one - particle states on the reduced momentum lattice is spanned on ket vectors
Another set of basis vectors is
with
We also use
Here
These vectors are defined also for any . We assume normalization of states
which implies completeness relations within :
In the similar way the Hilbert space for is defined:
Another set of basis vectors is
with
One can see that and . Here
We assume normalization of states
which implies completeness relations in :
One can consider inclusion of into with the following rules:
Notice that not all vectors for are independent:
Therefore, we may choose as basis vectors those with . In the similar way
while
Modified Buot symbol of operator is defined as the Buot symbol of operator :
| (72) | |||||
for , . Factor entering this expression results in vanishing of the modified Buot symbol for .
Here we express matrix elements of operator as follows:
| (73) | |||||
This expression works both for and for assuming .
This definition may be easily extended to the whole range of real values of and as follows:
| (74) |
With this extension we obtain the basic property of the modified Buot symbol that follows directly from the corresponding property of Buot symbol:
| (75) |
IV.2 The - dimensional construction with auxiliary lattice
Now we extend the above construction to the case of the - dimensional system. We still consider the even values of , and impose the following constraints on all operators.
where is unity vector in the - th direction while . Here . We define
We also require
for and any .
Now may be divided into the two pieces . Here is the remaining piece of .
By we denote restriction of operator to space spanned on vectors with .
Hilbert space of one - particle states on the reduced momentum lattice is spanned on ket vectors
Another set of basis vectors is
with
We also have
Here
We assume normalization of states
which implies completeness relations
One can consider inclusion of into with the following rules:
Notice that
while
Modified Buot symbol of operator is defined as the Buot symbol of operator :
| (76) | |||||
for , . Again, the modified Buot symbol is vanisning for .
Here we express matrix elements of operator as follows:
| (77) | |||||
This definition may be easily extended to the whole range of real values of and as follows:
| (78) |
With this extension we obtain the basic property of the modified Buot symbol that follows directly from the corresponding property of Buot symbol:
| (79) |
Let us rewrite the definition of the modified Buot symbol as follows
| (80) | |||||
For the operator of translation to the lattice spacing the modified Buot symbol is
| (81) | |||||
IV.3 Construction without auxiliary lattice of modified Buot symbol. The doubly modified Buot symbol
The above given construction gives expression for the modified Buot symbol for the operator defined on the lattice with lattice spacing equal to and the number of lattice sites . In order to build the modified Buot symbol for the lattice with unit lattice spacing and the number of lattice sites equal to we need to substitute . In the definition of momentum space we omit the subscript , that is we substitute :
while coordinate space is in place of :
Correspondingly, the previous becomes while becomes . Overall we have the following change of notations:
The modified Buot symbol is given by
| (82) | |||||
Here
and
The periodicity of Buot symbol receives now the form:
with .
Let us consider the modified Buot symbol of Eq. (82). Now we impose on operator the following conditions
where is unity vector in the - th direction while . Here . Now we define
We also require
for and any .
may be divided into the two pieces . Here is the remaining piece of . By we denote restriction of operator to space spanned on vectors with .
Hilbert space of one - particle states on the reduced momentum lattice is spanned on ket vectors
Another set of basis vectors is
with
We also have
Here
We assume normalization of states
which implies completeness relations
One can consider inclusion of into with the following rules:
Notice that
while
Doubly modified Buot symbol of operator is defined as the modified Buot symbol of operator :
| (83) | |||||
for , . One can see, however, that this symbol is not vanishing for , only.
Here we express matrix elements of operator as follows:
| (84) | |||||
This definition may be easily extended to the whole range of real values of and as follows:
| (85) |
With this extension we obtain the basic property of the modified Buot symbol that follows directly from the corresponding property of Buot symbol:
| (86) |
For the operator of translation to the two lattice spacings the modified Buot symbol is
| (87) | |||||
We can also represent the doubly modified Buot symbol as
| (88) | |||||
IV.4 Weyl symbol on the finite lattice defined through the doubly modified Buot symbol
The above given construction of the doubly modified Buot symbol of an operator was defined on the lattice with the sublattice of a lattice with the lattice spacing equal to . The distance between the adjacent lattice sites of this lattice is , while the number of the lattice sites within this sublattice is . In order to build the modified Buot symbol for the lattice with unit lattice spacing and the number of lattice sites equal to , let us first rewrite the above expression for the doubly modified Buot symbol in case of arbitrary lattice spacing :
| (89) | |||||
and
| (90) | |||||
for , with
while coordinate space is
We also define:
and
may be divided into the two pieces . Here is the remaining piece of . By we denote restriction of operator to space spanned on vectors with .
Hilbert space of one - particle states on the reduced momentum lattice is spanned on ket vectors
Another set of basis vectors is
with
We also denote
and
Now we replace by and by . This leads us to the definition of the doubly modified Buot symbol for the lattice with lattice sites and unit lattice spacing:
| (91) | |||||
and
| (92) | |||||
Here
This definition may be easily extended to the whole range of real values of and as follows:
| (93) |
Here
and
Now we are in a position to define the lattice Weyl symbol on the refined phase space as
| (94) | |||||
We can also say that
| (95) |
for the argument of from and
| (96) |
for . Similar to the definition of Buot symbol this definition is extended to the continuous values of and as
| (97) |
Weyl symbol of translation to one lattice spacing is given by
| (98) |
IV.5 Basic properties of Weyl symbol
Basic properties of Weyl symbol follow from the properties of the modified Buot symbols.
Operator defined on the ordinary lattice is to be extended to the lattice . Correspondingly, momentum space is extended to . In the present subsection we denote the basis vectors of these extended spaces by
Here
while
For operator extended to we impose constraints
(we assume here ), and
(we assume here ).
We have the following representation for :
| (99) |
where
| (100) | |||||
while
| (101) |
In order to derive the star property we start from expression for the Buot symbol of the product of two operators (defined for , ):
| (102) | |||||
We come to the star identity
The trace identities and the other important algebraic properties of Weyl symbol follow trivially from the corresponding properties of the Buot symbol.
IV.6 Alternative derivation of star identity
Let us give the alternative derivation of the star identity for Weyl symbol. It is assumed that while . We start from the star product of Weyl symbols and come back to the Weyl symbol of the product as follows.
| (103) | |||||
One can see that for both and are integer. At the same time if is half integer, both and are half integer. In both cases is integer. As a result
| (104) |
IV.7 Trace properties
Let us express trace of an operator through its Weyl symbol:
| (105) |
Now let us consider the trace of the product of two operators
| (106) | |||||
IV.8 Limit of inifinitely large lattice
Let us consider the limit of infinitely large lattice (). Then momentum space becomes continuous. Eq. (94) is reduced to the following expression for the Weyl symbol:
| (107) |
Because as a function of is continuous for the operators that originate from the tight - binding models (i.e. those composed of considered above), we obtain
| (108) | |||||
Recall that basis vectors of Hilbert space corresponding to definite momentum are defined for finite as:
Let us define the other set of basis vectors:
In terms of these basis vectors Weyl symbol of operator is given by
| (109) |
Extension to the real values of and in this limit obtains the form:
| (110) | |||||
One can see that our definition of Weyl symbol defined on the finite lattice is precisely reduced to the definition of [41] in the limit of infinite lattice.
V Dynamics of systems defined on finite lattice, and Hall conductivity
V.1 Keldysh technique of field theory
Here we follow closely the methodology of [83] and [43]. Let us consider the inhomogeneous system defined on the - dimensional lattice . Time remains continuous. The complete field Hamiltonian is denoted by . Let the operator be a functional of field operators . We suppose that is a local, i.e. at the moment of time it is a function of and defined at the same moment. Quantum average of this operator is given by
Here , and is density matrix at . Using time ordering we rewrite it as
For the lattice model an average of quantity is given by
Here and are independent Grassmann variables, and by we understand here a -dimensional lattice point. In the absence of interactions is given by , where is one-particle Hamiltonian. Integration over time is along the Keldysh contour . The contour starts at the initial moment of time , goes to the final moment , and returns back from to . The whole dynamics is concentrated between and .
The forward part of the contour carries the fields and . The fields on the backward part are and .
Variables of the two parts of the Keldysh contour are independent of each other. However, there exist boundary conditions relating them to each other: and . The integration measure contains , and , and a weight function responsible for the initial density matrix :
| (111) | |||||
Here (without hat) is defined as an operator in one - particle Hilbert space. Let us denote its eigenstates by . Its matrix elements are: is the probability that the one - particle state is occupied while is the probability that this state is vacant. Let us introduce Keldysh spinors
| (112) |
The expression for the average of an operator receives the form
| (113) |
Again, by we understand here a -dimensional vector. Here is in Keldysh representation
| (116) |
The correct expressions for the components of this matrix may be obtained either as continuum limit of the lattice regularized expressions or using operator formalism. The result is
| (117) |
Here is matrix that gives rise to initial one - particle distribution . In case of the distribution depending only on energy (and, in particular for thermal distribution of non - interacting particles) is a function of the one - particle Hamiltonian. The infinitely small contributions proportional to parameter symbolize the way those functions are understood as the so - called generalized functions (tempered distributions), for details see Sect. 5.1 of [84].
The Green’s function is defined as
| (118) |
Here index corresponds to components of Keldysh spinor (112). The Green function obeys equation
For the components of we have the following relation
| (119) |
while
| (120) |
Sometimes the new representation of Keldysh spinors is used that is related to the spinors defined above as follows
Green function in the new representation receives the triangle form
Here the Keldysh, Advanced and Retarded Green functions are introduced:
| (122) |
In our paper we will use yet another representation
| (123) |
It is related to the Green function defined by Eq. (118) as follows
| (124) |
where
and
In addition, we have
Here we denote
| (126) |
As a result
| (127) |
with
| (128) |
The elements of (which is inverse to ) are:
| (129) |
For more details on the basics of Keldysh technique briefly reviewed above the reader is advised to consult [85, 84].
V.2 Keldysh technique in terms of Weyl symbol, and conductivity
Basic notions of Wigner - Weyl calculus may be found, for example, in [33, 80]. Here we adopt them to the models defined on finite lattice. In the following the dimensional vectors (with space and time components) are denoted by large Latin letters. We denote matrix element of an operator by . Since we deal with the lattice models the space components of - vectors are discrete while the time components are continuous. We then define the Weyl symbol of an operator as the mixture of Weyl symbol (with respect to discrete space components) defined above, and the standard Wigner transformation with respect to the time component:
momentum is denoted by , and . Here is spatial momentum with components. Below Weyl symbol of Keldysh Green function is denoted by , while Weyl symbol of Keldysh is . We omit the subscript for brevity in this section.
Weyl symbols and obey Groenewold equation
| (131) |
Here the Moyal product is defined as
| (132) |
In the present paper we consider the situation when electromagnetic potential corresponds to constant components of field strength . Moreover, the gauge is chosen, in which the spatial part of potential is proportional to time but does not depend on spatial coordinates. Expansion in powers of will be used up to the leading order, proportional to electric field. Introduction of such a form of external gauge potential results in Peierls substitution . Here is - dimensional vector similar to . When index is lowered, its spatial components change the sign. The Moyal product may be decomposed as
| (133) |
with
| (134) |
This expression remains valid specifically for the case of the given external field that does not depend on spatial coordinates, but depends on time giving rise to external electric field. The case of the external field depending on spatial coordinates would be more involved due to the presence of spatial lattice.
Next, we use expansion of and in powers of and keep the terms up to the linear
| (135) |
Besides, below we omit for simplicity the superscript (0) of the zeroth order contribution to both and . For the case of Hamiltonian that does not depend on time and initial distribution depending on energy only we have
| (136) |
Matrix is inverse to (with respect to the Moyal product):
| (137) |
The Groenewold equation can be written as
| (138) |
In the zeroth order in the Groenewold equation is reduced to , while the first order reads . We come to
| (139) |
The derivation presented here is similar to the one of [80]. However, it differs essentially because we consider the model defined on the finite lattice. In the case of a uniform system Weyl symbol of operator does not depend on . It may depend on and and on . Electric current density on the lattice may be calculated as
Spatial components of momentum are .
| (140) |
Velocity operator is given by
In the presence of inhomogeneity the given expression for electric current density is already not valid. However, we can still calculate the response of the partition function to constant in space variation of electromagnetic potential. Such a variation gives expression for the electric current averaged over the whole lattice as
| (141) |
In the last expression we already deal with the Weyl symbols of operators.
The following quantity defined in phase space
may be considered as Weyl symbol of velocity operator. Therefore, we can write
| (142) |
Here contains trace over the Keldysh indexes as well as trace over spatial discrete coordinates.
Let us express average current through the Keldysh Green function written in triangle representation of Eq. (123).
Using Eq. (126) (, , , and ) we represent current density as
| (145) | |||||
The second term here is expressed through . At the same time the third term is its complex conjugate. We obtain
| (146) |
We represent electric current as
| (147) |
Due to the term proportional to the poles of () are shifted out of the real axis of . The integration contour may be closed at infinity. For that we need to use lattice regularization of time. The lattice regularization adds the factor that suppresses expressions inside the integral over (complex - valued) for . This results in vanishing of the sum of the first two terms in the above expression. We obtain
| (148) |
Applying Eqs. (135)-(139) we calculate the contribution to electric current proportional to external field strength :
| (149) | |||||
Assuming that includes only electric field, we represent this expression in two-dimensional systems as:
where the conductivity tensor may be given as follows:
| (150) |
In this expression we restore subscript index for the Weyl symbols. Here means anti-symmetrization. Conductivity may be expressed as a sum of symmetric and anti-symmetric parts . Here asymmetric part is the Hall conductivity while is the conventional conductivity.
V.3 Equilibrium limit of Hall conductivity
One of the basic properties of Weyl symbol is that the star may be inserted between the two Weyl symbols standing under the trace if the sum over the whole phase space is added (150). Our lattice version of Weyl symbol also obeys this property. Therefore, after averaging the conductivity over the whole volume of the system and over the overall time of the process, we obtain
| (151) |
The next step is to assume that entering the above expression does not depend on time. Recall that the external electromagnetic potential depends on time giving rise to electric field. But after we performed expansion in powers of electric field, this time dependence disappears from our expressions. For the initial thermal distribution we are able to represent the integral over the frequency as a sum over Matsubara frequencies. By we denote Euclidean - momentum, i.e. is Matsubara frequency, while for . Notice that . Substituting instead of we obtain the Matsubara Green function instead of the advanced or retarded Green function.
More specifically, for the system with the one - particle Hamiltonian one can define the real time Green function as
This Green function gives rise to Advanced, Retarded or time ordered Green function when the integration contour in plane of complex is shifted in a specific way. Namely, the time ordered Green’s function (Feynman propagator) is
| (152) |
The retarded Green’s function is defined as
| (153) |
The advanced Green’s function is given by
| (154) |
The Matsubara Green’s function is then defined as
| (155) |
We may write it in terms of imaginary time :
| (156) |
Here is the Matsubara frequency while is the inverse of temperature.
The conductivity of the - dimensional system (averaged over the system area) is given by
where
| (157) |
Here and are standard antisymmetric tensors. is inverse to Matsubara Green function :
In case of small temperatures the sum over Matsubara frequencies is reduced to an integral, which gives
| (158) |
(Here .) At any give finite value of the given expression is not robust to smooth variations of the system. However, when we increase the lattice size in such a way that it becomes much larger than any correlation length existing in the given system, then effectively the sum over momenta in the above expression may be replaced by an integral. In this case is very large but still finite. Eq. (158) remains well defined and finite and becomes topological invariant due to the presence of an integral over . Notice, that the sum over is important for the topological invariance of this quantity.
VI Conclusions
In the present paper we construct the Wigner - Weyl calculus for the tight - binding models defined on the finite lattices. Our starting point was the formalism proposed long time ago by Felix Buot. On the intuitive level his construction gives the hint on how to construct the formalism that allows to describe effectively dynamics of those models. However, the very definition of the symbol of operator proposed by Buot fails to reproduce relations obeyed by the Weyl symbol of operator defined in continuous theory. We, therefore, correct the Buot’s definition and define the symbol of operator that is defined not only for the momenta and coordinates that belong to the original discrete phase space . Our first symbol of operator is defined on the refined discrete momentum space that contains larger number of points. Moreover, the explicit formula for the extension of this definition to the continuous values of momenta and coordinates is given. The resulting symbol of operator obeys some of the basic properties that repeat those of the continuous Weyl symbol. We feel this appropriate to call this symbol the Buot symbol of operator defined on a finite lattice.
Although the Buot symbol of operator obeys several beautiful properties, it has certain disadvantages. Namely, for the simplest tight - binding models the Buot symbol of an elementary translation to one lattice spacing appears to oscillate fast both as a function of momenta and as a function of coordinates. The limit of this quantity is not regular, when the lattice size tends to infinity. The same refers to Buot symbol of unity operator. In principle, it might be possible to build the Wigner - Weyl field theory based on the Buot symbol. Then all quantities may be expressed through the Buot symbols of operators using the Moyal products. However, the limit of large lattice volume of the obtained expressions cannot be taken. For example, we cannot demonstrate in a transparent way how the Hall conductivity tends to a topological quantity in the limit of infinitely large lattice for the thermal equilibrium at small temperatures.
One of the purposes of the present construction (as mentioned in the Introduction) is to build the finite volume regularization of the lattice Wigner - Weyl calculus defined on the inifinite lattice proposed in [41]. The latter construction admits the topological expression for the Hall conductivity. However, the averaging over the lattice volume remains badly defined. We need the definition of Weyl symbol of operator for the finite lattice that tends in a regular way to the one of [41] in the limit of infinite lattice. Correspondingly, we need the rigorous finite volume regularization of the topological expressions for Hall conductivity proposed in [41]. To achive this purpose we construct here the modified Buot symbols, and arrive finally at the needed definition of the finite lattice Weyl symbol. This construction is performed in three steps. At the first step we refine momentum space, i.e. consider the auxiliary momentum lattice with times larger number of points. The modified Buot symbol for the given lattice is defined as the Buot symbol for the refined auxiliary one. The second step is the same procedure performed with respect to coordinate space. The resulting symbol of operator is called the doubly modified Buot symbol. It appears that this symbol contains the oscillating factors in momentum space and in coordinate space. Those factors are common for all operators. Our third step, therefore, is to omit these oscillating factors. The resulting construction is the needed Weyl symbol defined on finite lattice. Fortunately, this symbol obeys all needed properties listed in Sect. II.1. In particular, the elementary function of lattice momentum operator turns into the same function of discrete momentum, the Weyl symbol of unity is equal to real unity, etc. As a result, the limit of infinite lattice is regular, and expression for Hall conductivity approaches smoothly to the topological expression.
Using the constructed lattice Wigner - Weyl calculus we build the lattice version of Keldysh field theory for the description of the non - equilibrium processes. This theory is used to derive in a straightforward way the simple expression for the electric conductivity written in terms of the Weyl symbols of the two - point Keldysh Green function, and the Moyal products. In case of the thermal equilibrium we come to the needed expression that tends to the topological one of [41] in the limit of large lattice and small temperature. This way we achieve our purpose to build the finite volume regularization of the theory described in [41].
It is worth mentioning that our expressions were derived for the case of non - interacting nonhomogeneous systems. However, we expect that, at least, for the case of thermal equilibrium at small temperature the technique developed in [42] may be extended to the systems defined on finite lattice. As a result our expression for Hall conductivity may be, possibly, replaced by the one, in which the non - interacting Green function is replaced by the complete interacting one. The direct consideration of this possibility, however, remains out of the scope of the present paper.
Another possible extension of our results is related to rigorous description of the other non - dissipative transport phenomena. Namely, the proposed technique may be useful for the consideration of the finite volume (infrared) regularization and the consideration of the finite volume effects in Chiral Separation Effect, Chiral Torsional Effect, Spin Hall Effect, and so on. Recall that previously these effects were mainly considered for the homogeneous weak fields. Precise Wigner-Weyl calculus for the lattice models (including the models defined on finite lattices) may be useful for the rigorous description of various non - dissipative conductivities.
Acknowledgements.
The author is grateful for useful discussions of various issues related to Wigner - Weyl calculus and Keldysh technique to I.Fialkovsky, M.Suleymanov, C.Zhang, M. Lewkowicz, and C.Banerjee.References
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