Nonequilibrium transport through quantum-wire junctions and boundary defects for free massless bosonic fields

Nonequilibrium transport through quantum-wire junctions and boundary defects for free massless bosonic fields

Abstract

We consider a model of quantum-wire junctions where the latter are described by conformal-invariant boundary conditions of the simplest type in the multicomponent compactified massless scalar free field theory representing the bosonized Luttinger liquids in the bulk of wires. The boundary conditions result in the scattering of charges across the junction with nontrivial reflection and transmission amplitudes. The equilibrium state of such a system, corresponding to inverse temperature and electric potential , is explicitly constructed both for finite and for semi-infinite wires. In the latter case, a stationary nonequilibrium state describing the wires kept at different temperatures and potentials may be also constructed following Ref. [32]. The main result of the present paper is the calculation of the full counting statistics (FCS) of the charge and energy transfers through the junction in a nonequilibrium situation. Explicit expressions are worked out for the generating function of FCS and its large-deviations asymptotics. For the purely transmitting case they coincide with those obtained in Refs. [10, 11], but numerous cases of junctions with transmission and reflection are also covered. The large deviations rate function of FCS for charge and energy transfers is shown to satisfy the fluctuation relations of Refs. [2, 12]. The expressions for FCS obtained here are compared with the Levitov-Lesovic formulae of Refs. [29, 28].


1 Introduction

The transport phenomena in quantum wires (carbon nanotubes, semiconducting, metallic and molecular nanowires, quantum Hall edges) and, in particular, across their junctions, have attracted a lot of interest in recent times,  see e.g.  [16, 14]. To a good approximation, the charge carriers inside the wires may be described by the Tomonaga-Luttinger model [46, 42, 22, 26]. In the low energy limit, such a model reduces to a relativistic 1+1 dimensional interacting fermionic field theory that can also be represented by free massless bosonic fields. The junction between the leads couples together the conformal field theories (CFTs) describing at low energies the bulk volumes of the wires. Specific features of the coupling depend on how the junction is realized. Various models that couple two or more wires locally at their connected extremities were considered in the literature,  see e.g. [20, 36, 34, 35] where important results about transport properties of such models of wire-junctions were obtained. The low-energy long-distance effect of the interaction at the junction may be described with the use of boundary CFT, similarly as the effect of a magnetic impurity in the multi-channel Kondo problem [1]. Even if the coupling of the Luttinger liquid theories introduced by the junction breaks the conformal symmetry, the latter should be restored in the long-distance scaling limit. In the scaling limit, the effect of the junction will be represented, using the “folding trick” of ref. [49], by a conformal boundary defect in the tensor product of the bulk CFTs of individual wires [7]. Such a boundary defect preserves half of the conformal symmetry of the bulk theory. Examples of conformal boundary defects that describe the renormalization group fixed points of Luttinger liquid theories with a coupling localized at the junction were discussed in [20, 36, 34, 35]. It was also realized that the boundary CFT description of the junction of wires gives via the Green-Kubo formalism a direct access to the low temperature electric conductance of junctions [35, 39, 40] that measure small currents induced by placing different wires in slightly different external electric potentials. Getting hold of the transport properties of the quantum-wire junctions beyond the linear response regime is more complicated, see [20] for an early result using an exact integrability of a model of contact between two wires. The CFT approach seems also helpful here. It was shown in [10, 11, 12, 19] that for some boundary defects (those with pure transmission of charge or energy), not only the electric and thermal conductance but also the long-time asymptotics of the full counting statistics (FCS) of charge and energy transfers through the junction may be calculated for the wires initially equilibrated at different temperatures and different potentials. Moreover, steady nonequilibrium states obtained at long times from such initial conditions could be explicitly constructed. Physical restrictions for the applicability of the CFT approach in such a nonequilibrium situations were also discussed in some detail in those works, in particular in [11], see also [8, 18, 4, 15]. The incorporation of junctions corresponding to boundary defects with transmission and reflection into that approach poses more problems, although for a junction of two CFTs a general scheme has been recently laid down in [13], together with some examples.

The present paper arose from an attempt to calculate the FCS for nonequilibrium charge and energy transfers for simple conformal boundary defects with transmission and reflection. We describe each of wires by a compactified free massless -dimensional bosonic field, with the compactification radius related to the Luttinger model coupling constants that may be different for different wires. The product theory is a toroidal compactification of the massless -component free field,  i.e.,  on the classical level, its field takes values in the torus .  In such a theory, we consider the simplest conformal boundary defects that restrict the boundary values of the field at the junction to a subgroup isomorphic to the torus with . In the string-theory jargon, is called the D(irichlet)-brane [38]. First, we study the wires of finite length with the reflecting boundary condition at their ends not connected to the junction. The overall -symmetry of the theory is imposed, leading to the conservation of the total electric charge. We show that the boundary defect gives rise to an scattering matrix that relates linearly the left-moving and the right-moving components of the electric currents in various wires. The classical theory described above may be canonically quantized preserving the latter property. The exact solution for the quantum theory includes the formula for the partition function of the equilibrium state corresponding to inverse temperature and electric potential and for the equilibrium correlation functions of the chiral components of the electric currents. The thermodynamic limit may then be performed giving rise to a free-field theory that was constructed directly for in [32]. In that limit, the equilibrium correlation functions involving only left-moving (or only right-moving) currents factorize into the product of contributions from the individual wires. This property was used in [32],  following the earlier work [31], to construct a nonequilibrium stationary state (NESS) where the correlation functions of left-moving currents factorize into the product of equilibrium contributions from individual wires, each corresponding to a different temperature and a different potential. The NESS correlation functions involving also the right-moving currents are reduced to those of the left-moving ones using the scattering relation between the chiral current components. Following the approach of [10, 11], we show that such a state is obtained if one prepares disconnected wires each in the equilibrium state at different temperature and potential and then one connects the wires instantaneously and lets the initial state evolve for a long time [41].

The main aim of the present paper is the study of the FCS for charge and energy (heat) transfers through the junction modeled by the brane defect of the type described above. Similarly as in [11], the FCS is obtained from a two-time measurement protocol. First, the total charge and total energy is measured in each of the disconnected wires of finite length prepared in equilibria with different temperatures and potentials. Next the wires are instantaneously connected and evolve for time with the dynamics described by the field theory with the brane defect. After time ,  the wires are disconnected again and the second measurement of total charge and total energy in individual wires is performed. The FCS is encoded in the characteristic function of the probability distribution of the changes of total charge and total energy of individual wires. The above protocol is not practical for long wires as the total charge and and total energy of the wires, unlike their change in time, behave extensively with ,  but a similar charge and energy transfer statistics should be obtainable from an indirect measurement protocol where one observes the evolution of gauges coupled appropriately to the wires and registering the flow of charge and energy through the junction, see [29, 30]. In our model, we compute the generating function of FCS of charge transfers explicitly for any and and confirm that it takes for large the large-deviations exponential form that is independent of whether is sent to infinity first or,  e.g.,  kept equal to .  The equality of the large deviation forms for the two limiting procedures appears, however, to be less obvious than one could have expected. The choice leads to the simplest calculation of the large deviation rate function and was implicitly employed in [10, 11], where it was argued that it reproduces correctly the large deviations of the FCS for the junction of semi-infinite wires. We also compute explicitly the generating function of the FCS for heat transfers for and its large deviations form. The case of general and could be also dealt with but the corresponding formulae are considerably heavier and we did not present them here. The generating function of the joint FCS of the charge and energy transfers for and its large deviations form were also obtained. To our knowledge, the calculations of FCS presented in this paper are the first ones obtained for junctions with transmission and reflection modeled by conformal boundary defects. It should be mentioned, however, that in a different physical setup, the FCS of charge transfers across an inhomogeneous Luttinger liquid conductor connected to two leads with distinct energy distributions was obtained by a “nonequilibrium bosonization” in [24, 25, 33].

The present paper is organized as follows. In Sec. 2, we briefly recall the description of relativistic free massless fermions and bosons on an interval. We discuss the correspondence between the two theories and how it extends to the case of the Luttinger model of interacting fermions. Sec. 3 describes in detail the model of a junction based on a toroidal compactification of the multi-component massless bosonic free field with a boundary defect of the type mentioned above. We discuss first the classical theory on a space-interval of length and subsequently canonically quantize that theory in Sec. 4. In particular, we show how the scattering matrix relating the chiral components of the electric current arises from the brane describing the boundary defect. Sec. 5 constructs the equilibrium states of the quantized theory labeled by inverse temperature and electric potential . In Sec. 6, we discuss the Euclidean functional integral representation of the equilibrium state and in Sec. 7, its dual closed-string representation resulting from the interchange of time and space in the functional integral. The closed-string picture is particularly convenient in the thermodynamic limit of the equilibrium state that is analyzed in Sec. 8.  Sec. 9 discusses the NESS of the junction of semi-infinite wires kept in different temperatures and different electric potentials. By considering the nonequilibrium state for close temperatures and potentials, we obtain as a byproduct the formulae for the electric and thermal conductance of the junction. The central Sec. 10 is devoted to the analysis of FCS for charge and heat transfers through the junction. Subsecs. 10.1 and 10.2 treat the charge transport, Subsec. 10.3 that of heat, and Subsec. 10.4 the joint FCS for both. Sec. 11 compares the generating function of FCS for charge and heat transfers obtained in this paper with those given by the Levitov-Lesovik formulae for free fermions [29, 30] and free bosons [28]. In Sec. 12, we specify our general formulae to few simplest cases of junctions of two and three wires. Finally, Sec. 13 collects our conclusions and discusses the possible generalizations and open problems. Appendix A contains the calculations of the generating functional of FCS for charge transfers at general and .  Appendix B performs the computation of certain bosonic Fock space expectations that are needed to obtain the generating function of FCS for heat transfers through the junction. Appendix C calculates the quadratic contribution to the Levitov-Lesovik large-deviations rate function of charge transfers for free fermions.

Acknowledgements: The authors thank D. Bernard for discussions on nonequilibrium CFT and J. Germoni for Ref. [44]. A part of the work of K.G. was done within the STOSYMAP project ANR-11-BS01-015-02.

2 Field theory description of quantum wires

2.1 Classical fermions

Consider a fermionic 1+1-dimensional field theory describing noninteracting conduction electrons in a quantum wire of length . To a good approximation such electrons have a linear dispersion relation around the Fermi surface. For simplicity, we shall ignore here the electron spin. The classical action functional of the anticommuting Fermi fields of such a theory has the form

(2.1)

where ,  with the boundary conditions

(2.2)

We use the Fermi velocity to express time in the same units as length. The classical equations obtained by extremizing action (2.1) are

(2.3)

and their solutions take the form:

(2.4)
(2.5)

The space of classical solutions comes equipped with the odd symplectic form

(2.6)

leading to the odd Poisson brackets

(2.7)

The symmetry

(2.8)

corresponds to the Noether current

(2.9)

with the chiral components

(2.10)

and the conserved charge

(2.11)

The classical Hamiltonian is

(2.12)

2.2 Quantum fermions

Quantized Fermi fields and are given by expressions (2.4) with operators and their adjoints satisfying the canonical anticommutation relations

(2.13)

They act in the fermionic Fock space built upon the normalized vacuum state annihilated by and for (the annihilation operators of electrons and holes, respectively).  Upon quantization, fields and become the hermitian adjoints of and .  The quantum currents have the chiral components

(2.14)

and the conserved (electric) charge is1

(2.15)

The fermionic Wick ordering putting (electron and hole) creation operators and for to the left of annihilators and for ,  with a minus sign whenever a pair is interchanged, assures that the vacuum has zero charge.  The quantum Hamiltonian is

(2.16)

where the constant contribution is that of the zeta-function regularized zero-point energy

(2.17)

2.3 Classical bosons

Consider now a bosonic 1+1-dimensional massless free field defined modulo on the spacetime , with the action functional

(2.18)

We shall impose on the Neumann boundary conditions

(2.19)

Such a scalar field will be viewed as having the range of its values compactified to the circle of radius with metric . The classical solutions extremizing action (2.18) have the form

(2.20)

with

(2.21)

and .  The labeling of modes by even integers is for the later convenience. The symplectic form on the space of classical solutions is equal to

(2.22)

leading to the Poisson brackets

(2.23)

The symmetry

(2.24)

corresponds to the Noether current

(2.25)

with the chiral components

(2.26)

where the upper sign pertains to the left-moving component depending on and the lower one to the right-moving one depending on .  The classical Hamiltonian takes the form

(2.27)

2.4 Quantum bosons

The space of quantum states corresponding to the zero modes may be represented as ,  with viewed as the angle in . acts then as assuring the commutation relation .  An orthonormal basis of is composed of the states

(2.28)

with

(2.29)

The excited modes with the commutation relations

(2.30)

are represented in the bosonic Fock space built upon the vacuum state annihilated by with . The total bosonic space of states is .  We shall identify with its subspace and the state with the vacuum .  The chiral components of the quantum current are given by the right hand side of Eq. (2.26). The conserved charge takes the form

(2.31)

so that and it acts trivially in .  The quantum Hamiltonian requires a bosonic Wick reordering putting the creators for to the left of annihilators in the classical expression. Explicitly,

(2.32)

where the constant term is the contribution of the zeta-function regularized zero-point energy

(2.33)

2.5 Boson-fermion correspondence

In one space dimension there is an equivalence between quantum relativistic free fermions and free bosons that provides a powerful tool for the analysis of such systems [43, 27], see also [45] for the historical account. In the context of the fermionic system described in Sec. 2.2, such an equivalence involves the free bosonic field of Sec. 2.4 with the compactification radius and is realized by a unitary isomorphism that maps vacuum to vacuum, ,  and intertwines the action of currents and the Hamiltonians2.  In particular,

(2.34)

The Fermi fields are intertwined by with the bosonic vertex operators:

(2.35)
(2.36)
(2.38)
(2.39)

2.6 Luttinger model

The interaction of electrons near the Fermi surface gives rise to the addition of a perturbation to the free field Hamiltonian (2.16) that in the leading order takes the form of a combination of quartic terms in the free fermionic fields:

(2.40)

where an infinite constant is needed to make the operator well defined in the fermionic Fock space. Such a perturbation defines the Luttinger model of spinless electrons in one-dimensional crystal [46]. The crucial fact that enables an exact solution of such a model is that, under the bosonization map, the above perturbation becomes quadratic in the free bosonic field:

(2.41)
(2.42)

where on the bosonic side the Wick ordering takes care of the diverging part of the constant on the fermionic side. The perturbed bosonic Hamiltonian has then the form

(2.43)

in terms of the free field with the compactification radius .   corresponds to the classical Hamiltonian

(2.44)

where is the field canonically conjugate to .  The classical Lagrangian related to the above classical Hamiltonian is obtained by the Legendre transform:

(2.45)

for ,  where

(2.46)

Hence after the change of the spatial variable,  Lagrangian becomes that of the free bosonic field compactified on the radius that is different from if .  The factor gives the multiplicative renormalization of the wave velocity due to the interactions (we assume that ). The quantization of the free bosonic theory compactified at radius discussed in Sec. 2.4 provides the exact solution of the Luttinger model on the quantum level.

3 Bosonic model of a junction of quantum wires

In the spirit of the “folding trick” of [49, 37],  see Fig. 1,  we shall model a junction of quantum wires by a compactified free field with -components defined on the spacetime ,  with the action functional

(3.1)

and appropriate boundary conditions. The compactification radii may be different for different wires, corresponding to different quartic coupling constants and in the Luttinger models describing the electrons in the individual wires, see Eq. (2.46) of Sec. 2.6. We shall impose the Neumann reflecting boundary conditions at the free ends of the wires:

(3.2)

where .  Note that we use the rescaled spatial variables in the wires so that the lengths of the wires in physical variables are fixed to . This will not matter much because the length will be ultimately sent to infinity.

Figure 1: Folding trick

The “boundary defect” representing in the folding trick the junction of wires at will be described by the boundary condition requiring that the -valued field belongs to a “brane”:

(3.3)

where is a group homomorphism

(3.4)

specified by integers . We shall assume that is injective so that .  As may be seen from the Smith normal form of matrix , such a property is assured if and only if the matrix has rank and the g.c.d. of its minors is equal to 1,  see Proposition 4.3 of [44]. In particular, necessarily.  Consider matrices and defined by the relations

(3.5)

Matrix defines the projector on the subspace of spanned by the vectors that is orthogonal with respect to the scalar product

(3.6)

in . The boundary condition (3.3) implies that

(3.7)

where .  The stationary points of the action functional (3.1) satisfy, besides the imposed boundary conditions, the equations

(3.8)
(3.9)

Note that relations (3.7) and (3.9) imply mixed Dirichlet-Neumann boundary conditions at for massless free fields .  The solutions of the classical equations decompose in terms of the left- and right-movers:

(3.10)

with

(3.11)

where the upper sign relates to the and the lower one to ,  and

(3.12)

with real such that .  In particular, and .  The space of classical solutions comes equipped with the symplectic form

(3.13)

which determines the Poisson brackets of functionals on that space that may be directly quantized.

The particular case when in (3.4) is the identity mapping of ,  corresponding to the “space-filling” brane ,  describes the disconnected wires. In this case, (i.e. is the identity matrix) and field satisfies the Neumann boundary conditions both at and and only the even modes appear. One obtains in this case the product of theories considered in Sec. 2.3.

4 Quantization

4.1 Space of states

The quantization of the bosonic theory of Sec. 3 is again straightforward but a little more involved than for the disconnected wires. Let us first quantize the zero modes. According to the boundary conditions,

(4.1)

where ,  are angles parameterizing ,  so that

(4.2)

The corresponding Poisson brackets are

(4.3)

leading to the commutators

(4.4)

Keeping in mind that the angular variables are multivalued, the above commutators will be represented in the Hilbert space of functions of angles ,  square integrable in the Haar measure, by setting

(4.5)

An orthonormal basis of is given by the states

(4.6)

such that

(4.7)

and

(4.8)

For the excited modes, it is convenient to introduce a basis of vectors in such that

(4.9)

and the projected modes

(4.10)

with the inverse formulae

(4.11)

Note that relations (3.12) imply that for and for .  The Poisson brackets of the non-zero operators take the form

(4.12)

leading to the commutators

(4.13)

In the standard Fock space quantization, we take

(4.14)

where and are generated by vectors

(4.15)
(4.16)

with the scalar product determined by the relations

(4.17)
(4.18)

The total Hilbert space of states of the theory is

(4.19)

and in the following we identify

(4.20)

4.2 Currents, charge and energy

We shall be interested in the system that possesses global symmetry acting on fields by for .  Invariance of the theory requires that this action preserves the brane .  This holds if and only if the vector is in the image of projector ,  i.e. if or

(4.21)

The Noether (electric) current corresponding to the symmetry has then the form

(4.22)

in terms of the currents in individual wires with the left-right moving components

(4.23)

defining and