General method for calculating the universal conductance of strongly correlated junctions of multiple quantum wires

General method for calculating the universal conductance of strongly correlated junctions of multiple quantum wires

Armin Rahmani Department of Physics, Boston University, Boston, MA 02215 USA    Chang-Yu Hou Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands    Adrian Feiguin Department of Physics and Astronomy, University of Wyoming, Laramie, Wyoming 82071, USA    Masaki Oshikawa Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan    Claudio Chamon Department of Physics, Boston University, Boston, MA 02215 USA    Ian Affleck Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C., Canada, V6T 1Z1
July 4, 2019
Abstract

We develop a method to extract the universal conductance of junctions of multiple quantum wires, a property of systems connected to reservoirs, from static ground-state computations in closed finite systems. The method is based on a key relationship, derived within the framework of boundary conformal field theory, between the conductance tensor and certain ground state correlation functions. Our results provide a systematic way of studying quantum transport in the presence of strong electron-electron interactions using efficient numerical techniques such as the standard time-independent density-matrix renormalization-group method. We give a step-by-step recipe for applying the method and present several tests and benchmarks. As an application of the method, we calculate the conductance of the M fixed point of a Y junction of Luttinger liquids for several values of the Luttinger parameter and conjecture its functional dependence on .

pacs:

I Introduction

Advances in molecular electronics can extend the limits of device miniaturization to the atomic scales where entire electronic circuits are made with molecular building blocks. Nitzan and Ratner (2003); Tao (2006) Single molecule junctions connected to two macroscopic metallic leads have already been successfully fabricated, Nitzan and Ratner (1997) and there are several proposals such as laying quantum wires on top of each other for making junctions of multiple quantum wires. Kushmerick et al. (2002)

If we eventually manage to build entire electronic circuits with molecular building blocks, a paramount goal in the field of molecular electronics, junctions of three or more quantum wires will inevitably be a key ingredient. These junctions are comprised of several quantum wires, i.e., quasi-one-dimensional (1D) metallic structures with atomic scale sizes, that are connected to one another by a given molecular structure as shown schematically in Fig. 1. The structure and interactions at the junction depend on the particular system under study. What we mean by metallic in the above description of quantum wires is that they are capable of conducting electricity due to the presence of gapless excitations. A generic description for these 1D quantum wires is based on the Tomonaga-Luttinger-liquid theory. Tomonaga (1950); Luttinger (1963); Mattis and Lieb (1965); Haldane (1981) Structures involving Luttinger-liquid quantum wires have been the subject of numerous recent studies. Nayak et al. (1999); Lal et al. (2002); Rao and Sen (2004); Chen et al. (2002); Egger et al. (2003); Pham et al. (2003); Safi et al. (2001); Moore and Wen (2002); Yi (1998); Kim et al. (2004); Furusaki (2005); Giulianoa and Sodano (2005); Kazymyrenko and Douçot (2005); Das and Rao (2008); Agarwal et al. (2009); Bellazzini et al. (2009a, b); Safi (2009); Safi and Joyez (2011); Aristov et al. (2010); Aristov (2011) Experimentally, such quantum wires are realized with carbon nanotubes or through the cleaved edge overgrowth technique in GaAs heterostructures. Bockrath et al. (1999); Yao et al. (1999); Steinberg et al. (2008) Electrical current running in the wires can pass through this molecular structure at the junction.

Figure 1: (Color online) A schematic illustration of the generic system that is the subject of this paper. We have quantum wires connected to a molecular system of certain structure and interactions. We have currents running in wires and voltages applied to the endpoints of the wires.

At molecular length scales, quantum mechanics is important and the system represented above must be modeled accordingly. The simplest theoretical description of systems such as this is based on the tight-binding model, with anti-commuting creation and annihilation operators introduced for different atomic sites. An effective Hamiltonian can then be written in terms of these creation and annihilation operators and generically involves hopping terms and density-density interaction terms with .

Suppose we wish to study a rather arbitrary junction, the structure and interactions of which are represented by a tight-binding Hamiltonian. A very basic question regarding this system is how it conducts electricity. Consider a system of wires. In the presence of voltage biases applied to the endpoint of wire for , current will flow along each wire . By convention, a current is positive if it flows toward the junction and negative if it flows away from it as seen in Fig. 1. In general, i.e., at arbitrary biases and temperatures, this problem is very complicated and the currents flowing in the quantum wires are nonuniversal functions of the temperature, the voltages , and the microscopic details of the system. In this paper, we are concerned with the linear-response regime where universal behavior can emerge. We work at zero temperature and consider only the limit of infinitesimal biases. The currents in this regime will be a linear combination of the applied biases as seen in the following:

(1)

This linear relationship defines the linear conductance tensor which is the quantity of interest in this paper.

One of the most important ideas of modern physics is the remarkable universality that emerges near critical points. It turns out that due to the criticality of the bulk of quantum wires, i.e., having divergent correlation lengths, a large degree of universality also emerges in the behavior of quantum junctions. The universality can be understood in the framework of the renormalization group (RG). Wilson (1975) One can argue that in the limit of small biases and low temperatures, many of the microscopic details of the junction are irrelevant in the RG sense, which means their contribution to conductance, and other physical observables, decays to zero at large distances and low energies.

The junctions we are concerned with in this paper fall into the category of quantum impurity problems. The junction, with all the complex structure and interactions it contains, is localized at the endpoints of the wires. It can therefore be thought of as one (rather arbitrary) impurity inserted into a system, the bulk behavior of which is given by that of independent quantum wires. A classic example of quantum impurity problems is the Kondo model describing the behavior of conduction electrons interacting with a local magnetic moment. Kondo (1964) The powerful methods of boundary conformal field theory (BCFT) have proven useful in a multitude of quantum impurity problems. Affleck (1990, 2010) Thus, BCFT is the main analytical technique used in this paper.

Determining the conductance of quantum junctions in the presence of strong electron-electron interactions is a long-sought and challenging goal. The Landauer-Büttiker’s formalism, which is the method of choice in the calculation of quantum conductance, does not account for these interactions, which indeed play a key role in low dimensions. Functional renormalization-group methods have been helpful in studying the interaction effects in the vicinity of the junction, but their applicability is also dependent upon the presence of large noninteracting leads. Barnabé-Thériault et al. (2005a, b)

In recent years, efficient numerical methods, such as the density-matrix renormalization group (DMRG), White (1992) have been developed for studying strongly correlated quasi-1D quantum problems. Since the quantum junctions described above can be thought of as quasi-1D (by folding all the wires to one side so they run parallel to one another), these numerical methods could potentially be efficient tools for computing the conductance of junctions with an arbitrary number of wires and in the presence of strong interactions.

In fact, DMRG has already been applied to the study of quantum junctions. Guo and White (2006) However, when it comes to calculating the conductance of strongly correlated junctions, there are fundamental difficulties even when we are armed with powerful tools such as DMRG. One such difficulty arises from the fact that conductance is a property of an open quantum system. We define the conductance in terms of the current passing through the system and the underlying assumption is that we have reservoirs that can act as sources and drains for electrons. To study conductance, we either need to model the reservoirs carefully or send them to infinity. The latter is a simpler and more elegant way of formally dealing with quantum transport, but has the downside that for a numerical calculation of conductance, we would need to model large enough systems that faithfully approximate the semi-infinite ones.

Another difficulty with calculating the linear conductance is that, within the linear-response framework, conductance is formally related to dynamical correlation functions. It may then appear that one needs to use the much more computationally demanding time-dependent numerical methods such as time-dependent DMRG to calculate the conductance.

For junctions of two quantum wires, time-dependent DMRG has already been used for conductance calculations. Cazalilla and Marston (2002); Luo et al. (2003); White and Feiguin (2004) A brute force calculation with time-dependent methods in large systems is not, however, currently feasible for strongly correlated junctions of more than two quantum wires.

It is the objective of this paper to make such calculations possible with a combination of analytical and numerical techniques. More specifically, the main objective of this paper is to develop a formalism that would allow us to apply numerical methods such as time-independent DMRG and the related matrix product states to calculate the linear-response conductance of strongly-correlated junctions of an arbitrary number of quantum wires with rather generic structures and interactions in the junction. Verstraete et al. (2004) In this paper, we focus on the systems with spinless electrons, but our method can also be extended to systems with spin-1/2 electrons.

One particular application for the formalism we seek to develop in this paper is the problem of the M fixed point in a Y junction of spinless Luttinger liquids. The existence of this nontrivial fixed point was conjectured many years ago, but its nature, and more specifically its conductance, had remained an open question in quantum impurity problems. Chamon et al. (2003); Oshikawa et al. (2006)

The outline of this paper is as follows. In Sec. II, we summarize the main results of this paper. We present a key relationship [Eq. (2)] between the junction conductance and certain static correlation functions in a finite system. This relationship serves as the basis of the method developed here. We also present in Sec. II a step-by-step recipe for applying the method in practice as well as a summary of the new results on the Y junction, which were obtained with this method. In Sec. III, we present some explicit calculations in a noninteracting lattice model that help motivate the derivation of Eq. (2) and clarify the connection of the continuum results to lattice calculations. The results derived are in fact some special cases, which can be obtained with elementary methods, of the general relation Eq. (2), the derivation of which requires the machinery of BCFT. In Sec. IV, we briefly review the main analytical techniques, namely, bosonization and boundary conformal field theory, used in this paper and set up the notation. Section V is devoted to deriving Eq. (2) in the BCFT framework. In Sec. VI, we discuss in detail the method proposed in this work for conductance calculations and clarify practical issues regarding a lattice-model implementation. In Sec. VII, we present numerical benchmarks with DMRG for interacting systems and exact diagonalization for noninteracting systems to verify the correctness of the method. In Sec. VIII, we study a Y junction of quantum wires and obtain the previously unknown conductance of its M fixed point as a function of the Luttinger parameter . Finally, we conclude in Sec. IX by outlining the outlook for future applications and the impact of the results obtained in this paper. Some of the results of this paper have been briefly reported in Ref. 42.

Ii Main results

We developed a method to extract the universal linear conductance of quantum multi-wire junctions defined in Eq. (1) from a calculation of the ground state expectation values of operators involving currents and densities in an appropriately constructed finite system.

At the core of the method lies an important general relationship, which we recently derived in Ref. Rahmani et al., 2010 using the machinery of BCFT. The relationship is derived in Sec. V and simply states that

(2)

where is the distance from the boundary on the left in a system of length , with and finite , constructed from the junction of interest and an appropriate mirror image placed on the right endpoints of the wires as seen in Fig. 2. Here () is the right-moving (left-moving) current on wire (). Note that although Eq. (2) holds asymptotically (), it can be used to extract the conductance even with finite but large enough .

Figure 2: (Color online) A schematic illustration of the finite system, constructed from the junction of interest and an appropriate mirror image, which we use in our method to extract the conductance . The conductance is related to the ground state correlation function of chiral currents in this finite system through Eq. (2).

The relationship above is the key ingredient of our method for calculating the conductance. If we can compute the quantity , which is a ground-state expectation value in a finite system of length , as a function of , then we can multiply it by a universal function to get the left-hand side of Eq. (2) above. This quantity will then saturate to the universal conductance of the junction for large .

Apart from the derivation of the key Eq. (2), we provide a recipe for applying this continuum result to a lattice calculation. This requires specifying the procedure for constructing the lattice (tight-binding) Hamiltonian of the aforementioned finite system from the Hamiltonian of the junction of interest (that couples infinitely long wires). It also requires specifying lattice operators, i.e., in terms of the tight-binding fermionic creation and annihilation operators, the correlation functions of which are a good approximation to . This is important because the chiral current operators are defined for the continuum theory, and chiral creation and annihilation operators can not be directly modeled on the lattice. The recipe for applying the key relation Eq. (2) to a lattice computation is given in Sec. VI.

For quick reference and an illustration of the method, here we give a simple example and explain a step-by-step application of the method to the well-known problem of a weak link in the Luttinger liquid. The starting point for applying our method is always a tight-binding lattice Hamiltonian of spinless electrons for bulk wires and their connection at the junction, namely, . For a weak link in a Luttinger liquid, we can write

and the following Hamiltonian for the bulk of the wires:

(3)

Note that there is some arbitrariness in dividing the system into the junction and wires. The boundary Hamiltonian above is a minimal choice, but including more sites in does not affect the results as long as the system is large enough and the correlation functions discussed below are computed far away from the boundary. Given the system Hamiltonian, our method consists of the following steps.

  1. Construct the finite system as in Fig. 2. For this we need to construct a Hamiltonian where and respectively describe the junction on the left side of the system () in Fig. 2 and the mirror image at . The recipe for constructing these Hamiltonians is simple. The left boundary Hamiltonian is simply equal to and the bulk Hamiltonian has exactly the same form as the bulk Hamiltonian of the semi-infinite system but a finite number of terms, i.e., . The construction of the right Hamiltonian goes as follows. First we consider the same Hamiltonian as but acting on the other endpoint, namely and then we apply two transformations and on this Hamiltonian. simply takes the complex conjugate and changes . In this case, assuming a real hopping amplitude , we have

  2. Having constructed the Hamiltonian of the finite system, measure, by a numerical DMRG calculation, the following ground state expectation value:

    where is simply the current operator and is the charge carrier velocity for the Luttinger liquids described by . The above equation is valid for time-reversal symmetric systems like the example at hand. The general construction of the operator in terms of the lattice creation and annihilation operators is given in section VI.

  3. Fit the data for to the asymptotic functional form from Eq. (2), i.e., , and obtain from the overall coefficient.

Note that if , the fitting is tricky. These details will be discussed later.

The other important results of this paper concern a concrete application of the method to a previously unsolved quantum impurity problem. These results are presented in Sec. VIII. The main problem solved in that section by an explicit numerical DMRG calculation is determining the conductance of a nontrivial fixed point in an interacting Y junction of three quantum wires depicted in Fig. 3. The existence of this fixed point known as the M fixed point was conjectured in Ref. Chamon et al., 2003; Oshikawa et al., 2006, but the properties of the fixed point including its conductance remained unknown. In this paper, by combining the method developed in Secs. V and VI and numerical computations with time-independent DMRG, we calculate the universal conductance of this fixed point as a function of the Luttinger parameter (a parameter that quantifies the strength of electron-electron interactions in the wires). Based on the numerical results, we claim that the conductance of the M fixed point depends on the Luttinger parameter as

(4)

where . The above expression is a universal result which, for any nonvanishing hopping amplitude , holds independently of the value of . We also explicitly verify that the conductance exhibits universal behavior. Namely we find that at large length scales, the conductance is independent of the hopping amplitude at the junction.

Figure 3: A schematic illustration of the Y junction numerically studied in section VIII. The M fixed point corresponds to the time reversal symmetric case .

We also present in Sec. VIII a numerical verification of a theoretical prediction for the conductance of the chiral fixed point. The chiral fixed point is stable in the range and in the presence of a time-reversal symmetry breaking flux .

Iii Conductance and correlations in a noninteracting lattice model

There is a well-established framework, namely the Landauer-Büttiker formalism (see for example Ref. Nazarov and Blanter, 2009), for calculating conductances of multi-wire quantum junctions in the absence of electron-electron interactions. This framework is applicable both in the continuum and to lattice models. The key quantities in this framework are the incoming and outgoing scattering states and , which for a junction of wires can be represented by an () column vector for spinless (spinful) electrons. The scattering states are labeled by momentum quantum number , and the effect of the junction is encoded in an () unitary scattering matrix for spinless (spinful) electrons. The scattering matrix relates the incoming and outgoing scattering states as

(5)

In the Landauer-Büttiker formalism, conductance is simply related to the elements of the scattering matrix at the Fermi level. Because including spin is a simple extension of the method developed in this paper, we work with spinless electrons here. The conductance between two different wires is then given by

(6)

Let us now consider the simplest junction of two lattice wires. Each wire has hopping amplitude set to unity in the bulk and the two wires are connected by a hopping amplitude . More generically, one can consider a noninteracting junction of wires that are coupled quadratically as described below. Consider a lattice system of wires with electron annihilation operators , where is the wire index and is the site index on each wire. We can write the Hamiltonian in a compact form by defining :

(7)

where is a Hermitian matrix with diagonal elements (endpoint chemical potentials) and off-diagonal elements (hopping between endpoints). In the bulk of each wire, the nearest-neighbor hopping amplitude is set to unity. The scattering eigenstates are

(8)

where is a row vector of operators and is a column vector of scattering amplitudes for , and the standard matrix multiplication convention applies. By plugging the above states in the Schrödinger equation , we obtain and Oshikawa et al. (2006)

(9)

The scattering matrix relates the incoming and outgoing states and as . For the simple case of just two wires connected with hopping amplitude , we have

(10)

The above scattering matrix at half-filling then yields the following conductance :

(11)

As discussed in the Introduction, the key result of this paper is a general relationship between the conductance and certain static correlation functions in a finite system. The purpose of this section is to illustrate this relationship in the very special case of a simple noninteracting system where exact calculations can be done with elementary methods. Consider the following finite system shown in Fig. 4, which consists of our junction with hopping on the left. At a finite length away from the junction, the other endpoints of the wires are coupled with a hopping amplitude . Let us now calculate the current-current correlation function .

Figure 4: A simple noninteracting system of two wires constructed with a junction with hopping and mirror image with hopping . The quantity of interest is the expectation value with the current operator on wire .

To begin with, let us assume a simple special case where . In this case, our system is a loop, with no impurity, that is threaded with a flux . This gives rise to anti-periodic BC. In this special case, the Hamiltonian can be simply diagonalized and we get

(12)

where is the Fourier transform of with and . We focus on the half-filled case and assume is even. The ground state is then given by a Slater determinant and the fermionic correlation function

where and are the site index as in Fig. 5, can be written as

(13)

Each term in the above sum corresponds to one filled momentum level where the contributions of and levels are complex conjugates. The sum is just a geometric series, which can be calculated exactly and the result is

(14)

If (on wire ) is the current operator between sites and , will be between sites and in the chain. Let us write these operators explicitly as

Using the fermionic Green’s function Eq. (14) and Wick’s theorem, we can explicitly calculate the current-current correlation function . The real-space form of Wick’s theorem can be generically written in the form

(15)

To calculate , we write it as a sum of four quartic terms and reduce each term to a sum of products of single-electron Green’s functions. Notice that , a function of the distance alone. After some algebra, we can then write the current-current correlation function as

By plugging the explicit form of from Eq. (14) into Eq. (III), we then obtain the following exact expression for the current-current correlation function:

Consider the above correlation function away from the two endpoints and . We can then approximate the denominator of the second term as and write

(17)

As we will show later, the expression derived above for a simple noninteracting model has a universal form that survives electron-electron interactions.

Before proceeding, let us consider a limit of the above expression. If we take the limit of in Eq. (17) above, we are sending off the right junction to infinity and effectively describing a semi-infinite system. In this case, for a distance (lattice spacing set to unity), we obtain the following correlation function:

(18)

In the next step, we derive a similar expression for a semi-infinite system but with an impurity consisting of a site with hopping amplitude . It is illuminating to first give a short derivation of Eq. (18) formulated directly in the limit of the semi-infinite system. In this limit, we can treat the momenta in the continuum and work directly with the scattering wave functions, which in the special case of are just plane waves due to the absence of back-scattering. We have right-moving (left-moving) plane waves for () and at half-filling, the filled momentum states have . This leads to the following fermion Green’s function:

(19)

which is in fact the limit of of Eq. (14). Inserting the above expression into

where straightforwardly leads to Eq. (18).

So far, we have only considered . Let us now consider a semi-infinite system shown in Fig. 5 with an impurity of arbitrary hopping amplitude .

Figure 5: (Color online) The single-particle scattering states for a single-impurity infinitely long system.

Now, instead of the simple plane waves , we can use the following single-particle scattering states. Let us write the right-moving and left-moving scattering states separately for clarity. We have

In terms of the above scattering states, we can write the fermionic Green’s function for and as

(20)

Note that in the simple case , where we have no back-scattering and , Eq. (20) simply reduces to Eq. (19) above. By using the scattering matrix Eq. (10) or more directly from plugging in the scattering states into the equations

(21)
(22)

we obtain and . We also have and . By inserting the above transmission and reflection coefficients into the expression for the Green’s function Eq. (20) and after some algebra, we can write

(23)

This leads to the following expression for the current-current correlation function:

(24)

with defined in Eq. (23). We claim that the asymptotic behavior of the quantity is generically given by as in the special case Eq. (18). To check this claim we plot as a function of for several values of the hopping amplitude by straightforward numerical evaluation of the integral in Eq. (23). The results are shown in Fig. 6.

Figure 6: (Color online) The quantity calculated from Eq. (24) by a numerical integration of the integral in Eq. (23). For large , this quantity saturates to a constant value that only depends on the hopping amplitude . This indicates that the asymptotic form of is given by .

Since for each , saturates to a constant, a natural question is how this constant depends on the value of . A quite remarkable fact is that the saturation value is exactly proportional to the conductance Eq. (11) of this simple junction, which we calculated in the beginning of this section from the Landauer’s formalism. In other words, we have the following asymptotic behavior:

(25)

At this level, this observation appears rather mysterious. It is not very transparent from the expression for , which is a rather complicated double integral, why this ground state correlation function has such a simple scaling form. By comparing Eq. (25) above with Eq. (17), we claim that for a finite system shown in Fig. 4, the asymptotic behavior of the correlation function is given by

(26)

We do not attempt to prove the above expression here. The equation is indeed a very special case of the generic relationship Eq. (2) that we prove in section V in full generality, i.e., in the presence of electron-electron interactions for a rather arbitrary junction with an arbitrary number of wires and without assuming symmetries such as time-reversal.

The expression above, however, motivates the main idea of this paper, i.e., the fact that conductance can be extracted from ground state expectation values in a closed system, in a very elementary example. It is also an example where the relationship can be derived explicitly on the lattice instead of resorting to the continuum formalism, which helps clarify the application of the continuum CFT results to a lattice calculation.

Iv Review of bosonization and boundary conformal field theory

The main results of this paper are derived within the framework of boundary conformal field theory. In this section, we review the required steps for formulating the generic system we would like to study, namely a junction of quantum wires modeled as a tight-binding Hamiltonian with spinless electrons, in the language of the conformal field theory. The material in this section can be skipped by readers familiar with bosonization and CFT.

iv.1 Bosonization of quantum wires

The first step in this formulation is the bosonization procedure. Bosonization is a powerful nonlocal transformation that, in one space dimension, allows us to describe the low-energy limit of strongly-correlated fermionic systems as a noninteracting theory of bosons (see, for example, Ref. Sénéchal, 1999 for a review). Let us start by considering one infinitely long quantum wire with the following Hamiltonian:

(27)

where . The first two terms describe electron hopping and the last term is the density-density interaction. At half-filling, this model exhibits a charge density wave phase transition for large repulsive interactions . We then see that the ground state spontaneously breaks lattice translation symmetry and we get the two degenerate ground states shown in Fig. 7 for .

Figure 7: The degenerate ground states of the Hamiltonian Eq. (27) for . Filled circles represent occupied sites.

Also at very large attractive interactions, the electrons will clump together and form clusters of neighboring occupied sites. This results in phase separation which, at half-filling, happens at .

For a certain range of interactions , which includes the noninteracting fermionic system (), the system is in a gapless critical phase known as the Luttinger liquid. This is the regime where bosonization applies. Let us first bosonize the noninteracting system.

The noninteracting fermions have an dispersion and a ground state consisting of a filled Fermi sea up to the Fermi level . We can linearize the dispersion around the Fermi level and only consider the right-moving (left-moving) excitations close to ().

Let us define the following right-moving and left-moving fields:

(28)

We can then write the hopping part of the Hamiltonian as

(29)

Upon Fourier transforming, the above equation in real space reads as

(30)

where

(31)

The key step to bosonization is writing the Hamiltonian (both hopping and interaction part) in terms of the following chiral current operators:

(32)

instead of the fermionic creation and annihilation operators. One can easily verify using the Hamiltonian in Eq. (30) that these currents satisfy the following commutation relations:

(33)

The interaction term, which is quartic in creation and annihilation operators, will trivially become quadratic in terms of the currents. The nontrivial part of the bosonization procedure is to show that the hopping part of the Hamiltonian, which is quadratic in , will remain quadratic in terms of .

This important result can be shown using the commutation relation of the current operators. By Fourier transforming as , we can write the Fourier transform of the chiral current in Eq. (32) as

(34)

which leads to the following commutation relation Haldane (1981); Mahan (2000):

(35)

A similar equation with can be derived for . Using the commutation relation Eq. (35), one can show that the bosonized Hamiltonian

(36)

obeys the commutation relations in Eq. (33). We then argue that Hamiltonian (36) is the bosonized form of the noninteracting fermionic Hamiltonian (30).

By using the commutation relations between and , one can write the fermionic operators in terms of the currents by introducing bosonic fields . The result is

(37)

The interacting Hamiltonian density will now have diagonal terms of the form , coming from the hopping and off-diagonal terms of the form from the electron-electron interaction (back-scattering) and is quadratic in terms of the currents. By perfroming a linear transformation

(38)

that preserves the commutation relations, we obtain

(39)

It is convenient to define new bosonic fields and that are linear combinations of in Eq. (37) such that

(40)

In terms of these fields and by introducing the renormalized charge carrier velocity and the dimensionless Luttinger parameter , the low energy effective Hamiltonian density for a wire can be written as

(41)

The bosonized Hamiltonian Eq. (41) is the generic description of 1D metallic quantum wires that we work with in this paper. Notice that is the momentum conjugate to and we have the following commutation relations:

(42)

The above review covers the main ingredients of the bosonization scheme that we need for setting up the problem at hand in this paper.

iv.2 Boundary conformal field theory

Here, we briefly review the basics of CFT and BCFT that we need in the remainder of this paper. This review is largely based on Ref. Cardy, 2010. An important property of critical theories, i.e., theories where certain fields known as the scaling fields have critical correlation functions, is scale invariance. This simply means that the correlation functions of scaling fields satisfy

(43)

where the exponent is known as the scaling dimension of the operator and is an arbitrary scaling factor.

A powerful leap from scaling symmetry to conformal symmetry is by allowing the scaling factor to vary smoothly, i.e., considering more general transformation than with . If we have a local theory and the transformation locally resembles a scaling transformation (modulo a local rotation) then we expect

(44)

In three dimensions, the transformations that locally resemble scaling are limited but in two dimensions (2D) (or 1+1D quantum systems) any analytic mapping [ on the complex plane ] is conformal (preserves angles) and requiring conformal symmetry leads to highly nontrivial results. An important quantity in a CFT is the stress-energy tensor. Consider the action of a 1+1D quantum system that is defined on the complex plane . The stress-energy tensor is defined in terms of the variations of this action through

(45)

where is the variation of due to the infinitesimal transformation . Another important ingredient of CFT is the operator product expansion (OPE), which describes the nature of the singularities in as as

(46)

In any CFT, the primary fields are fields for which the most singular term in the OPE of is order where .

Let us consider primary fields with the following scaling transformation on the complex plane :

where and are called the complex scaling dimensions. Notice that the above equation is only a shorthand to describe the scaling behavior of correlation functions involving . An important result in CFT is that under a generic conformal transformation , the correlation functions of primary fields transform as Cardy (2010)

(47)

This is in fact the same as Eq. (IV.2) which we intuitively wrote down in the beginning of this section. We shall emphasize that the transformation (IV.2) only holds for the primary fields of the theory.

So far our discussion has been limited to CFTs on the entire complex plane . A CFT on a domain with boundaries is known as boundary conformal field theory (BCFT) where in addition to the bulk properties, we need to specify appropriate boundary conditions (BCs). An important task in BCFT is classifying the conformally invariant boundary conditions for a given bulk CFT. Cardy (1989)

The BCFT techniques have proved powerful in studying various quantum impurity problems such as the Kondo model and Luttinger liquids with impurities. Wong and Affleck (1994); Affleck (2010) The effect of the impurity, in this approach to quantum impurity problems, is to select the appropriate conformally invariant BC corresponding to the low-energy RG fixed point. At far away from the boundaries, the bulk CFT is expected to adequately describe the system, i.e., the boundary conditions do not matter. Also very close to the boundary, there is nonuniversal short-distance physics from the microscopic degrees of freedom. On distances from the boundary that are much larger than the microscopic length scales, but still much smaller than the domain size, the correlation functions are affected by the conformally invariant boundary conditions. Cardy (2010)

As pointed out by Cardy, the BCs in a BCFT are encoded in boundary states. Cardy (1989) To understand the notion of a boundary state, it is helpful to consider the partition function of a CFT on a cylinder with boundary conditions and on two sides of the cylinder as in Fig. 8Saleur (1998) One can write down the partition function using transfer matrices running parallel or perpendicular to the boundary.

Figure 8: Boundary states in the partition function of a CFT on a cylinder.

The transfer matrix in the direction parallel to the boundary depends on the boundary conditions and and can be written as (the lattice spacing is set to unity). The imaginary time runs parallel to the boundary and is the Hamiltonian with boundary conditions and . We then have the partition function

(48)

Alternatively one can construct the transfer matrix in the direction perpendicular to the boundary. The Hamiltonian for this transfer matrix is determined by the bulk only and is independent of the boundary conditions. To write the partition function we need to construct states and in the space where the transfer matrix acts such that the partition function Eq. (48) is also equal to

(49)

The two different representations of the partition function above illustrate the notion of the boundary state. Cardy (2010)

V A general relationship between the conductance and static correlation functions

Let us recall the two difficulties with calculating the conductance of a junction of multiple interacting quantum wires, which we discussed in the Introduction. First, conductance is a property of an open quantum system. This means that we think of the wires emanating from the junction as being attached to reservoirs that can serve as sources and drains for the electrons passing through the molecular structure at the junction. To study the conductance, we either need to faithfully model the reservoirs in our theoretical description or assume that we have infinitely long wires. In the latter case, we basically send the reservoirs to infinity. An appropriate boundary condition is assumed at infinity, which does not enter the theory of the junction explicitly. For a numerical calculation of conductance, it then seems that we would need to model large enough systems that approximate the semi-infinite ones.

The second difficulty is that, in the standard linear-response framework, the linear conductance is related to dynamical rather than static correlation functions. This can be seen explicitly from the Kubo formula for the conductance tensor Oshikawa et al. (2006)

(50)

where indicates imaginary-time ordering and the quantity is a dynamical current-current correlation function for currents and on wires and , respectively. It seems that, generically, a numerical calculation of this quantity requires time-dependent methods such, as for example, time-dependent DMRG.

With a brute force approach, the numerical calculation of conductance is not feasible for junctions of three or more wires. In this section, we derive a key relationship between the universal conductance and certain static correlation functions in a finite system. This important relationship is the basis of a method for numerically calculating the conductance that we will discuss in Sec. VI.

v.1 CFT for independent wires

Let us begin by formulating the problem in the BCFT framework reviewed in Sec. IV. We have quantum wires connected to a junction. Let us first consider the system of independent quantum wires in the absence of the junction. As argued in Sec. IV.1, the effective Luttinger-liquid Hamiltonian for one wire can be written as

(51)

Using the fact that is the conjugate momentum of , the Lagrangian density can be written as

Let us set the charge carrier velocity to unity for convenience. We can then write the Euclidean action as

(52)

Alternatively the action can be written in terms of the dual field as . It is convenient to represent the points on the plane with complex coordinate . The system described by the massless action Eq. (52) is a CFT on the complex plane . Physically, covering the whole complex plane implies 1D quantum wires extending from to at zero temperature ().

Let us begin by calculating the correlation functions of the bosonic fields from the action Eq. (52). We have

(53)

where the propagator is the inverse of the operator , i.e.,

(54)

and is given by

(55)

Up to an unimportant additive constant, we can then write the correlation function of the bosonic fields as

(56)

By differentiating the above Eq. (56) with respect to and , we obtain the following correlation functions:

(57)
(58)

Also we can similarly show that vanishes.

The chiral currents below, where we have put in a wire index , are then primary fields for this CFT:

(59)

with the notation

In the absence of a boundary, the chiral currents in different wires are uncorrelated and the only correlations are between chiral currents of the same chirality in the same wire. The only nonvanishing correlation functions of the chiral currents are then