Phys. Rev. B 85, 045120 (2012) - General method for calculating the universal conductance of strongly correlated junctions of multiple quantum wires

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 $g$ and conjecture its functional dependence on $g$ .

21 More

Received 22 August 2011

DOI:

Authors & Affiliations

Armin Rahmani¹, Chang-Yu Hou², Adrian Feiguin³, Masaki Oshikawa⁴, Claudio Chamon¹, and Ian Affleck⁵

¹Department of Physics, Boston University, Boston, Massachusetts 02215, USA
²Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, NL-2300 RA Leiden, The Netherlands
³Department of Physics and Astronomy, University of Wyoming, Laramie, Wyoming 82071, USA
⁴Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
⁵Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C., Canada, V6T 1Z1

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Issue

Vol. 85, Iss. 4 — 15 January 2012

Reuse & Permissions

Access Options

Buy Article »
Get access through a U.S. public or high school library »
Log in with a username/password provided by your institution »

Article Available via CHORUS

Download Accepted Manuscript

International Year Of Light

The Physical Review Journals Celebrate the International Year of Light

The editors of the Physical Review journals revisit papers that represent important breakthroughs in the field of optics. The articles covered are free to read throughout 2015.

Images

Figure 1
A schematic illustration of the generic system that is the subject of this paper. We have $M$ quantum wires connected to a molecular system of certain structure and interactions. We have currents $I_{α}$ running in wires $α = 1, ..., M$ and voltages $V_{α}$ applied to the endpoints of the wires.Reuse & Permissions
Figure 2
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 $G_{α β}$ . The conductance is related to the ground-state ${〈 J_{R}^{α} (x) J_{L}^{β} (x) 〉}_{GS}$ correlation function of chiral currents in this finite system through Eq. (2).Reuse & Permissions
Figure 3
A schematic illustration of the Y junction numerically studied in Sec. 8. The M fixed point corresponds to the time-reversal-symmetric case $φ = 0, π$ .Reuse & Permissions
Figure 4
A simple noninteracting system of two wires constructed with a junction with hopping $t$ and mirror image with hopping $- t$ . The quantity of interest is the expectation value $〈 J^{1} (x) J^{2} (x) 〉$ with $J^{α}$ the current operator on wire $α$ .Reuse & Permissions
Figure 5
The single-particle scattering states for a single-impurity infinitely long system.Reuse & Permissions
Figure 6
The quantity $m^{2} 〈 J^{1} (m) J^{2} (m) 〉$ calculated from Eq. (24) by a numerical integration of the integral in Eq. (23). For large $m$ , this quantity saturates to a constant value that only depends on the hopping amplitude $t$ . This indicates that the asymptotic form of $〈 J^{1} (m) J^{2} (m) 〉$ is given by $\sim \frac{1}{m^{2}}$ .Reuse & Permissions
Figure 7
The degenerate ground states of the Hamiltonian Eq. (27) for $V \to \infty$ . Filled circles represent occupied sites.Reuse & Permissions
Figure 8
Boundary states in the partition function of a CFT on a cylinder.Reuse & Permissions
Figure 9
The conformal transformation from the upper half-plane to a strip of width $ℓ$ . The physical system described by the half-plane is a semi-infinite junction at zero temperature with the imaginary time running from $- \infty$ to $+ \infty$ and the position along the wires running from 0 to $+ \infty$ . The physical system corresponding to the strip is a finite system of length $ℓ$ at zero temperature. The structure of the junctions at the two endpoints of this finite system must give rise to the appropriate boundary conditions, consistent with the conformal mapping.Reuse & Permissions
Figure 10
Given the bulk and boundary Hamiltonians of a semi-infinite system, the goal is to obtain the two boundary Hamiltonians $H_{0}$ and $H_{ℓ}$ of a finite system corresponding to the conformal transformation to the strip.Reuse & Permissions
Figure 11
A generic finite system with two boundaries. For noninteracting fermions, the boundary conditions can be written in the form of scattering matrices that relate the incoming and outgoing states. The correspondence of these states with the chiral fermions is illustrated at both boundaries.Reuse & Permissions
Figure 12
A labeling of system sites for exact diagonalization.Reuse & Permissions
Figure 13
The inset shows the noninteracting Y junction used for the numerical test of the method. The thick bonds at the junction have a different hopping amplitude $t$ than the thin bonds in the bulk of the wires, the hopping amplitude of which is 1. The static correlation function is for a finite system of length $ℓ = 99$ (100 sites in each wire) for $Φ = 0$ and five different values of the hopping amplitude $t$ . The lines drawn through the data points show the CFT prediction with the exact conductance Eq. (91). The strong agreement with the numerical data confirms that our method, i.e., extracting the conductance from fitting the correlators, yields accurate results.Reuse & Permissions
Figure 14
The conductance of the noninteracting Y junction calculated with our method versus the exact solution.Reuse & Permissions
Figure 15
The correlation functions are plotted for two values of hopping at the impurity $t = 0.8$ and $0.6$ for the Luttinger parameter $g = 2.0$ . The strong agreement with the CFT prediction of $G = g \frac{e^{2}}{h}$ supports the correctness of our method.Reuse & Permissions
Figure 16
The correlation functions are plotted for two values of hopping at the impurity $t = 0.8$ and $0.6$ for the Luttinger parameter $g = 6.0$ . Here, a larger healing length of $ℓ = 90$ is needed to observe the universal behavior and verify the CFT prediction.Reuse & Permissions
Figure 17
The correlation functions are plotted for hopping at the impurity $t = 0.3$ and the Luttinger parameter $g = 0.65$ . Since the universal conductance vanishes, we only observe the nonuniversal subleading corrections.Reuse & Permissions
Figure 18
The conjectured RG flow diagram for the Y junction of Fig. 3, given in Ref. 46, for the Luttinger parameter in the range $1 < g < 3$ .Reuse & Permissions
Figure 19
A plot of the $\ln 〈 J_{R}^{1} (x) J_{L}^{2} (x) 〉$ for $g = 1.5$ and $2.0$ and two different hopping amplitudes. We observe the universality of the conductance and good agreement with the theoretical predictions.Reuse & Permissions
<">

Physical Review B

condensed matter and materials physics

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

Armin Rahmani, Chang-Yu Hou, Adrian Feiguin, Masaki Oshikawa, Claudio Chamon, and Ian Affleck

Phys. Rev. B 85, 045120 – Published 20 January 2012

Abstract

Authors & Affiliations

Article Text (Subscription Required)

References (Subscription Required)

Issue

Access Options

International Year Of Light

Authorization Required

Other Options

Images

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19

Physical Review B

condensed matter and materials physics

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

Armin Rahmani, Chang-Yu Hou, Adrian Feiguin, Masaki Oshikawa, Claudio Chamon, and Ian Affleck

Phys. Rev. B 85, 045120 – Published 20 January 2012

Abstract

Authors & Affiliations

Article Text (Subscription Required)

References (Subscription Required)

Issue

Access Options

International Year Of Light

Authorization Required

Other Options

Download & Share

Images

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19