# zbMATH — the first resource for mathematics

Nonequilibrium transport through quantum-wire junctions and boundary defects for free massless bosonic fields. (English) Zbl 1331.82048
Summary: 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 $$\beta$$ and electric potential $$V$$, 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. 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 the literature, 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 and the expressions for FCS obtained here are compared with the Levitov-Lesovik formulae.

##### MSC:
 82C70 Transport processes in time-dependent statistical mechanics 82D77 Quantum waveguides, quantum wires 81Q37 Quantum dots, waveguides, ratchets, etc. 82D15 Statistical mechanical studies of liquids 81U05 $$2$$-body potential quantum scattering theory
Full Text:
##### References:
 [1] Affleck, I., Conformal field theory approach to the Kondo effect, Acta Phys. Pol. B, 26, 1869-1932, (1995) · Zbl 0966.81561 [2] Andrieux, D.; Gaspard, P.; Monnai, T.; Tasaki, S., The fluctuation theorem for currents in open quantum systems, New J. Phys., 11, 043014, (2009) [3] Araki, H.; Woods, E. J., Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas, J. Math. Phys., 4, 637-662, (1963) [4] Aschbacher, W.; Jakšic, V.; Pautrat, Y.; Pillet, C.-A., Transport properties of quasi-free fermions, J. Math. Phys., 48, 032101, (2007) · Zbl 1137.82331 [5] (Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, (1972), Dover) · Zbl 0543.33001 [6] Alekseev, A.; Schomerus, V., D-branes in the WZW model, Phys. Rev. D, 60, 061901, (1999) [7] Bachas, C.; de Boer, J.; Dijkgraaf, R.; Ooguri, H., Permeable conformal walls and holography, J. High Energy Phys., 0206, 027, (2002) [8] Bhaseen, M. J.; Doyon, B.; Lucas, A.; Schalm, K., Far from equilibrium energy flow in quantum critical systems [9] Bellazzini, B.; Mintchev, M.; Sorba, P., Bosonization and scale invariance on quantum wires, J. Phys. A, 40, 2485-2508, (2007) · Zbl 1113.81113 [10] Bernard, D.; Doyon, B., Energy flow in non-equilibrium conformal field theory, J. Phys. A, Math. Theor., 45, 362001, (2012), (8 pp.) · Zbl 1255.81212 [11] Bernard, D.; Doyon, B., Non-equilibrium steady-states in conformal field theory, Ann. Henri Poincaré, 16, 113-161, (2015) · Zbl 1312.82012 [12] Bernard, D.; Doyon, B., Time-reversal symmetry and fluctuation relations in non-equilibrium quantum steady states, J. Phys. A, Math. Theor., 46, 372001, (2013), (11 pp.) · Zbl 1278.82036 [13] Bernard, D.; Doyon, B.; Viti, J., Non-equilibrium conformal field theories with impurities, J. Phys. A, Math. Theor., 48, 05FT01, (2015), (16 pp.) · Zbl 1307.81059 [14] Blanter, Ya. M.; Büttiker, M., Shot noise in mesoscopic conductors, Phys. Rep., 336, 1-166, (2000) [15] Bruneau, L.; Jakšic, V.; Pillet, C.-A., Landauer-Büttiker formula and Schrödinger conjecture, Commun. Math. Phys., 319, 501-513, (2013) · Zbl 1277.82055 [16] Datta, S., Electronic transport in mesoscopic systems, (1995), Cambridge University Press [17] Dereziński, J.; Gérard, Ch., Mathematics of quantization and quantum fields, (2013), Cambridge University Press · Zbl 1271.81004 [18] Doyon, B., Lower bounds for ballistic current and noise in non-equilibrium quantum steady states, Nucl. Phys. B, 892, 190-210, (2015) · Zbl 1328.82042 [19] Doyon, B.; Hoogeveen, M.; Bernard, D., Energy flow and fluctuations in non-equilibrium conformal field theory on star graphs, J. Stat. Mech., P03002, (2014) [20] Fendley, P.; Ludwig, A. W.W.; Saleur, H., Exact nonequilibrium transport through point contacts in quantum wires and fractional quantum Hall devices, Phys. Rev. B, 52, 8934-8950, (1995) [21] Gaberdiel, M., Boundary conformal field theory and D-branes, lectures notes, (2003) [22] Giamarchi, T., Quantum physics in one dimension, (2003), Oxford University Press [23] K. Gawȩdzki, C. Tauber, unpublished. [24] Gutman, D. B.; Gefen, Y.; Mirlin, A. D., Bosonization out of equilibrium, Europhys. Lett., 90, 37003, (2010) [25] Gutman, D. B.; Gefen, Y.; Mirlin, A. D., Full counting statistics of Luttinger liquid conductor, Phys. Rev. Lett., 105, 256802, (2010) [26] Ishii, H., Direct observation of Tomonaga-Luttinger-liquid state in carbon nanotubes at low temperatures, Nature, 426, 540-544, (2003) [27] Kane, C. L., Lectures on bosonization, (2005), Boulder School for Condensed Matter and Materials Physics [28] Klich, I., Full counting statistics: an elementary derivation of Levitov’s formula, (Nazarov, Yu. V., Quantum Noise in Mesoscopic Physics, (2003), Springer), 397-402 [29] Levitov, L. S.; Lesovik, G. B., Charge distribution in quantum shot noise, JETP Lett., 58, 230-235, (1993) [30] Levitov, L. S.; Lee, H. W.; Lesovik, G. B., Electron counting statistics and coherent states of electric current, J. Math. Phys., 37, 4845-4866, (1996) · Zbl 0868.60099 [31] Mintchev, M., Non-equilibrium steady states of quantum systems on star graphs, J. Phys. A, Math. Theor., 44, 415201, (2011) · Zbl 1269.82076 [32] Mintchev, M.; Sorba, P., Luttinger liquid in non-equilibrium steady state, J. Phys. A, Math. Theor., 46, 095006, (2013) · Zbl 1267.82152 [33] Ngo Dinh, S.; Bagrets, D. A.; Mirlin, A. D., Nonequilibrium functional bosonization of quantum wire networks, Ann. Phys., 327, 2794-2852, (2012) · Zbl 1254.81105 [34] Oshikawa, M.; Chamon, C.; Affleck, I., Junctions of three quantum wires and the dissipative Hofstadter model, Phys. Rev. Lett., 91, 206403, (2003) [35] Oshikawa, M.; Chamon, C.; Affleck, I., Junctions of three quantum wires, J. Stat. Mech., 0602, P02008, (2006) · Zbl 07078078 [36] Nayak, C.; Fisher, M. F.A.; Ludwig, A. W.W.; Lin, H. H., Resonant multilead point-contact tunneling, Phys. Rev. B, 59, 15694-15704, (1999) [37] Oshikawa, M.; Affleck, I., Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line, Nucl. Phys. B, 495, 533-582, (1997) · Zbl 0933.82007 [38] Polchinski, J., String theory, vol. I: an introduction to the bosonic string, (2005), Cambridge University Press · Zbl 1075.81054 [39] Rahmani, A.; Hou, C.-Y.; Feiguin, A.; Chamon, C.; Affleck, I., How to find conductance tensors of quantum multi-wire junctions through static calculations: application to an interacting Y junction, Phys. Rev. Lett., 105, 226803, (2010) [40] Rahmani, A.; Hou, C.-Y.; Feiguin, A.; Oshikawa, M.; Chamon, C.; Affleck, I., General method for calculating the universal conductance of strongly correlated junctions of multiple quantum wires, Phys. Rev. B, 85, 045120, (2012) [41] Ruelle, D., Natural nonequilibrium states in quantum statistical mechanics, J. Stat. Phys., 98, 57-75, (2000) · Zbl 0988.82032 [42] Sassetti, M.; Kramer, B., Quantum wires as Luttinger liquids: theory, (Kramer, B., Advances in Solid State Physics, vol. 40, (2000), Springer), 117-132 [43] Sénéchal, D., An introduction to bosonization, (Sénéchal, D.; Tremblay, A.-M.; Bourbonnais, C., Theoretical Methods for Strongly Correlated Electrons, (2004), Springer), 139-186 [44] Sims, C. C., Computation with finitely presented groups, (1994), Cambridge University Press · Zbl 0828.20001 [45] (Stone, M., Bosonization, (1994), World Scientific Singapore) [46] Voit, J., One-dimensional Fermi liquids, Rep. Prog. Phys., 58, 977-1116, (1995) [47] Whittaker, E. T.; Watson, G. N., A course of modern analysis, (1990), Cambridge University Press · Zbl 0108.26903 [48] Witten, E., Non-abelian bosonization in two dimensions, Commun. Math. Phys., 92, 455-472, (1984) · Zbl 0536.58012 [49] Wong, E.; Affleck, I., Tunneling in quantum wires: a boundary conformal field theory approach, Nucl. Phys. B, 417, 403-438, (1994) · Zbl 1009.81596
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.