Time-dependent correlation functions in open quadratic fermionic systems

We formulate and discuss the explicit computation of dynamic correlation functions in open quadratic fermionic systems which are driven and dissipated by Lindblad jump processes that are linear in canonical fermionic operators. Dynamic correlators are interpreted in terms of a local quantum quench where the pre-quench state is a non-equilibrium steady state, i.e. a fixed point of the Liouvillian. As an example, we study the XY spin 1/2 chain and quadratic Majorana chains with boundary Lindblad driving, whose dynamics exhibit asymmetric (skewed) light cone behaviour. We also numerically treat the two-dimensional XY model and the XY spin chain with additional Dzyaloshinskii–Moriya interactions. The latter exhibits a new non-thermal phase transition which can be understood in terms of bifurcations of the quasi-particle dispersion relation. Finally, considering in some detail the periodic quadratic fermionic ring with dissipation at a single (arbitrary) site, we present analytical expressions for first order corrections (in the strength of dissipation) to the spectrum and non-equilibrium steady state (NESS) correlation functions.

[1]  Chu Guo,et al.  Solutions for bosonic and fermionic dissipative quadratic open systems , 2017 .

[2]  Jian-Jun Dong,et al.  The A-Cycle Problem In XY model with Ring Frustration , 2017, 1703.00595.

[3]  B. Shao,et al.  Quantum speed limit of the evolution of the qubits in a finite XY spin chain , 2017 .

[4]  F. Essler,et al.  Quench dynamics and relaxation in isolated integrable quantum spin chains , 2016, 1603.06452.

[5]  J. Caux.  The Quench Action , 2016, 1603.04689.

[6]  J. Cardy,et al.  Quantum quenches in 1  +  1 dimensional conformal field theories , 2016, 1603.02889.

[7]  B. Shao,et al.  Quantum speed limit and a signal of quantum criticality , 2016, Scientific Reports.

[8]  M. Fagotti.  Local conservation laws in spin-12 XY chains with open boundary conditions , 2016, 1601.02011.

[9]  P. Zanardi,et al.  Dynamical response theory for driven-dissipative quantum systems , 2015, 1512.07860.

[10]  T. Prosen.  Matrix product solutions of boundary driven quantum chains , 2015, 1504.00783.

[11]  A. Läuchli,et al.  "Light-cone" dynamics after quantum quenches in spin chains. , 2014, Physical review letters.

[12]  P. Zanardi,et al.  Quantum information-geometry of dissipative quantum phase transitions. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  J. Ignacio Cirac,et al.  Noise-driven dynamics and phase transitions in fermionic systems , 2012, 1207.1653.

[14]  T. Prosen,et al.  A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains , 2012, 1203.0943.

[15]  M. Ganahl,et al.  Observation of complex bound states in the spin-1/2 Heisenberg XXZ chain using local quantum quenches. , 2011, Physical review letters.

[16]  T. Prosen,et al.  Transport properties of a boundary-driven one-dimensional gas of spinless fermions. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Tomaž Prosen,et al.  Nonequilibrium phase transition in a periodically driven XY spin chain. , 2011, Physical review letters.

[18]  M. Znidaric.  Solvable quantum nonequilibrium model exhibiting a phase transition and a matrix product representation. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  A. Dzhioev,et al.  Super-fermion representation of quantum kinetic equations for the electron transport problem. , 2010, The Journal of chemical physics.

[20]  T. Prosen,et al.  Quantization over boson operator spaces , 2010, 1007.2921.

[21]  T. Prosen,et al.  Explicit solution of the Lindblad equation for nearly isotropic boundary driven XY spin 1/2 chain , 2010, 1007.2922.

[22]  M. Znidaric.  Exact solution for a diffusive nonequilibrium steady state of an open quantum chain , 2010, 1005.1271.

[23]  Tomaz Prosen,et al.  Spectral theorem for the Lindblad equation for quadratic open fermionic systems , 2010, 1005.0763.

[24]  Tomaz Prosen,et al.  Exact solution of Markovian master equations for quadratic Fermi systems: thermal baths, open XY spin chains and non-equilibrium phase transition , 2009, 0910.0195.

[25]  K. Sengupta,et al.  ENTANGLEMENT PRODUCTION DUE TO QUENCH DYNAMICS OF AN ANISOTROPIC XY CHAIN IN A TRANSVERSE FIELD , 2009, 0904.1059.

[26]  Tomaz Prosen,et al.  Operator space entanglement entropy in XY spin chains , 2009, 0903.2432.

[27]  H. Schoeller,et al.  A perturbative nonequilibrium renormalization group method for dissipative quantum mechanics , 2009, 0902.1449.

[28]  Tomaz Prosen,et al.  Quantum phase transition in a far-from-equilibrium steady state of an XY spin chain. , 2008, Physical review letters.

[29]  P. Calabrese,et al.  Evolution of entanglement entropy following a quantum quench : Analytic results for the XY chain in a transverse magnetic field , 2008, 0804.3559.

[30]  T. Prosen.  Third quantization: a general method to solve master equations for quadratic open Fermi systems , 2008, 0801.1257.

[31]  A. Millis,et al.  Nonequilibrium quantum criticality in open electronic systems. , 2006, Physical review letters.

[32]  B. Doyon.  Finite-temperature form factors in the free Majorana theory , 2005, hep-th/0506105.

[33]  D. Karevski.  RELAXATION IN QUANTUM SPIN CHAINS: FREE FERMIONIC MODELS , 2004 .

[34]  G. Milburn,et al.  Quantum technology: the second quantum revolution , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[35]  A. Starace,et al.  Anisotropy and magnetic field effects on the entanglement of a two qubit Heisenberg XY chain. , 2002, Physical review letters.

[36]  A. Kitaev.  Unpaired Majorana fermions in quantum wires , 2000, cond-mat/0010440.

[37]  H. Umezawa,et al.  A Unified formalism of thermal quantum field theory , 1994 .

[38]  S. Hammarling.  Numerical Solution of the Stable, Non-negative Definite Lyapunov Equation , 1982 .

[39]  David E. Evans.  Irreducible quantum dynamical semigroups , 1977 .

[40]  G. Lindblad.  On the generators of quantum dynamical semigroups , 1976 .

[41]  H. Capel,et al.  On a soluble model of an antiferromagnetic chain with Dzyaloshinsky interactions. II , 1975 .

[42]  P. Mazur,et al.  Time correlation functions in the a-cyclic XY model. I , 1973 .

[43]  Eytan Barouch,et al.  Statistical Mechanics of the XY Model. III , 1970 .

[44]  E. Lieb,et al.  Two Soluble Models of an Antiferromagnetic Chain , 1961 .

[45]  T. Niemeijer.  SOME EXACT CALCULATIONS ON A CHAIN OF SPINS $sup 1$/$sub 2$. II. , 1968 .

[46]  T. Niemeijer.  Some exact calculations on a chain of spins {1}/{2} II , 1967 .