Quenching through Dirac and semi-Dirac points in optical lattices: Kibble-Zurek scaling for anisotropic quantum critical systems

Abstract

We propose that Kibble-Zurek scaling can be studied in optical lattices by creating geometries that support, Dirac, Semi-Dirac and Quadratic Band Crossings. On a Honeycomb lattice with fermions, as a staggered on-site potential is varied through zero, the system crosses the gapless Dirac points, and we show that the density of defects created scales as 1/τ , where τ is the inverse rate of change of the potential, in agreement with the Kibble-Zurek relation. We generalize the result for a passage through a semi-Dirac point in d dimensions, in which spectrum is linear in m parallel directions and quadratic in rest of the perpendicular (d − m) directions. We find that the defect density is given by 1/τ mν || z || +(d−m)ν ⊥ z ⊥ where ν || , z || and ν ⊥ , z ⊥ are the dynamical exponents and the correlation length exponents along the parallel and perpendicular directions, respectively. The scaling relations are also generalized to the case of non-linear quenching. The Kibble-Zurek (KZ) scaling [1,2,3,4,5,6] of defect density in the final state of a quantum many-body system following a slow passage across a quantum critical point, has been an exciting area of recent research. The KZ argument predicts that the scaling of the defect density is universal and is given as n ∼ 1/τwhere τ is the inverse rate of change of a parameter, d is the spatial dimension and ν and z are the correlation length and dynamical exponents, respectively, associated with the quantum critical point [7,8] across which the system is swept. Following the initial predictions, a plethora of theoretical studies have been carried out [10,11,12,13,14,15,16,17,18] to explore the defect generation and the entropy production using different quenching schemes across critical points [10], quantum multicritical points [18], gapless phases [13], along gapless lines [16], etc. The possibility of experimental observations in a spin-1 Bose condensate [19] and also on ions trapped in optical lattices [20,21] has provided a tremendous boost to the related theoretical studies.Here, we propose that Kibble-Zurek scaling can be studied in optical lattices with fermionic atoms by creating geometries that support Dirac, semi-Dirac and Quadratic Band Crossings. For example, a Honeycomb lattice consists of two interpenetrating triangular lattices. If the two sublattices are controlled separately, one can create a staggered on-site potential, which creates a gap in the spectrum [22,23]. We will call the Honeycomb lattice with a staggered potential a gapped Graphene Hamiltonian in analogy with Graphene [22]. The system can be loaded with atoms when one of the sublattices has a much lower energy than the other. As the staggered potential is varied through zero, the sys- * Electronic address: dutta@iitk.ac.in † Electronic address: singh@physics.ucdavies.edu tem will cross through gapless Dirac point, and the creation of defect density can be studied as a function of the rate of change of potential. Similarly, two interpenetrating square-lattices can lead to Quadratic Band Cro...

Adiabatic multicritical quantum quenches: Continuously varying exponents depending on the direction of quenching

“…The KZ argument predicts that the scaling of the defect density (n) in the final state is universal and is given by n ∼ 1/τ νd/(νz+1) where τ is the inverse rate of driving across a QCP with the correlation length and dynamical exponents ν and z, respectively, and d is the spatial dimension. The possibility of the experimental verification of Kibble-Zurek scaling (KZS) in a spin-1 Bose condensate [16], in ions trapped in optical lattices [17,18], and also in ultracold fermionic atoms in optical lattices [19,20] has paved the way for the above mentioned theoretical studies.Although, the KZS for quenching through a quantum critical point is well-understood; the scaling of the defect density following an adiabatic quantum quench across a quantum multicritical point (MCP) is relatively less studied. A non-KZS behavior (n ∼ 1/τ 1/6 ) of the density of defects (wrongly oriented spins) for quenching across the MCP of the spin-1/2 transverse XY chain was reported for the first time in reference [8] which was later explained in reference [21] introducing an effective dynamical exponent z 2 (= 3) for Jordan-Wigner solvable spin chains [22] reducible to a collection of decoupled two-level systems in the Fourier space and applying Landau-Zener (LZ) transition formula [23].…”

“…The KZ argument predicts that the scaling of the defect density (n) in the final state is universal and is given by n ∼ 1/τ νd/(νz+1) where τ is the inverse rate of driving across a QCP with the correlation length and dynamical exponents ν and z, respectively, and d is the spatial dimension. The possibility of the experimental verification of Kibble-Zurek scaling (KZS) in a spin-1 Bose condensate [16], in ions trapped in optical lattices [17,18], and also in ultracold fermionic atoms in optical lattices [19,20] has paved the way for the above mentioned theoretical studies.…”

One- and two-dimensional quantum models: Quenches and the scaling of irreversible entropy

“…The edge states in a 2D topological insulator are described by an effective 1D Dirac Hamiltonian, whereas its bulk states are described by a 2D Dirac Hamiltonian and the QCP separating the gapped to gapless phases is a 2D massless Dirac point. These models have turned out to be immensely useful in the studies of the Kibble-Zurek scaling 43 , sudden quenches 44,45 , fidelity susceptibility and thermodynamic fidelity 46 , Loschmidt echo 47 and periodic steady state reached through a sinusoidal variation of the mass term 48 . Considering the massive 2D Dirac Hamiltonian one can rescale the units appropriately to obtain the 2 × 2 Hamiltonian describing the system close to a single valley as,…”

Non-equilibrium dynamics of one-dimensional isolated quantum systems

“…Azonban a legkisebb energia különbség (legkisebb gap) nullához tart, ahogy a rendszer közelít a fázisátalakulási ponthoz,így a külső paraméter változása nem lesz a folyamat egész ideje alatt "elég lassú". A kérdés, hogy milyen messze lesz a lassú dinamikával kapottállapot az alapállapottól a fázisátalakulási pont keresztezése után, intenzív vizsgálatok tárgyát képezte [50,56,[58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74]. Kibbleés Zurek [58] [59] megadott egyösszefüggést, amely a kétállapot "távolságát" a P (τ ) ∼ (density of deffects)…”

“…However when the system reaches the critical point, the smallest gap goes to zero, and the variation of the Hamiltonian cannot be slow enough to remain in the instantaneous ground state. The question, how far is the described system from the instantaneous ground state, is target of extensive investigations in the literature [50,56,[58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74].…”