Spectral Function of a Bosonic Ladder in an Artificial Gauge Field

: This is only the original submission, for the ﬁnal version download the ﬁle from the MDPI Open Access Journal Condensed Matter. We calculate the spectral function of a boson ladder in 2 an artiﬁcial magnetic ﬁeld by means of analytic approaches based on bosonization and Bogoliubov 3 theory. We discuss the evolution of the spectral function at increasing effective magnetic ﬂux, from the Meissner to the Vortex phase, focussing on the effects of incommensurations in momentum 5 space. At low ﬂux, in the Meissner phase, the spectral function displays both a gapless branch and 6 a gapped one, while at higher ﬂux, in the Vortex phase, the spectral functions display two gapless branches and the spectral weight is shifted at a wavevector associated to the underlying vortex spatial structure. While the Bogoliubov theory, valid at weak interactions, predicts sharp delta-like features in the spectral function, at stronger interactions we ﬁnd power-law broadening of the spectral functions due to quantum ﬂuctuations as well as additional spectral weight at higher momenta due to backscattering and incommensuration effects. These features could be accessed in ultracold atom 12 experiments using radio-frequency spectroscopy techniques. 13

to realize either artificial gauge fields and artificial spin orbit coupling have been put forward [14,15], 26 and an artificial spin-orbit coupling has been achieved in a cold atoms experiment [16]. A two leg boson 27 ladder under a flux is known to display a commensurate-incommensurate transition [1][2][3][4] between 28 a low flux commensurate Meissner-like phase and a high flux incommensurate vortex-like phase. 29 The transition has been characterized using equal time correlation functions [3,[17][18][19]. However, we expect a direct signature of the transition also in dynamical correlation functions. In one dimension, 31 the low energy modes are collective excitations [20,21], and in the two-leg ladder, there is a separation 32 between a total density ("charge") and a density difference ("spin") mode [2,4]. This is analogous to the 33 well-known spin charge separation in electronic systems [20] and two-component boson systems [22]. 34 Except at commensurate filling [23][24][25][26] the "charge" mode is gapless. By contrast, the "spin" mode 35 is gapped in the Meissner phase and gapless in the Vortex phase, the transition as a function of flux 36 being in the commensurate-incommensurate class [5,6]. Thus, the two phases are characterized by 37 very different dynamical correlation functions. Among those correlation functions, one could for 38 example consider the "spin-spin" dynamical structure factor. This would display a well defined in the Meissner state the minimum of the dispersion remains at q = 0. A particular feature of the 46 single-particle spectral function is that it is incoherent [22,28] i.e. the low energy excitation branches 47 emerge as power law singularities instead of delta function singularities. From the experimental 48 point of view, single-particle spectral functions are accessible via radiofrequency (RF) spectroscopy 49 techniques [29,30]. In the present paper, we calculate the boson spectral function in the different phases 50 of the boson ladder at incommensurate filling in order to fully characterize the transition under flux.

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In the following we use the notations and definitions of Ref. [19]. We consider a model of bosons 53 on a two-leg ladder in the presence of an artificial U(1) gauge field [13,31]: where σ =↑, ↓ represents the leg index or the internal mode index [32][33][34], b j,σ annihilates a boson on 55 leg σ on the j−th site, n jα = b † jα b jα , t is the hopping amplitude along the chain, Ω is the tunneling 56 between the legs or laser induced tunneling between internal modes, λ is the flux of the effective 57 magnetic field, U is the repulsion between bosons on the same leg. The low-energy effective theory 58 for the Hamiltonian (1), where Ω t is treated as a perturbation, is obtained by using Haldane's 59 bosonization.

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Hamiltonian as describes the antisymmetric density (or spin) fluctuations. In Eq.
The phase diagram of the Hamiltonian has been determined by looking at the behavior of the chiral 71 current, i.e. the difference between the currents in upper and lower leg, which is defined as As a function of the flux λ, the chiral current first increases linearly with λ while being in the Meissner 73 phase and above a critical value of λ it starts to decrease in the Vortex phase [2]. In this phase the 74 rung current starts to be different from zero. In Fig.1    and Ω, at the filling value n = 1. The black solid line that joins solid dots is the phase boundary between the Meissner and the Vortex phase, while the dashed red line is the prediction for this boundary in the non-interacting system. In the insets we show the different behavior of the spin-current J s (λ) for two values of interchain coupling Ω when there is the Meissner/Vortex transition and where there is not, respectively panel b) for Ω = 1.25 and a) for Ω = 1.75 DMRG simulation results at L = 64 in PBC.
Beyond the chiral and rung current the Meissner to Vortex phase transition can be traced out by 79 looking at the behavior of the spectral function which is more sensitive to incommensurations.

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For the case of lattice bosons the spectral function is defined as: and can be experimentally accessed by, e.g. via radiofrequency (RF) spectroscopy techniques [29,30].

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In the following we will focus on the positive-frequency part of the spectral function, given by the first

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Within the bosonization technique the boson annihilation operator to the lowest order approximation can be represented as: where a is the lattice spacing, A 0 is a non-universal constant and σ stands for ↑ in the upper chain and ↓ for the lower chain. Knowing the Green's function for the single particle operators b one gets the spectral function as: where α is the theory cutoff taken equal to the lattice spacing. The result of the integral yields The approximation (11) only yields the behavior of the spectral function at ω lower than the gap ∆ s in the θ s modes. The actual correlation function can be obtained from the Form factor expansion [37][38][39][40][41]. The lowest contribution, from a soliton-antisoliton pair yields

Spectral function in the Vortex phase for weak interchain hopping 94
In the vortex phase the boson field to the lowest order reads: where q 0 (λ) is the incommensurate wavevector of the vortex phase.

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Thus we find the Matsubara Green's function of the bosons in the form and the spectral function is obtained by the integral: where q 0 (λ) is absorbed into k. The Fourier transform of the Matsubara Green's function (12) can be 97 calculated by the method outlined in [22,28] and after analytic continuation iν → ω + i0 + it reads: where the function F 1 (a, b 1 , b 2 , c; z 1 , z 2 ) is an Appell hypergeometric function [42], which has a series 99 representation in terms of two complex variables z 1 and z 2 when |z 1 | < 1 and |z 2 | < 1.

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In order to wash out at least one of the divergencies near the two poles small values of K c < 1/2 are u s /u c =0.5 Figure 4. Schematic representation of the spectral function A σ (q, ω) as a function of ω and q for a fixed applied field λ inducing a finite q 0 (λ) for u * s /u c = 0.5. Finite spectral weights are present only in the colored region. In the blue region there is only the contribution from the singularity at ω 2 (q, λ), while in the green one the contribution from ω 1 (q, λ) adds up. 117 We adopt here an alternative bosonization scheme[4], valid at weak interactions but arbitrary 118 inter-leg tunnel coupling Ω. In this regime, one can bosonize starting from the exact single-particle 119 excitation spectrum [4] which displays a single minimum in the Meissner phase and two minima in the 120 Vortex phase. In the Meissner state, the result (9) is recovered, but by construction of the bosonization 121 scheme, the contribution of gapped modes at higher energy is not accessible.

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In the Vortex phase, at low energy the field operators are approximated as [4] where u Q and v Q are the single-particle amplitudes which diagonalize the non-interacting ladder 124 Hamiltonian, calculated at the minima ±Q of the lower branch dispersion relation, and β j± = 125 ∑ q e −iqja β q±Q with β k being the destruction operator of the lower single-particle excitation branch.
Then, the field operators are bosonized as β j± = √n e iθ ± (x j ) and the Luttinger liquid Hamiltonian 127 takes the usual quadratic form in the symmetric, antisymmetric sectors corresponding to the operators 128 θ s(a) = (θ + ± θ − )/ √ 2. The associated Luttinger parameters are called K s , v s K a , v a .

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The Green's function, calculated e. g. for the upper leg σ = 1/2 reads From bosonization we obtain while β j± (t)β † 0∓ ( where µ is the chemical potential. The field operator is approximated by where h σ νj and Q σ νj are the Bogoliubov mode wavefunctions with energy ω ν and γ ν are the quasiparticle 141 creation and destruction field operators, satisfying bosonic commutation relations (see [27] for the full 142 expressions).

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The spectral function in the Bogoliubov approximation is illustrated in Figure...

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We have obtained the spectral functions of a two-leg boson ladder in an artificial gauge field. The 147 bosonization approach, describing the regime of sufficiently strong interactions, predicts that in the it is located near ω = 0, ±q 0 (λ). In both cases, the spectral weight is incoherent and characterized 150 by power law singularities at ω = u c |q| (Meissner phase) or ω = u c |q ± q 0 (λ)| (Vortex phase) with 151 known exponents, and a specific incommensuration effect due to the shift of the spectral weight for