Abstract
We consider the circuit complexity of free bosons and free fermions in 1+1 dimensions. Motivated by the results of [1, 2, 3] who found different behavior in the complexity of free bosons and fermions, in any dimension, we consider the 1+1 dimensional case where, thanks to the bosonisation equivalence of the Hilbert spaces, we can consider the same state from both the bosonic and the fermionic perspectives. This allows us to study the dependence of the complexity on the choice of the set of gates, which explains the discrepancy. We study the effect in two classes of states: i) bosonic-coherent / fermionic- gaussian states; ii) states that are both bosonic- and fermionic-gaussian. We consider the complexity relative to the ground state. In the first class, the different complexities can be related to each other by introducing a mode-dependent cost function in one of the descriptions. The differences in the second class are more important, in terms of the structure of UV divergencies and the overall behavior of the complexity.
Highlights
Of complexity may be an important component in our understanding of the properties of emergent spacetime, and possibly point to a solution of the information loss paradox, by giving additional insight, beyond what can be obtained from entanglement entropy, into the information-theoretic properties associated to spacetime and in particular to the region inside black hole horizons
There are two different proposals: one takes the volume of a maximal spatial slice of the geometry [5], the other takes the action evaluated on the Wheeler-DeWitt patch [10, 11]
In the case of the entanglement entropy, one can compute it in a 2d CFT, for an interval of length, and obtain the famous exact result [17]
Summary
We will review the basics of the 2D bosonisation formalisms, for free bosons and fermions, which we will use in the rest of the paper. The bosonisation can be proved exactly for a system on a finite size interval [−L/2, L/2] In such a system, the unbounded momentum k satisfies k. If there are M chiral fermions with periodic conditions (δb = 0) in the system, an index η could be used to denote different types of fermions, and the mode decomposition for each fermion type is given as ψη(x) =. The bosonic chiral fields are given by the mode decomposition φη(x) = −. We can take (2.4) as a definition of the bosonic modes; the proof of bosonisation amounts to showing that, with this definition, the bosonic commutation relations are satisfied if the fermionic ones are:. The so-called Klein factor Fη in (2.7) has the role of compensating the mismatch in fermion number (this factor is often omitted in many presentations of bosonisation). We will consider the simplest case with only two chiral fermions and one chiral bosons, the species index will be omitted
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