PHYSICAL REVIEW D 98, 126001 (2018)

Circuit complexity in fermionic theory

† ‡ Rifath Khan,* Chethan Krishnan, and Sanchita Sharma Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India

(Received 15 May 2018; published 4 December 2018)

We define and calculate versions of complexity for free fermionic quantum field theories in 1 þ 1 and 3 þ 1 dimensions, adopting Nielsen’s geodesic perspective in the space of circuits. We do this both by discretizing and identifying appropriate classes of Bogoliubov-Valatin transformations, and also directly in the continuum by defining squeezing operators and their generalizations. As a closely related problem, we consider cMERA tensor networks for : viewing them as paths in circuit space, we compute their path lengths. Certain ambiguities that arise in some of these results because of cutoff dependence are discussed.

DOI: 10.1103/PhysRevD.98.126001

I. INTRODUCTION definition of complexity, even though clearly it involves some arbitrary choices: what is a good reference state? The formation of black holes via gravitational collapse in What are good choices for the gate unitaries?….1 For our anti–de Sitter space is expected to be dual to thermalization purposes in this paper, it is worth noting that this mini- in the dual conformal field theory. This leads one to think of mization of the number of gates can be reinterpreted ala` a thermal Conformal Field Theory (CFT) state as a Nielsen as a geodesic length minimization in the space of gravitational configuration that can be approximated by circuits [3]. an eternal black hole. If this is so, one needs to have a CFT To make a sensible holographic definition of complexity explanation for the fact that the Einstein-Rosen bridge (or from the CFT, one needs to generalize these constructions wormhole) that shows up inside the horizon of an eternal in three substantive ways. First, we need a definition that black hole is a time-dependent configuration, and that its generalizes these issues to the setting of quantum field “size” increases towards the future. It has recently been theory rather than in the setting of qubits and 0 þ 1- proposed by Susskind and others [1] that the quantity one dimensional quantum mechanics, as is usual. Second, since should compare against the size of the Einstein-Rosen “ ” at least in known concrete examples of holography the wormhole is the complexity of the CFT state: the idea boundary theory has a gauge invariance, it seems necessary being that the complexity of a state can increase even after that we will need to define complexity in the context of it has thermalized in some appropriate sense. gauge theories, in particular non-Abelian gauge theories. The trouble however is that the definition of complexity Third, perhaps after some field theory constructions have is often very murky. Loosely it captures the number of gates been undertaken for the free theory, one would need to required to prepare a state. One way to make things a bit “ ” come up with a way to make such a definition viable, for more concrete is to consider circuit complexity, where strongly coupled theories. After all, weakly coupled gravi- one assigns complexity to quantum states as follows. First tational physics is captured by a strongly coupled field one picks a reference state and a set of unitary operators that theory. are called gates. Then the complexity of any particular state Recently, two related but different attempts at the first of is captured by the minimal number of gates that one must these problems was made in the context of scalar field act with on the reference state in order to get to that theories in [4,5]. In this paper, we will consider both these particular state. This is believed to be a fairly reasonable approaches and adapt and generalize these approaches to include various classes of fermionic field theories. One of our motivations for this undertaking is the recent emer- *[email protected] † [email protected] gence of a class of strongly coupled fermionic gauged ‡ [email protected] quantum mechanical theories called Sachdev-Ye-Kitaev (SYK)-like tensor models, as well as their generalizations Published by the American Physical Society under the terms of to higher dimensions which are bona fide field theories. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1Note also that this definition is usually implemented in and DOI. Funded by SCOAP3. discrete qubit systems and often with discrete time evolution [2].

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Of course, this paper does not deal with neither gauge to a discussion of the meaning of a cutoff dependent theories nor strongly coupled systems: it should be viewed quantity like the complexity, and to speculations on the as a preliminary exploratory attempt. We hope that our possibility that subleading terms in complexity could be of explicit calculations will be useful in shedding some light physical interest, in analogy with (holographic) entangle- on the questions at the intersection of holography and ment entropy. In particular, these UV divergences could be complexity, and perhaps in sharpening them. reinterpreted as AdS IR divergences in a holographic The structure of the calculation in [4] was to discretize context. It must be born in mind here that we are dealing spacetime and then work effectively with quantum with free quantum field theories in our work. mechanical wave functions for the discretized oscillators. In a concluding section we make various speculative By working with a class of Gaussian wave functions that comments. In particular, we discuss the possibility of doing interpolated between the decoupled reference state and the similar calculations for gauge theories and perturbative true ground state (chosen as the target state) they were able string theory. Especially since the world sheet theory is free to define complexity for the state by minimizing the after gauge fixing, the latter is tractable and is currently Nielsen-like path length in the space of unitaries that did under investigation. We also briefly allude to some issues such an interpolation. Since we are working with fermions, which were suppressed in the main text, the choice of an analogue of the Gaussian wave function will involve metric and penalty factors in Nielsen’s geometric definition Grassmann variables and will be too awkward for many among others. Various Appendices contain some review purposes. But we will see that it is easy enough to work material as well as technical details. directly with (finite dimensional) fermionic Hilbert spaces that contain the reference and target states, and that they II. LATTICE OF FERMIONIC OSCILLATORS lead to natural notions of circuit length minimization. The Lets us start with the Dirac Lagrangian in d þ 1 slight subtlety due to doubling in the lattice will not spacetime dimensions: affect the main points we make. As a result we find an ¯ expression for the complexity, which is controlled by the L ¼ Ψðiγ:∂ − mÞΨ: ð2:1Þ formula in (2.17), and which shows up in various guises in the various cases we consider. This fermionic form is The conjugate momentum is distinct from the form of complexity in [4], where the ∂L analogous expression is a log instead of our arctan. Π ¼ ¼ iΨ† ð2:2Þ On the continuum side, we will find that a construction ∂ð∂0ΨÞ entirely parallel to that of [5] for scalars is possible for 1 1 and the Dirac Hamiltonian is Dirac fermions in þ dimensions. The strategy of using Z squeezing operators and their generalizations as entangler d − Ψ¯ γi∂ Ψ ΨΨ¯ operators turns out to be a viable strategy for fermions as H ¼ d x½ i i þ m : ð2:3Þ well. Our analysis of the 1 þ 1-dimensional continuum Dirac case works entirely parallel to the results for the We regulate it by placing it on a lattice with lattice spacing δ scalar in [5]. Instead of the metric on the hyperbolic plane, and the Hamiltonian becomes 1 we discover the metric on a CP sphere, and instead of an X X Ψðn⃗þ xˆ Þ − Ψðn⃗Þ SUð1; 1Þ isometry, we find an SUð2Þ. We further generalize δd − Ψ¯ ⃗ γi i H ¼ i ðnÞ δ our approach to Majorana theory in 1 1 dimensions, as ⃗ i þ n well as to massive and massless Dirac theories in 3 1 þ ¯ dimensions. In all these cases we are able to identify þ mΨðn⃗ÞΨðn⃗Þ : ð2:4Þ convenient squeezing operators that take us to natural target 2 states that approximate the ground state. The xi here are “unit” vectors along the axes of the In the final section before the conclusion, we discuss the (d-dimensional) lattice. The summation over i is only over cMERA tensor network for fermions, as an alternate path in the spatial directions, and therefore range over d values. the space of unitaries. We first do this for 1 þ 1 dimensions By introducing ω ≡ δdm and Ω ≡ δd−1 we can bring it to and then move on to 3 þ 1 dimensions (where we introduce the form a new cMERA-like path), both in the massive and massless Xh X i cases. In both cases we find that as the cutoff tends to ¯ i ¯ −iΩΨðn⃗Þ γ (Ψðn⃗þ xˆiÞ − Ψðn⃗Þ) þ ωΨðn⃗ÞΨðn⃗Þ : infinity, the state tends to the ground state. Since when the n⃗ i cutoff is finite the target state for the cMERA is not quite the ð2:5Þ target state of the previous paragraph (even though they both tend to the ground state as Λ → ∞), it is not possible to meaningfully compare the lengths of the paths at finite 2The quotes around the unit emphasize that the length of the cutoff by looking at their leading divergences. This leads us vector is the lattice spacing.

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† Note that this is a system of coupled fermionic oscillators, We will define complex Grassmann variables bi and bi , but since the coupling is quadratic, we can solve the system ψ − − ψ þ ψ − ψ þ i ffiffiffii i † i þffiffiffii i completely by identifying the normal modes. We will see bi ¼ p bi ¼ p ; ð2:10Þ various versions of this general idea throughout this paper. 2 2 and quantize by imposing the anticommutation relations † δ A. A toy model fbi;bj g¼ ij, see Appendix A for our conventions on The above form of the Hamiltonian suggests that we fermionic oscillators. The Hamiltonian takes the form should study a set of coupled oscillators of the form † † † † H ¼ ω½b1;b1þω½b2;b2 − iΩðb1b2 þ b1b2Þ: ð2:11Þ ¯ ¯ ¯ 1 H ¼ ωðΨ1Ψ1 þ Ψ2Ψ2ÞþiΩΨ1ρ ðΨ1 − Ψ2Þ: ð2:6Þ To bring this to the diagonalized normal mode form, we use the standard technology of Bogoliubov-Valatin (B-V) The basic idea is this: we wish to consider the simplest transformations (see Appendix B for a self-contained oscillator system which has a coupling analogous to the review). The result is discretization (2.5) of the field theory. A candidate is the ˜ † ˜ ˜ † ˜ system with two lattice points parallel to the work for H ¼ λ½b1; b1þλ½b2; b2; ð2:12Þ scalars considered in [4]. But if we work with d þ 1- dimensional Dirac fermions, the coupling term will force us where to consider the d spatial directions. We wish to avoid this 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ 4ω2 Ω2 technical complication, and therefore in writing the above, ¼ 2 þ : ð2:13Þ we have restricted ourselves to a 1 þ 1-dimensional theory. ’ ˜’ After we gain some intuition by working with this system, The b s and b s are related by we will consider more general cases. In particular, we will ˜ † ˜ † b1 ¼ ηb2 þ iρb ; b2 ¼ ηb1 − iρb ; ð2:14Þ see that the above Hamiltonian (2.6), when interpreted as a 1 2 theory for Majorana fermions, is essentially a precise where ρ and η are given in Eq. (2.16). parallel to the bosonic two oscillator case discussed in ρa [4]. We have adopted the notation for in B. Complexity of the toy model ground state two dimensions in this subsection. In two dimensions, gamma matrices allow a purely A B-V transformation matrix that does the job of imaginary representation in which the can be taken diagonalizing the Hamiltonian in the previous section is 2 3 as Majorana. The gamma matrices take the form 0 η iρ 0 6 7 0 − 0 6 η 00−iρ 7 0 i 1 i T ¼ 6 7; ð2:15Þ ρ ¼ ; ρ ¼ : ð2:7Þ 4 − ρ 00 η 5 i 0 i 0 i 0 iρη 0 The general Majorana in two dimensions Ψ (here i i where is part of the name of the spinor and not a spinor index) can sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi be written as 1 ω 1 ω ρ ¼ − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; η ¼ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: ψ − 2 4ω2 þ Ω2 2 4ω2 þ Ω2 Ψ ¼ i ; ð2:8Þ i ψ þ i ð2:16Þ

ψ This form suggests that we define where the i are real Grassmann variables. This means that our minimal Hamiltonian (2.6) is made of four real 1 Ω ≡ −1 Majorana fermions and takes the form r 2 tan 2ω ; ð2:17Þ

þ − þ − þ þ − − so that H ¼ 2iωðψ 1 ψ 1 þ ψ 2 ψ 2 ÞþiΩðψ 1 ψ 2 − ψ 1 ψ 2 Þ: ð2:9Þ ρ ¼ sin r; and η ¼ cos r: ð2:18Þ Note that the Hamiltonian is real (Hermitian) because the fermions are Majorana and anticommute. Now we will To define complexity, we need to define a reference state rewrite this Hamiltonian in a form that is useful for future from which we reach the target states via appropriate generalizations.3 unitary transformations. A natural reference state is to choose j00i, defined via 3 Note that because the fermions are Majorana, we can use 00 0 1 2 Hermitian conjugation and transposition interchangeably. bij i¼ ; for i ¼ ; ; ð2:19Þ

126001-3 KHAN, KRISHNAN, and SHARMA PHYS. REV. D 98, 126001 (2018) and the target state which is the ground state of our theory is Notice that the middle two components of the state vector defined by are zero for both reference and target state. If these two components are changed along the path then we need to ˜ ˜ ˜ bij0 0i¼0; for i ¼ 1; 2: ð2:20Þ bring them back down to zero, when we get to the target state. So it stands to reason that (for appropriately defined From the B-V matrix, it follows that Euclidean notions of distance in the space of unitaries) this would only increase the length, so we restrict ourselves to ˜ ˜ j0 0i¼cos rj00iþi sin rj11i: ð2:21Þ the path which only changes the first and fourth compo- nents. These transformations clearly fall inside a Uð2Þ To go from the reference state to the target state by a unitary group: transformation, we first write the states in matrix form with the basis jiji. The unitary transformation is a 4 × 4 unitary matrix: cos r 1 ¼ Uð2×2Þ : ð2:23Þ 2 3 2 3 i sin r 0 cos r 1 6 7 6 7 6 0 7 6 0 7 6 7 ¼ Uð4×4Þ6 7: ð2:22Þ ð2×2Þ 2 2 4 0 5 4 0 5 Now U is a × unitary matrix. After extracting a Uð1Þ phase eiy and writing the remaining SUð2Þ as an S3, i sin r 0 we can parametrize a general Uð2Þ matrix as

eiy cos ρ cos τ þ ieiy sin θ sin ρ eiy cos θ sin ρ þ ieiy cos ρ sin τ U ¼ : ð2:24Þ ieiy cos ρ sin τ − eiy cos θ sin ρ eiy cos ρ cos τ − ieiy sin θ sin ρ

4 The ranges of the various coordinates are i 0 0 i O0 ¼ ;O1 ¼ ; θ τ ∈ 0 2π ρ ∈ 0 π 2 0 i i 0 y; ; ½ ; Þ; and ½ ; = : ð2:25Þ 01 i 0 For finite value of the parameter along the path, the state O2 ¼ ;O3 ¼ : ð2:29Þ −10 0 −i will be

jΨðσÞi ¼ UðσÞjRi; ð2:26Þ Using where jRi is the reference state. This means that in the TrðO O Þ¼−2δ ; ð2:30Þ explicit matrix above, we treat ρ, τ, θ, y as functions of σ. a b ab We can view this unitary as a path-ordered exponential R we can extract the velocities via σ I ⃖ Y ðsÞOI UðσÞ¼Pe 0 ð2:27Þ 1 I − (∂ )( −1 ) if we write5 Y ðsÞ¼ 2 Tr½ sUðsÞ U ðsÞOI : ð2:31Þ

YIðsÞO ¼ (∂ UðsÞ)U−1ðsÞ: ð2:28Þ I s Using these, we define the length of the path (using a Euclidean Metric) as Here the OI are the generators of Uð2Þ, and they can be taken as Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 I J D½u¼ ds δIJY ðsÞY ðsÞ: ð2:32Þ 4We will work with y ∈ ½−π; πÞ when we want to go across the 0 y ¼ 0 point without changing charts. 5 This equation is the analogue of the time-dependent Schro- The path taken by the states will be a curve in y, ρ, τ, θ dinger equation, written in a form that is usually written when one I’ solves it via time-ordered exponentials. The left-hand side is the coordinates. The Y s will be linear in derivatives of them. analogue of the Hamiltonian. So the solution, (2.27), can be Explicitly calculating this metric, we find the natural metric directly exported here by analogy. To write (2.28), we merely on Uð1Þ × SUð2Þ: note that since the right-hand side is made from a general Uð2Þ matrix, the left-hand side must be writable in terms of the 2 2 ρ θ2 ρ2 2 ρ τ2 2 generators of Uð2Þ whose coefficients we call YI. ds ¼ sin ð Þd þ d þ cos ð Þd þ dy : ð2:33Þ

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Extremizing this length gives us the geodesic equations conditions is globally minimized for ϕ ¼ 0,6 and the corresponding geodesic is given by y00 ¼ 0 0 ρ 0 ρ00 − sin ρ cos ρθ02 þ sin ρ cos ρτ02 ¼ 0 yðsÞ¼ ; ðsÞ¼ ; τ τ θ τ00 − 2 tan ρρ0τ0 ¼ 0 ðsÞ¼ 1s; ðsÞ¼unfixed: ð2:42Þ θ00 þ 2 cot ρθ0ρ0 ¼ 0; ð2:34Þ The length of this minimum path can be directly calculated and the result is where the derivatives are with respect to the curve param- Z ∂ ∂ ∂ 1 1 Ω eter. Looking at the metric it is evident that ∂y, ∂θ, ∂τ are the −1 D½U¼ dsτ1 ¼ r ¼ tan : ð2:43Þ killing vectors. Using these one sees that 0 2 2ω 2 2 y0 ¼ const; cos ρτ0 ¼ const; sin ρθ0 ¼ const This minimum length is the complexity of the target state. I ð2:35Þ In the analogous discussion in [4], the generators (the M in their notation) used were nonstandard and that resulted in are the constants of motion. It is easy to check that this a more complicated form of the metric and resulting same result can be obtained by directly integrating the geodesic equations. To solve the geodesic equations, more relevant equations of motion. Killing vectors were identified. Here on the other hand, we Now we can use the boundary conditions to solve the explicitly see the S3 × R form of the metric, and we only system. Demanding needed to use the obvious Killing vectors to find the explicit solution. Uðs ¼ 0Þ¼I ð2:36Þ immediately yields C. Squeezing operators as gates In this section, we will present an alternate approach for ( 0 ρ 0 τ 0 θ 0 ) 0 0 0 θ yðs ¼ Þ; ðs ¼ Þ; ðs ¼ Þ; ðs ¼ Þ ¼ð ; ; ; 0Þ: discussing complexity, which has a natural role in the ð2:37Þ continuum case. We will define an entangling operator K, which is also known as squeezing operator in some Furthermore, we know that contexts: † † − ˜ † ˜ † − ˜ ˜ jψ Ti¼Uðs ¼ 1ÞjRi: ð2:38Þ K ¼ b2b1 b2b1 ¼ b1b2 b1 b2 : ð2:44Þ

Here jψ Ti is the target state (and our target state is the The first equality can be viewed as the definition of the ground state). A 2 × 2 unitary matrix that takes the operator, the second equality is the result of a calculation, reference state to the target state is of the general form where we have used the definition of the tilde’d operators via the B-V transformation from the previous subsection. cos risin re−iϕ It is also useful to define a unitary Uðs ¼ 1Þ¼ i sin r cos re−iϕ U ¼ e−iKr; ð2:45Þ iϕ=2 −iϕ=2 − ϕ 2 cos re i sin re ¼ e i = ð2:39Þ ∈ R 0˜ 0˜ i sin reiϕ=2 cos re−iϕ=2 where r . The target state j i can be reached from the reference state j00i via ϕ ∈ −π π for some arbitrary ½ ; Þ. This translates to the end 0˜ 0˜ 00 point boundary condition in terms of coordinates j i¼Uj ið2:46Þ for some appropriately chosen r. It is easy to see this via a (yðs ¼ 1Þ; ρðs ¼ 1Þ; τðs ¼ 1Þ; θðs ¼ 1Þ) ¼ðy1; ρ1; τ1; θ1Þ; ˜ similarity transformation of bi using U that gives bi: ð2:40Þ † ˜ iKr ˜ −iKr U b1U ¼ e b1e where 2 3 1 ˜ − ˜ † − r ˜ ir ˜ † 4 ˜ ¼ b1 irb2 2 b1 þ 3! b2 þ 4! r b1 y1 ¼ −ϕ=2; τ1 ¼ r; ρ1 ¼ ϕ=2; θ1 ¼ π=2 − r: ˜ ˜ † ¼ b1 cos r − ib sin r: ð2:47Þ ð2:41Þ 2

Because the metric is Euclidean, one can convince oneself 6Note that when this happens, θ is no longer determinate, that the distance with the above initial and final boundary including at the boundaries.

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† ˜ Setting U b1U ¼ b2, we see that the transformation U does approach to the definition of complexity. In the present indeed take the reference state to the target state that we case, we see that the two approaches match. have worked with in the previous sections if we take the value of r to be what we found before. D. 1 + 1-dimensional Majorana on a lattice Now let us consider an arbitrary path generated by the So far we have considered just a pair of fermionic squeezing operator which takes the reference state to some oscillators, albeit with a coupling that was motivated by more general target state: our eventual interest in field theory. Now we will consider discretized versions of the field theory, and consider full Ψ σ U σ 00 e−iKYðσÞ 00 ; : j ð Þi ¼ ð Þj i¼ j i ð2 48Þ lattices, with periodic and antiperiodic boundary conditions. We will work with Majorana fermions for concreteness. such that The Hamiltonian of 1 þ 1-d Majorana theory in terms 0 0 ⇒ Ψ 0 00 of the complexified variables is (see Sec. III E for a Yð Þ¼ j ð Þi ¼ j i derivation): Yð1Þ¼r ⇒ jΨð1Þi¼j0˜ 0˜i: ð2:49Þ Z † † † H ¼ dxð−iΨ∂1Ψ − iΨ ∂1Ψ þ m½Ψ ; ΨÞ: ð2:55Þ We evaluate the length of this path using the Fubini-Study (FS) metric and minimizing it to get the complexity. In 8 δ other words, the allowed circuits we are considering are the By placing it on a circular lattice with lattice spacing , the ones generated by K. Hamiltonian for N oscillators is The Fubini-Study metric7 is XN−1 Ψ − Ψ Ψ† − Ψ† qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ − Ψ ð nþ1 nÞ − Ψ† ð nþ1 nÞ H ¼ i n δ i n δ σ σ ∂ Ψ σ 2 − Ψ σ ∂ Ψ σ 2 0 dsFSð Þ¼d j σj ð Þij jh ð Þj σj ð Þij : ð2:50Þ n¼ Ψ† Ψ The circuit length of a path traced by intermediate states þ m½ n; n : ð2:56Þ jΨðσÞi ¼ UðσÞj00i is Z We can quantize by imposing the anticommutation rela- σ l Ψ σ f σ tions ðj ð ÞiÞ ¼ dsFSð Þ: ð2:51Þ σ i † † † fΨn; Ψmg¼δnm fΨn; Ψmg¼0 ¼fΨn; Ψmg: ð2:57Þ The complexity is the length of minimal path: Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The Hamiltonian will be written as 1 2 C ¼ min lðds Þ¼ min dσ ð∂σYðσÞÞ : ð2:52Þ −1 FS σ XN fYðsÞg fYð Þg 0 ω Ψ† Ψ − Ψ Ψ Ψ†Ψ† H ¼ ð ½ n; n ið n nþ1 þ n nþ1ÞÞ; ð2:58Þ This gives the geodesic to be a straight line path. Under a n¼0 simple affine parametrization σ the geodesic is where ω ¼ mδ. We will look at the lattice with periodic σ σ (Ramond) and antiperiodic (Neveu-Schwarz) boundary Yð Þ¼ r: ð2:53Þ conditions. The complexity for the target state is the length of this straight line path 1. Ramond boundary condition Z qffiffiffiffiffiffiffiffi The Ramond boundary condition is imposed by 1 σ 2 C ¼ d ðrÞ ¼ r: ð2:54Þ Ψ Ψ 0 nþN ¼ n: ð2:59Þ

This matches with the complexity derived in the previous The discrete Fourier transform for this boundary condition section. The idea here is that the unitary transformation was is generated by one generator K, the squeezing operator. We XN−1 interpret the squeezing operator as creating entanglement 1 2πikn Ψ ffiffiffiffi N Ψ between the two oscillators, and we use this as another n ¼ p e k ð2:60Þ N k¼0

7Note that the Fubini-Study metric is in the Hilbert space, while the distance we calculated in the previous subsection is in 8We use the word circular to refer to both periodic and the space of unitaries. antiperiodic boundary conditions simultaneously.

126001-6 CIRCUIT COMPLEXITY IN FERMIONIC FIELD THEORY PHYS. REV. D 98, 126001 (2018) and the inverse discrete Fourier transform is This can be written as

−1 1 XN N−1 ffiffiffiffi −2πikn X2 Ψk ¼ p e N Ψn: ð2:61Þ † H ¼ Hk þ ω½Ψ0; Ψ0; ð2:68Þ N k¼0 k¼1 From the sum rule of nth roots of unity where Hk is the Hamiltonian for the two oscillators Ψk and −1 Ψ ∈ 1 N−1 Ψ Ψ 1 XN N−k for k ½ ; 2 . Defining b1 ¼ k and b2 ¼ N−k 2πinðkþk0Þ δ − 0 ¼ e N ð2:62Þ k; k N n¼0 † † 2πk † † Hk ¼ ω½b1;b1þω½b2;b2þ2 sin ðb1b2 − b1b2Þ: one can show that the anticommutation relations N translate to ð2:69Þ Ψ Ψ† δ Ψ Ψ 0 Ψ† Ψ† f k; k0 g¼ kk0 f k; k0 g¼ ¼f k; k0 g: ð2:63Þ After B-V transformation ˜ † ˜ † One can check that the Fourier transformed variables also b1 ¼ ρb1 þ ηb2; b2 ¼ ηb1 − ρb2: ð2:70Þ satisfy a Ramond-like boundary condition in Fourier space: Here Ψ Ψ kþN ¼ k: ð2:64Þ s Using all of these, the Hamiltonian in terms of the Fourier ρ ¼ ffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; p p 2 2 2 transformed variables can be written as 2 ωðω − s þ ω Þþs qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN−1 ω ω − 2 ω2 2 † −2πik † † 2πik ð s þ Þþs H ¼ ω½Ψ ; Ψ − i Ψ Ψ− e N þ Ψ Ψ− e N : η ¼ pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð2:71Þ k k k k k k 2 2 ω2 k¼0 s þ

ð2:65Þ 2πk where s ¼ sin N . The ground state is defined by ˜ 0˜ 0˜ 0 ∈ 1 2 Since the range of k is periodic, we can use Ψ−k ¼ ΨN−k to bij i¼ for i f ; g. Writing bring this to a more convenient form: 0˜ 0˜ α j i¼ ijjiji; ð2:72Þ XN−1 † −2πik † † 2πik ω Ψ Ψ − Ψ Ψ N Ψ Ψ N H ¼ ½ k; k i k N−ke þ k N−ke : where now i; j ∈ ð0; 1Þ, using B-V transformations we can k¼0 show that the ground state takes the form ð2:66Þ ˜ ˜ j0 0i¼α00j00iþα11j11ið2:73Þ This form can be directly translated to the normal modes by an adaptation of our earlier B-V construction. We can see with α01 ¼ α10 ¼ 0. Furthermore that the oscillators at k and N − k are getting mixed, independently from the rest of the oscillators. The ground ρα00 ¼ ηα11 ð2:74Þ state will be the tensor product of the ground state of each such pair, together with the ground states of the unpaired and oscillators at the boundary. The pairing is different for N odd and even, so we will do these two cases separately. 2 2 jα00jþjα11j¼1: ð2:75Þ The reference state is defined as ΨkjRi¼0 which is the Ψ 0 ∈ 0 − 1 same thing as njRi¼ for n; k ½ ;N . This state So here also the Uð4Þ → Uð2Þ reduction (analogous to the has no entanglement between any two oscillators on the minimal oscillator toy model) happens. The transformation lattice. from reference state to target state for all N oscillators When N is odd, rewriting the Hamiltonian in the paired together can therefore be viewed as an element of form yields ⊗N−1 2 2 N−1 Uð Þ : ð2:76Þ X2 ω Ψ† Ψ ω Ψ† Ψ ω Ψ† Ψ H ¼ ½ 0; 0þ ½ k; kþ ½ N−k; N−k N−1 N−1 This is a tensor product of 2 factors of Uð2Þ because 2 k¼1 2π pairs of oscillators are getting mixed at a time when N is 2 k Ψ†Ψ† − Ψ Ψ odd. The complexity for the ground state of the þ sin ð k N−k k N−kÞ : ð2:67Þ N Hamiltonian in (2.68) is then immediately calculated to be

126001-7 KHAN, KRISHNAN, and SHARMA PHYS. REV. D 98, 126001 (2018) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uðNX−1Þ=2 The Hamiltonian takes the form t C ¼ ðr Þ2; ð2:77Þ k XN−1 k¼1 ω Ψ† Ψ − Ψ Ψ Ψ†Ψ† H ¼ ð ½ n; n ið n nþ1 þ n nþ1ÞÞ 0 where n¼ −1 XN −2π 1 2π 1 † iðkþ2Þ † † iðkþ2Þ ¼ ðω½Ψ ;Ψk−iðΨkΨ−k−1e N þΨ Ψ− −1e N ÞÞ: 1 −1 1 2πk k k k r ¼ − tan sin : ð2:78Þ k¼0 k 2 ω N ð2:86Þ Similarly when N is even the Hamiltonian in paired form is Using periodicity in k, we can use

N−2 Ψ Ψ X2 −k−1 ¼ N−k−1 ð2:87Þ ω Ψ† Ψ ω Ψ† Ψ H ¼ ½ 0; 0þ ½ N ; N þ Hk; ð2:79Þ 2 2 k¼1 to rewrite the Hamiltonian as

−1 XN −2π 1 with the Hk here being the same as in the odd case. The † iðkþ2Þ N−2 H ¼ ðω½Ψ ; Ψ − iðΨ Ψ − −1e N only difference is that in the even case there are 2 pairs k k k N k 0 and so the transformation from reference state to target state k¼ 2π 1 † † iðkþ2Þ for all N oscillators can be taken to be in Ψ Ψ N þ k N−k−1e ÞÞ: ð2:88Þ ⊗N−2 Uð2Þ 2 : ð2:80Þ When N is even, the Hamiltonian in the paired form is

N−1 The complexity for the ground state of the Hamiltonian in X2 (2.79) is now ω Ψ† Ψ ω Ψ† Ψ H ¼ ½ k; kþ ½ N−1−k; N−1−k vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 u k¼ uðNX−2Þ=2 2π 1 t † † 2 þ 2 sin k þ ðΨ Ψ − Ψ Ψ − −1Þ C ¼ ðrkÞ ; ð2:81Þ N 2 k N−k−1 k N k k¼1 N−1 X2 ≡ where rk is the same as in (2.78). Hk: ð2:89Þ k¼0 2. Neveu-Schwarz boundary condition As in the Ramond case, this can again be interpreted as a Let us now impose a Neveu-Schwarz boundary con- pairwise mixing Hamiltonian, and the corresponding term dition, of the Hamiltonian is Hk. At each k, defining b1 ¼ Ψk and b2 ¼ Ψ −1− (the index k is suppressed in the operators b), Ψ −Ψ N k nþN ¼ n; ð2:82Þ † † Hk ¼ ω½b1;b1þω½b2;b2 on our Majorana lattice. The discrete Fourier transform for 2π 1 † † this boundary condition is þ 2 sin k þ ðb b − b1b2Þ: ð2:90Þ N 2 1 2 N−1 1 1 X 2πinðkþ Þ ffiffiffiffi 2 Ψn ¼ p e N Ψk ð2:83Þ Comparing with the Ramond case, the only difference is N k¼0 1 that k there is replaced by k þ 2 here. Following a similar approach one can show here too that the Uð4Þ → Uð2Þ and the inverse transform reduction happens. The transformation from the reference N ⊗2 N−1 1 state to the target state for all N oscillators is in Uð2Þ . 1 X −2πinðkþ Þ ffiffiffiffi 2 Ψk ¼ p e N Ψn: ð2:84Þ The complexity for the ground state of the Hamiltonian in N n¼0 (2.89) is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Although Ψ satisfies the Neveu-Schwarz boundary con- u n uN=X2−1 dition, Ψ satisfies an analogue of the Ramond boundary t 2 k C ¼ ðr Þ ; ð2:91Þ condition in Fourier space: k k¼0 Ψ Ψ kþN ¼ k: ð2:85Þ where

126001-8 CIRCUIT COMPLEXITY IN FERMIONIC FIELD THEORY PHYS. REV. D 98, 126001 (2018) Z 1 −1 1 2π 1 ¯ ¯ r ¼ − tan sin k þ : ð2:92Þ H ¼ dx½−iΨγx∂ Ψ þ mΨΨ: ð3:2Þ k 2 ω N 2 x

T Similarly when N is odd the paired form of the Here Ψ ¼ðΨ1; Ψ2Þ is the two component complex t x Hamiltonian is fermion and γ ¼ σ3 and γ ¼ iσ2. By a Fourier transform Z N−2 X2 dk † Ψ ffiffiffiffiffiffi Ψ ikx ω Ψ Ψ −1 iðxÞ¼ p iðkÞe H ¼ ½ N−1; N þ Hk; ð2:93Þ 2π 2 2 k¼0 we can write the Hamiltonian as with the same Hk as in the even case (for the Neveu- N−1 Z Schwarz boundary condition). This time there are 2 pairs † † and so the transformation from the reference state to the H ¼ dk½kΨ1ðkÞΨ2ðkÞþkΨ2ðkÞΨ1ðkÞ ⊗N−1 target state for all N oscillators is in Uð2Þ 2 . The Ψ† Ψ − Ψ† Ψ complexity for the ground state of the Hamiltonian in þ m 1ðkÞ 1ðkÞ m 2ðkÞ 2ðkÞ: ð3:3Þ (2.93) is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The canonical anticommutators in momentum space u uðNX−2Þ=2 become t 2 C ¼ ðrkÞ ; ð2:94Þ † 0 † 0 0 k¼0 fΨ1ðkÞ; Ψ1ðk Þg ¼ fΨ2ðkÞ; Ψ2ðk Þg ¼ δðk − k Þ: ð3:4Þ where rk is the same as in (2.92). The reference unentangled IR state jRi can be defined by

† III. CONTINUUM FIELD THEORY Ψ1ðkÞjRi¼0; Ψ2ðkÞjRi¼0 ∀ k ∈ R: ð3:5Þ Everything we did so far was by discretizing spacetime into a lattice. This was the strategy adopted for the scalar This is the ground state of the ultralocal Hamiltonian case in [4], and what we have shown is that a similar Z Z strategy works (modulo minor—for our purposes—subtle- ¯ † † H ¼ dxðmΨΨÞ¼ dkmðΨ ðkÞΨ1ðkÞ − Ψ ðkÞΨ2ðkÞÞ: ties like fermion doubling) for fermions as well. Now we m 1 2 will work directly in the continuum case following the ð3:6Þ corresponding approach for scalars undertaken in [5]. The idea here is basically a generalization of the squeezing The ground state of this fermionic theory can be obtained operator approach we discussed in passing in the lattice by direct analogy with the discrete case via a Bogoliubov- case. We can reach from an unentangled reference state R 9 j i Valatin transformation which is to the entangled target state jΨi via a unitary transformation R † σ ˜ ˜ − f σ Ψ1ðkÞjΨi¼0; Ψ2ðkÞjΨi¼0 ∀ k ∈ R; ð3:7Þ i σ GðσÞd jΨi¼Pe i jRi; ð3:1Þ where where in many cases, we will find that the GðσÞ can be realized via an appropriate squeezing operator, which ˜ Ψ1ðkÞ¼(A Ψ1ðkÞþB Ψ2ðkÞ) creates quantum entanglement below some UV cutoff scale k k Λ σ σ σ ˜ . Here parametrizes our path such that at ¼ i we have Ψ2ðkÞ¼( − BkΨ1ðkÞþAkΨ2ðkÞ) ð3:8Þ our reference state jRi and at σ ¼ σf we get the target state jΨi. The path ordering P is not required for commuting are the normal mode coordinates and generators GðσÞ, as will often be the case if we manage to − find appropriate squeezing operators. We will begin with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik Dirac fermions in 1 þ 1 dimensions, which is a standard Ak ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð3:9Þ 2 2 2 − 2 context where the cMERA tensor network is discussed k þð k þ m mÞ [6,7], with an eye towards our discussions in the next section.

A. Dirac in 1 + 1 dimensions 9Note that this same ground state is precisely the one that is obtained in the standard approach to quantization of Dirac We consider the Dirac Hamiltonian in 1 þ 1 dimensions fermions, which involves the introduction of Dirac u and v given by modes. We demonstrate this in Appendix C.

126001-9 KHAN, KRISHNAN, and SHARMA PHYS. REV. D 98, 126001 (2018) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − 2 2 and then minimize the length. The circuit length of a path qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim k þ m Bk ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð3:10Þ traced by intermediate states jΨðσÞi ¼ UðσÞjRi is 2 p 2 2 2 k þð k þ m − mÞ Z σf lðjΨðσÞiÞ ¼ ds ðσÞ: ð3:17Þ Noting that A2 B2 1, we introduce FS k þ k ¼ σi 1 k Using (3.1) we can write the FS metric as r ¼ − tan−1 ð3:11Þ k 2 m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ σ 2 σ − σ 2 dsFSð Þ¼d hG ð ÞiΨðσÞ hGð ÞiΨ σ : ð3:18Þ so that ð Þ In the context of our discussion in the previous sub- A ¼ − cosðr Þ and B ¼ sinðr Þ: ð3:12Þ k k k k section, we can consider a general path generated by the squeezing operator as a unitary transformation that takes Note the similarity of this expression r with analogous k the state through expressions in the discrete case. R As our target state, following the scalar case in [5],we − σ Λ i dkKðkÞYkð Þ consider the approximate ground state jmð Þi which is UðσÞ¼e jkj≤Λ jRi: ð3:19Þ defined as Here ˜ ðΛÞ ˜ † ðΛÞ Ψ1ðkÞjm i¼0; Ψ ðkÞjm i¼0 ∀ k∶jkj ≤ Λ Z 2 σ ðΛÞ † ðΛÞ Y σ y s0 ds0 Y s r ; : Ψ1ðkÞjm i¼0; Ψ2ðkÞjm i¼0 ∀ k∶jkj > Λ: kð Þ¼ kð Þ and kð fÞ¼ k ð3 20Þ si ð3:13Þ where the last condition arises from our specific choice of We can reach the target state from the reference state via a target state. The order of k and σ integrals can be exchanged unitary transformation of the form because KðkÞ at different k are all commuting. By direct R calculation one sees that Λ −i dkKðkÞrk Z jmð Þi¼e jkj≤Λ jRi; ð3:14Þ σ 0 2 σ 2 σ hGð Þi ¼ ; and hG ð Þi ¼ Vol dkykð Þ: ð3:21Þ where the reference state is jRi¼jΩi ∀ k and KðkÞ is the squeezing operator which we define as Here we write δð0Þ¼Vol. The complexity is the length of † † the minimal path: KðkÞ¼i(Ψ1ðkÞΨ2ðkÞþΨ1ðkÞΨ2ðkÞ): ð3:15Þ

C ¼ min lðdsFSÞ We will see that this construction shares many of the fGðsÞg sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi features of the scalar case discussed in [5], despite the fact Z Z s f 2 that here the entanglement is happening between the two ¼ min dσ Vol dkð∂σYkðσÞÞ : ð3:22Þ fY ðσÞg ≤Λ fermionic modes, and not between modes at antipodal k si jkj momenta. Note also that the target ground state that we have defined is not the cMERA ground state [6,7] even We recognize a flat Euclidean geometry associated with though in the Λ → ∞ limit they both tend to the true coordinate YkðσÞ, implying that the geodesic is a straight ground state. The cMERA can be viewed as a nongeodesic line path. Under a simple affine parametrization σ the path in our language, as we will discuss in the next section. geodesic is

σ − si B. Ground state complexity YkðσÞ¼ YkðsfÞ: ð3:23Þ sf − si Before we do all that, let us evaluate the complexity using our squeezing operator above. With the specific The complexity for the target state is the length of this choice of the squeezing operator we have made in the straight line path, previous section, the calculation goes entirely parallel to the sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi one in [5], but we review it here largely to establish our Z notation. First, we calculate the distance with the Fubini- 2 C ¼ Vol dkðrkÞ; ð3:24Þ Study metric, k≤Λ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 and rk is given by (3.11). As we see, the essential difference ds σ dσ ∂σ Ψ σ − Ψ σ ∂σ Ψ σ ; 3:16 FSð Þ¼ j j ð Þij jh ð Þj j ð Þij ð Þ between complexity in the scalar case and the fermionic

126001-10 CIRCUIT COMPLEXITY IN FERMIONIC FIELD THEORY PHYS. REV. D 98, 126001 (2018)

0 2 δ − 0 case is in the form of the rk, both in the discrete case and the ½KþðkÞ;K−ðk Þ ¼ K0ðkÞ ðk k Þ continuum case. We find that the parallel between the form 0 0 K0 k ;K k K k δ k − k of the complexity in the discrete [4] and continuum [5] ½ ð Þ þð Þ ¼ þð Þ ð Þ 0 0 cases for the scalar, holds for the fermionic case as well. ½K0ðkÞ;K−ðk Þ ¼ −K−ðkÞδðk − k Þ: ð3:29Þ These integrals are divergent in the cutoff Λ. In the massive case, the above expression can be explicitly This is easily seen to be a set of decoupled SUð2Þ algebras evaluated to at each k, once one rescales the generators appropriately with δð0Þ¼Vol. Now as in [5] let us consider a general m Λ2 Λ C2 ¼ − Vol log þ 1 tan−1 path of the form: 2 m2 m R dkgðk;σÞ 1 Λ Ψ σ jkj<Λ − mVol − logð16Þtan−1 j ð Þi ¼ e jRið3:30Þ 4 m generated by gðk; σÞ, given as −2itan−1ðΛÞ 2itan−1ðΛÞ − iLi2 −e m þ iLi2 −e m gðk;σÞ¼α ðk;σÞK ðkÞþα−ðk;σÞK−ðkÞþωðk;σÞK0ðkÞ: 2 þ þ 1 −1 Λ þ ΛVol tan : ð3:25Þ ð3:31Þ 2 m Its behavior at large Λ takes the form The unitarity condition for the above transformation α σ −α σ ω σ −ω σ implies þð Þ¼ −ð Þ and ð Þ¼ ð Þ. We can π2 4 2 4 decompose the unitary transformation above using [8] 2 Λ m m − m O 1 Λ C ¼ 8 Vol þ π log Λ π þ ð = Þ : R R dkβ ðk;σÞK ðkÞ dk log β0ðk;σÞK0ðkÞ UðσÞ¼e jkj<Λ þ þ e jkj<Λ ð3:26Þ R dkβ−ðk;σÞK−ðkÞ When m ¼ 0 the integral is simple and the complexity × e jkj<Λ ; ð3:32Þ simplifies: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the coefficients are π2Λ Vol 2α Ξ C ¼ 8 : ð3:27Þ β sinh ¼ 2Ξ Ξ − ω Ξ ; cosh sinh ω −2 β Ξ − Ξ C. Ground state complexity and SUð2Þ generators 0 ¼ cosh 2Ξ sinh ; Our discussion so far used a single generator KðkÞ to reach the target state. This can be viewed as a specific with Ξ defined via choice of gate in the circuit complexity language. Since we ω2 were able to construct a parallel between the entangling 2 Ξ ¼ þ α α−: ð3:33Þ operator in the scalar case in [5] and the case 4 þ above, it is interesting to ask whether a construction Analogous to [5] now we can observe that K− annihilates analogous to the more general SUð1; 1Þ generators dis- cussed in [5] is possible here. It turns out that the answer is the reference state while the exponential of K0 only yes, except that the generators satisfy an SUð2Þ algebra changes the reference state up to a phase now. Our calculations in this section are direct adaptations δð0Þ of those in the Letter of [5], but we include some tricks here 0; − K−jRi¼ K0jRi¼ 2 jRi: ð3:34Þ that simplify the calculation. The basic idea is to note that the target state can also be So jΨðσÞi can be written as reached by using a more general set of generators: R dkβ ðk;σÞK ðkÞ Ψ† Ψ jΨðσÞi ¼ N e Λ þ þ jRið3:35Þ KþðkÞ¼i 1ðkÞ 2ðkÞ † K−ðkÞ¼iΨ1ðkÞΨ2ðkÞ with † † R Ψ1ðkÞΨ1ðkÞ − Ψ2ðkÞΨ2ðkÞ −δð0Þ 2 dk log β0ðk;σÞ K0ðkÞ¼ 2 : ð3:28Þ N ¼ e jkj≤Λ : ð3:36Þ

These commute with the number preserving operators To compute the Fubini-Study metric from here is a bit of † † ðn1 − n2Þ where n1 ¼ Ψ1Ψ1 and n2 ¼ Ψ2Ψ2. These gen- work, and we will introduce a small trick to accomplish it erators satisfy the following commutations relations: painlessly. Let us define

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˜ ˜ ˜ − Kþ ¼ K− K− ¼ Kþ K0 ¼ K0 ð3:37Þ Further in this section we suppress some notations: the integral over k is just denoted with the integral symbol, and The crucial fact that makes them useful is that they the argument k is often not explicitly written. To calculate satisfy identical commutation relations as the untilde’d the Fubini-Study metric, we have operators: Z Z Vol β0 ∂ Ψ σ − 0 β 0 Ψ σ σj ð Þi ¼ þ þ Kþ j ð Þi: ð3:44Þ ˜ ˜ 0 2 ˜ δ − 0 2 β0 ½KþðkÞ; K−ðk Þ ¼ K0ðkÞ ðk k Þ ˜ ˜ 0 ˜ 0 ½K0ðkÞ; K ðk Þ ¼ K ðkÞδðk − k Þ þ þ This leads to ˜ ˜ 0 − ˜ δ − 0 ½K0ðkÞ; K−ðk Þ ¼ K−ðkÞ ðk k Þ: ð3:38Þ Z Z β0 Vol 0 0 hΨðσÞj∂σjΨðσÞi ¼ − þ β hΨðσÞjK jΨðσÞi σ 2 β þ þ The gðk; Þ can be written as a linear combination of these Z 0 Z generators as well: Vol β0 − 0 β 0ζ ¼ 2 β þ þ þihRjKijRi ˜ ˜ ˜ Z 0 Z g k;σ α˜ k;σ K k α˜ − k;σ K− k ω˜ k;σ K0 k : ð Þ¼ þð Þ þð Þþ ð Þ ð Þþ ð Þ ð Þ −Vol β0 0 β 0ζ ¼ þ þ þ0 ; ð3:39Þ 2 β0

Comparing this with Eq. (3.32), we get where ζij is defined via

α˜ α α˜ α ω˜ −ω † þ ¼ − − ¼ þ ¼ : ð3:40Þ U KiU ¼ ζijKj: ð3:45Þ

As the tilde’d generators also satisfy identical commutation This leads to σ relations, Uð Þ can be decomposed in the same fashion in Z Z terms of these generators as 2 β0 2 Vol 0 0 jhΨðσÞj∂σjΨðσÞij ¼ þ β ζ 2 β þ þ0 R R Z Z 0 β˜ σ ˜ β˜ σ ˜ dk þðk; ÞKþðkÞ dkðlog 0ðk; ÞÞK0ðkÞ β0 UðσÞ¼e jkj<Λ e jkj<Λ 0 β 0ζ R × β þ þ þ0 : ð3:46Þ ˜ ˜ 0 dkβ−ðk;σÞK−ðkÞ × e jkj<Λ ; ð3:41Þ It is useful to note that where the coefficients satisfy exactly the same formulas as ’ ζ ζ before, but now with tilde d quantities. Using (3.40) we can þ0 ¼ −0: ð3:47Þ show that these are related to the coefficients in (3.33) by The second piece in the Fubini-Study metric follows from a β˜ −β β˜ β similar, but slightly lengthier calculation: ¼ 0 ¼ 0: ð3:42Þ Z Z 2 † Vol β0 It is useful to write the decomposition of U ðσÞ with the ∂ Ψ σ 2 0 β0 ζ j σj ð Þij ¼ 2 β þ þ þ0 latter generators: Z Z 0 β0 0 β 0ζ R R × þ þ þ0 β˜ σ ˜ β˜ σ ˜ β † dk −ðk; ÞK ðkÞ dkðlog 0ðk; ÞÞK0ðkÞ 0 U ðσÞ¼e jkj<Λ þ e jkj<Λ Z R 0 0 ˜ ˜ β β ζ ζ dkβ ðk;σÞK−ðkÞ þ Vol þ þ −− þþ: ð3:48Þ × e jkj<Λ þ R R − dkβ−ðk;σÞK−ðkÞ − dkðlog β0ðk;σÞÞK0ðkÞ ¼ e jkj<Λ e jkj<Λ R So the final form of the metric is − dkβ ðk;σÞK ðkÞ × e jkj<Λ þ þ ð3:43Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ σ β0 β0 ζ ζ dsFS ¼ d Vol þ þ −− þþ; ð3:49Þ This last form helps us in substantially reducing the mindless labor involved in the calculations here as well as in the analogous results in [5]. where

126001-12 CIRCUIT COMPLEXITY IN FERMIONIC FIELD THEORY PHYS. REV. D 98, 126001 (2018) β β 2 1 At σ ¼ 1 the β for BðkÞ becomes þ − þ ζ−−ζ ¼ 1 þ ¼ : ð3:50Þ þþ β β 2 0 j 0j β − þ ¼ i tan rk: ð3:59Þ β 1 β 2 Making use of the identity j 0j¼ þj þj , the metric The length of this path, on plugging the expression (3.58) finally simplifies to into the CP1 metric from the end of last section, yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 0 Z β β0 π σ þ þ 2 dsFS ¼ d Vol dk 2 2; ð3:51Þ l ¼ Vol dk sin r : ð3:60Þ ≤Λ ð1 þjβ j Þ k jkj þ 2 jkj≤Λ σ where the prime denotes the derivative with respect to . This can be explicitly evaluated to be Thus the metric has the form of the Fubini-Study metric on 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S ka CP and the complexity arises as its geodesic. Note 2 2 2 2 π 1 Λ þ m − Λ the sign difference from the results of [5]. l ¼ Vol Λ þ m log pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð3:61Þ 2 2 Λ2 þ m2 þ Λ D. Another generator which goes as Now that we have the SUð2Þ generators in our hand, we 1 m can construct another single generator BðkÞ out of them that 2 π2 Λ l ¼ 4 Vol þ m log 2Λ þ ð3:62Þ is distinct from KðkÞ, which takes us from the initial reference state to the target state, just like K k does [5]. ð Þ Λ Λ This takes the form for large , with the dots denoting subleading powers in . This distance is more than the one presented in (3.26) which is obvious in the m → 0 limit. BðkÞ¼−2i sinðr Þ½K − K− − 4 cosðr ÞK0: ð3:52Þ k þ k For comparison, for the straight line path using squeez- 2 The unitary transformation which does the job is ing operator KðkÞ the SUð Þ coefficients are R π α ¼ −ir σ; α− ¼ −ir σ; ω ¼ 0; Ξ ¼ ir σ Λ i4 dkBðkÞ þ k k k jmð Þi¼e jkj≤Λ jRi: ð3:53Þ ð3:63Þ Let the intermediate state be β R which gives the þ for KðkÞ as π i4 dkBðkÞYkðσÞ jΨðσÞi ¼ e jkj≤Λ jRi: ð3:54Þ β − σ þ ¼ i tanðrk Þ: ð3:64Þ

Plugging this in the Fubini-Study metric gives The complexity of this path has been discussed in a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ previous subsection. π σ 2 ∂ σ 2 In Appendix D, we explicitly show that the minimal path dsFS ¼ d Vol dksin rkð σYkð ÞÞ : ð3:55Þ when the squeezing operator is K k is a geodesic of the 2 jkj≤Λ ð Þ CP1 metric, but the minimal path of the BðkÞ operator So the path with the least length is is not.

YkðσÞ¼σ: ð3:56Þ E. Majorana fermions in 1 + 1 dimensions Now we move on to field theories other than the 1 þ 1-d We can compare the BðkÞ form with the SUð2Þ gen- Dirac fermion, which as we demonstrated, has very close erators of the last section, and read off parallels to the scalar field theory discussed in [5]. First we πσ πσ turn to the Majorana theory. α α − 1 1 þ ¼ 2 sin rk; − ¼ 2 sin rk; Previously we discussed the discrete version of þ - πσ dimensional Majorana field theory. Now we consider the ω − πσ Ξ i continuum version of it. The Lagrangian is ¼ i cos rk; ¼ 2 ; ð3:57Þ Z β ψ¯ γμ∂ − ψ which translates to the þ: L ¼ dx ði μ mÞ ; ð3:65Þ i sin r sin πσ β k 2 where the gamma matrices are the same ones we used in þ ¼ πσ − πσ : ð3:58Þ i cos 2 cos rk sin 2 Sec. II [see Eq. (2.7)], and

126001-13 KHAN, KRISHNAN, and SHARMA PHYS. REV. D 98, 126001 (2018)

μ 0 1 γ ∂μ ¼ ρ ∂0 þ ρ ∂1: ð3:66Þ We define the reference state to be

The field ψ is a two-component spinor with its components ΨðkÞjRi¼0 ð3:76Þ real Grassmann variables classically: Ψ 0 1 which is the same as defining ðxÞjRi¼ . This state has cΨ no entanglement in position space and is also the ground ψ ¼ : ð3:67Þ Ψ2 state of the ultralocal Hamiltonian, ie., the above Hamiltonian without the terms arising from the derivatives The Lagrangian in terms of them becomes in position space. Z In the ground state the oscillators at k and −k are to be 1 1 2 2 1 1 2 2 entangled. The target state can be reached by using the L ¼ dxðiðΨ ∂0Ψ þ Ψ ∂0Ψ þ Ψ ∂1Ψ − Ψ ∂1Ψ Þ squeezing operator KðkÞ defined here as þ mΨ1Ψ2 − mΨ2Ψ1Þ: ð3:68Þ KðkÞ¼i(Ψ†ðkÞΨ†ð−kÞþΨðkÞΨð−kÞ): ð3:77Þ We developed our B-V technology more directly in the language of complex Grassmann variables, so we rewrite This operator entangles the oscillators at k and −k. Let us the Lagrangian in terms of rewrite the Hamiltonian as Z Ψ1 − iΨ2 Ψ1 þ iΨ2 ∞ Ψ ¼ pffiffiffi ; Ψ† ¼ pffiffiffi : ð3:69Þ H ¼ dkðm½Ψ†ðkÞ; ΨðkÞ þ m½Ψ†ð−kÞ; Ψð−kÞ 2 2 0 þ 2kðΨ†ðkÞΨ†ð−kÞ − ΨðkÞΨð−kÞÞÞ: ð3:78Þ Here Ψ, Ψ† are complex Grassmann variables. The Lagrangian becomes In the Dirac case the integration limits ran from −∞ to ∞ Z but here it is from 0 to ∞. In other words, we have the † † † † L ¼ dxðiðΨ ∂0Ψ þ Ψ∂0Ψ þ Ψ∂1Ψ þ Ψ ∂1Ψ Þ freedom to define

− Ψ† Ψ † m½ ; Þ ð3:70Þ Ψ1ðkÞ ≡ ΨðkÞ; Ψ2ðkÞ ≡ Ψ ð−kÞð3:79Þ and the Hamiltonian here, and when we do it, we get the Dirac Hamiltonian that Z we wrote down earlier but with the integration over k † † † 10 H ¼ dxð−iΨ∂1Ψ − iΨ ∂1Ψ þ m½Ψ ; ΨÞ: ð3:71Þ limited to the positive k real axis. In other words, we are reinterpreting the entanglement between k and −k modes here as entanglement between two different fields, but at Quantization proceeds by imposing the canonical anticom- same k (restricted to be positive). Since much of the mutation relations calculations in the Dirac case goes through at each k † separately, this means that the Majorana calculation reduces fΨðxÞ; Ψ ðx0Þg ¼ δðx − x0Þð3:72Þ loosely to half of Dirac. Apart from the restriction in the and all other anticommutators are zero. Doing a Fourier range of k, the ground states for both cases are the same and transform the reference states are also the same. So the rk is the same as before and every derivation here is the same as before Z dk with the only difference being in the lower integration limit ΨðxÞ¼ pffiffiffiffiffiffi ΨðkÞeikx ð3:73Þ −Λ 2π which is 0 for this case and not as in the previous case. The complexity for the ground state of the Majorana theory they turn to is then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ Ψ Ψ† 0 δ − 0 Λ f ðkÞ; ðk Þg ¼ ðk k Þð3:74Þ 2 C ¼ Vol dkðrkÞ: ð3:80Þ 0 and all other anticommutators are zero. The Hamiltonian in Fourier variables is As r 2 is an even function in k, we have Z ð kÞ † † H ¼ dk( − kΨðkÞΨð−kÞþkΨ ðkÞΨ ð−kÞ 10 These Ψ1 and Ψ2 refer to the Dirac notation analogous to that Ψ1 Ψ2 Ψ† Ψ ) in Sec. III A and should not be confused with the real and þ m½ ðkÞ; ðkÞ : ð3:75Þ (with lower indices) used in this section.

126001-14 CIRCUIT COMPLEXITY IN FERMIONIC FIELD THEORY PHYS. REV. D 98, 126001 (2018) pffiffiffi 2 H Ha Hb CDirac ¼ CMajorana; ð3:81Þ ðkÞ¼ ðkÞþ ðkÞð3:88Þ where CDirac and CMajorana are the complexities for the with ground state of Dirac and Majorana theories in 1 þ 1 k3 k3 dimensions. Ha † † − − ðkÞ¼ 2 ½a1;a1þ 2 ½a2;a2 ðk1 ik2Þa1a2 † † F. Fermions in higher dimensions þðk1 þ ik2Þa1a2

Now we will generalize some of the above consider- k3 k3 Hb † † − † † ations to higher dimensions. The fermionic field theoretic ðkÞ¼ 2 ½b1;b1þ 2 ½b2;b2þðk1 ik2Þb1b2 Hamiltonian in d þ 1 spacetime dimensions is − ðk1 þ ik2Þb1b2: ð3:89Þ Z d ¯ i H ¼ d xΨðxÞð−iγ ∂i þ mÞΨðxÞ; ð3:82Þ After the by-now-familiar B-V transformation we get

X2 where i runs from 1 to d. After doing a Fourier transform on ω † † HðkÞ¼ k ð½a˜ ; a˜ þ½b˜ ; b˜ Þ; ð3:90Þ each of the components of the spinor 2 i i i i i¼1 Z d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d k ⃗⃗ 2 2 2 Ψ Ψ ik:x where ωk ¼ k1 þ k2 þ k3. The B-V transformation for a aðxÞ¼ d aðkÞe : ð3:83Þ ð2πÞ2 type oscillators is

0 ω 1 The Hamiltonian in Fourier variables is 0 1 † ðk3þ kÞ a2 þ a1 ˜ ðk1þik2Þ Z a1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB C B C B † ω C B C − ðk3þ kÞ d (Ψ¯ γi Ψ ) a˜ 2 1 B a1 a2 C H ¼ d k ðkÞð ki þ mÞ ðkÞ : ð3:84Þ B C k3 B ðk1þik2Þ C B † C ¼ 1 − B C ð3:91Þ B ˜ C 2 ω † ðk3þω Þ @ a1 A k B a2 þ a k C @ 1 ðk1−ik2Þ A ˜ † a2 † ðk3þωkÞ a1 − a2 1. Massless theory in 3 + 1 dimensions ðk1−ik2Þ Let us start by considering a massless theory in 3 þ 1 and the B-V transformation for b type oscillators is dimensions in chiral (aka Weyl) basis: 0 1 0 1 † ðk3þω Þ k b b1 k 0 I2 0 σ ˜ 2 þ ðk1−ik2Þ γ0 γk : 3:85 b1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB C ¼ ¼ k ð Þ B C B † ω C I2 0 −σ 0 B ˜ C − ðk3þ kÞ B b2 C 1 B b1 b2 − C 1 − k3 B ðk1 ik2Þ C B † C ¼ B C: ð3:92Þ B ˜ C 2 ω † ðk3þω Þ For massless particles the two Weyl spinors decouple from @ b1 A k B b2 þ b k C @ 1 ðk1þik2Þ A each other. The Hamiltonian is ˜ † b2 † ðk3þω Þ b1 − b k Z Z 2 ðk1þik2Þ 3 ¯ 3 H ¼ d kðΨðkÞγik ΨðkÞÞ ¼ d kHðkÞ; ð3:86Þ i Here there is no natural ultralocal Hamiltonian, because mass is zero. We can choose any reference state which has where i runs from 1 to 3. Each component of the spinor no entanglement in the physical (i.e., x) space. Let us define ΨðkÞ can be interpreted as an oscillator at k. At each k there the reference state jRi to be the state which is annihilated by are four fermionic oscillators with the Hamiltonian HðkÞ: † † 3 Ψ1ðxÞ, Ψ2ðxÞ, Ψ3ðxÞ and Ψ4ðxÞ ∀ x ∈ R . This is the same thing as defining the reference state to be annihilated by k3 k3 k3 H Ψ Ψ† Ψ† Ψ Ψ† Ψ a k b k i 1 ∀ k ∈ R3 ðkÞ¼ 2 ½ 1; 1þ 2 ½ 2; 2þ 2 ½ 3; 3 ið Þ and ið Þ for ¼ , 2 and . We define the target state to be the approximate ground k3 Ψ Ψ† − − Ψ†Ψ − Ψ†Ψ state jTðΛÞi defined as þ 2 ½ 4; 4 ðk1 ik2Þ 1 2 ðk1 þ ik2Þ 2 1 − Ψ†Ψ Ψ†Ψ ðΛÞ ˜ ðΛÞ þðk1 ik2Þ 3 4 þðk1 þ ik2Þ 4 3: ð3:87Þ a˜ iðkÞjT i¼0 biðkÞjT i¼0 ∀ k∶jkj ≤ Λ † ðΛÞ 0 † ðΛÞ 0 ∀ ∶ Λ In the above equation the dependence of the fields on k is ai ðkÞjT i¼ bi ðkÞjT i¼ k jkj > : suppressed. The two oscillators Ψ1ðkÞ and Ψ2ðkÞ are ð3:93Þ decoupled from Ψ3ðkÞ and Ψ4ðkÞ. † For each k, let us define a1 ¼ Ψ1, a2 ¼ Ψ2, b1 ¼ Ψ3, It is possible to see that the target state can be reached from † b2 ¼ Ψ4 so that the reference state by the unitary transformation

126001-15 KHAN, KRISHNAN, and SHARMA PHYS. REV. D 98, 126001 (2018) R 3 Λ −i d krðkÞKðkÞ This is again a flat Euclidean geometry associated with jTð Þi¼e jkj≤Λ jRi; ð3:94Þ coordinate XkðσÞ and so the geodesic is where the squeezing operator KðkÞ here is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 † † ˜ ˜ XkðσÞ¼ k1 þ k2rðkÞσ: ð3:102Þ KðkÞ¼iðk1 − ik2Þ(a˜ 1ðkÞa˜ 2ðkÞþb1ðkÞb2ðkÞÞ ˜ † ˜ † þ iðk1 þ ik2Þða˜ 1ðkÞa˜ 2ðkÞþb1ðkÞb2ðkÞÞ † † The complexity is the length of this path ¼ iðk1 − ik2Þða2ðkÞa1ðkÞþb2ðkÞb1ðkÞÞ † † sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ þ iðk1 þ ik2Þða2ðkÞa1ðkÞþb2ðkÞb1ðkÞÞ ð3:95Þ ω 2 2 3 pk3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ k C ¼ Vol d k arctan 2 2 : ð3:103Þ and rðkÞ is jkj≤Λ k1 þ k2 1 ω − pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pk3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ k rðkÞ¼ 2 2 arctan 2 2 : ð3:96Þ This integral can in principle be explicitly evaluated, but we k1 þ k2 k1 þ k2 will not present it here. Instead we go on to the massive case, where we will present all the gory details. It can also be checked that the unitary transformation ’ ’ ˜ ’ ˜ ’ takes the ai s and bi stoai s and bi s via the similarity 2. Ground state complexity and transformations SU(2) ×SU(2) generators

† † † † Let us consider an extended set of paths as we did before Ua U ¼ a˜ 1 Ua U ¼ a˜ 2 2 1 in Sec. III C. We introduce SUð2Þ × SUð2Þ generators and † † ˜ † † ˜ Ub2U ¼ b1 Ub1U ¼ b2: ð3:97Þ consider the paths generated by them. More general paths are possible as we will see in the next subsection, but this is As the annihilation operators of the reference state are a sufficiently interesting generalization that contains our transformed to the annihilation operators of the target state squeezing operator. The squeezing operator used in the via the similarity transformation, the unitary transformation previous subsection can be written as takes the reference state to the target state. Now let us consider an arbitrary path generated by the − ˜ † ˜ † ˜ ˜ squeezing operator KðkÞ¼iðk1 ik2Þða1ðkÞa2ðkÞþb1ðkÞb2ðkÞÞ R R ˜ ˜ ˜ † ˜ † σ 3 þ iðk1 þ ik2Þða1ðkÞa2ðkÞþb1ðkÞb2ðkÞÞ −i ds d kykðsÞKðkÞ jΨðσÞi ¼ e 0 jkj≤Λ jRi † † R ¼ iðk1 − ik2Þða2ðkÞa1ðkÞþb2ðkÞb1ðkÞÞ −i d3kY ðσÞKðkÞ † † k ≤Λ k ¼ e j j jRi: ð3:98Þ þ iðk1 þ ik2Þða2ðkÞa1ðkÞþb2ðkÞb1ðkÞÞ: ð3:104Þ

Note that by identifying a judicious squeezing operator, κ iθ − κ −iθ here too we have bypassed the need to do any path ordering Let ðk1 þ ik2Þ¼ e and ðk1 ik2Þ¼ e and by because all KðkÞ commute. absorbing the phases into the ladder operators the squeez- Evaluating the length of this path using the Fubini-Study ing operator simplifies to metric gives sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi † † Z Z KðkÞ¼iκða˜ 1ðkÞa˜ 2ðkÞþa˜ 1ðkÞa˜ 2ðkÞÞ 1 Ψ σ σ 2 3 2 2 ∂ σ 2 ˜ † ˜ † ˜ ˜ lðj ð ÞiÞ ¼ d Vol d kðk1 þ k2Þð σYkð ÞÞ : þ iκðb1ðkÞb2ðkÞþb1ðkÞb2ðkÞÞ 0 jkj≤Λ † † ¼ iκða ðkÞa ðkÞþa2ðkÞa1ðkÞÞ ð3:99Þ 2 1 † † þ iκðb2ðkÞb1ðkÞþb2ðkÞb1ðkÞÞ Doing a “coordinate” transformation ¼ κKaðkÞþκKbðkÞ; ð3:105Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 XkðσÞ¼ k1 þ k2YkðσÞð3:100Þ a † † b where K ðkÞ ≡ iða2ðkÞa1ðkÞþa2ðkÞa1ðkÞÞ and K ðkÞ≡ † † the length of the path becomes iðb2ðkÞb1ðkÞþb2ðkÞb1ðkÞÞ. Notice from the Hamiltonian in the previous section Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ 1 the a type oscillators and b type oscillators completely 3 2 lðjΨðσÞiÞ ¼ dσ 2Vol d kð∂σXkðσÞÞ : ð3:101Þ decouple from each other. We can introduce the following 0 jkj≤Λ generators:

126001-16 CIRCUIT COMPLEXITY IN FERMIONIC FIELD THEORY PHYS. REV. D 98, 126001 (2018)

a † † b † † KþðkÞ¼ia2ðkÞa1ðkÞ;KþðkÞ¼ib2ðkÞb1ðkÞ a b K−ðkÞ¼ia2ðkÞa1ðkÞ;K−ðkÞ¼ib2ðkÞb1ðkÞ † − † † − † a a2ðkÞa2ðkÞ a1ðkÞa1ðkÞ b b2ðkÞb2ðkÞ b1ðkÞb1ðkÞ K0ðkÞ¼ 2 ;K0ðkÞ¼ 2 : ð3:106Þ

Together these six generators generate a SUð2Þ × SUð2Þ algebra of generators. This is unsurprising: the Hamiltonian algebra. Each of these generators commute with the number of the massless 3 þ 1 theory in the chiral basis can be seen preserving operators ðn1 − n2Þ and ðn3 − n4Þ, where to be two copies of the 1 þ 1 Dirac theory that we studied earlier, after the phase redefinitions etc. that we did. So the † † † † n1 ¼ a2a2;n2 ¼ a1a1;n3 ¼ b2b2;n4 ¼ b1b1: path length can be understood as path lengths in two separate factorized Hilbert spaces, each of which contains ð3:107Þ the SUð2Þ structure. Let us consider the path generated by an arbitrary linear combination of these six generators 3. Massive theory in 3 + 1 dimensions R Now we go on to consider the massive theory in 3 þ 1 dkgðk;σÞ jΨðσÞi ¼ e jkj≤Λ jRi; ð3:108Þ dimensions in the Dirac basis: k where jRi is the reference state defined in the previous I2 0 0 σ γ0 γk : 3:113 σ ≡ a σ b σ ¼ ¼ k ð Þ section and here gðk; Þ g ðk; Þþg ðk; Þ and 0 −I2 −σ 0

a σ αa σ a αa σ a g ðk; Þ¼ þðk; ÞKþðkÞþ −ðk; ÞK−ðkÞ The Hamiltonian in terms of the Fourier variables is a a þ ω ðk; σÞK0ðkÞ Z Z 3 Ψ¯ γi Ψ 3 H b σ αb σ b αb σ b H ¼ d k ðkÞðki þ mÞ ðkÞ¼ d k k; ð3:114Þ g ðk; Þ¼ þðk; ÞKþðkÞþ −ðk; ÞK−ðkÞ þ ωbðk; σÞKbðkÞ: ð3:109Þ 0 where i runs from 1 to 3. Interpreting the components of the spinor Ψ k as oscillators, the Hamiltonian for the four The Ka’s commute with Kb’s (for i,j ¼þ; −; 0), so ð Þ i j oscillators at k is R R R dkg k;σ dkga k;σ dkgb k;σ ≤Λ ð Þ ≤Λ ð Þ ≤Λ ð Þ m m m m e jkj ¼ e jkj e jkj : ð3:110Þ H Ψ† Ψ Ψ† Ψ Ψ Ψ† Ψ Ψ† k ¼ 2 ½ 1; 1þ 2 ½ 2; 2þ 2 ½ 3; 3þ 2 ½ 4; 4 † † Proceeding in the same way as in Sec. III C we get þðk1 − ik2ÞðΨ1Ψ4 þ Ψ3Ψ2Þ R R Ψ†Ψ Ψ†Ψ dk βa ðk;σÞKa ðkÞ dk βb ðk;σÞKb ðkÞ þðk1 þ ik2Þð 2 3 þ 4 1Þ Ψ σ N aN b Λ þ þ Λ þ þ j ð Þi ¼ e e jRi † † † † þ k3ðΨ1Ψ3 − Ψ2Ψ4 þ Ψ3Ψ1 − Ψ4Ψ2Þ: ð3:115Þ ð3:111Þ At each k let us define the operators with θ − θ − θ † θ † R R i2Ψ i2Ψ i2Ψ i2Ψ −δð0Þ a −δð0Þ b b1 ¼ e 1 b2 ¼ e 2 b3 ¼ e 3 b1 ¼ e 4; 2 dk log β ðk;σÞ 2 dk log β ðk;σÞ N a ¼ e jkj≤Λ 0 N b ¼ e jkj≤Λ 0 ð3:116Þ ð3:112Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −1 k2 2 2 where θ ¼ tan ð Þ and define κ ¼ k1 þ k2. In terms βa’ βb’ k1 and the i s and i s are defined exactly analogous to the of these the Hamiltonian for the four oscillators at k 1 1 þ -d case. becomes Computing the Fubini-Study metric will give two copies 2 of the metric on S and the length of the whole path is 4 m X minimized if the length of each individual path (i.e., βa and H † † † − þ k ¼ 2 ½bi ;bi þ k3ðb1b3 b1b3Þ βb i 1 þ) is minimized. As these are just two copies of what we ¼ † † † † had before in Sec. III, we conclude that the straight line þ k3ðb4b2 − b4b2Þþκðb2b3 − b2b3Þ path taken by our squeezing operator is the shortest path in κ † † − the space of paths generated by the entire SUð2Þ × SUð2Þ þ ðb1b4 b1b4Þ: ð3:117Þ

126001-17 KHAN, KRISHNAN, and SHARMA PHYS. REV. D 98, 126001 (2018) 1 After the usual B-V transformation this becomes − jkj rðkÞ ¼ arctan ω : ð3:124Þ Z jkj m þ k 1 X4 3 ˜ † ˜ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ¼ d k ω ½b ; b ; ð3:118Þ 2 2 2 2 k i i Here jkj¼ k1 þ k2 þ k3. i¼1 This unitary transformation takes the bi to a linear pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ˜ 2 2 2 2 combination of bi via the similarity transformations where ωk ¼ k1 þ k2 þ k3 þ m . The B-V transformation is ˜ ˜ ˜ ˜ † κb1 þ k3b2 † k3b1 − κb2 Ub1U ¼ Ub2U ¼ 0 1 0 1 jkj jkj ˜ κ − † ω − b1 b1 b2k3 þ b4ð k mÞ ˜ ˜ ˜ ˜ B C B C † κb3 þ k3b4 † k3b3 − κb4 B ˜ C B κ † ω − C Ub3U ¼ Ub4U ¼ : Bb2 C B b2 þ b1k3 þ b3ð k mÞ C B C B C jkj jkj B ˜ C B− κ † ω − C Bb3 C B b3 þ b4k3 þ b2ð k mÞC ð3:125Þ B ˜ C B † C Bb4 C 1 B −b4κ − b3k3 þ b ðω − mÞ C B C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB 1 k C B ˜ † C ¼ B † † C: Crucially, there are no creation operators in the linear Bb1 C 2ω ω − m B κ − ω − C B C kð k ÞB b1 b2k3 þ b4ð k mÞ C combinations on the right-hand side, hence this unitary † Bb˜ C B † † C transformation takes the reference state to the target state. B 2 C B b2κ þ b1k3 þ b3ðωk − mÞ C B † C B C Now we can as usual consider an arbitrary path gen- Bb˜ C B− †κ † ω − C @ 3 A @ b3 þ b4k3 þ b2ð k mÞA erated by the squeezing operator † b˜ † † 4 −b4κ − b3k3 þ b1ðωk − mÞ R R R σ 3 3 −i ds d kykðsÞKðkÞ −i d kYkðσÞKðkÞ Ψ σ 0 jkj≤Λ jkj≤Λ ð3:119Þ j ð Þi¼e jRi¼e jRi: ð3:126Þ We will take the reference state jRi is defined as the ground state of the ultralocal Hamiltonian Path ordering is not necessary because all KðkÞ commute. Z Z Evaluating the length of this path using the Fubini-Study 3 ¯ 3 ¯ metric gives Hm ¼ d xðmΨðxÞΨðxÞÞ ¼ d kðmΨðkÞΨðkÞÞ Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ Z 1 X4 3 2 2 3 m † lðjΨðσÞiÞ ¼ dσ 2Vol d kjkj ð∂σY ðσÞÞ : d k b k ;b k : 3:120 k ¼ 2 ½ i ð Þ ið Þ ð Þ 0 jkj≤Λ i¼1 ð3:127Þ This state is annihilated by biðkÞ for i ¼ 1; 2; 3; 4 and ∀ k ∈ R3. This state has no entanglement in x space. The change of variable analogous to the massless case takes The target state is the approximate ground state jTðΛÞi the form defined as XkðσÞ¼jkjYkðσÞð3:128Þ b˜ k TðΛÞ 0 ∀ k∶ k ≤ Λ ið Þj i¼ j j and the length of the path takes the usual form ðΛÞ 0 ∀ ∶ Λ biðkÞjT i¼ k jkj > : ð3:121Þ Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ 1 Ψ σ σ 2 3 ∂ σ 2 The basic observation of this section is that the target lðj ð ÞiÞ ¼ d Vol d kð σXkð ÞÞ : 0 jkj≤Λ state can be reached from the reference state by the unitary transformation ð3:129Þ R 3 This is again a flat Euclidean geometry associated with Λ −i d kr k KðkÞ Tð Þ e jkj≤Λ ð Þ R U R ; 3:122 j i¼ j i¼ j i ð Þ coordinate XkðσÞ and so the geodesic is where the squeezing operator KðkÞ is XkðσÞ¼jkjrkσ: ð3:130Þ

† † † † KðkÞ¼ik3ðb1b3 þ b1b3Þþik3ðb4b2 þ b4b2Þ The complexity is the length of this path, and is given by † † † † sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ iκðb b þ b2b3Þþiκðb b þ b1b4Þð3:123Þ Z 2 3 1 4 jkj 2 C ¼ 2Vol d3k arctan ð3:131Þ jkj≤Λ m þ ωk and rðkÞ is

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This integral can be explicitly evaluated: cMERA-like path that is a natural generalization of these  works to 3 þ 1 dimensions. 2 1 3 2m C ¼ πVol 12im Li2 1 − 18 m − iΛ A. Dirac cMERA in 1 + 1 dimensions 2 2 2 m 2 We will use the version of cMERA that is described for − 24m Λ þ 2m log þ m m − iΛ the Dirac theory in 1 þ 1 dimensions in [7]. Λ The cMERA path jΨðuÞi is −1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi × tan p 2 2 R R Λ u þ m þ m −i du0 dkKðkÞy ðu0Þ jΨðuÞi ¼ Pe −∞ k jRi; ð4:1Þ þ m2ð12Λ þ iπ2mÞþ48ðΛ3 þ im3Þ  Λ 2 where jRi is the reference state defined in Eq. (3.5), KðkÞ is × tan−1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð3:132Þ Λ2 þ m2 þ m the squeezing operator given in Eq. (3.15) and [7], ke−u Its behavior at large Λ takes the form Θ Λ u − ykðuÞ¼gðuÞ Λ ð e jkjÞ; ð4:2Þ π3 6m 12m2 C2 Λ3 − Λ2 Λ where ΘðxÞ is the step function (it is 1 for x ≥ 0 and zero ¼ 6 Vol π þ π2 elsewhere) and gðuÞ is 4m3 Λ 2m3 þ log − þ Oð1=ΛÞ : ð3:133Þ 1 Λ u Λ u π 2 3π − ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie m e m gðuÞ¼ arcsin p þ 2 2 2 : ð4:3Þ 2 Λ2e2u þ m2 Λ e u þ m When m ¼ 0, the exact form of the integral is simple to write down: Note that the integration range of k in (4.1) is superficially not restricted and is from −∞ to þ∞. Effectively there are restrictions, arising from the step function. It is important π3Λ3Vol 2 to be careful about this type of thing in what follows, C ¼ 6 : ð3:134Þ especially when changing the order of integration in u and k in the double integral. The path is parametrized from −∞ Analogous to the previous cases, one can ask whether to 0 by the parameter u ¼ ln σ, where σ is our conventional there exists a bigger class of generators in which our path parameter that runs from 0 to 1. The parametric values squeezing operator is a specific linear combination. It is of the reference state and target state are −∞ and 0 straightforward to see that this is the case, and that the respectively. natural algebra generated by the creation annihilation The target state reached by cMERA is operators is an SOð8Þ, and therefore a natural geometry R R that arises in the space of these more general paths is an S7. −i dkKðkÞYkð0Þ −i dkKðkÞRk We elaborate on this in Appendix E. jΨð0Þi ¼ Pe jkj≤Λ jRi¼Pe jkj≤Λ jRi; ð4:4Þ IV. FERMIONIC CMERA where11 The discussion we have had in the previous section is Z very close in spirit to the so-called cMERA tensor network, 0 Λ 0 Λ k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi which one can view in our language just as an alternate Ykð Þ¼Rkð Þ¼ duykðuÞ¼ arcsin p 2 2 −∞ 2Λ Λ m choice of path connecting the reference state to the target þ state. The cMERA circuit can be viewed as entangling the þ rk; ð4:5Þ neighboring oscillators in x space and doing a scale transformation iteratively. where rk is given by Eq. (3.11). Note that changing the ≤ Λ In what follows we will define the target state at finite order of integration has brought out the promised jkj in cutoff to be some approximate ground state using cMERA. the k-integral. This target state reproduces the true ground This approximate ground state need not be the same approximate ground state as defined previously. But as 11Note that some of the expressions we present here are the cutoff is taken to infinity both will tend to the true superficially in tension with the results in [7]. This is because (we — – — ground state of the theory. Our discussion of cMERA will believe) the results see in particular (121) (123) in [7] should be compared only up to terms suppressed by Λ. A clean way to follow the papers [6,7]. We will compute the length of the see this is to note that the ϕk in (121) and (122) in [7] do not have cMERA path for fermions. The discussions in [6,7] are for the same Λ dependence: in particular, (122) vanishes when 1 þ 1-dimensional fermions, we will also consider a jkj¼Λ, but (121) does not.

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Λ → ∞ → β σ state in the limit as Rk rk in this limit. Note that j þj¼tan ðRk Þ: ð4:12Þ at finite values of the cutoff, it does not lead to the target state that we defined in our 1 þ 1-dimensional discussion in For the cMERA path, from (4.6) and (4.7) we find the previous section. This is unlike in the case of the bosonic case that was considered in [5] where the cutoff arcsin pffiffiffiffiffiffiffiffiffiffik k arcsin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiΛσ 2 2 Λ2σ2 2 target state had an extra parameter available (it was called β k þm − þm j þj¼tan 2 2Λσ M in [5,7]) and this could be used to make the two cutoff target states identical. We give a conceptual explanation for Θðσ − jkj=ΛÞ; ð4:13Þ the absence of this extra scale in the fermionic case at the β end of Appendix A, as a feature implicit in the structure of where the step function arises because in calculating þ we bosonic vs fermionic oscillators. The fact that the states are want the entire u (or σ) dependence to be on the integrand not the same at finite cutoff means that it is not meaningful in (4.6). See Fig. 1. β to compare quantities at finite cutoff. We elaborate on some The j þj of the first (upper) plot here is for the geodesic aspects of this observation in Appendix F. path and the second plot is for the cMERA path. In both At intermediate points on the cMERA path cases the target state is the cMERA target state. The theta R function is responsible for cutting off the curves in the −i dkKðkÞYkðuÞ second figure at the k-axis: in practice this means that Ψ jkj≤Λeu j ðuÞi ¼ UðuÞjRi¼e jRið4:6Þ β Λσ the cMERA path þ has no support on k> , whereas the straight line geodesic path has support on all k. This was with observed in the bosonic case in [5] as well. Z u k Λeu 0 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi YkðuÞ¼ du ykðu Þ¼ u arcsinp 2 2 2 þrk: B. 3+1 dimensions −∞ 2Λe Λ e u þm We can generalize an analogous construction to 3 þ 1 ð4:7Þ dimensions (which we will call a cMERA-like path), and calculate the length of a similar construction for fermions in Note again the k-integration range, again getting fixed by 3 þ 1 dimensions in the Dirac basis. The path jΨðuÞi we the change of integration order. Evaluating the length of take is this cMERA path using Fubini-Study metric gives R R −i u du0 d3kKðkÞy ðu0Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z Ψ −∞ k 0 j ðuÞi ¼ Pe jRi; ð4:14Þ 2 u Λ 2 lcMERA ¼ 3 Vol dujðgðuÞje : ð4:8Þ −∞ where jRi is the ground state of the ultralocal Hamiltonian (3.120), KðkÞ is the squeezing operator given in Eq. (3.123) In some of the calculations, it is useful to note that (4.6) can and be written as R jkje−u u Θ Λ u − −i dkKðkÞYkðuÞΘðe −jkj=ΛÞ ykðuÞ¼gðuÞ ð e jkjÞ ð4:15Þ jΨðuÞi ¼ e jkj≤Λ jRi Λ R −i dkKðkÞYθðuÞ ≡ e jkj≤Λ k jRi: ð4:9Þ and gðuÞ is 2 Λeu In any event, it is possible to calculate this length explicitly, gðuÞ¼− arcsin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Λ u 2 2 2u we present the result in Appendix G where we summarize e Z þ Λ e pffiffiffiffi various explicit formulas for complexities and circuit m Zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lengths. Here we merely note that when m ¼ 0, þ 2 2 2 3p 2 2 ð4:16Þ 2ðΛ e u þ m Þ4 mZ þ Λ e u π ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi − p 2 2 2 gðuÞ¼ 4 : ð4:10Þ with Z ≡ Λ e u þ m þ m. To obtain gðuÞ we follow the prescription of [7] and use So the length of the cMERA path in this case is jkj2 ∂ Λr rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g u ≡ − k ; : 2 ð Þ Λ ∂ ð4 17Þ π ΛVol jkj jkj jkj¼Λeu lcMERA ¼ 6 : ð4:11Þ where rk is given by (3.124). β Now let us plot the j þj for the following two paths. The In the following, we will discuss the massless case for straight line path taking the reference state to the cMERA simplicity because that is enough to make our points. The target state is integrals in the above expressions are explicitly doable and

126001-20 CIRCUIT COMPLEXITY IN FERMIONIC FIELD THEORY PHYS. REV. D 98, 126001 (2018) we have calculated them also for the massive case. In V. FUTURE DIRECTIONS AND SPECULATIONS particular, we have checked that the ground state is attained In this paper, we have calculated various natural notions by the above path even in the massive case. of complexity in the space of unitary circuits for free In the m ¼ 0 case the gðuÞ reduces to fermionic quantum field theories. It seems possible that π these results will be of some use in understanding the g u − : 4:18 holographic significance of complexity (if any). We ð Þ¼ 2Λ u ð Þ e will conclude in this section by listing various future directions beyond the ones we briefly touched upon in −∞ u The path is parametrized from to 0 by the parameter . the Introduction. The parametric values of the reference state and target state −∞ One of the questions that might be of interest is to are and 0 respectively. The target state reached by this understand the physical content hiding behind a cutoff path is dependent quantity like complexity. We have discussed the R question of ambiguities that arise due to the cutoff in the 3 −i d kKðkÞYkð0Þ jΨð0Þi ¼ Pe jkj≤Λ jRi; ð4:19Þ target states and in the path. It is an interesting questions what IR sensible physical quantities we can extract from these without knowing the full UV completion of these where Y 0 is kð Þ theories. There are some parallels here to the case of Z entanglement entropy. Entanglement entropy is also not an 0 π 1 0 k − observable in the conventional sense like complexity, but it Ykð Þ¼ duykðuÞ¼ 2 : ð4:20Þ −∞ 4 Λ k

At finite u,wehave 0.4 R 3 −i d kKðkÞYkðuÞ jΨðuÞi ¼ Pe jkj≤Λeu jRi R 0.3 3 u −i d kKðkÞYkðuÞΘðe −jkj=ΛÞ ¼ Pe jkj≤Λ jRi; ð4:21Þ

0.2 where

Z 0.1 u jkj −u YkðuÞ¼ due gðuÞ: ð4:22Þ Λ ln jkj=Λ k 2 4 6 8 10 Evaluating the length of this cMERA-like path using the Fubini-Study metric gives rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 0.4 8 0 5 5u πΛ 2 lcMERA ¼ 7 Vol dujgðuÞje : ð4:23Þ −∞ 0.3

So the length of the cMERA-like path is 0.2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8π3VolΛ3 lcMERA ¼ 63 : ð4:24Þ 0.1

One interesting feature of this construction is that one k can see by comparing with (3.134) that at finite cutoff, the 2 4 6 8 10 supposedly minimal complexity found in (3.134) is higher β FIG. 1. The j þj of first plot is for the geodesic path and the than the one found here. This is because the target states in second plot is for the cMERA path. In both cases the target state both cases reach the true ground sate only at infinite cutoff, is the cMERA target state. The above figure shows the plots of β Λ 10 1 σ 0 2 and it is not meaningful to compare the lengths to the two j þj vs k for ¼ and m ¼ for the values of ¼ . , 0.4, target states (which are distinct at finite cutoff). We present 0.6, 0.8, 1.0, increasing the peak of the curve signifies increasing an example in 1 þ 1 dimensions that clarifies and illustrates σ. We see that while the cMERA path only acts on momenta jkj σ β this type of cutoff dependence in Appendix F. Λ < , the geodesic path alters j þj for all the different momenta.

126001-21 KHAN, KRISHNAN, and SHARMA PHYS. REV. D 98, 126001 (2018) does capture interesting physical information that is sen- related) models which have a controllable large-N expan- sible in the IR, even though it is UV divergent [9].Itis sion. In particular, one possible case where a strongly tempting to also speculate whether supersymmetric theories coupled gauge theory could be solvable is in the context of with their better defined UV behavior lead to any special the so-called SYK-like tensor models [13]. These theories simplifications when calculating these quantities. But note are not field theories, but they offer the possibility of exact that such simplifications do not seem to happen for solvability [14], and therefore it might be possible to entanglement entropy. calculate complexity for them exactly. Between the insights Two obvious interesting directions from a holographic one can get from free fermionic field theories we discussed perspective is to introduce gauge invariances and inter- here, and these strongly coupled gauged fermionic (and actions. Perturbatively adding interactions seems reason- possibly holographic) quantum mechanics, it will be ably straightforward, but ultimately for holographic interesting to see if one can make any headway into an purposes, we would like to get to a strongly coupled understanding of complexity in strongly coupled holo- (conformal) gauge theory where we will probably need new graphic gauge theories. ideas. The question of gauge invariance on the other hand seems like one where substantial progress might not be too ACKNOWLEDGMENTS hard to make: perhaps an approach along the lines of [10] We thank Shadab Ahamed, Vishikh Athavale, Arpan which uses Wilson line based variables to define entangle- Bhattacharyya, Aninda Sinha, Sudhir Vempati and Aditya ment entropy might be useful here as well for non-Abelian Vijaykumar for discussions and/or correspondence. C. K. gauge theories. In the case of asymptotically free theories, it thanks the Dublin Institute for Advanced Studies (DIAS) seems reasonable that the UV behavior we find here for for hospitality during the final stages of this work. complexity will capture some of the relevant physics there as well. At least in an appropriate free theory limit, it seems Note added.— plausible that the free field approach we have used might After this result was obtained by us (but before our preprint appeared on the arXiv), a paper [15] also work for gauge theories once one takes care of ghost appeared on the arXiv which considers discretized complex- modes as well. A discussion of complexity in Abelian ity for Dirac fermions. Even though we work with Majorana gauge theories was done in [11]. spinors and they seem to be using Dirac, it is reasonable for A simple situation which is quite interesting concep- the results to be similar: we present concrete arguments why tually, and is a natural generalization of the work we have this is so (but in the continuum case), in Sec. III. One of our done in this paper, is to consider perturbative (bosonic and resultsq forffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the discrete case (2.17) can be rewritten as super) string theory on the world sheet. Unlike in the case −1 1 ffiffiffiffiffiffiffiffiffiffiffiffiω cos 2 þ p , which we suspect is the same as of entanglement entropy where target space issues com- 4ω2þΩ2 plicate some of the questions [12], at least in the free limit, Eq. (73) in [15]. See also the related earlier work [16]. the circuit complexity of perturbative string theory should be conceptually easier to characterize and compute. This is APPENDIX A: OSCILLATOR CONVENTIONS because the reference states and target states can be understood purely within the world sheet theory as long We review some elementary facts to establish notation as one makes sure that the central charge is zero. Some (as well as to set up a dictionary to go between notations). work along this direction will be reported elsewhere. The phase space of the bosonic simple harmonic One thing we have ignored in this paper is the question oscillator is comprised of two real variables x and p. of penalty factors for directions in circuit space [4] and the The number of degrees of freedom can be defined as half of possibility of more general Finsler/non-Riemannian met- the dimensionality of the phase space, and so the bosonic rics. At the moment, it is not clear to us what metrics are oscillator has exactly 1 degree of freedom. By combining more natural than others, so we have restricted our attention the two phase space variables one defines the (complex, to simple Riemannian choices. A related question that aka non-Hermitian) creation and annihilation operators a clearly needs a better understanding is the question of what and a†, qualifies as a natural choice of interesting gates when rffiffiffiffi rffiffiffiffi defining complexity. We have used some natural choices ω p ω p a ¼ x þ i pffiffiffiffiffiffi ;a† ¼ x − i pffiffiffiffiffiffi ; ðA1Þ suggested by the problem itself in the discrete and 2 2ω 2 2ω continuum cases, but it will be nice to find a canonical way to choose these gates. which satisfy the standard commutation relations We had two motivations for writing this paper. One was to understand complexity in as a ½a; a†¼1; ðA2Þ prerequisite for understanding holographic complexity [1]. A second reason was the recent emergence of a type of as a result of the canonical commutator, ½x; p¼i. In terms strongly coupled fermionic theories called SYK (and of these the oscillator Hamiltonian can be written as

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1 APPENDIX B: BOGOLIUBOV-VALATIN H ¼ ðp2 þ ω2x2Þ¼ωða†a þ 1=2Þ: ðA3Þ 2 TRANSFORMATIONS FOR FERMIONS ffiffiffiffi Bogoliubov-Valatin (B-V) transformations12 preserve pω’ † We could have absorbed the s in the definition of a, a the anticommutation relations while diagonalizing the above into the x and p, without changing the discussion. pffiffiffi Hamiltonian. Consider the quadratic Hamiltonian for The factor of 2 in the definition can be absorbed only by some n>1: introducing a compensating factor on (say) the right-hand side of the canonical commutator: it changes the symplectic Xn 1 1 α † γ † † γ structure. H ¼ ij½bi ;bjþ2 ijbi bj þ 2 jibibj; ðB1Þ By analogy, the phase space of the simple fermionic i;j¼1 oscillator will contain two real (but now Grassmann) ψ ψ † variables 1 and 2. The number of degrees of freedom where bi and bj are the annihilation and creation operators as given by half the phase space dimensionality is then 1. respectively and follow the anticommutation relations. To We define linear combinations of them which are again ensure that the Hamiltonian is Hermitian, the αi;j and γi;j non-Hermitian: must satisfy

1 1 † α α γ −γ : B2 b ¼ pffiffiffi ðψ 1 − iψ 2Þ;b¼ pffiffiffi ðψ 1 þ iψ 2Þ: ðA4Þ ij ¼ ji ij ¼ ji ð Þ 2 2 Now, (B1) can be written as We want them to satisfy the anticommutator 1 H ¼ Ψ†MΨ; ðB3Þ fb; b†g¼1: ðA5Þ 2

where Ψ and Ψ† are This is accomplished by the canonical anticommutator (note that there is no i on the right-hand side): b Ψ ¼ Ψ† ¼ b† bT ðB4Þ ðb†ÞT fψ i; ψ jg¼δij; ðA6Þ and b, ðb†ÞT are column vectors of size n: The Hamiltonian can be taken again in analogy with the 2 3 2 3 bosonic case † b1 b1 6 7 6 7 6 7 6 † 7 − ωψ ψ ω † − 1 2 6 b2 7 6 b2 7 H ¼ i 1 2 ¼ ðb b = Þ; ðA7Þ 6 7 † T 6 7 b ¼ 6 . 7 ðb Þ ¼ 6 . 7: ðB5Þ 4 . 5 4 . 5 where the zero point constant now has (famously) the † bn b opposite sign. n A significant distinction between the fermionic and bosonic oscillators for our purposes is that for bosonic The matrix M has the form oscillators a scaling of the form αγ M ¼ : ðB6Þ −1 γ† −αT x → λx; p → λ p ðA8Þ

Our goal is to diagonalize the Hamiltonian and identify the for λ ∈ C − f0g preserves the commutation relations. Such normal modes, while preserving the anticommutators. This a scaling is not possible in the fermionic case if one wants is accomplished by the B-V transformations T: to preserve the anticommutation relations. This operational fact is at the basis of our observation that we could not Ψ Φ construct a reference state that depended on some arbitrary ¼ T : ðB7Þ scale M (like it was possible for bosons in [5,7]) when dealing with fermions. It seems possible that this fact is of Here, Φ is the column matrix of normal modes: some deep significance, even outside the context of complexity, though we are not aware of a systematic 12We will only discuss the fermionic case here, an analogous exploration of it. discussion can be made for bosons as well, see [17].

126001-23 KHAN, KRISHNAN, and SHARMA PHYS. REV. D 98, 126001 (2018) ˜ does not qualify as a B-V matrix,13 it has to be of the Φ cb ¼ ˜† ðB8Þ form (B14). ðb ÞT with APPENDIX C: BOGOLIUBOV-VALATIN 2 3 2 3 VS DIRAC MODES ˜ ˜† b1 b1 6 7 6 7 The spinor ΨðxÞ was Fourier transformed in Sec. III A as 6 ˜ 7 6 ˜† 7 6 b2 7 6 b2 7 ˜ 6 7 ˜† T 6 7 Z b ¼ 6 . 7 ðb Þ ¼ 6 . 7: ðB9Þ dk 4 . 5 4 . 5 ΨðxÞ¼ pffiffiffiffiffiffi ΨðkÞeikx: ðC1Þ . 2π b˜ ˜† n bn A more standard basis is to expand in the basis of Dirac To do the diagonalization, T must satisfy modes, Z dk 1 † ffiffiffiffiffiffi ffiffiffiffiffiffi † ikx D ¼ T MT; ðB10Þ ΨðxÞ¼ p p ðakuk þ b− v−kÞe ; ðC2Þ 2π 2E k where D is the diagonal matrix of normal mode frequen- where a , b are the annihilation operators of the ground cies. This is immediate from k k state and uk and vk are the spinors satisfying the equation 1 1 1 Ψ† Ψ Φ† † Φ Φ† Φ ð=k þ mÞu ¼ 0 and ð=k − mÞv ¼ 0 ðC3Þ H ¼ 2 M ¼ 2 T MT ¼ 2 D : ðB11Þ k k with the normalization A general linear transformation is of the form ¯ 2 ¯ −2 ˜ ˜† uu ¼ m and vv ¼ m: ðC4Þ bi ¼ Aijbj þ Bijbj ðB12Þ These can be explicitly calculated to be which can be written in the matrix form pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi E þ m − E − m ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi b AB ˜ uk ¼ p and v−k ¼ p : ðC5Þ b E − m E þ m † ¼ ˜ : ðB13Þ ðb ÞT B A ðb†ÞT Comparing (C1) and (C2) we get This means that T looks like Ψ1 k 1 ð Þ ffiffiffiffiffiffi † ¼ p ðakuk þ b−kv−kÞðC6Þ AB Ψ2ðkÞ 2E T ¼ : ðB14Þ B A from which Using the above explicit form of T, a direct calculation Ψ Ψ shows that the condition for T to unitary is identical to the ak ¼ Ak 1ðkÞþBk 2ðkÞ condition that the anticommutators are preserved under † b− ¼ −BkΨ1ðkÞþAkΨ2ðkÞðC7Þ (B13). It is useful when doing this calculation to remember k manipulations like the following: for column vectors x and follows, with the Ak and Bk given in Sec. III A. It is clear y if x ¼ Ay, i.e., xi ¼ Aijyj, then from this that the ground state obtained there via a Bogoliubov-Valatin transformation in the main body of † † † † † † † † xj ¼ðAyÞj ¼ðy A Þj ¼ yi Aij ¼ yi Aji ¼ Ajiyi ; ðB15Þ the paper is just the usual ground state (as it should be). We have phrased this discussion in 1 þ 1 dimensions, and therefore, ðx†ÞT ¼ Aðy†ÞT. Note that the transposition but an entirely similar discussion holds in higher dimen- in this last equation is necessary because we want to sions as well. The only difference is that in higher interpret it as a column vector equation, as in the second dimensions (say 3 þ 1d) the Dirac u, v modes are not definition in (B5). uniquely fixed (ie., the u and v come with an index as is In conclusion, a B-V transformation for fermions is a familiar in 3 þ 1d where the index has two values), and so unitary matrix T of the form (B14), qhich can diagonalize the Hamiltonian [as captured by (B10)]. The crucial point is 13Note that the Hamiltonian being Hermitian, it can always be that any unitary matrix that does the job of diagonalization diagonalized by a unitary.

126001-24 CIRCUIT COMPLEXITY IN FERMIONIC FIELD THEORY PHYS. REV. D 98, 126001 (2018) there are some more arbitrary choices that need to be made consider a natural, yet more general class of generators in the choice of their specific form. that can be used as the primitive gates. The squeezing operator we constructed in the main body of the paper is a APPENDIX D: GEODESICS ON CP1 specific linear combination of these generators and can be viewed as a restricted class of paths. The metric we derived in Sec. III C is the Fubini-Study 1 The basic observation is that the most general (quadratic) metric on CP : generators that we can construct at each k from the creation and annihilation operators are built from β0 β 0 ds2 ¼ þ þ dσ2; ðD1Þ 1 β 2 2 † † † ð þj þj Þ bi bj ;bibj;bi bj; ðE1Þ β β0 d þ β where þ is dσ . Since þ is a complex variable, we can for i, j ¼ 1, 2, 3, 4. Using the antisymmetry property, β 14 † † † † write þ ¼ x þ iy, to bring the metric to a real form: − − ≠ bi bj ¼ bj bi and bibj ¼ bjbi for i j and bibi ¼ † † 2 2 b b ¼ 0, we can count that there are 6 þ 6 þ 16 ¼ 28 2 dx þ dy i i ds ¼ 2 2 2 : ðD2Þ total generators. This is a strong suggestion that the algebra ð1 þ x þ y Þ of these generators forms an SOð8Þ algebra, which also has The geodesics on this metric satisfy the usual Euler- the same number of generators. Lagrange equations To explicitly see the algebra, all one has to do is define

2 α β γ Γ † Γ − † d x dx dx 2i−1 ¼ bi þ bi 2i ¼ iðbi bi ÞðE2Þ þ Γα ¼ 0; ðD3Þ dσ2 βγ dσ dσ and note that it follows from the anticommutation relations where xα ≡ ðxðσÞ;yðσÞÞ. For the straight line path given in that the Γ’s satisfy the Clifford algebra β (3.64), þ takes the form fΓa; Γbg¼2δab ðE3Þ β − σ þ ¼ i tan rk : ðD4Þ for SOð8Þ. The generators of SOð8Þ are then as usual 0 − σ So we get x ¼ , y ¼ tan rk . This can be immediately defined via checked to satisfy the above geodesic equation. Now for alternate path generated by BðkÞ, the form of β þ Σ − i Γ Γ is given in (3.58): ab ¼ 4 ½ a; b; ðE4Þ πσ i sin r sin Σ β k 2 where a, b runs from 1 to 8. ab is an antisymmetric tensor, þ ¼ πσ πσ : ðD5Þ i cos 2 − cos rk sin 2 which again yields the 28 independent generators of SOð8Þ. Γ † The fact that the a defined from the bi and bi satisfy the This gives Clifford algebra guarantees that the generators correspond- Γ Γ πσ πσ 2 πσ ing to ða bÞ are proportional to the identity (when a ¼ b, sin r sin 2 cos 2 sin r cos rsin 2 else they are zero) and therefore only contribute to a x ¼ 2 2 πσ 2 πσ y ¼ 2 2 πσ 2 πσ : cos rsin 2 þ cos 2 cos rsin 2 þ cos 2 decoupled phase. 1 1 ðD6Þ In analogy with the þ -dimensional case where we found an SUð2Þ algebra and an associated CP1 geometry Plugging this into the left-hand side of the geodesic on which it naturally acts, this indicates that the natural 7 equation above, we find that the equation is not satisfied. geometry arising from the paths here is an S where the We see that the straight line path generated by squeezing SOð8Þ acts linearly. It will be interesting to see whether the operator KðkÞ is the geodesic of the CP1 metric but the path squeezing operator we have defined in the text leads to a taken by BðkÞ is not. geodesic or nongeodesic path when considered as a path in this geometry. Based on genericity alone, there is no reason APPENDIX E: SOð8Þ why the squeezing operator path is geodesic, but it will be nice to (dis)prove this explicitly. The calculation will be the In this Appendix, we consider the massive 3 þ 1- generalization of the one we did for the SUð2Þ case in 1 þ 1 dimensional fermions discussed in the main text and dimensions, but unlike the three generators there, now we have 28 generators. So one has to either go for more brute 14Said another way, manifesting the complex structure of the force than we have been able to muster, or find an alternate metric is not necessary to answer metric questions. clever way to do this, or a suitable combination of both.

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APPENDIX F: Cutoff Dependence from a UV sensitive quantity like complexity: note that entanglement entropy is also typically UV divergent, but Notice that in Sec. IV B the Λ-dependent length of the its subleading behavior is proportional to the area. So it cMERA-like path is shorter than that of the straight line will be interesting to understand what kind of UV- path calculated in Sec. III. This is a result of the fact that at insensitive information one can extract from quantities finite cutoff, the target states do not match and therefore it is like complexity. not meaningful to compare them, even though they do match at infinite cutoff. Let us define a one parameter family of target states for APPENDIX G: COMPLEXITIES AND CIRCUIT the 1 þ 1-dimensional Dirac theory, with a cutoff Λ such LENGTHS: SELECTED SUMMARY that the target state approaches the ground state of the field 1. Dirac complexity in 1 + 1 dimensions theory in the Λ → ∞ limit: R The following is a real quantity: 2 k ðΛÞ −i KðkÞrðkÞð1þα 2Þ jTα i¼e jkj≤Λ Λ jRi; ðF1Þ m Λ2 Λ C2 ¼ − Vol log þ 1 tan−1 2 m2 m where rðkÞ is given in Eq. (3.11) and α ∈ R. In the limit Λ → ∞ the coefficient of KðkÞ in the exponent becomes 1 −1 Λ − mVol − logð16Þtan rðkÞ thereby reproducing the ground state of the field 4 m theory. −2itan−1ðΛÞ 2itan−1ðΛÞ We will use the same reference state as the one we used − iLi2ð−e m ÞþiLi2ð−e m Þ in the main body of the paper and calculate the length of the 2 minimal path from this reference state to this new target 1 −1 Λ þ ΛVol tan ðG1Þ state. To simplify the integrals we will do this in the m ¼ 0 2 m case. Again using the Fubini-Study metric and following the by now standard derivation, we get the straight line path which gives in the large Λ expansion to be the minimal path. The length of this minimal path is π2 4m 2m 4m sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ 2 Λ − O 1 Λ 2 2 C ¼ 8 Vol þ π log Λ π þ ð = Þ : ðG2Þ 2 1 α k lα ¼ Vol rðkÞ þ 2 : ðF2Þ jkj≤Λ Λ

For m ¼ 0 2. Circuit length with the B generator in 1 + 1 dimensions sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π2 2 1 Λ 1 α α2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lα ¼ 8 Vol þ 3 þ 5 π 2 Λ2 2 − Λ 2 Λ m ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ m C ¼ Vol þ log p 2 2 ðG3Þ 1 2 2 Λ Λ 2 1 2 þ m þ 1 α α2 ¼ C þ 3 þ 5 ; ðF3Þ which gives in the large Λ expansion qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where C ¼ π ΛVol is the complexity that we calculated π2 m 8 C2 ¼ Vol Λ þ m log þ Oð1=ΛÞ : ðG4Þ in the text. 4 2Λ It is easy to see that this length can be bigger or even smaller (e.g., α ¼ −5=3) than the complexity C, depending on the value of α. What is happening here is that the cutoff 3. Massive Dirac complexity in 3 + 1 dimensions dependence in the definition of the target state is making The following is again a real quantity: the comparison between path lengths at finite cutoff  meaningless between states that are distinct at finite cutoff. 2 1 3 2m C ¼ πVol 12im Li2 1 − This is what is manifest in Sec. IV B. There also the target 18 m − iΛ states were different and so we do not expect the cMERA- 2m like lengths to be directly comparable to the complexity at − 24m Λ2 þ 2m2 log þ m2 m − iΛ finite cut off. In Sec. IVA it turned out that cMERA length was longer than the complexity and in Sec. IV B it is the Λ × tan−1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ m2ð12Λ þ iπ2mÞ opposite. Λ2 2 þm þ m  This is a demonstration of the fact that UV sensitive Λ 2 quantities are not stable under change of cutoff. It is 48 Λ3 3 −1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð þ im Þ tan p 2 2 ðG5Þ interesting to ask what kind of information one can extract Λ þ m þ m

126001-26 CIRCUIT COMPLEXITY IN FERMIONIC FIELD THEORY PHYS. REV. D 98, 126001 (2018) and it yields in the large Λ expansion π3 6m 12m2 4m3 Λ 2m3 C2 ¼ Vol Λ3 − Λ2 þ Λ þ log − þ Oð1=ΛÞ : ðG6Þ 6 π π2 π 2m 3π

4. cMERA circuit length in 1 + 1 dimensions For the massive case, the explicit expression for the cMERA circuit length is rffiffiffiffiffiffiffiffi rffiffiffiffiffi rffiffiffiffiffi Vol pffiffiffiffi m pffiffiffiffiffiffi iΛ iΛ l ¼ 2 Λ cot−1 þ 3 im tanh−1 − tan−1 ðG7Þ cMERA 6 Λ m m and it has the large Λ form π2 pffiffiffiffiffiffiffiffiffiffi 9 8 2 Λ − 3 2 Λ m m O 1 Λ lcMERA ¼ 6 Vol m þ 2 þ π þ ð = Þ : ðG8Þ

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