Entanglement transitions with free fermions Joseph Merritt and Lukasz Fidkowski Department of Physics University of Washington Seattle WA 98195-1560 USA

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Entanglement transitions with free fermions

Joseph Merritt and Lukasz Fidkowski

Department of Physics, University of Washington, Seattle, WA 98195-1560, USA

(Dated: October 2022)

We use Majorana operators to study entanglement dynamics under random free fermion unitary

evolution and projective measurements in one dimension. For certain choices of unitary evolution,

namely those which swap neighboring Majorana operators, and measurements of neighboring Ma-

jorana bilinears, one can map the evolution to the statistical model of completely packed loops

with crossings (CPLC) and study the corresponding phase diagram. We generalize this model us-

ing the language of fermionic Gaussian states to a general free fermion unitary evolution acting on

neighboring Majorana operators, and numerically compute its phase diagram. We ﬁnd that both

the Goldstone and area law phases persist in this new phase diagram, but with a shifted phase

boundary. One important qualitative aspect of the new phase boundary is that even for the case of

commuting measurements, the Goldstone phase persists up to a ﬁnite non-zero measurement rate.

This is in contrast with the CPLC, in which non-commuting measurements are necessary for real-

izing the Goldstone phase. We also numerically compute the correlation length critical exponent at

the transition, which we ﬁnd to be near to that of the CPLC, and give a tentative symmetry based

explanation for some diﬀerences in the phase transition line between the CPLC and generalized

models.

I. INTRODUCTION

Recently the study of ‘hybrid’ quantum circuits, in-

volving both unitary dynamics and projective measure-

ments, has received a great deal of attention [1–3]. By

focusing on the ensemble of quantum trajectories of pure

states deﬁned by the various measurement outcomes one

can study new types of non-equilibrium phase transitions,

with the canonical example being the ‘entanglement tran-

sition’. In the entanglement transition, the ensemble-

averaged entanglement entropy changes from scaling as

an area law to scaling as a volume law as the measure-

ment rate is decreased. A closely related concept is that

of a ‘puriﬁcation’ transition, where instead of pure quan-

tum state trajectories one studies the purifying behavior

of an initial maximally mixed state.

In the special case of free fermion dynamics [4] - i.e.

unitaries which are exponentials of bilinears of the cre-

ation and annihilation operators, and measurements only

of Fock space mode occupation numbers - the mixed (or

‘volume law’) phase is known to be unstable to any non-

zero measurement rate [5]. However, such free fermion

dynamics can still accommodate an interesting phase

transition from a purifying phase to a so-called ‘Gold-

stone’ phase [6]. The latter still exhibits purifying be-

havior, but on time scales parametrically longer in system

size. More precisely, in the Goldstone phase the entropy

for a system of length Lafter a time of order Lscales like

log L(whereas in the purifying phase this entropy would

be close to 0, i.e. the state would have approximately pu-

riﬁed long before this time scale was reached). This phase

transition can be realized in a speciﬁc free fermion model,

one that can be solved [7] by exact mapping to a known

statistical mechanical model, the completely packed loop

model with crossings (CPLC) [6].

A natural question one may ask is, how generic is the

CPLC phase diagram in the context of free fermion hy-

brid dynamics? In other words, do the area law and

Goldstone phases persist when the dynamics is deformed

slightly away from the speciﬁc point dual to the CPLC

model? Is the phase transition still continuous, and is it

in the same universality class as that of the CPLC model?

In this work, we investigate these questions by extend-

ing the CPLC-dual free fermion model to a more general

family of free fermion models. Speciﬁcally, following [7],

the CPLC-dual model has a convenient description in

terms of a one dimensional chain of Majorana fermions,

as follows: the unitary gates swap a neighboring pair of

Majoranas (γ1→γ2, γ2→ −γ1), and the measurements

measure the occupation number of a free fermion mode

deﬁned by a neighboring pair of Majoranas. These gates

are implemented with respect to one pairing of Majo-

ranas and its complementary pairing in an alternating

fashion, as described in detail below. The phase diagram

is a function of two parameters, pand q, which control,

respectively, the rate of measurements and the asymme-

try between the two complementary pairings of Majo-

ranas. Our generalized model replaces the swap gate,

which may be thought of as a π

2rotation in the SO(2)

that rotates γ1into γ2, by a rotation by a random an-

gle inside this SO(2). The measurement gates are as in

the CPLC model, and the phase diagram is once again a

function of the two parameters pand q.

Our generalized model no longer admits an easily solv-

able statistical mechanical dual, although in principle

some statistical mechanical dual should exist, as dis-

cussed below. To study it, we therefore instead leverage

the free fermion nature of the dynamics to perform eﬃ-

cient Monte Carlo simulations using the Gaussian state

formalism. The essential feature of both the CPLC and

our generalized models which makes this possible is the

fact that, for a particular quantum trajectory, the many-

body quantum state of 2NMajoranas remains Gaussian,

meaning that it can be eﬃciently encoded in a correla-

arXiv:2210.05681v3 [cond-mat.str-el] 15 Nov 2022

tion matrix with O(N2) entries. This allows us to avoid

having to simulate dynamics in a Hilbert space exponen-

tially large in N. Ultimately this is just a consequence

of the free fermion nature of the dynamics.

We ﬁnd that the general features of the CPLC phase

diagram persist in the generalized model. The area law

and Goldstone phases remain, but the phase transition

between them shifts slightly. An important qualitative

diﬀerence is that the Goldstone phase persists down to

ﬁnite p < 1 for q= 0,1 in the generalized model, in

contrast to the CPLC. This implies that the unitaries

in the generalized model are more scrambling in some

sense than those of the CPLC, because they can support

the Goldstone phase with commuting projectors (i.e. at

q= 0,1). The CPLC, on the other hand, requires non-

commuting measurements [8–11] to support the Gold-

stone phase. Our result is consistent with the fact that

a volume entanglement law cannot be maintained in free

fermi systems at ﬁnite measurement rates [5,12], and

comports with the results found in [13] in the case of con-

tinuous monitoring, and bears resemblance to the results

of [14] and [15] in the case of non-unitary free fermion

evolution (see also [16] for additional exploration of free

fermion phases and phase transitions with weak measure-

ments, and [17,18] for further study of the phase transi-

tion from an area law phase to a logarithmic phase). We

would also like to note that the transitions found in the

CPLC have also been studied using entanglement neg-

ativity and other measures in the context of monitored

dynamics in [19].

We also perform a ﬁnite size scaling analysis that al-

lows us to extract a correlation length critical expo-

nent ν≈2.4 for the generic transition between the two

phases. The accuracy of our analysis is not suﬃcient

to deﬁnitively conclude that this corresponds to a diﬀer-

ent universality class from the CPLC model, which has

νCPLC ≈2.75 [6].

The rest of this paper is structured as follows. In Sec-

tion II we review the CPLC model and construct the

duality mapping between this model and a free fermion

hybrid dynamics. In particular, we highlight the connec-

tion between the ‘spanning number’ in the CPLC model

and the entropy in the quantum model. In Section III we

discuss more general free fermion models, and introduce

the Guassian state formalism that allows us to eﬃciently

simulate them. In Section IV we present the results of

our Monte Carlo numerical simulations of the more gen-

eral free fermion models. In Section Vwe summarize our

results and consider future directions. In particular, we

discuss a qualitative change in the shape of the phase

boundary between the CPLC-dual and generalized mod-

els, and propose a symmetry-based explanation of this

diﬀerence.

II. EXACTLY SOLVABLE MODEL OF FREE

FERMION HYBRID DYNAMICS

This Section outlines a particular implementation of

the duality between the CPLC and a quantum model of

Majorana worldlines, ﬁrst proposed in [7]. We consider a

one dimensional chain of Nspinless fermions, and write

the operator algebra in terms of 2NMajorana fermions

γk(k= 1,...,2N). These are related to the usual cre-

ation and annihilation operators aj, a†

j(j= 1, . . . , N)

by:

γ2j−1=aj+a†

γ2j=i(aj−a†

aj=1

2(γ2j−1−iγ2j)

a†

j=1

2(γ2j−1+iγ2j)

We have iγ2j−1γ2j= (−1)nj= 1 −2njwhere nj=a†

jaj

is the occupation number operator at site j, taking eigen-

values 0 and 1.

Hybrid unitary-measurement circuit

We take periodic boundary conditions, so that a sub-

script of N+ 1 below is to be interpreted as 1. The

time step is labeled by a positive integer, and the proto-

col depends on the parity of this time step. pand qare

two real numbers between 0 and 1 which serve as con-

trol parameters. For convenience, let us ﬁrst deﬁne the

two-Majorana unitary gate Ur,r+1, (r= 1,...,2N) by

Ur,r+1 =1

√2(1 −γrγr+1) (1)

This gate acts as follows:

Ur,r+1γrU†

r,r+1 =γr+1

Ur,r+1γr+1U†

r,r+1 =−γr

while commuting with all γj,j6=r, r + 1.

Odd time steps: We perform 2-Majorana gates on

all pairs (2j−1,2j) (j= 1, . . . , N) of nearest neighbor

Majoranas. For each such pair (2j−1,2j) the gate is

chosen randomly from 3 possibilities: (1) with probability

pwe act with U2j−1,2j; (2) with probability (1 −p)qwe

measure iγ2j−1γ2j; and (3) with probability (1−p)(1−q)

we do nothing, i.e. act with the identity gate.

Even time steps: We perform 2-Majorana gates on

all pairs (2j, 2j+ 1) (j= 1, . . . , N) of nearest neighbor

Majoranas. For each such pair (2j, 2j+ 1) the gate is

chosen randomly from 3 possibilities: (1) with probability

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EntanglementtransitionswithfreefermionsJosephMerrittandLukaszFidkowskiDepartmentofPhysics,UniversityofWashington,Seattle,WA98195-1560,USA(Dated:October2022)WeuseMajoranaoperatorstostudyentanglementdynamicsunderrandomfreefermionunitaryevolutionandprojectivemeasurementsinonedimension.Forcertainchoices...

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Entanglement transitions with free fermions Joseph Merritt and Lukasz Fidkowski Department of Physics University of Washington Seattle WA 98195-1560 USA

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