Topological invariants for SPT entanglers Carolyn Zhang Department of Physics Kadanoff Center for Theoretical Physics University of Chicago Chicago Illinois 60637 USA

2025-05-06 0 0 1021.38KB 26 页 10玖币

侵权投诉

Topological invariants for SPT entanglers

Carolyn Zhang

Department of Physics, Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA

(Dated: October 7, 2022)

We develop a framework for classifying locality preserving unitaries (LPUs) with internal, unitary symmetries

in ddimensions, based on (d−1) dimensional “ﬂux insertion operators” which are easily computed from the

unitary. Using this framework, we obtain formulas for topological invariants of LPUs that prepare, or entangle,

symmetry protected topological phases (SPTs). These formulas serve as edge invariants for Floquet topological

phases in (d+ 1) dimensions that “pump” d-dimensional SPTs. For 1D SPT entanglers and certain higher

dimensional SPT entanglers, our formulas are completely closed-form.

I. INTRODUCTION

In recent years, there has been much progress in not

only classifying topological quantum phases, but also in un-

derstanding methods for detection and preparation of such

phases. Methods for preparing, or “entangling,” a state in a

topological phase are deeply connected to the entanglement

properties of the phase[1]. For example, it was shown that

a state in a long-range entangled phase, such as a 2D topo-

logical order, cannot be prepared by any ﬁnite depth quantum

circuit (FDQC). Instead, such a state can only be prepared

either by a circuit whose depth scales with the system size[2–

4] or by supplementing an FDQC with measurements and

post-selection[5–9]. On the other hand, short-range entangled

phases such as symmetry-protected topological phases (SPTs)

can be prepared by FDQCs, but these circuits must contain lo-

cal gates that break the symmetry[1,10].

Interestingly, broad classes of SPT phases can be entangled

by FDQCs that, though containing gates that break the sym-

metry, respect the symmetry as a whole[1,11–13]. Some-

times, an SPT cannot be entangled by an FDQC that respects

the symmetry as a whole, but can be entangled by a more

general locality preserving unitary (LPU) which respects the

symmetry[14–16]. When the locality is strict, without expo-

nentially decaying tails, these nontrivial LPUs are also known

as quantum cellular automata (QCA)[17–19].

LPUs have also recently received attention because they

describe the stroboscopic boundary dynamics of many-body

localized, periodically driven systems, also known as Flo-

quet systems[17,20,21]. LPUs with symmetry describe the

boundary dynamics of these kinds of Floquet systems when

the drive is constrained to respect the symmetry[22–25]. Non-

trivial Gsymmetric Floquet systems in dspatial dimensions

can “pump” (d−1) dimensional GSPTs to the boundary every

period[22,23,26]. For these systems, the stroboscopic bound-

ary dynamics is described by a Gsymmetric (d−1) dimen-

sional SPT entangler. These kinds of boundary unitaries have

been classiﬁed and studied in exactly solvable models[23,24]

and matrix product unitaries[27–31].

Although LPUs with various kinds of symmetry have been

classiﬁed, there exist very few explicit formulas for topologi-

cal invariants of these LPUs. For bosonic systems in 1D with-

out any symmetry, there is an explicit formula that takes as

input an LPU and produces the GNVW index, that classiﬁes

LPUs without any symmetry[17,20]. In this work, we will

provide similar formulas for topological invariants of LPUs

with symmetry. These formulas also serve as boundary invari-

ants for many-body localized Floquet systems with symmetry.

For simplicity, we will consider only LPUs with strict locality.

In general, nontrivial Gsymmetric strict LPUs fall into two

classes: those that entangle SPTs and those that do not entan-

gle SPTs. For example, all nontrivial strict LPUs with discrete

symmetries in 1D entangle SPTs[22], while nontrivial strict

LPUs with U(1) symmetry in 1D are not related to SPTs[25].

In this work, we will obtain formulas for topological invari-

ants of Gsymmetric strict LPUs that are SPT entanglers. To

specify that we are restricting to this particular subset of G

symmetric strict LPUs, we will refer to them as Gsymmet-

ric SPT entanglers in the remainder of this work. In partic-

ular, we will focus on symmetric entanglers for bosonic “in-

cohomology” SPTs. These SPTs are classiﬁed by elements

of Hd+1(G, U(1)), and we will show that for the dimensions

and symmetries we consider, the topological invariants com-

puted from our formulas completely specify this element. In

short, in this work, we present formulas that take as input a G

symmetric SPT entangler and produce topological invariants

that completely specify the SPT phase it entangles.

We have two guiding principles. First, of course, our for-

mulas must produce the same result for two equivalent LPUs,

that entangle the same SPT phase. Roughly speaking, our for-

mulas must be insensitive to modiﬁcation of the input uni-

tary by any strictly local, symmetric unitary. This means that

they will also be insensitive to Gsymmetric FDQCs, which

are FDQC constructed out of such symmetric local unitaries.

Second, we try to make our formulas as closed-form as possi-

ble. This means that whenever possible, they only involve the

truncation of operators that are products of on-site operators,

such as on-site symmetry operators. In particular, when Gis

unitary, we do not truncate the SPT entangler.

These two guiding principles, along with the fact that we

classify Gsymmetric SPT entanglers rather than SPT states,

differentiate the work we present here from previous related

work. In particular, most work related to classifying SPTs via

their entanglers do not assume that the entanglers respect the

symmetry[32–37]. This extra assumption allows us to make

our invariants more explicit. Since broad classes of SPTs can

be entangled by symmetric entanglers, we do not lose much

generality in making this assumption. Ref. 13 also assumed

that the SPT entangler is symmetric as a whole. Using the

SPT entangler truncated to a ﬁnite disk, they obtained a corre-

sponding “anomalous edge representation of the symmetry.”

arXiv:2210.02485v1 [cond-mat.str-el] 5 Oct 2022

They then showed how to get the cocycle labeling the SPT

phase from the anomalous edge representation of the sym-

metry. We discuss the relation between our methods and the

anomalous representation of the symmetry on the edge in Ap-

pendix. C. Unlike Ref. 13, we do not truncate the SPT entan-

gler to compute our invariants, when the symmetry is unitary.

In some cases, when the entangler is actually a nontrivial QCA

(even in the absence of symmetry), it cannot be truncated at

all. Furthermore, our invariants are actually gauge invariant

quantities: unlike the cocycles computed in Ref. 13, which

are only deﬁned up to a coboundary, our invariants have no

ambiguity.

The rest of this paper is organized as follows. We include

in this section a summary of our main results and an illustra-

tive example of an invariant for 1D SPT entanglers. In Sec. II,

we describe our general framework for classifying SPT entan-

glers with ﬂux insertion operators. We then apply this frame-

work to SPT entanglers related to 1D SPTs in Sec. III and 2D

SPTs with discrete, abelian, unitary symmetries in Sec. IV.

We include some results regarding fermionic SPT entanglers

in Sec. V, before concluding with interesting open questions

in Sec. VI. We defer most of the proofs, including the explicit

derivations of relations between our invariants and known SPT

invariants, to the appendices.

A. Summary of results

Our main result is a framework for classifying LPUs with

symmetry, which can be applied to both SPT entanglers and

LPUs that are not related to SPTs. Using this framework, we

obtain topological invariants for various kinds of SPT entan-

glers. These topological invariants can be divided into two

main groups.

Our ﬁrst group of invariants apply to 1D SPT entanglers

with discrete symmetries. When the symmetry is unitary

and discrete, we obtain closed form formulas for topolog-

ical invariants that take as input only a global SPT entan-

gler Uand global symmetry operators. These formulas are

given in Eq. (3.5) for abelian symmetries and (3.10) for non-

abelian symmetries. We also have an invariant for time rever-

sal SPT entanglers, written in Eq. (3.15), but it is not com-

pletely closed form because it involves truncating the entan-

gler. These invariants can be easily leveraged to obtain in-

variants of SPT entanglers in higher dimensions described by

decorating domain walls with 1D SPTs, written in Eq. (3.20).

Our second group of invariants apply to 2D SPT entanglers

with discrete, unitary, abelian symmetries, beyond those with

domain walls decorated by 1D SPTs. The explicit formulas

for these invariants are given by Eqs. (4.3) and (4.10), and

are not completely closed form in that they involve truncation

of certain non-onsite operators. Again, these invariants can

be leveraged to obtain invariants of SPT entanglers in higher

dimensions described by decorating domain walls with 2D

SPTs.

We also obtain a closed form formula, given by Eq. 5.3, for

the Z2invariant classifying SPT entanglers with only fermion

parity symmetry in 1D. Nontrivial SPT entanglers in this case

entangle the Kitaev wire, and differ from the others considered

in this work in that they are nontrivial QCAs[4,21].

B. Example: Z2×Z2SPT entangler in 1D

To give a ﬂavor of the kinds of formulas for topological

invariants studied in this work, we begin with a simple ex-

ample. In this example, we present a set of formulas that

compute topological invariants for SPT entanglers in 1D with

Z2×Z2symmetry. The classiﬁcation of 1D SPTs with this

symmetry is Z2: there is one trivial phase and one nontriv-

ial phase. We will show that our closed form formulas com-

pute a set of U(1) phases {c(g, h)}=nω(g,h)

ω(h,g)o, where

ω(g, h)∈H2(Z2×Z2, U(1)) labels the SPT phase1. The

set of phases {c(g, h)}for all g, h ∈Z2×Z2completely

deﬁnes the SPT phase.

The physical setup consists of a ﬁnite, periodic 1D chain

with an even number of spin-1/2’s. The two global Z2sym-

metries, generated by unitary operators Ug1and Ug2, are spin

ﬂips on all the even sites and all the odd sites respectively:

Ug1=Y

reven

σx

rUg2=Y

rodd

σx

r,(1.1)

where σx

ris the Pauli xoperator on site r. An example of a

symmetric, gapped Hamiltonian with a trivial ground state is

given by

H0=−X

σx

r.(1.2)

The ground state of H0is the state with all the spin-1/2’s in

the +1 eigenstate of σx

r. A Z2×Z2symmetric SPT entangler

is given by

U=Y

eiπ

4(−1)rσz

rσz

r+1 .(1.3)

Notice that Uis symmetric under both global Z2symme-

tries, but its individual gates eiπ

4(−1)rσz

rσz

r+1 are not symmet-

ric under either symmetry. This is expected, because for Uto

be a nontrivial SPT entangler, it must contain gates that break

the symmetry.

To conﬁrm that Uindeed entangles the Z2×Z2SPT, notice

that Utransforms H0into HSPT, whose ground state is the

well-known cluster state:

HSPT =U†H0U=−X

σz

r−1σx

rσz

r+1.(1.4)

Our formula for c(g, h)takes as input Uand two restricted

symmetry operators UA,g and UB,h, where g, h ∈Z2×Z2.

UA,g and UB,h are restrictions of global symmetry operators

Ugand Uhto intervals Aand Brespectively. Because each of

1See Appendix Afor a review of group cohomology

FIG. 1: c(g, h)in Eq. (1.6) is deﬁned using symmetry

operators for sufﬁciently large, overlapping intervals Aand

Bon a 1D lattice.

the global symmetry operators is a product of on-site opera-

tors, these restrictions can be done unambiguously. In partic-

ular,

UA,g1=Y

r∈A

reven

σx

rUB,g2=Y

r∈B

rodd

σx

r.(1.5)

It is important that we choose Aand Bto be sufﬁciently

overlapping intervals in 1D, as illustrated in Fig. 1. For con-

creteness, let A= [0,2a]and B= [a, 3a]where ais an odd

integer and a1(we will later precisely deﬁne the relevant

length scales). Then c(g, h)is given by

c(g, h) = Tr U†UA,gUUB,hU†U†

A,gUU†

B,h,(1.6)

where Tr is a trace normalized by the dimension of the total

Hilbert space, so that Tr( ) = 1.

Let us check that Eq. (1.6) produces the correct c(g, h)for

U=and for the SPT entangler in Eq. (1.3). It is easy to

see that, because UA,g and UB,h commute, c(g, h)=1for all

g, h ∈Z2×Z2if U=. On the other hand, if Uis the SPT

entangler deﬁned in (1.3), then Uattaches a σz

rof the opposite

sublattice near the left and right endpoints of a restricted sym-

metry operator. For example, U†UA,gU=σz

−1UA,gσz

2a+1.

It follows that c(g1, g2) = −1 = c(g2, g1)and c(g, h) = 1

otherwise, which matches with the set {c(g, h)}deﬁning the

1D Z2×Z2SPT.

Notice that Eq. (1.6) satisﬁes our two guiding principles.

First, it is insensitive to modiﬁcations of Uby local, symmet-

ric unitaries. If U→UrUwhere Uris fully supported in

Aor A, then we can commute Urthrough UA,g and U†

A,g to

cancel with its inverse. If Uris fully supported deep in Bor

B, then we can use UrU=U(U†UrU)and then commute

U†UrUthrough UB,h and U†

B,h. For sufﬁciently large and

overlapping Aand B, any local unitary is supported deep in-

side A, A, B, or B, so Eq. (1.6) is insensitive to U→UrUfor

any local, symmetry unitary. This ensures that Eq. (1.6) pro-

duces a topological invariant. Second, it is completely closed

form in that it only takes as input the global SPT entangler U

and restrictions of Ugand Uh, which are products of on-site

operators. Formulas like Eq. (1.6) are the main result of this

paper.

FIG. 2: A depth 3 FDQC in 1D consists of 3 layers where

each layer is a product of commuting local unitary operators.

Here, the x-axis is the spatial dimension and the y-axis is

time. Each black circle represents a lattice site and each

rectangle is a local unitary operator. To restrict the FDQC to

the region A, we simply delete all the local unitaries with

support outside of A. The restricted FDQC is a product of all

the colored rectangles.

II. FRAMEWORK FOR CLASSIFYING LPUS WITH

SYMMETRY

In this section, we will present our framework for classify-

ing LPUs with symmetry. This framework is based on a set

of (d−1) dimensional operators that we call ﬂux insertion

operators, that form an anomalous representation of the sym-

metry. These operators are useful for our purposes because

they can be easily computed from the SPT entangler when the

symmetry is on-site, and completely classify the entangler.

A. Preliminaries: SPT phases and SPT entanglers

For the most part, we will consider only bosonic systems

in this work. Speciﬁcally, we consider a lattice of bosonic

spins on a general ddimensional lattice Λwith a symmetry

G, which may contain antiunitary elements such as time rever-

sal. The action of Gon the lattice spins is given by {UgKg},

where Kgis complex conjugation for antiunitary elements and

Kg= 1 for unitary elements.

To deﬁne SPT phases and SPT entanglers, it is useful to

ﬁrst deﬁne what we mean by FDQC. An FDQC is a unitary U

that can be written as a ﬁnite product of layers Un, n ∈[1, N],

where each layer Unis a product of commuting local unitary

operators (“gates”):

U=UNUN−1· · · U1Un=Y

Un,r.(2.1)

Here, each gate Un,r is strictly supported within a bounded

distance λof the site r∈Λ. By deﬁnition, Nmust be ﬁnite,

in that it does not grow with the system size. The generic form

of a 1D FDQC is illustrated in Fig. 2. FDQCs can be used to

approximate, by Trotter decomposition, ﬁnite time evolution

by any local Hamiltonian.

An important property of FDQCs is that they can be re-

stricted. To restrict an FDQC to a region A, we simply re-

move all the gates with support outside of A. For example, a

restriction of a 1D FDQC to an interval Ais shown in Fig. 2.

AG-symmetric FDQC is an FDQC in which every local

gate Un,r is symmetric. In other words, every Un,r commutes

with every element of {UgKg}. They describe ﬁnite time evo-

lution with a local Hamiltonian that respects the symmetry at

all points in time.

One can also consider a more general unitary operator

which we call an LPU, that simply maps local operators to

nearby local operators. Recall that in this work we only

consider strict LPUs, which map strictly local operators to

nearby strictly local operators, without exponentially decay-

ing tails. We can associate with any LPU an “operator spread-

ing length” ξ, which is the maximum distance it can spread a

local operator. Speciﬁcally, for any operator Orsupported on

site r,U†OrUis supported within a disk of radius ξcentered

at r. It is easy to see that for FDQCs, which form a subset of

LPUs, ξ= 2Nλ. A G-symmetric LPU respects the symmetry

as a whole, but may not have a decomposition into an FDQC

built out of local, symmetric gates.

AG-symmetric strict LPU Uis an SPT entangler if it sat-

isﬁes

U|ψ0i=|ψSPTi,(2.2)

where |ψ0iis a symmetric product state of the form

|ψ0,ri⊗|Λ|, satisfying UgKg|ψ0,ri=|ψ0,rifor all g∈G.

Here, |ψSPTiis a (possibly trivial) SPT state. In this paper,

we say that |ψSPTiis in a nontrivial SPT phase if Uis lo-

cality preserving, but there is an obstruction to making Ua

G-symmetric FDQC. Note that this is different from the more

standard deﬁnition that |ψSPTiis in a nontrivial SPT phase if

it can be connected to |ψ0iby an FDQC, but only if the FDQC

contains gates that break the symmetry. In this paper, we will

consider Uthat are more general symmetric LPUs, which may

not be FDQCs even after forgetting about the symmetry. This

is because for some SPT phases, the entangler can only be

made symmetric if it is a nontrivial LPU[14–16].

We deﬁne two SPT entanglers as equivalent if they differ

by a Gsymmetric FDQC:

U0∼U:U0=GFDQC ·U. (2.3)

This means that the SPT states they entangle are equivalent,

because

|ψ0

SPTi=GFDQC · |ψSPTi,(2.4)

which is the usual deﬁnition of equivalence for SPT states[1].

Note that the converse does not necessarily hold: two equiva-

lent SPT states, that differ by a Gsymmetric FDQC, may have

inequivalent entanglers.

1. Flux insertion

One way to detect the anomaly of an SPT is by inserting

symmetry ﬂux and measuring the degrees of freedom bound

to the ﬂux. For example, in an integer quantum Hall state,

the Hall conductance is computed from the quantized U(1)

charge bound to 2πﬂux insertion[38].

A,g

UA,g

A,g

∼

(a)

(b)

WU†

A,g

FIG. 3: For illustration purposes, we specialize here to 2D.

(a) A closed ﬂux insertion operator WU

A,g (pink annulus) acts,

on the SPT state |ψSPTi=U|ψ0i, as the symmetry

transformation restricted to a 2D patch A(grey disk). (b)

When WU

A,g is an FDQC, we can restrict it to WU

A,g, which is

supported on an open 1D interval A. This operator inserts

symmetry ﬂux at the boundary of A(black crosses) and has

the same action on |ψSPTias the symmetry defect operator

A,g =WU†

A,gUA,g.DU

A,g is a symmetry defect operator

because it acts like the symmetry transformation near the top

boundary of A(which we denote by A), but leaves |ψSPTi

invariant near the lower boundary of A(which we denote by

A).

When Gis a unitary, on-site symmetry, we can insert ﬂux

using ﬂux insertion operators. A “closed” ﬂux insertion oper-

ator WU

A,g is deﬁned as follows. For any region A⊂Λdeep

in the bulk of the SPT, WU

A,g is a (d−1) dimensional operator

supported near the boundary of Athat has the same action on

the SPT state as UA,g =Qr∈AUr,g, where Ur,g is the on-site

representation of the symmetry:

A,g|ψSPTi=UA,g|ψSPTi.(2.5)

Aclosed ﬂux insertion operator does not insert any symme-

try ﬂux, but is useful for deﬁning an open ﬂux insertion oper-

ator, which does insert symmetry ﬂux. Assuming that WU

A,g

is a FDQC, we can restrict WU

A,g to a region A⊂∂A, where

∂A is the support of WU

A,g. The resulting operator, which we

denote by WU

A,g, is strictly supported on a (d−1) dimensional

manifold A. We can also deﬁne WU

A,g =WU

A,gWU†

A,g, which

is roughly supported on A=∂A \A.2Using these deﬁni-

2Strictly speaking, WU

A,g is supported on a slightly larger region and over-

laps slightly with A

tions, we have

A,g|ψSPTi=WU†

A,gUA,g|ψSPTi.(2.6)

Here, DU

A,g =WU†

A,gUA,g is a symmetry defect operator: it

acts as UA,g near Aand the interior of A, and WU†

A,g dresses

the operator so that it leaves the ground state invariant near A.

A,g inserts symmetry ﬂux through the boundaries of the re-

striction, as illustrated in Fig. 3. For example, in Fig. 3,WU

A,g

still acts like UA,g on |ψSPTinear the upper boundary of A,

but it leaves the ground state unchanged in the lower bound-

ary of A. This means that it creates an extrinsic defect line

along the upper boundary of A, which terminates at symme-

try ﬂuxes.

B. Flux insertion operators from LPUs

We will now introduce our main tool for studying SPT en-

tanglers, which is a particular choice of ﬂux insertion oper-

ators. Note that in the study of SPT phases, any WU

A,g that

has the action deﬁned by (2.5) on the SPT ground state is a

valid ﬂux insertion operator. However, we can deﬁne a par-

ticular WU

A,g that satisﬁes (2.5) that is easy to compute using

the SPT entangler. This deﬁnition uses the SPT entangler U,

the symmetry operator UA,g, and a slightly smaller symmetry

operator UAin,g , as follows:

A,g =UA,gUU†

Ain,gU†.(2.7)

Speciﬁcally, Ain is a subset of Acontaining all the points

lying deeper than ξwithin A:Ain ={r∈A: dist(r, A)>

ξ}. Since Uis G-symmetric and locality preserving, it can

only modify UAin,g within ξof the boundary of Ain, which is

a strip of width 2ξinside A. Denoting this strip by ∂A, we

have

UUAin,gU†=UAin,gU∂A,g,(2.8)

where U∂A,g is an operator fully supported in ∂A.

Let us now check that WU

A,g is a (closed) ﬂux insertion oper-

ator. It is easy to see that WU

A,g is supported near the boundary

of A, in ∂A. To check that WU

A,g has the same action on the

SPT state as UA,g, note that

A,g|ψSPTi=UA,gU U†

Ain,g|ψ0i

=UA,gU|ψ0i

=UA,g|ψSPTi.

(2.9)

To get the second line, we used the fact that |ψ0iis invariant

under U†

Ain,g.

1. Properties of {WU

A,g }

The set of operators {WU

A,g}is easy to compute because

it only involves restricting the global symmetry operator Ug,

∂A

Ain

FIG. 4: WU

A,g is deﬁned using Ugrestricted to two regions A

(large disk) and Ain (middle disk), which contains points

deeper than ξinside A.WU

A,g is fully supported on ∂A,

which is a strip of width 2ξinside A.

which can be done unambiguously for unitary, on-site sym-

metries. It has several important properties:

1. Every element in {WU

A,g}is a (d−1) dimensional strict

LPU.

2. {WU

A,g}forms a representation of G.

3. WU

A,g satisﬁes

U†

hWU

A,gUh=WU

A,h−1gh.(2.10)

4. U0∼Uaccording to Eq. (2.3)3if and only if {WU0

A,g} ∼

{WU

A,g}for any A⊂Λ, where

{WU0

A,g} ∼ {WU

A,g}:WU0

A,g =V†WU

A,gV(2.11)

for every g∈G, where Vis a Gsymmetric FDQC fully

supported within ξof the boundary of Ain.

The ﬁrst three properties help us classify different possi-

ble {WU

A,g}while the fourth property justiﬁes using {WU

A,g}

to classify SPT entanglers and, more generally, Gsymmetric

LPUs.

We will now prove each of the four properties.

Proof of Property 1: We will ﬁrst show that WU

A,g is supported

on a (d−1) dimensional manifold, matching the description

of a ﬂux insertion operator in Sec. II A 1. This follows

directly from the deﬁnition of WU

A,g in (2.5) together with

(2.8). Note that (2.8) relies on Ubeing G-symmetric and

3Stricly speaking, we will only prove that U0

∼Uup to multiplication by

lower dimensional Gsymmetric QCA. However, we conjecture that this

stronger statement holds, as we discuss in appendix B.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

TopologicalinvariantsforSPTentanglersCarolynZhangDepartmentofPhysics,KadanoffCenterforTheoreticalPhysics,UniversityofChicago,Chicago,Illinois60637,USA(Dated:October7,2022)Wedevelopaframeworkforclassifyinglocalitypreservingunitaries(LPUs)withinternal,unitarysymmetriesinddimensions,basedon(d1)dimensio...

展开>> 收起<<

Topological invariants for SPT entanglers Carolyn Zhang Department of Physics Kadanoff Center for Theoretical Physics University of Chicago Chicago Illinois 60637 USA.pdf

共26页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Topological invariants for SPT entanglers Carolyn Zhang Department of Physics Kadanoff Center for Theoretical Physics University of Chicago Chicago Illinois 60637 USA

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: