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Entanglement in Interacting Majorana Chains and Transitions of von Neumann Algebras
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Entanglement in Interacting Majorana Chains and Transitions of von Neumann Algebras
Pablo Basteiro , Giuseppe Di Giulio ,Johanna Erdmenger, and Zhuo-Yu Xian
Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Excellence Cluster ct.qmat,
Julius Maximilians University Würzburg, Am Hubland, 97074 Würzburg, Germany
(Received 17 January 2024; accepted 26 March 2024; published 16 April 2024)
We consider Majorana lattices with two-site inte ractions consisting of a general function of the
fermion bilinear. The models are exactly solvable in the limit of a large number of on-site fermions. The
four-site chain exhibits a quantum phase transition co ntrolled by the hopping parameters and manifests
itself in a discontinuous entanglement entropy, obtained by constraining the one-sided modular
Hamiltonian. Inspired by recent work within the AdS/CFT correspondence, we identify transitionsbetween types of von Neumann operator algebras throughout the phase diagram. We find transitions of
the form II
1?III?I∞that reduce to II 1?I∞in the strongly interacting limit, where they connect
nonfactorized and factorized ground states. Our resu lts provide novel realizations of such transitions in a
controlled many-body model.
DOI: 10.1103/PhysRevLett.132.161604
Introduction. —Entanglement in many-body systems and
quantum field theories (QFTs) has recently been explored
via a novel take on local operator algebras and axiomatic
QFT [1–6](see[7,8] for a review). A fruitful platform for
these analyses is the anti –de Sitter/conformal field theory
correspondence [9–11](also known as holography), which
relates strongly coupled QFTs in ddimensions with gravity
theories on negatively curved spacetimes in d?1dimen-
sions. In this context, operator algebras, and especially von
Neumann algebras, have recently been leveraged to rigor-
ously describe the entanglement structure of holographicsystems [12–20]. A direct consequence of these investiga-
tions is the algebra typification of the two phases in the
Hawking-Page transition [12,13] , characterized by factor-
ized and nonfactorized Hilbert spaces, respectively. Inaddition to these new developments, the study of algebraic
properties emerging in quantum systems in the limit of
infinitely many degrees of freedom, which enables phe-nomena such phase transitions and facilitates the study of
entanglement, is a long-standing line of investigation [21].
This state of the art motivates us to study transitions of
operator algebras arising in interacting many-body quan-
tum systems whose Hilbert space structure allows for acontrolled analysis of entanglement. A useful object which
helps in classifying types of algebras is the one-sided
modular Hamiltonian associated to a given subregion [6].
For free fermionic systems, this object is uniquelydetermined by the two-point correlation functions restricted
to the subregion [22]. We extend these results to interacting
fermionic systems by constraining the form of the one-sided modular Hamiltonian in the limit of a large number ofon-site fermions. For a wide class of interacting Majoranalattices, we exploit this extension to identify the operatoralgebras underlying our models, together with their tran-sitions between different regimes of the phase diagram.This paves the way for addressing the classification ofalgebras, and possible transitions thereof, in previouslysuggested models for discrete holography, such as O?N?-
invariant aperiodic spin chains [23].
More precisely, in this Letter, we introduce a lattice
model with NMajorana fermions on each site, interacting
via a general potential involving multibody hoppings. Inthe large- Nlimit, all higher-point functions factorize. Thus,
we solve the system exactly by obtaining the two-pointcorrelation function for generic interaction potentials. Weshowcase the wide applicability of our techniques byconsidering instances of the model including both finiteand infinite chains with nearest-neighbor hopping.
We report three main results: First, we derive the entropy
of a two-site chain with generic interaction potential. This
entropy is fully determined by the correlations between the
two sites, which are dictated by the interaction potential viaself-consistency. This result can be interpreted both as thethermal entropy of the chain at a finite temperature or as theentanglement entropy of a two-site subregion in a largerchain. Remarkably, we find that the entropy itself does notdepend explicitly on the chosen potential. Second, for afour-site chain, we identify two phases of the systemcharacterized by strong and weak correlations within anytwo-site subsystem of the chain relative to all othercorrelations in the system. In particular, we identify a
Published by the American Physical Society under the terms of
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and DOI. Funded by SCOAP3.PHYSICAL REVIEW LETTERS 132, 161604 (2024)
0031-9007 =24=132(16) =161604(7) 161604-1 Published by the American Physical Societyregime where the correlation structure indicates the fac-
torization of the ground state. For a vast class of potentials,
we show that these phases are connected by a quantumphase transition. Moreover, we determine the entanglement
entropy of the half-chain by imposing a constraint on the
one-sided modular Hamiltonian. In correspondence withthe phase transition, the entanglement entropy exhibits adiscontinuity above a critical value of the interaction
strength. Third, exploiting the exact solvability of our
model, we identify transitions between the local operatoralgebras underlying the four-site chain. When the correla-
tions within a given subregion are the most relevant in the
system, we find a von Neumann algebra of type I
∞, which,
in general, encodes finite entanglement. In our model,
entanglement vanishes, thus signaling factorization. In the
opposite regime, the algebra is of type II 1, associated with
infinite entanglement entropy, while still allowing for the
definition of a trace functional. The intermediate domain is
described by type III algebras, where the entanglement isinfinite and a trace functional cannot be defined. Strikingly,in the strongly interacting limit and for an exponential
potential in the fermion bilinear, we find that the transition
between types II
1and I∞algebras occurs in correspondence
with the identified quantum phase transition. This transition
then connects a nonfactorized ground state to a ground state
factorized into a product state.
Hamiltonian and Schwinger-Dyson equations. —We
consider a lattice of Lsites with NMajorana fermions ψj
x
at each site, with anticommutation relations fψj
x;ψkyg?
δjkδxy. The microscopic Hamiltonian is given by
H?N
2XL
x;y?1hxy/C182
iNXN
j?1ψj
xψjy/C19
; ?1?
with a general interaction potential hxy,w h i c h ,w i t h o u t
loss of generality, obeys hxy?ξ??hyx??ξ?andhxx?ξ??0.
The theory is invariant under a global O?N?rotation
ψj
x→P
kOjkψkx,w h e r e Ois an orthogonal matrix. It is a
lattice counterpart of the Gross-Neveu model [24] with a
general interaction potential, and only the bubble diagramscontribute to two-point functions at leading order in 1=N;
see Supplemental Material [25].W h e n h
xy?ξ?∝ξq,t h e
model is equivalent to the replicated Brownian Sachdev-
Ye-Kitaev (SYK q) model in disorder averaging [26]. In this
work, hxy?ξ?always includes a linear term in ξ.
We solve the model (1)by introducing the effective
action of two auxiliary bilocal fields, the Green ’s function
Gxy?τ1;τ2??1=NP
jψj
x?τ1?ψj
y?τ2?and the self-energy
Σxy?τ1;τ2?, introduced as a Lagrange multiplier, in the
spirit of [27–29]. Here, τdenotes Euclidean time. We
consider the canonical ensemble at temperature 1=βand
write the thermal partition function Z?RD?GD?Σe?SE??G; ?Σ/C138
with the effective action?SE=N?log PF ??τδxy?Σxy?
?1
2X
x;yZβ
0dτ1dτ2Gxy?τ1?τ2?Σxy?τ1?τ2?
?1
2X
x;yZβ
0dτhxy??2iGxy?0??; ?2?
where Gxy?τ???Gyx??τ?and we have assumed time-
translational invariance. From Z, we may derive the
properties of the ground state (when β→∞) or the
thermodynamics of the system at finite temperature.
In the large- Nlimit, which we take first in all following
computations, the saddle point approximation of the pathintegral leads to the Schwinger-Dyson (SD) equations
G
0xy?τ12??X
zZ
dτ3Σxz?τ13?Gzy?τ32??δxyδ?τ12?; ?3?
Σxy?τ12??2ih0xy??2iGxy?0??δ?τ12?; ?4?
with τij≡τi?τjandΣxy??Σyxdue to the conditions on
hxymentioned above. This model is exactly solvable in the
large- Nlimit, in the sense that all higher-point functions
factorize into products of two-point functions; i.e., we havelarge- Nfactorization.
We solve Eqs. (3)and(4)for general τby leveraging the
fact that the self-energy is solely determined by ?2iG
xy?0?.
We can, thus, solve for G?0?by means of self-consistency
(SC) conditions. Using the form of (3)in Fourier space,
SC imposes
Gxy?0??1
βX
n???iωn?Σ?ωn?/C138?1/C138xy; ?5?
with ωn?2π?n?1=2?=βandΣgiven by (4). The Green ’s
function G?τ?is then obtained by inserting G?0?back
into(4)and(3).
Two-site chain. —To provide an explicit application of
our general techniques, we now focus on a system governed
by a general Hamiltonian Hof the form (1)withL?2sites
and at a finite temperature 1=β. Without loss of generality,
we can absorb βinto the general form of the Hamiltonian
(1), i.e., set β?1. Thus, the density matrix reads
ρ?e?H
Z;H ?Nh/C182
iNX
jψj
1ψj2/C19
; ?6?
where Z?Tr?e?H?andh?ξ?≡h12?ξ??h21??ξ?. We can
exactly solve the SD equation (3)to obtain the Green ’s
function G?τ?[25].A t τ?0, the solution reads
Gxy?0??1
2/C181 ?i tanh ?h0?X?/C138
i tanh ?h0?X?/C138 1/C19
;?7?
with X≡?2iG12?0?. We can read off the SC equation
X??tanh ?h0?X?/C138; ?8?PHYSICAL REVIEW LETTERS 132, 161604 (2024)
161604-2which can alternatively be derived from (5). In general, we
have?1Two relevant regimes of (8)are when jXj→0andjXj→1,
corresponding to weak and strong correlations between the
two sites, respectively.
From the path integral in the large- Nlimit, we can study
the thermodynamic properties of the system [25].I n
particular, we find the entropy density S=N≡sto be
s?X???1?X
2log1?X
2?1?X
2log1?X
2;?9?
where Xsatisfies the SC equation (8). The fact that the
entropy density function (9)does not explicitly depend on
the interaction potential signifies the first main result of this
work. This independence is a feature of the entropy only,
while other quantities, like the free energy, which is given
byF?N?h?X??s?X?/C138 ?O?1?, indeed depend on the
form of the potential. Also, this independence of the
entropy on the interaction potential is spoiled by 1=N
corrections [25]. Remarkably, from (9), we see that
S→Nlog2?O?1?when jXj→0and S→0when
jXj→1[25]. Let us emphasize that, since the large- N
limit is always taken first, the entropy is finite only when
X?1and is otherwise linearly divergent with N.
Four-site chain. —We can think of the two-site system as
being part of larger chains and take advantage of the results
derived above to study the entanglement structure. As an
example, we consider an open chain of length L?4at zero
temperature. Although our techniques are valid for any
potential, we consider here a specific instance hx;x?1?ξ??
μx?1?eJξ?=?2J??hx?1;x??ξ?, where J> 0is the inter-
action strength and μ1?μ3≡μa,μ2≡μbdenote hopping
parameters. All remaining entries of hxyare zero. For
convenience, we introduce the hopping ratio r≡μa=μb.T o
access the ground state properties in the large- Nlimit, the
hierarchy of parameters N?μbβ,μaβ?1needs to be
taken into account. We solve the SD equation (3)and obtain
G?τ?[25], which at τ?0reads
Gxy?0??1
20
BBB@1 isinθ 0 icosθ
?isinθ 1 icosθ 0
0?icosθ 1 isinθ
?icosθ 0?isinθ 11
CCCA;?10?
where the parameter θis determined by the SC constraint
derived from (5):
tanθ
2?G12?0?
2G23?0??h0
12??2iG12?0??
h0
23??2iG23?0???reJ?sinθ?cosθ?:?11?
This transcendental equation may be solved numerically
and has a unique solution for JJ>J c, where the critical value can be proven analytically
to be Jc????
2p
[25]. This multivaluedness of the SC
equation indicates that the system exhibits a discontinuous
behavior as a function of r, now to be seen as a controlparameter. We identify the thermodynamically dominant
solutions by minimizing the free energy Fobtained from
the effective action (2) [25] . This free energy is shown in
Fig.1for different values of the interaction strength below,
at, and above the critical point. We see that the free energy
exhibits nonanalyticity at r?1=2for interaction strengths
J>J c. Thus, the system is characterized two phases for
J>J cand it undergoes a first-order quantum phase
transition [30] across r?1=2, the existence of which
constitutes the second main result of this work. At thecritical point J?J
c, this transition is of second order.
Let us stress that this phase transition is present for a large
class of potentials other than the exponential [25]. The two
phases differ by the order parameter tan θ(11), which
characterizes the correlation structure via (10). Two limit-
ing regimes of this structure when r→0andr→∞are
shown in the insets in Fig. 1.
We now turn our attention to the study of entanglement in
the four-site model and consider a connected two-sitesubregion A, which we take to be, e.g., the sites x?1,2 .
The reduced density matrix ρ
Aof this system can be written
as a thermal density matrix of a two-site chain of theform (6),w i t h Hnow to be thought of as the one-sided
modular Hamiltonian. Its explicit form is not known in our
case, so we take as an ansatz the general form given in (1).
For this ansatz to describe a proper reduced density matrix,
ρ
Ashould reproduce the expectation values of local operators
in the subregion. In particular, it must reproduce thecorrelations given by the Green ’s function (10) restricted
to the subregion A. We must, therefore, impose a constraint
for the one-sided modular Hamiltonian at large N:
G
xy?0??1
NX
jTr?ρAψj
xψjy?;x ; y ∈A; ?12?
FIG. 1. Free energy of the four-site chain with potential
hx;x?1?ξ??μx?1?eJξ?=?2J?for different interaction strengths
J. We observe a nonanalyticity at r≡μa=μb?1=2forJabove
the critical value Jc????
2p
, signaling a phase transition. The two
phases of the system are characterized by the correlation structuregiven by (10), whose limiting cases for r→0andr→∞are
shown in the two embedded diagrams.PHYSICAL REVIEW LETTERS 132, 161604 (2024)
161604-3where Gxy?0?is given in (10). Equation (12) uniquely
determines the modular Hamiltonian only when J?0[22].
Nevertheless, we can still use it to compute the entanglement
entropy for J≥0. Indeed, we have shown that, for a density
matrix of the form given in (6), the entropy density of the
two-site system can be computed for any form of the one-sided modular Hamiltonian ansatz and is given by (9).
Because of the constraint (12),s?X?needs to be evaluated on
the SC solution X??2iG
12?0??sinθobtained from (11),
withX∈?0;1?since θ∈?0;π=2?.I nt h i sw a y ,e v e nw i t h o u t
knowing the explicit form of ρA,w ef i n dt h a t s?X??SA=N
is the entanglement entropy of subregion A. The resulting
entanglement entropy density as a function of r[recall X
depends on rvia(11)] is shown in Fig. 2for different values
ofJ. The phase transition reflects itself in the entanglement
entropy evaluated on SC solutions as it becomes discon-
tinuous for J>J c.
von Neumann algebras. —Motivated by recent results in
holography [12,13,15 –18], we study the classification of
operator algebras associated to subsystems of our four-site
model. Operator algebras can generally be classified into
three types, denoted as type I, II, and III [1,5,8] . Based on
the standard trace Tr, a type I algebra encapsulates a finiteentanglement entropy. Using Tr, entanglement entropy is
infinite in both type II and type III algebras. To further
distinguish the algebras, a key ingredient is the trace
functional , denoted by lowercase “tr”(to differentiate it
from uppercase Tr), which is defined to be a positive, linear,
and cyclic functional on the algebra [7,8]. In particular,
type II algebras allow for the definition of such a
trace functional, while type III algebras do not. For the
technical construction of operators in the algebra, weclosely parallel [7]. In our system of consideration andin the N→∞limit, operators in A
Aconsist of products of
finitely many Majoranas located in subregion A. In par-
ticular, these will act trivially on countably infinitely manyindices jof the Majorana color space.
Based on the results for the ground state entanglement of
our model, together with the operators in A
Adefined above,
we identify the operator algebras associated to subregion A
in different regimes of the correlation measure X?
?2iG12?0?. This classification is shown in the phase
diagram inset in Fig. 2. When X→1, the entropy SA→0
[25]. This is consistent with our physical intuition, since we
expect the subsystems to completely factorize in this limit.
Thus, we find that AAis a type I ∞algebra when X→1
[which implies r→∞by(11)]. The index in I ∞alludes to
the infinite dimensionality of the local Hilbert space.
When X< 1, the ground state is no longer factorized,
and the entropy (9)is infinite, therefore ruling out AAbeing
of type I. To specify the type, we resort to the definition of a
trace functional tr on AA. When the maximally entangled
state jΨiis in the Hilbert space generated by the algebra
AA, a well-defined trace functional is given by tr ?a?≡
hΨjajΨi, with a∈AA[7]. Importantly, when X?0,w e
find that the entanglement entropy in our ground state
is infinite and maximal up to subleading corrections in
1=N[25]. This implies that our ground state can be mapped
tojΨiby applying finitely many Majorana operators.
Therefore, we conclude that the functional tr defines aproper trace when X?0[r?0by(11)], thus unveiling
thatA
Ais of type II 1only at this point; cf. Fig. 2.
As for the regime 00entropy (9)is infinite but not maximal at leading order as
N→∞. Therefore, our ground state cannot be mapped to
the maximally entangled state jΨiby finitely many local
operators, and, therefore, the algebra AAdoes not admit the
definition of a trace [7]. This implies that AAis of type III.
Recall that the first and second equalities in (11)impose SC
for generic potentials, and, therefore, the analysis above isvalid in the general interacting case.
In the free case J?0, where the entanglement
Hamiltonian ought to be quadratic in the fermions, the
classification of the algebras can be attained by studying
the spectrum of the modular operator Δ?lim
N→∞ρA?
ρ?1
?A[5,31 –33]. Given this setup, we are able to compute the
large- Nspectrum of Δ[25], finding Spec ?Δ??f λngn∈Z.
Here, the parameter λis related to the correlations within
the subsystem as λ?? ?1?X?=?1?X?/C138. When the modu-
lar operator has precisely this form, the associated operatoralgebras are said to be of I
∞when λ?0, type II 1forλ?1,
and type III λforλ∈?0;1?.S u c ht y p eI I I λalgebras are known
to arise for free fermions on a lattice [34].
At finite J, these considerations lead to transitions
between operator algebras of type II 1?III?I∞in the
phase diagram; cf. Fig. 2. Given that the limit of the entropy
forr→0andr→∞is the same for other types ofFIG. 2. Entanglement entropy (9)of subregion Aon solutions
to(11)as a function of r≡μa=μbfor different couplings J. Solid
lines denote all SC solutions, while a sample of physical solutionsminimizing Fin Fig. 1is marked with dots. The phase transition
is signaled by the discontinuity for J>J
c. Inset: von Neumann
type of the algebra AAin different regimes of the phase diagram.
Each type is denoted by a different color, and the black dot (line)represents a phase transition of second (first) order.PHYSICAL REVIEW LETTERS 132, 161604 (2024)
161604-4potentials, we expect similar phase diagrams to hold for
in those cases when Jis finite. Our setup ’s analytical
tractability enables us to study the limit J→∞, where the
solution to the SC equation (11)for the class of interaction
potentials with exponential behavior is X?Θ?r?1=2?,
withΘthe Heaviside step function. In this limit, the
entanglement entropy is SA?Θ?1=2?r?Nlog2?O?1?
[25]. This results in a direct transition of algebras II 1?I∞
atr?1=2, which coincides with the phase transition
undergone by the system. The transitions between differenttypes of local operator algebras across the phase diagramprovide the third result of our work.
Closed periodic chains. —To showcase the generality of
our methods, we now consider closed periodic chains.
In particular, we focus on a closed chain consisting of L
sites with a Hamiltonian of the form (1)with staggered
interaction
h
xy?ξ??δx?1;y?hb?ξ?δmod 2x;0?ha?ξ?δmod 2x;1/C138
?δx?1;y?ha??ξ?δmod 2x;0?hb??ξ?δmod 2x;1/C138;?13?
where haandhbare generic functions and we have the
periodic identification L?x?x. Notice that, by defining
cells consisting of adjacent sites interacting by hb, we can
leverage translational invariance with respect to these cells tosolve the model in momentum space [25].W ef i n dt h a tt h e
Green ’s function is determined by an implicit dependence on
its own entries. In particular, Gis an implicit function of only
the correlations within a given cell G
2x;2x?1?0?and those
connecting adjacent cells G2x?1;2x?0?.T h i sd e p e n d e n c e
manifests itself via the parametrization
1?v
1?v?h0a??2iG2x?1;2x?0??
h0
b??2iG2x;2x?1?0??; ?14?
where v∈??1;1/C138. By the aforementioned translational
invariance, the correlations entering (14) are independent
ofx. Explicit expressions for G2x?1;2x?0?andG2x;2x?1?0?at
zero temperature and in the limit L→∞can be found in
terms of vitself and read
G2x?1;2x?0??i
2g??v?;G 2x;2x?1?0??i
2g?v?; ?15?
g?v??2sgn?v?
π/C20E?1?1=v2?
1?1=v?K?1?1=v2?
1?v/C21
;?16?
withK?ξ?andE?ξ?the complete elliptic integrals of the first
and second kind, respectively. In the spirit of our methods,we can impose self-consistency inserting (15) into (14)
and solving for v. The model is, thus, completely solved
once this value of vhas been found. Notice that v→?v
exchanges G
2x?1;2x?0?andG2x;2x?1?0?by virtue of (15),a n d
this amounts to exchanging haandhb.S i n c e g?1??1,t h e
limits v→1andv→?1correspond to maximal correla-
tions between nearest-neighbor sites within and across cells,respectively. For periodic chains with L?4, we can parallelthe previous discussions on the entanglement structure and,
consequently, on the typification of the operator algebras
across the phase diagram.
Conclusions and future work. —We determine the phase
structure and entanglement for a large class of Majorana
models with O?N?symmetry in the large- Nlimit. Despite that
we mostly focus on models defined on few sites, these areenough to exhibit a rich phase diagram and entanglementstructure yet also sufficiently tractable in the large- Nlimit such
as to explicitly compute key quantities like free energy and
entanglement entropy. The von Neumann algebras underlyingthis entanglement structure are summarized in Fig. 2.
Remarkably, all three types of algebra are featured throughout
the phase diagram of our model. Therefore, our class of exactlysolvable models gives rise to nontrivial operator algebra
transitions in a highly controllable way. This allows us to
track the parameter regimes in which the correlations signalthe factorization of the ground state into a product state, as
s h o w ni nF i g . 1. While here we use correlations for character-
izing factorization, a state-based approach consists of usingentanglement orbits [18]. In spite of the different approaches
used, we see a similar relation between factorization and the
value of entanglement entropy. Analyzing these similarities is
a promising line of future investigations.
Intriguingly, the algebra transition we find in the strong
coupling limit J→∞coincides with a phase transition that
connects a factorized and a nonfactorized state, similarly to
the holographic Hawking-Page phase transition [12,13] .
Differently from our case (i.e., II
1?I∞), this is a transition
between algebras of type I ∞and III 1as a function of the
temperature. Remarkably, we observe an analogous algebra
transition, although our model differs from previous worksin the context of holography [27–29,35 –38], which con-
sider on-site random interactions leading to a nonzero
entropy at zero temperature.
Further relations to holography can be obtained by
including random disorder into our model by attachingan SYK model to each lattice site. This enlarges the phasediagram, and we expect the competition between spatially
inhomogeneous hoppings and the locally random disorder
to change the renormalization group properties of criticalpoints. Additionally, it is promising to investigate the
mentioned connections with Brownian SYK [26,39 –41].
Since our setup allows for general spatially disordered
interactions, it is also relevant for further infinite disordered
chains [42,43] , where the hopping parameters are distrib-
uted according to a binary aperiodic sequence. These so-
called aperiodic spin chains have recently been considered
[23,44 –46]as a step toward establishing a holographic
duality on discrete spaces. Finally, in the free case, our
model can be interpreted as an instance of a Kitaev
chain [47], which has physical realizations in terms of
superconducting quantum wires. It would be intriguing toinvestigate the changes to this physical picture in the
interacting case.PHYSICAL REVIEW LETTERS 132, 161604 (2024)
161604-5We are grateful to Souvik Banerjee, Moritz Dorband,
Elliott Gesteau, Shao-Kai Jian, Changan Li, Ren? eM e y e r ,
Alexey Milekhin, Sara Murciano, and Jie Ren for fruitful
discussions. This work was supported by Germany ’s
Excellence Strategy through the Würzburg-DresdenCluster of Excellence on Complexity and Topology inQuantum Matter —ct.qmat (EXC 2147, Project-ID
No. 390858490), and by the Deutsche Forschungsgemein-schaft (DFG) through the Collaborative Research Center
“ToCoTronics, ”Project-ID No. 258499086, SFB 1170, as
well as a German –Israeli Project Cooperation (DIP) grant
“Holography and the Swampland. ”Z.-Y . X. also acknowl-
edges support from the National Natural Science Foundationof China under Grant No. 12075298. We are grateful to thelong-term workshop YITP-T-23-01 held at the YukawaInstitute for Theoretical Physics, Kyoto University, where
a part of this work was done.
giuseppe.giulio@uni-wuerzburg.de
[1] F. J. Murray and J. von Neumann, On rings of operators,
Ann. Math. 37, 116 (1936) .
[2] F. J. Murray and J. von Neumann, On rings of operators. II,
Trans. Am. Math. Soc. 41, 208 (1937) .
[3] J. von Neumann, On rings of operators. III, Ann. Math. 41,
94 (1940) .
[4] F. J. Murray and J. von Neumann, On Rings of Operators.
IV,Ann. Math. 44, 716 (1943) .
[5] M. Takesaki, Theory of Operator Algebras I-III (Springer,
New York, 1979), 10.1007/978-1-4612-6188-9.
[6] R. Haag, Local Quantum Physics (Springer, Berlin, 1996).
[7] E. Witten, APS medal for exceptional achievement in
research: Invited article on entanglement properties ofquantum field theory, Rev. Mod. Phys. 90, 045003 (2018) .
[8] J. Sorce, Notes on the type classification of von Neumann
algebras, Rev. Math. Phys. 36, 2430002 (2024) .
[9] J. Maldacena, The large-N limit of superconformal field
theories and supergravity, Int. J. Theor. Phys. 38, 1113 (1999) .
[10] E. Witten, Anti-de Sitter space and holography, Adv. Theor.
Math. Phys. 2, 253 (1998) .
[11] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge
theory correlators from noncritical string theory, Phys. Lett.
B428, 105 (1998) .
[12] S. Leutheusser and H. Liu, Causal connectability between
quantum systems and the black hole interior in holographicduality, Phys. Rev. D 108, 086019 (2023) .
[13] S. Leutheusser and H. Liu, Emergent times in holographic
duality, Phys. Rev. D 108, 086020 (2023) .
[14] E. Witten, Why does quantum field theory in curved
spacetime make sense? And what happens to the algebraof observables in the thermodynamic limit?, in Dialogues
Between Physics and Mathematics (Springer International
Publishing, New York, 2022), p. 241 –284.
[15] E. Witten, Gravity and the crossed product, J. High Energy
Phys. 10 (2022) 008.
[16] V. Chandrasekaran, G. Penington, and E. Witten, Large N
algebras and generalized entropy, J. High Energy Phys. 04
(2023) 009.[17] V. Chandrasekaran, R. Longo, G. Penington, and E. Witten,
An algebra of observables for de Sitter space, J. High
Energy Phys. 02 (2023) 082.
[18] S. Banerjee, M. Dorband, J. Erdmenger, and A.-L. Weigel,
Geometric phases characterise operator algebras andmissing information, J. High Energy Phys. 10 (2023)
026.
[19] S. E. Aguilar-Gutierrez, E. Bahiru, and R. Espíndola, The
centaur-algebra of observables, J. High Energy Phys. 03
(2024) 008.
[20] N. Engelhardt and H. Liu, Algebraic ER ?EPR and
complexity transfer, arXiv:2311.04281 .
[21] R. Haag, N. M. Hugenholtz, and M. Winnink, On the
equilibrium states in quantum statistical mechanics, Com-
mun. Math. Phys. 5, 215 (1967) .
[22] I. Peschel, Calculation of reduced density matrices from
correlation functions, J. Phys. A 36, L205 (2003) .
[23] P. Basteiro, R. N. Das, G. D. Giulio, and J. Erdmenger,
Aperiodic spin chains at the boundary of hyperbolic tilings,SciPost Phys. 15, 218 (2023) .
[24] D. J. Gross and A. Neveu, Dynamical symmetry breaking in
asymptotically free field theories, Phys. Rev. D 10, 3235
(1974) .
[25] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.132.161604 for more
details on solutions to the SD equations, thermodynamiccomputations, the modular operator, the strongly coupled
limit, and the closed periodic ring.
[26] P. Saad, S. H. Shenker, and D. Stanford, A semiclassical
ramp in SYK and in gravity, arXiv:1806.06840 .
[27] S. Sachdev and J. Ye, Gapless spin-fluid ground state in a
random quantum Heisenberg magnet, Phys. Rev. Lett. 70,
3339 (1993) .
[28] A. Kitaev, A simple model of quantum holography (2015),
talks at KITP, April 7 and May 27.
[29] J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-
Kitaev model, Phys. Rev. D 94, 106002 (2016) .
[30] S. Sachdev, Quantum phase transitions, Phys. World 12,3 3
(1999) .
[31] J. von Neumann, On infinite direct products, Compos.
Math. 6, 1 (1939), http://www.numdam.org/item/?id=CM_
1939__6__1_0 .
[32] R. T. Powers, Representations of uniformly hyperfinite
algebras and their associated von Neumann rings, Ann.
Math. 86, 138 (1967) .
[33] H. Araki and E. J. Woods, A classification of factors, Publ.
RIMS 4, 51 (1968) .
[34] G. K. Pedersen, C
-Algebras and their Automorphism
Groups (Elsevier, New York, 2018).
[35] S. Sachdev, Bekenstein-Hawking entropy and strange met-
als,Phys. Rev. X 5, 041025 (2015) .
[36] Y. Gu, X.-L. Qi, and D. Stanford, Local criticality, diffusion
and chaos in generalized Sachdev-Ye-Kitaev models,J. High Energy Phys. 05 (2017) 125.
[37] J. Maldacena and X.-L. Qi, Eternal traversable wormhole
arXiv:1804.00491 .
[38] T. Numasawa, Four coupled SYK models and nearly AdS
2
gravities: Phase transitions in traversable wormholes and in
bra-ket wormholes, Classical Quantum Gravity 39, 084001
(2022) .PHYSICAL REVIEW LETTERS 132, 161604 (2024)
161604-6[39] D. Stanford, S. Vardhan, and S. Yao, Scramblon loops,
arXiv:2311.12121 .
[40] A. Milekhin and J. Xu, Revisiting Brownian SYK and its
possible relations to de Sitter, arXiv:2312.03623 .
[41] S.-K. Jian and B. Swingle, Phase transition in von Neumann
entropy from replica symmetry breaking, J. High Energy
Phys. 11 (2023) 221.
[42] A. P. Vieira, Aperiodic quantum XXZ chains:
Renormalization-group results, Phys. Rev. B 71, 134408
(2005) .
[43] R. Juhász and Z. Zimborás, Entanglement entropy in
aperiodic singlet phases, J. Stat. Mech. (2007) P04004.[44] A. Jahn, Z. Zimborás, and J. Eisert, Central charges of
aperiodic holographic tensor-network models, Phys. Rev. A
102, 042407 (2020) .
[45] A. Jahn, M. Gluza, C. Verhoeven, S. Singh, and J. Eisert,
Boundary theories of critical matchgate tensor networks,J. High Energy Phys. 04 (2022) 111.
[46] P. Basteiro, G. Di Giulio, J. Erdmenger, J. Karl, R. Meyer,
and Z.-Y. Xian, Towards explicit discrete holography:Aperiodic spin chains from hyperbolic tilings, SciPost Phys.
13, 103 (2022) .
[47] A. Y. Kitaev, Unpaired Majorana fermions in quantum
wires, Phys. Usp. 44, 131 (2001) .PHYSICAL REVIEW LETTERS 132, 161604 (2024)
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