Characterizing Long-Range Entanglement in a Mixed State Through an Emergent Order on the Entangling Surface

Topologically-ordered phases of matter at non-zero temperature are conjectured to exhibit universal patterns of long-range entanglement which may be detected by a mixed-state entanglement measure known as entanglement negativity. We show that the entanglement negativity in certain topological orders can be understood through the properties of an emergent symmetry-protected topological (SPT) order which is localized on the entanglement bipartition. This connection leads to an understanding of ($i$) universal contributions to the entanglement negativity which diagnose finite-temperature topological order, and ($ii$) the behavior of the entanglement negativity across certain phase transitions in which thermal fluctuations eventually destroy long-range entanglement across the bipartition surface. Within this correspondence, the universal patterns of entanglement in the finite-temperature topological order are related to the stability of an emergent SPT order against a symmetry-breaking field. SPT orders protected by higher-form symmetries -- which arise, for example, in the description of the entanglement negativity for $\mathbb{Z}_{2}$ topological order in $d=4$ spatial dimensions -- remain robust even in the presence of a weak symmetry-breaking perturbation, leading to long-range entanglement at non-zero temperature for certain topological orders.

Strongly-interacting quantum phases of matter at zero temperature can exhibit universal patterns of long-range entanglement [1][2][3], which may be used to store and manipulate quantum information.Many of these quantum phases, such as topological orders in two and three spatial dimensions, cannot function as self-correcting quantum memories at non-zero temperature [4][5][6][7][8][9][10][11].It remains of interest to characterize the patterns of long-range entanglement in quantum many-body systems that can survive the presence of thermal fluctuations.
A thermal density matrix  ∼  − of a quantum manybody system described by a local Hamiltonian  is said to exhibit topological order if it cannot be prepared from a classical mixed state using finite-depth local quantum circuits [8].It was recently proposed [12] that topological order in a thermal state can be detected by the entanglement negativity [13][14][15][16], a mixed-state entanglement measure which has been studied extensively in quantum many-body systems ; in particular, topologically-ordered mixed states were conjectured [12] to possess a universal, constant contribution to the entanglement negativity which quantifies the long-range entanglement that cannot be removed by a finite-depth, local quantum circuit.This correction, termed the topological entanglement negativity, coincides with the topological entanglement entropy [42,43] at zero temperature, though it is known that the latter fails as a diagnostic for topological order in a mixed state.Outstanding questions remain regarding (i) the universality of the topological entanglement negativity within a finite-temperature topological order, and (ii) its behavior across a thermal phase transition in which the topological order is destroyed.
In this work, we make progress towards answering these questions by demonstrating a connection between the entanglement negativity in certain topological quantum orders, and the properties of an emergent, symmetry-protected topological (SPT) order [44,45] localized on the entanglement bipartition.Specifically, we show that the entanglement negativity is determined by the "strange correlator" [46] for this emergent SPT Transpose FIG.1: The thermal density matrix for certain topological orders after partial transposition on a subregion A -denoted   A -can be related to an emergent symmetry protected topological (SPT) order on the boundary of A. Eigenvalues of   A are related to "strange correlators" +|  | , where |+ is a symmetric trivial state and | is an SPT ordered state.We argue that long-range order in these strange correlators gives rise to a non-zero topological entanglement negativity.This schematic correspondence for the Z 2 topological order in  spatial dimensions is shown.order (see Fig. 1).We argue that the stability of the SPT phase, as characterized by the presence of long-range order in these strange correlators, is intimately related to the robustness of the topological entanglement negativity in the finite-temperature topological order.In addition, we show that proliferating topological excitations near the entanglement bipartition can drive a phase transition in which long-range entanglement across the bipartition is destroyed.We derive universal scaling forms for the topological entanglement negativity by relating this "disentangling" thermal phase transition to a zero-temperature phase transition between the emergent SPT order and a trivial state.
The connection that we identify between an emergent SPT order and a topologically-ordered mixed state may be understood heuristically as follows.In certain gapped topological orders at zero temperature, the "partially-transposed" density matrix with respect to a subsystem -an essential operation arXiv:2201.07792v1[cond-mat.str-el]19 Jan 2022 < l a t e x i t s h a 1 _ b a s e 6 4 = " I H / h P J c F y 4 X Q r B w 5 5 G y < l a t e x i t s h a 1 _ b a s e 6 4 = " I H / h P J c F y 4 X Q r B w 5 5 G y R e X F e 5 6 1 L z m L m A P y A 8 / Y J v 0 q W Y A = = < / l a t e x i t > (E topo = 0) < l a t e x i t s h a 1 _ b a s e 6 4 = " e 3 8 h g 1 o A w d 5 n i w j t j 9 i f s 9 I V I + 8 5 A n 9 g f P 0 C X 5 a d 7 g = = < / l a t e x i t > ⇢ ⇠ e H @ /T @ < l a t e x i t s h a 1 _ b a s e 6 4 = " r U g I b P l J FIG. 2: Schematic phase diagram for the topological negativity ( topo ) of a thermal state with tunable temperatures in the bulk ( bulk ) and bipartition boundary (  ) in the 3d toric code when forbidding point-like excitations as well as 4d toric code when forbidding either type of excitations.When  bulk <  bulk, the bulk is topologically ordered, and tuning   drives a "disentangling" phase transition where longrange entanglement across the bipartitioning surface is destroyed.This transition is related to the zero-temperature phase transition between an SPT order and a trivial order. in the calculation of the entanglement negativity of that region -can be regarded as a wave function in which gapped excitations of the topological order have proliferated near the entanglement bipartition, with relative configurations of the excitations weighted by their statistical braiding phase.In a dual description, this wave function describes an SPT order where the protecting symmetry is inherited from the gauge symmetry of the bulk topological order, when restricted to the entanglement bipartition.When the bulk topological order is at a non-zero temperature, the excitations in this state are no longer localized to the entangling surface, and the partiallytransposed density matrix may be related to an emergent SPT state which is acted upon by a symmetry-breaking field.While most SPT orders are immediately destroyed by such a perturbation, if the symmetry to be broken is a one-form symmetry [48], the SPT order can remain robust below a finite strength of the symmetry-breaking field, resulting in a stable long-range entanglement below a finite critical temperature   .This scenario occurs in the Z 2 topological order in  = 4 spatial dimensions as well as in  = 3 dimensions with Z 2 charge excitations forbidden.
While we focus on Z 2 topological order in various dimensions throughout this work, our characterization of long-range entanglement in a mixed state through the stability of an emergent SPT order also holds for Z  topological orders as well as certain fracton orders [47].The applicability of this correspondence to other topological orders, as well as the behavior of the negativity across phase transitions in which the entire bulk topological order is destroyed by thermal fluctuations remain important open questions of our work.
Entanglement negativity-Given a density matrix ρ acting on a bipartite Hilbert space H = H A ⊗ H B , the entanglement negativity between A and B is defined by taking the partial transpose of the density matrix with respect to the Hilbert space of the A subsystem.The negativity   is defined with respect to this partially-transposed density matrix ρ A as   = log ρ A 1 , where  1 denotes the 1-norm of the matrix , i.e. the sum of all absolute eigenvalues of .Given a local Hamiltonian Ĥ and a thermal density matrix ρ ∼  − Ĥ , the negativity   between a subsystem A of linear size  A and its complement B in  space dimensions can be written as   =  local −  topo [12,49]. local captures short-range entanglement along the bipartition boundary, and exhibits area-law scaling  local ∼  −1  −1 A to leading order in  A while  topo is the topological entanglement negativity, a universal constant contribution that is believed to characterize the long-range entanglement in topological order.
Entanglement Negativity at Zero Temperature and an Emergent SPT Order-We now demonstrate the emergence of an SPT order on the entanglement bipartition in a topologically-ordered state at zero temperature, and the relevance of this order for the topological entanglement negativity.We focus on the Z 2 topological order in various spatial dimensions, and find that the emergent SPT order is protected by a Z 2 × Z 2 symmetry.The SPT order hosts two distinct symmetry charges corresponding to two species of gapped excitations in Z 2 topological order, namely the charge () and flux (), and the protecting symmetry is given by the action of emergent conservation laws in the topological order along the entanglement bipartition.The emergent SPT order completely characterizes the negativity spectrum (i.e. the eigenspectrum of the partially transposed density matrix ρ A ) in that various eigenvalues correspond to various operator choices in the strange correlators [46] that diagnoses the SPT orders.
Here we outline the approach for taking a partial transpose and discuss the emergence of SPT wave functions localized on the bipartition boundary.Consider a stabilizer Hamiltonian Ĥ = −  =1 θ  , where each stabilizer θ  is a tensor product of Pauli operators over lattice sites.Distinct terms in the Hamiltonian mutually commute, and each stabilizer has eigenvalues   = ±1.As a result, the density matrix for this system at zero temperature is ρ

𝑗
with   ∈ {0, 1}.Alternatively, the spectrum  can be expressed as an expectation value of operators evaluted in the Hilbert space of the {  } variables: where |+ = 2 − /2 {  } {  } , and the Pauli operator   acts within the {  } Hilbert space as Only the choice of stabilizer eigenvalues   = 1 ∀  gives a non-zero eigenvalue of the density matrix, as expected.
We now consider the entanglement negativity within the ground state.By dividing the system into disjoint subsystems A and B, taking a partial transpose on A gives: ( We now focus on the stabilizer Hamiltonian for the toric code, which describes Z 2 topological order in  spatial dimensions.Since only those stabilizers on the bipartition boundary can anticommute with each other when restricted in A, the negativity spectrum of the toric code at zero temperature depends only on the boundary part of the Hamiltonian, namely, Ĥ = −   ∈  Â −    ∈  B  , where { Â }, { B  } are the Pauli- and -type stabilizers corresponding to the gapped Z 2 charge and flux excitations of the toric code, while   ,   denote the locations of those stabilizers acting across the bipartitioning boundary.Following Eq.1, one finds (, ), with   ,   ∈ {0, 1} and the sign (, ) may be written as where  ∈   denotes a sum of   variables adjacent to   , and  ∈    denotes a sum over the   variables adjacent to   .This is because any two adjacent Â and B  must anticommute when restricted on a subregion.Using Eq. 2, one can define the state | = 1 √ N {  ,  } (, ) |,  with normalization constant N , and the negativity spectrum can be derived as [51] The wavefunction | exhibits a non-trivial SPT order with respect to Z 2 × Z 2 symmetries, which arise from restricting the conservation laws obeyed by the Z 2 charge and flux excitations to the entanglement bipartition.Each conservation law (e.g. the global Z 2 charge conservation    ∈  = 1 for the  = 2 toric code) gives rise to a symmetry of the wavefunction (e.g.  → 1 −   on all sites).In addition, | is the ground-state of the Hamiltonian   = −  ∈     ∈   −  ∈     ∈    , where the first term is the product of a Pauli-X on site  in   and the Pauli-Zs on its neighboring sites in   .The second term is defined similarly.The SPT wave function (, ) has a Z 2 × Z 2 symmetry may be understood in a "decorated domain wall" picture [50]; the phase  ∈  (−1)   ∈   implies decorating the domain walls of   charges using   charges, and  ∈  (−1)   ∈    implies decorating the domain walls of   charges using   charges.
The robust braiding of the symmetry defects in this SPT order can be observed in strange correlators, and gives rise to a topological entanglement negativity for the original Z 2 toric code.Below we demonstrate these features using the 2d toric code as an example.The model is defined on a 2d lattice with every bond accommodating a spin-1/2 degree of freedom.The Hamiltonian reads Ĥ = −  Â −  B , where Â is the product of four Pauli-X's on bonds emanating from a site , and B is the product of four Pauli-Z's on bonds locating on the boundary of a plaquette .Considering a subsystem A with a closed boundary of size , one can label the star and plaquette stabilizers on the boundary as Â1 , B1 , Â2 , B2 , • • • , Â , B , and the negativity spectrum is

|𝜓
where | is the 1d cluster state with the parent Hamiltonian   = − odd   −1    +1 − even   −1    +1 under the periodic boundary condition.| exhibits an SPT order protected by the Z 2 × Z 2 symmetry generated by odd    and even    .As a consequence, only even number of   excitations (  = −1) and   excitations (  = −1) can give non-vanishing strange correlators.This number parity conservation reflects the number parity conservation of anyon charges in the topological order.Key features in the entanglement negativity are encoded in the SPT order.
For the eigenvalue of   A with two excitations   =   = −1, the corresponding strange correlator gives On the other hand, exciting both   and   in such a way that the excitations must be exchanged in order to return to the ground state produces a sign (−1) for the strange correlator.A simple example is given by considering One can easily generalize the above discussion to arbitrary pattens of excitations, and conclude that operators  which are invariant under the Z 2 × Z 2 symmetry exhibit order in the strange correlator +|  | ≠ 0, and that the spectrum of   A only contains two eigenvalues ± +| , where plus/minus sign corresponds to trivial/non-trivial braiding of two species of excitations.Knowledge of the entire negativity spectrum allows us to derive the zero temperature entanglement negativity [51]   =  log 2 −  topo with  topo = log 2 being the topological entanglement negativity that reflects the underlying topological order. topo may be understood as a universal reduction in the negativity due to the fact that only operators  which are invariant under the Z 2 × Z 2 symmetry exhibit order in the strange correlator +|  | ≠ 0.
Entanglement Negativity Transition at Finite Temperature-For stabilizer models at finite temperature, the partial transpose still acts non-trivially on the boundary of A region, resulting in ρ A ∼  − ( ĤA + ĤB ) ( − Ĥ )  A , where ĤA / ĤB contains stabilizers supported only on A/B, and Ĥ denotes the interaction between A and B. The negativity spectrum from the boundary part remains in the form of a strange correlator: For the thermal density matrix of the 2d toric code under partial transposition, the corresponding state |() on the entangling surface is a 1d cluster state purturbed by an onsite field, which destroys the SPT order.This can be seen by computing the negativity spectrum, which takes the form of correlation functions in 1d Ising model, with different choices of {   } and {  } corresponding to different spin insertion and coupling strength between neighboring spins: Since the Ising order cannot survive at any finite temperature or any finite symmetry-breaking field, the negativity spectrum is shortrange correlated, and the non-trivial braiding structure between {  } and {  } no longer exists.As a result,  topo = 0 at any non-zero temperature in the thermodynamic limit [12], corresponding to the vanishing topological order in the thermal Gibbs state.
3d toric code -Now we discuss the negativity spectrum and the emergent SPT wave function localized on the entangling surface for the 3d toric code with the Hamiltonian  = −  Â −  B , where Â is the product of six Pauli-Xs on bonds emanating from a site , and B is the product of four Pauli-Z on bonds on the boundary of a plaquette .Therefore, the stabilizers on the 2d bipartition boundary consists of Â living on sites and B  living on links.Using the formalism introduced above, the negativity spectrum of ρ A at zero temperature reads | , where | is the ground state of the Hamiltonian   = −     ∈    −          .The first term    ∈    is a product of Pauli-X on the site  and four Pauli-Zs acting on links whose boundary contains the site .The second term        is a product of Pauli-X on the link   and two Pauli-Zs on the boundary of the link.| exhibits an SPT order protected by a Z 2 0-form × Z 2 1-form symmetry.Here the 0-form symmetry is implemented by    and the 1-form symmetry is implemented by   ∈ C    , where C is any 1d closed loop.Note that such an SPT order defined on a two-dimensional three-colorable graph has been discussed in Ref. [52].
Due to the Z 2 0-form and Z 2 1-form symmetry, eigenvalues of   A are non-vanishing only when the number of excitations in   is even, and excitations in    exist along a closed loop.Similar to the 2d toric code at zero temperature, the negativity spectrum only contains two distinct eigenvalues with an opposite sign ± +| , where the minus sign corresponds to the non-trivial braiding between two species of excitations.Specifically, a loop formed by    excitations enclosing an odd number of   excitation will give a minus one sign.In this case, the degeneacy of negativity spectrum originates from the long-range correlation in the following two kinds of strange correlators: As the 2d Ising model exhibits a spontaneous symmetrybreaking order up to a finite critical temperature   only in the absence of symmetry-breaking fields (i.e.  = 0), the existence of the long-range strange correlator and the long-range entanglement negativity requires taking   → ∞.This is consistent with the observation that forbidding point-like excitations in the Gibbs state of 3d toric code supports a finite-T topological order [9,12,53,54].Within the SPT order picture, setting   → ∞ while tuning   corresponds to enforcing the zero-form symmetry while adding a perturbation to break the 1-form symmetry.Crucially, the 1-form symmetry will be emergent at low energy as long as the Hamiltonian remains gapped and away from a quantum critical point as shown by Hastings and Wen based on the quasi-adiabatic continuation [55].Therefore, the boundary SPT order is protected by this emergent symmetry up to a finite perturbation strength, corresponding to the persistence of long-range entanglement up to a finite critical temperature.When prohibiting the excitations away from the entangling surface and only allowing the loop-like excitations on the entangling surface, the transition in long-range entanglement negativity corresponds to the transition from an SPT order to a trivial state, which turns out to be the order-disorder transition in the 2d Ising model.Specifically, we derive the exact entanglement negativity   in the 3d toric code with a  ×  bipartition boundary at all temperatures in the Supplemental Material [51]: where   = −2 2   is the ground state energy and  () = {  }         is the partition function for the Ising model on a square lattice of size  × .This result indicates that the negativity relates to the free energy in the 2d Ising model.As approaching the zero temperature, the partition function  take the form of    −  with   = 2 being the number of ground states, giving the zero-temperature negativity   ( = 0) =  2 log 2 − log   .As a result, one finds topological entanglement negativity  topo simply reveals the number of symmetry broken sectors through the expression  topo = log   = log 2.Moreover, since the transition in the Ising model occurs at a finite temperature   , above which log  () no longer has the subleading term log   in the thermodynamic limit (due to the restoration of the Ising symmetry), the topological entanglement negativity exhibits a discontinuity in the thermodynamic limit:  topo = log 2, 0 for  <   and  >   , corresponding to the presence and absence of long-range entanglement across the bipartition surface.In particular, as  →  +  , the Ising partition function is dominated by the largest and the next-largest eigenvalues of the row transfer matrix, i.e.  ≈   0 +   1 =   0 (1 + ( 1 / 0 )  ) ≈   0 (1 +  −/  ) with  being the correlation length for the 2-points function in the 2d Ising model [56].This suggests the following scaling form of topological negativity where the critical properties in  are within the 2d Ising universality class.
Above we have discussed a disentangling transition in longrange entanglement is destroyed by thermalizing the boundary while the bulk remains fixed at zero temperature.Now we consider the general situation (Fig. 2) with a tunable bulk temperature  bulk = 1/ bulk and a tunable boundary temperature   = 1/  , namely, ρ ∼  − bulk ( ĤA + ĤB )  −  Ĥ .We still impose the condition   → ∞ so that the point-like excitations are prohibited in the thermal state.As  bulk <  bulk,c , the bulk is topologically ordered, and the corresponding looplike excitations are well-defined and obey an emergent, local constraint after an appropriate coarse-graining.This local constraint results in an emergent one-form symmetry localized on the entanglement bipartition and gives rise to an emergent SPT order in the description of the partially-transposed density matrix; this implies that the topological negativity  topo = log 2 at small   .More precisely, we show that the negativity relates to the annealed average of the 2d boundary theory over the 3d bulk fluctuation, and for  bulk <  bulk,c , the bulk fluctuations can be integrated out, leaving behind a coarse-grained 2d boundary theory whose universal properties remain the same as in  bulk = 0 [51].Therefore, at a fixed  bulk <  bulk,c , increasing the boundary temperature   leads to a disentangling transition from long-range entanglement ( topo = log 2) to short-range entanglement ( topo = 0) where the universality belongs to the aforementioned 2d Ising universality.Alternatively, the long-range entanglement can vanish by destroying the bulk topological order when increasing  bulk at a fixed   .This is because for  bulk >  bulk,c , the loop-like excitations are in the confined phase where the emergent gauge symmetry no longer exists.This in turn invalids the description of SPT localized on the bipartition boundary, which indicates the absence of long-range entanglement.
4d toric code -Finally, we discuss the 4d toric code where spins reside on each face (i.e.2-cell) of a 4 dimensional hypercube.The Hamiltonian is Ĥ = −   Â −    B , where Â is the product of 6 Pauli-X operators on the faces adjacent to the link , and B is the product of

𝑝
respect the one-form symmetry.
This implies that these operators are closed loops supported on the edges of the direct lattice and its dual lattice, and the braiding between a loop in the direct lattice and a loop in the dual lattice gives a −1 sign factor in the negativity spectrum as a consequence of the emergent SPT order.Note that the sign structure of braiding between loops in this SPT order has been discussed in Ref. [58].At finite temperature, negativity spectrum from the boundary Gibbs state is given by the strange correlator with   = − log[tanh(  )].Since the deconfined phase of the gauge theory persists up to a finite /  , /  , [59][60][61][62][63], the SPT order in |() persists up to a finite symmetrybreaking field that corresponds to   , below which the longrange entanglement negativity exists.Note that despite the perturbation, the SPT order is protected by the emergent 1form symmetry through out the entire deconfined phase [55,[63][64][65].
Using the emergent SPT order, we now discuss the nature of transition in long-range entanglement.As considering zero temperature in the bulk, tuning the boundary temperature drives a disentangling transition that corresponds to the presence/absence of long-range entanglement across the bipartition surface.The universal properties of this transition is governed by the transition from an SPT order to a trivial state.In particular, for the case where one type of excitations on the entangling surface is prohibited, e.g. say   → ∞ so the matter fields are absent in Eq.8, we determine the entanglement negativity in the Supplemental Material to be [51] Such an expression resembles Eq.6 for the 3d toric code with point-like charges forbidden, but here   = −3 3   and  () = {  }    ∈    denote the ground state energy and the partition function in the 3d classical pure Z 2 gauge theory.This expression implies that the transition in negativity is mapped to a confinement-deconfinement transition of the Z 2 gauge theory.Moreover, across the critical temperature   in the thermodynamic limit, log  exhibits a discontinuity in its universal subleading term, i.e. log  ( −  ) − log  ( +  ) = 2 log 2, indicating  topo = 2 log 2, 0 respectively for  <   and  >   .This can be seen by mapping the finite-temperature 3d classical Z 2 gauge theory to the ground subspace of 2 + 1D quantum Z 2 gauge theory  = −    ∈     −       with the Gauss law imposed on every vertex   ∈+    = 1.Tuning  drives a transition from a deconfined phase with four degenerate ground states to a confined phase with a single ground state, thus resulting in a discontinuity of log 4 in log  at the critical point of the 3d classical Z 2 gauge theory, which belongs to the 3d Ising universality.
Finally, for the more general case when the bulk is thermal (but one type of excitations is still prohibited), the schematic phase diagram of bulk topological order and the long-range entanglement still follows Fig. 2. In particular, when the bulk is topologically ordered ( bulk <  bulk,c ), tuning the boundary temperature drives the disentangling transition, where the critical properties of long-range entanglement are still governed by the SPT-trivial order transition as we argue in the Supplemental Material [51].On the other hand, for the case where both types of excitations are allowed on the boundary, since the negativity spectrum is described by 3d Ising gauge theory coupled to matter, we expect the transition in the negativity remains governed by a deconfinement transition.This is also suggested based on the replica calculation[51], but we are unable to derive a closed-form expression of negativity.
Summary and discussion-In this work, we point out an intriguing connection between topological order and SPT order via partial transpose.The gapped excitations in Z 2 topological order manifest as symmetry charges in the SPT order localized on the entangling surface, and these symmetry charges exhibit long-range correlation and robust braiding structure which reflect the underlying topological order.In particular, stability of topological order at finite temperature corresponds to stability of SPT order under symmetry breaking fields.The robustness of the SPT is possible if the broken symmetry is a one-form symmetry, in agreement with the fact that a finite-T topological order is allowed when supporting loop-like excitations.This provides a novel understanding in the existence of finite-T topological order in the 4d toric code, as well as the 3d toric code with point-like charges forbidden.In addition, assuming the excitations only occur on the bipartition boundary, we completely determine the nature of the transition for topological order via a mapping to certain statistical models.
We mainly focus on the Z 2 topological order, so it is natural to ask whether the emergent SPT order picture applies to more general types of topological order.In this regard, we briefly discuss two other generalizations, the details of which will be presented in the forthcoming work [47].First, our result can be generalized from Z 2 to Z  gauge group, in which case a partial transpose acting on a Z  topological order leads to a Z  × Z  SPT order localized on the entangling surface.Such a calculation is more involved since a partial transpose acting on a stabilizer string with Z  structure not only induces a nontrivial sign, but also acts non-trivially on stabilizers such that the resulting local operators no longer commute.This noncommuting structure means that the partially transposed Gibbs state is not diagonal in the eigenbases of stabilizers.Nevertheless, SPT order is encoded in the matrix elements, and one can still analytically solve for the negativity spectrum.Second, the physics of emergent SPT orders induced by partial transpose is applicable to fracton topological order.Specifically, for Xcube model [66] and Haah's code [67], i.e. the representative of type-I and type-II fracton orders, a partial transpose results in an SPT order that is protected by subsystem symmetries [68] and fractal symmetries respectively [69].
where we have used and N is a normalization constant such that sum of all eigenvalues is one.Now we apply the above formalism to the toric code Hamiltonian: Ĥ = −   Â −    B  , where Â and B  denote the stabilizers consisting of Pauli-X and Pauli-Z operators respectively.First we divide the entire system into two subsystems A and B, and write the Hamiltonian as Ĥ = ĤA + ĤB + Ĥ , where ĤA ( ĤB ) denotes the terms in Ĥ suppported on A (B), and Ĥ denotes the interaction betwene A and B. Since only the stabilizers acting on the bipartition boundary can anticommute when restricted in a subregion, the partial transpose acts on the boundary part of ρ as ρ A ∼  − ( ĤA + ĤB )  − Ĥ  A , and the non-trivial feature of spectrum of ρ A solely comes from the boundary part  − Ĥ  A , which we derive below.Let   and   label the collection of lattice sites corresponding to the location of   and   stabilizers acting on the boundary (e.g.see Fig. 3), applying Eq.A9 gives the following spectrum where the state | lives in the Hilbert space spanned by {  ,   } , and can be written as (via Eq.A6) where the sign ({  ,   }) =  ∈  (−1)   ∈   (via Eq.A3) with  ∈ indicating a summation over the sites  ∈   that are adjacent to the site  ∈   .This is because any two adjacent boundary stabilizer Â and B  must anticommute when restricted on a subregion.Alternatively, the sign  ∈  (−1)   ∈   can be written as  ∈  (−1)   ∈    .Therefore, Eq.A10 and Eq.A11 show that the eigenspectrum of  −   A are given by various choices of {   ,   }, which correspond to various choices of Pauli-Z's insertion in the strange correlators.
Strange correlators as conventional correlators in classical statistical models: By explicitly computing the strange correlators, one finds that they can be expressed as multi-spin correlation functions of   spins (  = 1 − 2  = ±1) in a classical statistical model: where H is consisting of the onsite-field terms as well as the interactions with a coupling strength specified by   : H =    1−  2 −       ∈    with   = − log(tanh(  )).Alternatively, the negativity spectrum can also be expressed as multi-spin correlation functions of   = 1 − 2  = ±1 in the corresponding "dual" classical model: where H =    1−  2 −       ∈   with   = − log(tanh(  )).This formalism allows us to derive the statistical models that determines the negativity spectrum for -dim toric code.

2d toric code
The boundary of the 2d toric code involves alternating  1 ,  1 ,  2 ,  2 • • •   ,   stabilizers, and therefore one can define a 1d lattice with   defined on the -th site and   defined on the link between the  and +1-sites.

3d toric code
The boundary of the 3d toric code involves   on lattices and    on links in a 2d lattice.The effective classical model describing the negativity spectrum is given by a 2d classical Ising model: H =     bulk.We show that the negativity spectrum encodes long-range braiding between two types of charges below   , and we derive the exact result of entanglement negativity at all temperatures.
When forbidding the bulk excitations, the negativity spectrum   A is solely given by  −   A , which can be written as correlation functions in the 2d Ising model under a symmetry-breaking field (via Eq.A12): Here the constraint   ∈     = 1 is imposed on every plaquette , which is a consequence of the local constraint in the 3d toric code that the product of six   stabilizers on the boundary of each cube equals identity as the bulk excitations are prohibited.{   }, {   } determine the choice for correlators and the sign of interactions between neighboring spins.The above expression suggests that a finite-temperature order can exist only when the symmetry-breaking field   = 0 (i.e.  → ∞), corresponding to prohibiting point-like excitations in Gibbs states at any temperatures.In this limit, the negativity spectrum is Due to the Ising symmetry in the Boltzmann weights, non-vanishing negativity spectrum requires the quantity    1−  2 having even number of   spins, which amounts to the constraint that    = 1.Now let's analyze the sign structure of negativity spectrum.First consider the case with no charges, i.e.   ,    = 1, the corresponding eigenvalue   A ∼ {  }          is surely positive.Next, we consider    = 1∀   , which gives the eigenvalue

𝑖
{  }          .Choosing   =   = −1 gives the two-point correlation {  }              .This is again positive since it can be written as {  }       cosh(  ) +     sinh(  ) , where one can expand the product   and notice that only terms without containing   spin variables will survive after the summation {  } .Such an argument applies to the expectation value of 2-point functions for any integer .We now consider a case with negative eigenvalue by flipping {   } and {   } at the same time.Notice that the constraint   ∈     = 1 implies different allowed {   } configurations are generated by flipping    on four links emanating from the site .To have a negative eigenvalue of   A , one can set   =   = −1 and    = −1 on four links emanating from the site .The corresponding eigenvalue is negative as can be seen by making a local spin flip at the site , giving rise to   A ∼ − {  }              .Pictorially, one can connect the two lattice sites ,  with   =   = −1 with the -string, and construct a -loop corresponding to four links with    = −1.A minus sign results from the -string piercing through the -loop.

Exact entanglement negativity
Utilizing the analysis of negativity spectrum above, we here derive the entanglement negativity for 3d toric code when pointlike charges forbidden (same limit as considered above), and show that the transition of the topological order at finite temperature can be understood as a spontaneous symmetry breaking transition of the 2d Ising model.To start with, we utilize the negativity spectrum to write down the one-norm of the partially transposed Gibbs state: ρ A 1 =     with where {   } denotes a summation over {   } subject to the local constraint   ∈     = 1.For the denominator   , it is straightforward to sum over {  } and {   } to find   = 2  2 {   }         .For the numerator   , using the fact that   ∈     = 1 and the local gauge invariance of the Gibbs weight, namely,   → −  and    → −   on four links emanating from the site , one can remove    in the Gibbs weight and find ρ with {   } subject to the constraint that   ∈     = 1 on the 2d bipartition boundary.Using the local gauge symmetry in the numerator:    → −   for four links emanating from a site  with    → −   for all replicas  = 1, 2, • • • , , one can remove    in the Gibbs weight, the numerator can be simplified as Note that such the aforementioned gauge symmetry exists only for even  while for odd , sending    to −   for all replicas is not allowed (due to the violation of the constraint  =1    = 1).On the other hand, natively taking  → 1 would lead to {   }         for the numerator, which would give ρ A 1 = 1 (i.e.negativity would be zero).Using the constraint   ∈     = 1, one can introduce the Ising variables via    =     so that with  = {  }          being the partition function of 2d Ising model that behaves as  = 2 − 2  for  <   and  − 2  for  >   with  being the free energy density.Therefore, the non-zero log 2 topological entanglement negativity results from the spontaneous symmetry breaking of the Ising model that emerges from the local constraint of the boundary    stabilizers, namely,   ∈     = 1.Now we consider the Gibbs state with tunable temperature in the bulk and the boundary, i.e. ρ ∼  − bulk ( A + B )  −   AB .The 2d boundary theory is coupled to the 3d Ising gauge theory in the bulk, and the aforementioned constraint of the boundary plaquettes no longer exists due to the fluctuating   stabilizers in the bulk.However, when the bulk is in the low-temperature deconfined phase ( bulk <  bulk, ), the Wilson loop operator satisfies the perimeter law, i.e.  () ∼  − | | where | | is the perimeter of the closed loop .It is known that one can find a renormalized (fattened) Wilson loop operator such that it satisfies the zero-law (i.e. the Wilson loop does not decay at all).Since the product of plaquette across the bipartition boundary is equivalent to the product of two Wilson loops in the bulk, one expects to find an emergent constraint on those boundary plaquettes satisfied on a larger length scale so that one can find a coarse-grained 2d Ising model, which displays an order-disorder transition as tuning the boundary temperature.On the other hand, for  bulk >  bulk, , due to the confinement of the Wilson loop in the bulk, the emergent constraint on the boundary plaquettes no longer exists.Therefore, there is no emergent 2d Ising model description that exhibits an ordered phase, contributing to topological entanglement negativity.The 4d toric code exhibits a topological order below a certain critical temperature   .As a simplification, here we consider only the bipartition-boundary part of the density matrix by forbidding any excitations in the bulk.The spectrum of the partially transposed boundary part of the Gibbs state is characterized by strange correlators that can be written as correlation functions in a 3d Ising gauge theory coupled to matter field: Here {   } determines the spin insertion in the correlator and {  } determines the sign of interaction between spins in the 3d lattice.To understand the structure of negativity spectrum, it is useful to denote   = −1 with an occupied link in the lattice and denote   = −1 with an occupied link in the dual lattice piercing through the plaquette  on the original lattice.Due to the local constraints in 4d toric code, the bipartition-boundary stabilizers   and   are subject to the constraints that the product of 6 plaquettes   on the boundary of each cube is one, and the product of 6 links   emanating from each vertex is one.The above constraints amount to imposing the condition that only closed loops of   (denoted as -loops) in the direct lattice and closed loops of   (denoted as -loops) in the dual lattice are allowed.Eq.D1 shows that the negativity spectrum is characterized by a classical 3d Ising gauge theory coupled to matter fields, which therefore exhibits a deconfinement-confinement transition at a certain critical temperature.As a result, such a transition corresponds to the transition for topological order in the toric code.Note that it is interesting that while the thermal partition function tr  − ĤAB can be written as a product of two partition functions for two independent pure gauge theories (therefore exhibiting two transitions with critical temperatures   ∼  (  ) and   ∼  (  )), the partially transposed Gibbs state exhibits a single deconfined transition which is determined by both   and   .It is also interesting that setting   =   while vaying the temperature corresponds to a transition along the well-known self-dual line in the gauge theory [59][60][61][62][63] 2. Exact entanglement negativity Here we consider the limit   → ∞, i.e.   = 0, and discuss the sign structure of negativity spectrum and the derivation of entanglement negativity.In this case, the negativity spectrum is given by the pure gauge theory ∈    . (D2) Here we first discuss the braiding structure between -loops and -loops.Consider the case with a single -loop, the corresponding eigenvalue is {  }  ∈        ∈    , which is nothing but a Wilson loop in the 3d Ising gauge theory.Such a quantity exhibits a long-range correlation below a certain critical temperature   (deconfined phase) in the sense of a perimeter-law  − |  | > 0, where |  | denotes the length of the close loop   .Now we consider adding a -loop in the dual lattice that braids with the -loop by flipping   .It follows that the corresponding eigenvalue can be written as − {  }  ∈        ∈    , which remains long-range correlated in the deconfined phase with the perimeter-law scaling − − |  | > 0. The above analysis indicates that the braiding and sign structure survives in the long-distance below   .Now we discuss the calculation of negativity.First, the one norm of ρ A with ρ ∼  − Ĥ is the sum of all absolute eigenvalues: where the denominator is simply fixed by the requirement that sum of all eigenvalues of   A is one.Here {   } {  } refers to summing over {   } and {  } subject to the constraints that the product of 6 plaquettes   on the boundary of each cube is one, and the product of 6 links   emanating from each vertex is one.To resolve the local constraint of {  }, one introduces the dual variables {  } on links via   =  ∈    , and therefore {  } can be replaced by summing over independent   variables.Using a calculation analogous to 3d toric code, we find with  −2 = tanh().Therefore,  ∼    −   , where   is the ground state energy of the 2+1D quantum Ising gauge theory, and   is the corresponding ground state degeneracy.Crucially, tuning  in such a model induces a confinement-deconfinement transition at a critical   .The regime  <   corresponds to the deconfined phase with   = 4, while the regime  >   corresponds to the confined phase with   = 1.As a result, there exists a universal subleading term in log  that characterizes the number of topological sector in the gauge theory, and  topo = 2 log 2 or 0 for  <   and  >   .
3. Replica calculation for general   and   when forbidding bulk excitations In the discussion above, we consider the limit   → ∞ to derive the entanglement negativity.For the negativity at any   and   , we employ a replica trick by studying tr ρ A  for even integer , from which negativity can be obtained by taking  → 1 limit, namely,   = lim even →1 tr ρ A  = lim even →1 Z /, where  is the thermal partition function  = tr  − Ĥ and Z is the n-th moment for the boudary part of the Gibbs state, i.e.Z = tr  and evaluating such a quantity for even  followed an analytic continuation to  → 1 gives entanglement negativity.Although we are unable to compute such an quantity analytically, this expression suggests that the transition in negativity relates to the deconfinement-confinement transition of 3d Ising gauge theory coupled to matter fields.

Entanglement negativity when bulk is thermal
We here discuss the entanglement negativity when bulk excitations are allowed, but one type of excitations is prohibited (via   → ∞).In this case, we consider the partially transposed Gibbs state ρ A =  − ( ĤA + ĤB )  − Ĥ

Z 2
t e x i t s h a 1 _ b a s e 6 4 = " T 7 1 X / w L T S z c 0 a R 6 D V J n O J y R A 7 e g = " > A A A C D n i c b V C 7 S g N B F J 3 1 G e M r a m k zG A J W Y T c E t Q z Y W E b I C 5 M Q Z i c 3 y Z D Z n W X m r h i W / Q I b f 8 X G Q h F b a z v / x s m j 0 M Q D A 4 d z z m X u P X 4 k h U H X / X b W 1 j c 2 t 7 Y z O 9 n d v f 2 D w 9 z R c c O o W H O o c y W V b v n M g B Q h 1 F G g h F a k g Q W + h K Y / v p 7 6 z X v Q R q i w h p M I u g E b h m I g O E M r 9 X K F T s B w 5 P v J X d o r d R A e M K E 1 F d m 5 o c 1 I q n Q f d N r L 5 d 2 i O w N d J d 6 C 5 M k C 1 V 7 u q 9 N X P A 4 g R C 6 Z M W 3 P j b C b M I 2 C S 0 i z n d h A x P i Y D a F t a c g C M N 1 k d k 5 K C 1 b p 0 4 H S 9 o V I Z + r v i Y Q F x k w C 3 y a n y 5 t l b y r + 5 7 V j H F x 1 E x F G M U L I 5 x 8 N Y k l R 0 W k 3 t C 8 0 c J Q T S x j X w u 5 K + Y h p x t E 2 m L U l e M s n r 5 J G q e h d F M u 3 5 X y l X p n X k S G n 5 I y c E 4 9 c k g q 5 I V V S J 5 w 8 k m f y S t 6 c J + f F e X c + 5 t E 1 Z 1 H h C f k D 5 / M H L H + d G A = = < / l a t e x i t > Topological order < l a t e x i t s h a 1 _ b a s e 6 4 = " k s f k G B A F H z P p k e h i s R T d g p S 8 F 7 Y = " > A A A C A X i c b V D N S g M x G M z W v 1 r / V r 0 I X o J F q J e y K 0 U 9 F r x 4 r O C 2 h X Y p 2 W z a h m a z S / K t W J Z 6 8 V W 8 e F D E q 2 / h z b c x 2 / a g r R 8 E h p n J l 8 w E i e A a H O f b K q y s r q 1 v F D d L W 9 s 7 u 3 v 2 / k F T x 6 m i z K O x i F U 7 I J o J L p k H H A R r J 4 q R K B C s F Y y u c 7 1 1 z 5 T m s b y D c c L 8 i A w k 7 3 N K w F A 9 + 6 g S d o E 9 g I o y H P K I y d y p J / i s Z 5 e d q j M d v A z c O S i j + T R 6 9 l c 3 j G l q V g A V R O u O 6 y T g Z 0 Q B p 4 J N S t 1 U s 4 T Q E R m w j o G S R E z 7 2 T T B B J 8 a J s T 9 W J k j A U / Z 3 z c y E m k 9 j g L j j A g M 9 a K W k / 9 p n R T 6 V 3 7 G Z Z I C k 3 T 2 U D 8 V G G K c 1 2 E y K 0 Z B j A 0 g V H H z V 0 y H R B E K p r S S K c F d j L w M m u d V 9 6 J a u 6 2 V 6 1 5 9 V k c R H a M T V E E u u k R 1 d I M a y E M U P a J n 9 I r e r C f r x X q 3 P m b W g j W v 8 B D 9 G e v z B 2 C V l x 0 = < / l a t e x i t > (d dimensions) < l a t e x i t s h a 1 _ b a s e 6 4 = " k q G T f 0 a a R x 2 K E 0 G 2 G F D 3 p V b 0 7 f Y = " > A A A C G 3 i c b V C 7 S g N B F J 2 N r x h f U U u b w S B Y h d 0 Q 1 D J g Y x k x m w S z I c x O b p I h s w 9 m 7 o p h y X / Y + C s 2 F o p Y C R b + j Z N H Y R I P D B z O u c M 9 9 / i x F B p t + 8 f K r K 1 v b G 5 l t 3 M 7 u 3 v 7 B / n D o 7 q O E s X B 5 Z G M V N N n G q Q I w U W B E p q x A h b 4 E h r + 8 H r i N x 5 A a R G F N R z F 0 A 5 Y P x Q 9 w R k a q Z M v e Q H D g e + n 9 + N O i V I P R Q B 6 Q f M Q H j G l d 9 U a j V Q X 1 L i T L 9 h F e w q 6 S p w 5 K Z A 5 q p 3 8 l 9 e N e B J A i F w y r V u O H W M 7 Z Q o F l z D O e Y m G m P E h 6 0 P L 0 J C Z C O 1 0 e t u Y n h m l S 3 u R M i 9 E O l X / / k h Z o P U o 8 M 3 k J L V e 9 i b i f 1 4 r w d 5 V O x V h n C C E f L a o l 0 i K E Z 0 U R b t C A U c 5 M o R x J U x W y g d M M Y 6 m z p w p w V k + e Z X U S 0 X n o l i + L R c q b m V W R 5 a c k F N y T h x y S S r k h l S J S z h 5 I i / k j b x b z 9 a r 9 W F 9 z k Y z 1 r z C Y 7 I A 6 / s X 7 a a h n w = = < / l a t e x i t > Z 2 ⇥ Z 2 SPT order < l a t e x i t s h a 1 _ b a s e 6 4 = " u W 4 l Q f 7 e B T t p 6 B n v i / D K s t T e 8 j Q = " > A A A C A 3 i c b V D N S g M x G M z W v 1 r / V r 3 p J V i E e r D s S l G P B S 8 e K 7 h t o V 1 K N p u 2 o c n u k n w r l q X g x V f x 4 k E R r 7 6 E N 9 / G b N u D V j 8 I D D O T L 5 k J E s E 1 O M 6 X V V h a X l l d K 6 6 X N j a 3 t n f s 3 b 2 m j l N F m U d j E a t 2 Q D Q T P G I e c B C s n S h G Z C B Y K x h d 5 X r r j i n N 4 + g W x g n z J R l E v M 8 p A U P 1 7 I N K e O p 2 g d 2 D k h k O u W R R 7 t U T f N K z y 0 7 V m Q 7 + C 9 w 5 K K P 5 N H r 2 Z z e M a W p W A B V E 6 4 7 r J O B n R A G n g k 1 K 3 V S z h N A R G b C O g R G R T P v Z N M M E H x s m x P 1 Y m R M B n r I / b 2 R E a j 2 W g X F K A k O 9 q O X k f 1 o n h f 6 l n / E o S Y F F d P Z Q P x U Y Y p w X Y j I r R k G M D S B U c f N X T I d E E Q q m t p I p w V 2 M / B c 0 z 6 r u e b V 2 U y v X v f q s j i I 6 R E e o g l x 0 g e r o G j W Q h y h 6 Q E / oB b 1 a j 9 a z 9 W a 9 z 6 w F a 1 7 h P v o 1 1 s c 3 Q t m X j w = = < / l a t e x i t > (d 1 dimensions) < l a t e x i t s h a 1 _ b a s e 6 4 = " X a d t 1 o w 1 F 1 i l / b p P f y E H x z y m L s 8 = " > A A A B + X i c d Z D L S s N A F I Y n 9 V b r L e r S z W A R X I V E v C 0 L b l x W M G 2 h D W U y n b R D Z 5 I w c 1 I s o W / i x o U i b n 0 T d 7 6 N k z Y F F T 0 w 8 P P 9 Z 8 6 c + 7 8 G 2 e T C C p a 0 F B U d d P d 5 c e C a + O 6 H 0 5 u Y 3 N r e y e / W 9 j b P z g (), where () = ±1 is a nontrivial sign determined by the number of pairs of stabilizers that anticommute when restricted in A. Specifically, introducing θ | A to denote the part of θ that acts within A and the matrix  that encodes the commutation relation amongθ | A :    = 0, 1 for [ θ | A , θ  | A ] = 0 and { θ | A , θ  | A } = 0,respectively, the sign introduced by partial transpose is ({  }) = (−1) <         .
It follows that the classical model describing the negativity spectrum is given by the 1d classical Ising model: H =    =1 1−  2 −    =1      +1 .Alternatively, one can consider the dual description by H , which is again a 1d Ising model.

FIG. 3 :
FIG. 3: The location of boundary stabilizers in d-dim toric code, where blue circles and red squares label the lattice sites corresponding to   and   stabilizers.(a) 1d bipartition boundary in 2d toric code.(b) 2d bipartition boundary in 3d toric code.(c) 3d bipartition boundary in 4d toric code.

4 .
4d toric code The boundary of the 4d toric code involves   on links and   on plaquettes in a 3d lattice.The effective classical model describing the negativity spectrum is a 3d classical Ising gauge theory: H =    1−  2 −       ∈    .Alternatively, one can consider the dual description by H , which is again a 3d classical Ising gauge theory (but defined on the dual lattice): H =    1−  2 −       | ∈    , where  | ∈    is the interaction between four  spins on plaquettes that share the same bounday link .

Appendix D :
Structure of negativity spectrum and entanglement negativity in 4d toric code 1. Structure of negativity spectrum
|() , but crucially, a +|     | / +| = 1 and +|   ∈ C    | / +| = 1.At a finite temperature, the boundary part of the negativity spectrum is  −   A ∼ +|   ) is the ground state of the perturbed SPT Hamiltonian   = −  [   ∈    + sinh(  )  ] −   [       + sinh(  )   ].The strange correlators can be written as correlation functions in the 2d Ising model: 6 Pauli-Z operators on the faces around the boundary of the cube .Since this model only possesses loop-like excitations, it exhibits a topo-logical order up to a finite critical temperature.The boundary of a 4d hypercube is a 3d lattice, where the boundary stabilizers are Â living on links and B living on plaquettes.The negativity spectrum of 4d toric code at zero temperature reads   A = +|   | is the ground state of 3d cluster state Hamiltonian[57]:   = −    : ∈    −    : ∈    .The 3d cluster state exhibits an SPT order protected by the Z 2 1-form ×Z 2 1-form symmetry so that the corresponding symmetry transformation acts on closed deformable two-dimensional surfaces.Specifically, any symmetry transformation can be obtained by taking the product of the following symmetry generators   = :  ∈   and   = : ∈   .It follows that the non-zero eigenvalules of   A require the opera- 1−  2 −            .Alternatively, one can consider the dual description by H , which is a 2d Ising gauge theory: H =     1−   2 −        | ∈      , where   | ∈      is the interaction between four  spins on links that share the same bounday site .

∼
− Ĥ +     ∈      = − log[tanh(  )] and ∼ indicates that we have omitted a prefactor [cosh(  )]   with   being the number of links in the 3d lattice.Due to the local constraint for   that the product of six   on the boundary of a cube is one, any allowed {  } can be reached by flipping four   sharinge a link , which is equivalently to flipping the spin   .Therefore, one can introduce independent   = ±1 variables living on links to resolve the local constraint on {  } and findZ ∼ ∑︁{   }, {  } +   ∈     denoting the replica index, we can sum over {   } subject to the constraint that the product of six   on links emanating from a vertex is one.This effectively couples spins on different replicas: {   }  (   )   , ∈     = 1 , where the constraint is that for any given plaquette , the product of spins on its boundary across all replicas is one.Therefore, one finally simplifies Z = tr  −   A  as tr  −   A  ∼ ∑︁ −   1−    2