Contents

Supersymmetric Many-Body Systems from Partial Symmetries

- Integrability, Localization and Scrambling -

Pramod Padmanabhan, Soo-Jong Rey,

Daniel Teixeira, Diego Trancanelli


Fields, Gravity & Strings, CTPU

Institute for Basic Science, Daejeon 34037 KOREA School of Physics and Astronomy & Center for Theoretical Physics

Seoul National University, Seoul 06544 KOREA Department of Basic Sciences, University of Science and Technology

Daejeon 34113 KOREA Institute of Physics, University of São Paulo,

05314-970 São Paulo BRAZIL pramod23phys@gmail.com, rey.soojong@gmail.com, dteixeira@usp.br, dtrancan@if.usp.br


1 Introduction

In quantum mechanics, a symmetry is implemented by requiring a system to be invariant under a certain set of transformations. These transformations must form a group, whose action on the associated Hilbert space is realized by unitary or anti-unitary operators, as stated by Wigner’s theorem [1]. If the symmetry is obeyed only by part of the system, one can still implement it by the use of sets of transformations, but these have to be taken to form an inverse semigroup rather than a group. This generalization leads to the notion of partial symmetries.

An inverse semigroup is a pair formed by a set and an associative binary operation , such that every element has a unique inverse. So, for a given , there exists a unique such that

(1.1)

There is no single identity on an inverse semigroup, but only idempotents or projectors which can be thought of as partial identities. The elements of are called partial symmetries. Since they do not act on the entire Hilbert space, but only on parts of it, there is no unitary representation of these operators. Such structures do arise in quantum mechanics but have often been discarded without getting much consideration. It turns out, however, that inverse semigroups are relevant for physics as they provide the precise description for invariances that underlie certain physical systems [2]. To convince the readers, we illustrate this point in two instances of physical interest.

The first instance concerns a complete classification of tilings of . It is known that, for every such tiling, there exists an inverse semigroup associated with it [3]. Whereas periodic crystal structures - familiar to -dimensional crystallography - are well described by group theory, aperiodic structures like quasicrystals [4, 5, 6] - see Fig. 1 - are associated to aperiodic tilings described by inverse semigroups [3, 8, 9, 10]. The classic examples of quasicrystals are the Fibonacci tiling in one dimension and the Penrose tiling in two dimensions.

Figure 1: An example of quasicrystal, which is a crystal with ordered but aperiodic structure, lacking in particular translational invariance. Picture adapted from [7].

Further studies on aperiodicity include the motion of particles in quasicrystal potentials like the Fibonacci Hamiltonian [11], works on gap-labelling theorems for such systems and, more generally, for substitution sequence Hamiltonians [12, 13].

The second instance concerns operators in Hilbert space. In a finite-dimensional vector space , a complex matrix is always decomposable to polar factorization, or , where are non-negative Hermitian matrices and is a unitary matrix. In an infinite-dimensional Hilbert space , a bounded linear operator is decomposable to polar factorization, or , where are non-negative self-adjoint operators and is an element of an inverse semigroup. Here, should be in general an element of an inverse semigroup, not of a unitary group, as exemplified by the creation and annihilation operators (with ) that involve upper and lower shifts of the number basis of .111For finite-dimensional Hilbert spaces, the counterpart of these shift operators is provided by the set of nilpotent operators. In this paper, we will dwell on some particular examples of them. Physically speaking, conjugate to the Hermitian number operator , there is no phase operator that satisfies both Hermiticity and the commutation relation .

In this paper, we shall work with a particular type of inverse semigroup, known as the symmetric inverse semigroup (SIS), which can be thought of as the analog of the permutation group in this context. Just like any group can be embedded into a symmetric group, according to the Wagner-Preston representation theorem [14], any inverse semigroup can be embedded into a SIS such that it is isomorphic to some sub SIS [2]. This means that we can think of the SISs as building blocks of an arbitrary inverse semigroup. Thus, by working with SISs, there is no loss of generality. This result is the counterpart of Cayley’s representation theorem in group theory stating that any group is isomorphic to the group of permutations or some subgroup of it.

When exploring a physical system, it is always a good idea to endow it with extra symmetries that might constrain the dynamics and bring the evaluation of observables under computation control. A symmetry that is proven notoriously capable of doing so is supersymmetry. This is a relation between bosons and fermions, which was first introduced in the context of relativistic quantum field theory as an extension of the Poincaré group, see for example [15, 16, 17, 18, 19] and [20] for a review. In this paper, we shall focus on -dimensional supersymmetric systems, viz. supersymmetric quantum mechanics [21, 22]. This system has proven to be a very useful arena, where problems with wide range of potentials could be solved exactly [23].

A supersymmetric system is equipped with a multiplet of fermionic charges, called supercharges, , that generate the supersymmetry transformations. By construction, the vacuum expectation value of top auxiliary components of supermultiplets is the order parameter of spontaneous supersymmetry breaking. After eliminating these auxiliary components, the system’s Hamiltonian is given by , implying that the vacuum expectation value of the Hamiltonian is positive semidefinite. If the supersymmetry is unbroken, is zero and . If supersymmetry is spontaneously broken, is non-zero and . Thus, the ground state energy provides an easily calculable order parameter of spontaneous supersymmetry breaking.

Another related order parameter is the Witten index [24], , the number of bosonic zero-energy ground states minus the number of fermionic zero-enegy ground states. If this quantity is non-zero, then there ought to be unequal numbers of bosonic and fermionic zero-energy ground states. If the quantity is zero, then there are either equal pairs of bosonic and fermionic zero-energy ground states or the ground states have positive energy. In the former case, the system preserves supersymmetry. In the latter case, the system breaks supersymmetry. So, provided the system’s energy spectrum is discrete, the Witten index is a quantized quantity that cannot be continuously changed by supersymmetry preserving deformations.

Topological phases of matter are described by topological invariants, examples of which include the quantum double models of Kitaev described by the genus of a surface [25], or the symmetry protected states of matter described by the cohomology of corresponding symmetry groups [26]. These invariants are often computable as the partition functions of systems or as the trace of corresponding transfer matrices [27]. Along these lines, we propose to use the Witten index, which can be thought of as a twisted partition function of systems or as a supertrace 222We use the twisted sum or the supertrace as the Hilbert space is graded. of the corresponding transfer matrix formed out of supersymmetric Hamiltonian, to also be a useful indicator for interesting phases of many-body systems, protected by the global supersymmetry.333The Witten index also has a deep connection with the topology of the bundle spaces upon which the Hamiltonian acts. This is the content of the Atiyah-Pataudi-Singer index theorem [28, 29, 30, 31]. Generalizations of this index have also been considered [32].

Following these lines of reasonings, we are naturally led to construct supersymmetric systems with partial symmetries, or supersymmetric SISs. We will do just that in the following, focusing in particular on supersymmetric many-body systems on a lattice, where the internal degrees of freedom are built from SIS algebra elements. In this way, we implement a supersymmetric algebra on the Hilbert space spanned by these elements, which can be thought of as realizing supersymmetric algebras out of partial symmetries. Many-body supersymmetry preserving zero-energy ground states can be constructed as eigenstates of many-body Hamiltonians by considering supersymmetry in the non-relativistic setting, as in statistical physics, see [33, 34] and citations therein. Supersymmetric many-body systems have been considered in the past [35], where their lattice formulation was done by realizing the global supercharges on the total Hilbert space of the system [36, 37, 38, 39, 40, 41]. Exact solutions for these models were obtained in [42]. Entanglement entropy in such systems have also been considered [43].

An important aspect of the models we construct with partial symmetries is that they are generically quantum integrable: all of these models have as many local integrals of motion as the number of sites, as we shall see. This is the case when we construct our supersymmetric system with a unique grading of the Hilbert space spanned by the elements of the SIS algebra.444However, we can work in a scenario where the system is constructed out of two different gradings of the Hilbert space and in such a case we obtain non-integrable systems. This is discussed in App. B. Given this, we ask the question about scrambling properties in such supersymmetric systems, which forms the subject of the remaining part of the main text.

Recently, the topic of scrambling has proven to be of great interest in many fields, from the physics of black holes to quantum information. Concretely, scrambling can be used to identify the onset for quantum chaos [44] and the propagation of entanglement [45]. From a gravitational/holographic perspective, black holes are conjectured to be the fastest scramblers in Nature [46] and, in the same spirit, it has been conjectured that chaos cannot grow faster than in Einstein gravity [47, 48]. These works provide several hints connecting fast scrambling and thermalized phases, which are embedded into a large program concerned with the full understanding of thermalization in complex quantum systems.

On the opposite extreme, and the focus of this work, are systems where time evolution results in a localized phase and no thermalization occurs at all. Roughly speaking, thermalization is the process by which a system evolves to a point in which it can be well described by few thermodynamical quantities. This seems to conflict with the quantum mechanical intuition that evolution is unitary and the system is not expected to lose information about the initial state. The resolution to this puzzle is the eigenstate thermalization hypothesis (ETH) [49], which says that a quantum system thermalizes if it does so irrespective of the initial state it was prepared in, thereby also giving a proper definition of quantum thermalization. This brings up the possibility of the existence of systems that have states where such a process of equilibration does not occur and, consequently, the ETH is not satisfied. A classic example of such a system includes integrable models whose local integrals of motion prevent the transport of conserved quantities hindering the reach of equilibrium. The presence of disorder is another source of localization and loss of thermalization. In situations in which the phenomenon is described by one-particle effects, this is called Anderson localization (AL) [50], while in cases in which it is described by the many-particle effects, it is known as many-body localization (MBL). A recent review on thermalization and localization is provided by [51].

The supersymmetric models we present here fall into the MBL class. The presence of “local” integrals of motion in these systems prevent the transport of the conserved quantities and thus we expect to find MBL states in these systems. Since any such supersymmetric system constructed out of a unique grading of the SIS algebra is expected to have this property, we hereby developed a method of constructing supersymmetric MBL states using the SIS algebras. The system we study is a long-ranged interacting one and we emphasize that this is done just for simplicity. In App.  A, we construct short-ranged interacting supersymmetric systems built out of a unique grading of the SIS algebra, which continue to have local integrals of motion and are thus expected to possess MBL states.

We shall deduce the MBL property of these systems by studying the out-of-time-order correlators (OTOC) [44, 45, 47]. OTOCs have been recently used for analyzing localized systems, in particular in distinguishing MBL and AL phases in [52, 53, 54, 55, 56]. This leads to connections between the OTOC’s, scrambling, and localization. We briefly summarize the procedure used for this purpose.

To probe scrambling using OTOCs, one has to consider two local, arbitrary operators at different times, say and , and the squared commutator between these probe operators in some state of interest, which can be taken to be a thermal state with temperature

(1.2)

or, alternatively, the out-of-time order correlator

(1.3)

that is contained in Eq. (1.2). The behavior of these quantities as a function of time has crucial information about the system. For example, thermalized, MBL and AL phases can be diagnosed, respectively, by an exponential decay, a power-law decay and a constant behavior of the OTOC. We will refer to the systems presenting less than exponential OTOC as slow scramblers. It should be noted that OTOC have been considered long ago in the context of semi-classical methods in superconductivity [57].

With this setup we organize the paper as follows. In Sec. 2, we review the basics of partial symmetries and SIS and construct single-particle supersymmetric systems out of SISs. In Sec. 3, we construct supersymmetric models in dimensions based on SIS with supercharges leading both to free and to interacting Hamiltonians. In Sec. 4, we apply this setup to investigate scrambling in interacting many-body systems. We will present a toy model with quenched disorder that exhibits a supersymmetric MBL phase and argue that systems generated using supersymmetric realizations of SIS should be slow scramblers. An outlook for future work and a discussion of the scope of these supersymmetric SIS models is given in Sec. 5.

We also include several appendices, where we discuss other possible supersymmetric systems based on SISs. In App. A, we show other supercharges that exhibit a spectrum which is more complicated than the ones considered in the main body of the paper. App. B sets the ground for possible ways of constructing non-integrable supersymmetric many-body systems with partial symmetries. We construct supersymmetric systems that have entangled eigenstates in App. C. Finally, in App. D we sketch how to obtain para-supersymmetric quantum mechanical systems from partial symmetries.

2 Partial symmetries and supersymmetric quantum mechanics

In this section, we begin with the simplest quantum system with partial symmetries: a one-particle or one-site quantum mechanical system defined on a finite-dimensional Hilbert space.

Let . Consider the set of all partial bijections on together with the usual composition rule, which is binary and associative. This pair forms a SIS, denoted by . We can also form a class of SIS by choosing subsets of order , , and considering the set of partial bijections on this subset. We will refer to the resulting SIS as . Taking the elements of as basis, we will construct a Hilbert space, whose inner product is defined by the natural pairing of the basis. In what follows, we will work with such SISs and Hilbert spaces.

2.1 Diagrammatics for SISs

We shall first introduce a convenient diagrammatic way of representing the elements that renders the binary operations transparent and defines an algebra over .

To illustrate this in a transparent manner, let us start with the simplest example, , whose diagrammatics are shown in Fig. 2. The partial symmetry elements of are denoted by , with , and obey the following composition rule

(2.1)

The indices and can be thought of, respectively, as the domain and range of the partial symmetry operation. The product between these elements is null when the range of the first element is different from the domain of the second element it is being composed with. Note that this product is non-commutative.

Elements of

Composition on

Figure 2: Diagrammatic representation of . The composition rules are obtained by tracing arrows connecting two elements. If one cannot trace a continuous arrow, the product is 0.

The next step up in complexity is illustrated by , which is made up of nine elements, with . These partial symmetries of are depicted in Fig. 3.

Figure 3: Elements of .

Further moving up, we can construct a SIS with arbitrary and , . For the sake of definiteness, we will focus on , but the formulation is straightforwardly generalizable to any .

We next associate Hilbert spaces to every SISs. The Hilbert spaces we consider are spanned by the partial symmetries of the chosen SIS, meaning that we are working with the algebra of this SIS. Therefore, the SIS acquires a vector space structure and an inner product is naturally defined by

(2.2)

For instance, an arbitrary element of the Hilbert space spanned by the elements of is given by

(2.3)

This is equivalent to working in the regular representation of the chosen SIS.

2.2 Supersymmetric systems from SISs

We now turn to the construction of a single-site, one-particle supersymmetric system. The first step is to realize the supersymmetry algebra from the SISs. Start with and define

(2.4)

As and are nilpotent, they can be thought of as supercharges. Out of these supercharges, we construct the Hamiltonian in the usual manner

(2.5)

In the regular representation, this is just the identity operator and hence leads to a trivial spectrum. The Hilbert space has a -graded structure, and , as shown in Fig. 4. These two halves may be dubbed as the “bosonic” and the “fermionic” halves of the space corresponding to the fermion number operator, . In this case this turns out to be the projector as well. Consequently, the and operators are the “bosonic” and “fermionic” parts of the Hamiltonian.

Note that this system admits no zero-energy ground state, for if were such a state, it must satisfy that and there is no such state in . The Witten index for this system is zero.

I

II

Figure 4: The grading of the Hilbert space spanned by the partial symmetries of . and are the “bosonic” and “fermionic” parts of this grading. The grading operator is given by which gives eigenvalue +1 on and -1 on .

We can construct a system with non-empty zero-energy ground states, the first nontrivial example being using the partial symmetries of with dim. We choose the one-site supercharges as

(2.6)

and we straightforwardly confirm that they are nilpotent. The Hamiltonian is now given by

(2.7)

where

(2.8)

I

II III

Figure 5: The grading of the Hilbert space spanned by the partial symmetries of . and are the “bosonic” and “fermionic” parts of this grading. The grading operator , [see Eq.(2.12)], gives +1 on and -1 on .

The Hilbert space is graded, as shown in Fig. 5. Note that this grading is just an arbitrary choice. We could have made the gradings which split the space as II, I+III or III, I+II, corresponding to cyclic permutations of in the choice of the supercharges. Such possibilities and their consequences for many-body systems are further discussed in App. B. It consists of two subspaces, the “bosonic” and “fermionic” subspaces, and respectively. First, there is the three-dimensional subspace comprised of the normalized zero-energy ground states

(2.9)
(2.10)
(2.11)

We introduce the fermion number operator as

(2.12)

Clearly, this has eigenvalue 1 upon acting on the states in , so that the ground states are all fermionic. This makes . Second, there is the three-dimensional subspace consisting of the excited fermionic states

(2.13)
(2.14)
(2.15)

We denote this space by , where the label stands for “excited fermionic”. Finally, there is the three-dimensional subspace , consisting of bosonic states (the fermion number operator acting on these states gives zero). These states are given by

(2.16)
(2.17)
(2.18)

The supercharges pair up bosonic states in and excited fermionic states in , as dictated by the product rules of the underlying SIS. They are excited states with positive energy eigenvalues. Note that

(2.19)

so the act as projection operators on bosonic and fermionic subspaces, respectively. Correspondingly, the energy eigenvalue of the excited states is 1. The fermionic zero-energy ground states are unpaired. Summing up, we have and .

The Witten index is computed as

(2.20)

thus counting correctly the three fermionic zero-energy states in .

The grading of is provided by the Klein operator . It is easy to check that and , satisfying the usual properties of the fermionic number operator in a supersymmetric theory.

As a remark we only consider operators that are even under the grading, they then form superselection sectors of the theory which are essentially the “bosons” and the “fermions”. We cannot have physical states that are a superposition of the two sectors, as they do not have a well defined fermion number. However, it is possible to construct many-body supersymmetric systems with the above grading where entangled states exist, a subject discussed in App. C.

2.3 Supersymmetric deformations and Witten index

We are interested in classifying deformations of the Hamiltonian while preserving supersymmetry. We do not expect these deformations to mix different zero-energy ground states nor to lift them to excited states. As such, we require the Witten index to be invariant under supersymmetry preserving deformations added to the Hamiltonian.

We will classify the supersymmetry preserving deformations into two parts, those that are obtained by deforming the supercharges to resulting in deformed supersymmetric Hamiltonians, , and those that are added to the original supersymmetric Hamiltonian as just perturbations that can possibly lift the ground state degeneracy. We show that the Witten index is left invariant by both these types of deformations.

Deformed supercharges

Consider deformed supercharges of the form

(2.21)

with . The resulting deformed Hamiltonian is given by

(2.22)

with

(2.23)

with both , being orthogonal projectors as in the undeformed Hamiltonian, which is recovered with . Now we can compute the zero modes for this deformed system and we find them to be precisely

(2.24)
(2.25)
(2.26)

These are fermionic ground states under the old fermion number operator, , Eq. (2.12), and the corresponding Klein operator, . Thus we see that we again have -3 as the Witten index for the deformed supersymmetric Hamiltonian.

Supersymmetry preserving perturbations

The simplest class of supersymmetry preserving deformations corresponds to local perturbations that can be added to the Hamiltonian while at the same time preserving the supercharges, and , and commuting with the Klein operator, . The only operator that preserves the supercharges in Eq. (2.6) and the Klein operator is provided by the , which is the Hamiltonian itself. Clearly this does neither mix the three zero modes in Eqs. (2.9)-(2.11) nor does it create an energy gap among these states.

We could ask for nontrivial deformations which do not preserve the supercharges but still keep the Witten index unchanged. It turns out that the system is stable to any perturbation built as a linear combination of , and and their hermitian conjugates. This can be seen by their action on the zero-energy ground states in Eqs. (2.9)-(2.11). Accordingly, the system keeps the Witten index intact. Such deformation extends to more nontrivial partial symmetries.

For , with arbitrary , we find that the Witten index is given by

(2.27)

and remains invariant under the class of deformations specified above.

This number for the Witten index can be seen by considering the following supercharge for an arbitrary

(2.28)

and its conjugate, . The zero modes are now given by

(2.29)

which counts of the zero modes and for each of them we have choices in cyclic permutations of . Here is the -th root of unity. Thus we have zero modes which are all fermionic.

This index is again left invariant by deformed supercharges of the form

(2.30)

The argument goes just as in the case of . Apart from these deformed supercharges, the system is also invariant under the local perturbations which are a linear combination of , and their hermitian conjugates, as these operators do not lift the degeneracy of the ground states just as in the case.

3 Supersymmetric systems on a chain

So far, we constructed a system whose supercharge is defined on a single site, so it could be thought of as a one-particle supersymmetric quantum mechanics. Our next step is to extend the construction to a many-body supersymmetric system on a chain, described by a globally defined supercharge. For simplicity, we will choose the homogenous chain such that all sites are equivalent. The Hilbert space of the -site lattice is , where each site supports one and the same Hilbert space spanned by the partial symmetries of . In total, . The Hamiltonian is determined once the supercharges are specified.

We will first need to choose the grading of , which again can be chosen from many possibilities. We continue adopting the convention that, locally, the sector I is “bosonic” and the sectors II+III are “fermionic”. We will now present several examples of many-body systems that can be obtained by selecting different supercharges for this choice of grading.

3.1 Non-interacting supersymmetric chain

As a warm-up, we start by considering a simple system in which different lattice sites do not interact with one another. This corresponds to taking the supercharge as

(3.1)

where, by definition, ’s are anticommuting and nilpotent variables, , that are built out of ’s and ’s. We can concretely realize these variables using the ’s in Eq. (2.6) as follows

(3.2)

where the fermion number operator was defined in Eq. (2.12). The variable Eq. (3.2) can be thought of as the well-known non-local Jordan-Wigner transformation of the local variables. The purpose of this procedure is to ensure that the ’s on adjacent sites anticommute. We will use these variables repeatedly in the rest of this paper.

The Hamiltonian defined by the supercharge Eq. (3.1) is

(3.3)

The total fermion number operator in this case is given by . Therefore, the -grading operator is given by

(3.4)

It is easy to see that this operator commutes with the Hamiltonian , thus forming superselection sectors as in the one-particle case. It also anticommutes with the supercharges and only when is odd. Henceforth we assume that is odd.

Clearly, the -site chain Hamiltonian Eq. (3.3) describes a non-interacting many-body system. Since for all , the system is easily solved by labeling the eigenstates of with the eigenvalues of the operators on each of the sites. The spectrum of the one-site Hamiltonian, , was studied in the previous section. The states can be thought of as being bosonic or fermionic and the operators and project onto the bosonic and fermionic subspaces at site , respectively. For example, we can label the -site eigenstates in the following way

(3.5)

For simplicity, we assume the chain to be homogeneous and set for all sites . The form of in terms of commuting orthogonal projectors, and (recall that and ), results in integer eigenvalues between and . Then, we can write the energy spectrum of the chain as

(3.6)

These spectra exhaust all the possible states of as

(3.7)

which, as we have seen, is the dimension of the total Hilbert space .

The ground states of are given by Eqs. (2.9)-(2.11). There are of them for the chain, all fermionic, matching the Witten index . These zero-energy ground states have the form

(3.8)

The local excited states include both bosonic and fermionic ones. The three bosonic states on every site are given by Eqs. (2.16)-(2.18) and the normalized fermionic ones are given by Eqs. (2.13)-(2.15). The many-body excited states are then built by filling up the sites with these local bosonic and fermionic excited states. Consequently, the system is fully solved.

At finite temperature , the partition function is given by

(3.9)

3.2 Long-range interacting supersymmetric chain

We now demonstrate how a model of an interacting -site system can be constructed. The model is associated with long-ranged supercharges and Hamiltonian. We can, however, also construct interacting models but with local supercharges and Hamiltonians. We relegate them to App. A. We emphasize that the MBL property studied for the supercharges in this section is also shared by the supercharges in App. A as those systems continue to possess the local integrals of motion that are possessed by the long-range interacting supercharges in this section. We study the long-ranged interacting supercharges merely for the simplicity of computations.

Consider the following choice of supercharge

(3.10)

which is just a product of the local supercharges at each site. This is clearly a nilpotent operator and hence generates a supersymmetry algebra. The resulting Hamiltonian is given by

(3.11)

Though interacting, the resulting Hamiltonian is integrable; there are local integrals of motion given by .

We can organize the Hilbert space in terms of the cohomologies of nilpotent and The subspace of zero-energy states is spanned by solutions of We see that they are labelled by the product states of the following types. The first type of ground states have at least one local zero-energy state on an individual site:

(3.12)

where the ellipses denote any of single-site boson or fermion excited states. There are many such states. The second type of ground states is built from the mixture of single-site boson and fermion excited states with at least one local fermion excited state:

(3.13)

The ellipses are occupied by single-site bosons and there are such states. Combining the two types of ground states, the Hilbert subspace has the dimension dim .

The excited states belonging to are all of the form

(3.14)

The number of such states is precisely for and , all with eigenvalue 1. Although they are entangled eigenstates of the operator , note that they are not entangled as eigenstates of the Hamiltonian. This can be understood as arising due to the fact that Eq. (3.14) is a superposition of a bosonic and a fermionic state (except in the even case, when this state is an eigenstate of the fermion number operator). The Hamiltonian of this long-range interacting system then has only product states as eigenstates.555It is however possible to construct supercharges resulting in supersymmetric Hamiltonians that do preserve the Klein operator, have entangled eigenstates and a Witten index different from . An example will be discussed in App. C.

The total number of eigenstates is the number of ground states plus the number of excited states, which is equal to , the total dimension of the Hilbert space, dim . Note that the spectrum of this system is independent of and is given by the two eigenvalues 0 and 1.

The partition function can be easily computed for this system and is found to be

(3.15)

3.3 Supersymmetric deformations and Witten index

The -grading Klein operator is

(3.16)

The supercharges and anticommute with .

The Witten index is defined as the trace of the Klein operator. We can count this index from the ground states we identified above and find precisely for arbitrary . This can be easily seen by considering the form of the states enumerated in Eq. (3.12) and Eq. (3.13). In each of these product states, there are an equal number of bosonic and fermionic states. The only state which is unpaired is the product state made of one-particle ground state at every site. As each of these local zero modes are fermionic (recall from Eqs. (2.9)-(2.11)), all these states are fermionic. One can easily confirm that the excited states are paired between bosonic and fermionic states, with multiplicity one.

As in the one-particle case, we are interested in classifying supersymmetry preserving deformations in the many-body setting. Such deformations are defined by continuous perturbations of the Hamiltonian that commute with the supercharge given by Eq. (3.10), its adjoint , and the grading Klein operator in Eq. (3.16). We split these up into those that can be added as perturbations to the supersymmetric Hamiltonian and those that are obtained by deforming the supercharge as in the one particle case.

Local and quasi-local supersymmetry preserving perturbations

On a chain, we can deform the system in a variety of manners. First, we can deform the system on each site. Such deformations are given by

(3.17)

where is a site-dependent function. Obviously, the system is invariant under these single-site deformations as they do not change the Witten index .

Next, we can also deform the system over two sites. These deformations take the form

(3.18)

where the two-site coefficient function decreases sufficiently fast when the two-site distance becomes large and is a real parameter characterizing such deformations. Such deformations commute with the Klein operator and preserve supersymmetry. It is easily seen that this quasi-local operator does not mix the eigenstates of this system and, in fact, it is diagonal in this basis. Thus, the Witten index is clearly left invariant under the deformation of quasi-local operators. Note that these are deformations that are added to the original supersymmetric Hamiltonian and are not obtained from a deformed supersymmetry algebra.

Continuing in a similar manner, we can also deform the system over multiple sites; these deformations are supported on several sites and take the form,

(3.19)

where and the site-dependent coefficient function is taken to be suitably quasi-local. Such operators are again diagonal in the eigenbasis of this system and hence the Witten index is left invariant. These operators account for all the allowed deformations to this system.

Deformed supercharges

We can introduce a deformation of the supercharge as follows

(3.20)

where each of the local deformed supercharges are given by Eq. (2.21) with the coefficients in these supercharges being now site dependent. The deformed Hamiltonian resulting from this has the same kind of spectrum as the undeformed supersymmetric Hamiltonian in Eq. (3.12)-Eq. (3.13). The only difference is that the local zero modes, bosons and fermions are replaced by the deformed counterparts like those given in Eqs. (2.24)-(2.26). These states maintain their grading under the Klein operator and thus it is clear that the Witten index stays unchanged to these deformations.

4 The spreading of quantum information

So far, we focused on the spectrum and Witten index of the supersymmetric system on a chain. Here, we dwell on the time evolution of many-body entanglement. This is captured by correlations functions of various time-orderings. More specifically, we will compute out-of-time-order correlators (OTOC) and study whether the system scrambles and equilibrates, and, if so, how it does it. We do this for a prototype model, consisting of an interacting disordered system built from Eq. (3.10) that exhibits a many-body localized phase which is supersymmetric. The solvability of this model, as we have seen for the spectral analysis in the previous section, is a remarkable feature brought by the supersymmetric nature of the SIS we utilized in the construction. This will allow us to proceed with analytic computations for the OTOC.

4.1 Slow scrambling

First of all, we introduce a quenched disorder in the system by dressing the supercharge in Eq. (3.10) as

(4.1)

where are real-valued time-independent random variables that can be thought of as analogous to a static random on-site potential. One could generalize this choice by restricting the product to subsets of sites, rather than including all sites, as we shall present in Appendix A. For now, however, we work with this simpler choice. The results should not depend on this choice.

The many-body interacting Hamiltonian built out of this supercharge is

(4.2)

To probe scrambling behavior of this model, we should compute the correlator in Eq. (1.2) and study its time dependence, as discussed in the Introduction. We choose local supercharges as local operators, and , with . Moreover, we will set for simplicity, since we are primarily interested in highly excited states,

(4.3)

We prefer to compute rather than the OTOC (usually considered in this context), but of course the results are independent of this choice.

It is possible to show that the time evolved operator is given by

(4.4)

and, consequently,

(4.5)

All the information about disorder is contained in the argument of the cosine, as a result of the “on-site disorder”. In such circumstances, it is reasonable to absorb the effect of randomness into a single variable, defined with its probability measure as

(4.6)

where is a constant. Performing the disorder average leads to

(4.7)
Figure 6: Normalized for both Gaussian (left) and uniform (right) ensembles in the unit .

The usual Hamiltonian employed in the study of many-body localization is the one for a system of qubits that contains, among other contributions, on-site magnetic fields given by static random variables which are uniformly distributed. With this in mind, we also average over a uniform ensemble between for comparison, leading to

(4.8)

The behavior of Eq. (4.7) and Eq. (4.8) are shown in Fig. 6. Notice that no -dependence appears in the time-dependence, apart from the trivial one in the normalization factor of the commutator.

Instead of the quenched quantity we have computed, , one could first average over realizations of the couplings and then take the expectation value. While the two procedures generally lead to different results, one can readily perform the computation on the reverse order and see that in this simple case they provide the same answer, that is, the averages commute. Under what conditions the two procedures are (in)equivalent is an interesting question, that we leave for future studies.

In both choices of the ensemble, the early-time behavior is given by

(4.9)

which is valid for any nonzero disorder. While we have shown this result for Gaussian and uniform distributions, it seems to hold for more general choices as well, with different proportionality constants set by the random disorder coupling. We discuss the meaning of this behavior in Sec. 4.3.

4.2 OTOC-EE theorem from partial symmetries

Recently, a connection between the decay of the OTOC taken in a thermal equilibrium state and the growth of a certain entanglement entropy was proposed for a system quenched by an arbitrary operator [53]. To formulate the exact statement, assume a system described by a Hamiltonian initially in thermal equilibrium at temperature and split it into two regions, and . Let be the second Rényi entropy of , a quench operator that acts on the system at time with the property , , and a complete set of operators for . In this setup, the theorem of [53] establishes the following equality

(4.10)

We can state a modified result in terms of our formalism that will supply the story we are developing with further insights. This is accomplished by requiring the quench operator to act only on the subspace spanned by the partial symmetries of . In other words, we demand to form a complete set for the quench operators we may consider. This is certainly a restriction, since an arbitrary operator acting on the full Hilbert space cannot in general be expressed in terms of partial symmetries. However, this restriction will also provide hints for expecting slow scrambling in any supersymmetric model constructed out of SIS algebras.

As an example, we will verify the partial symmetry version of the OTOC-EE theorem for Eq. (4.2). To this end, we assume again, for simplicity, that the system is at infinite temperature such that the right-hand side of Eq. (4.10) is reduced to

(4.11)

As a first step, suppose the initial state to be a maximally mixed state, where . We then quench the system at the first site with

(4.12)

which amounts to sending . Then, let the system evolve for a time under , leading to . Next, we write the Hilbert space as a bipartite decomposition, , where we take the -subsystem as a single site, for definiteness.666We emphasize that the actual partition or choice of the quench operator is irrelevant to state this theorem. The second Rényi entropy on region is defined by

(4.13)

To trace out , we first need to compute the eigenvectors of the local Hamiltonian . Recall from previous sections that these are given by

(4.14)

With this in mind, it is straightforward to show that

(4.15)

In order to compute the OTOCs in Eq. (4.11), we use the hypothesis that the partial symmetries form a complete set on , i.e. any quench operator in our model can be expressed in terms of at site . Thus, it is a straightforward, albeit tedious, exercise to show that