Network structure and dynamics of effective models of non-equilibrium quantum transport
Abigail N. Poteshman, Mathieu Ouellet, Lee C. Bassett, Danielle S. Bassett
NNetwork structure and dynamics of effective modelsof non-equilibrium quantum transport
Abigail N. Poteshman ‡ , Mathieu Ouellet , Lee C. Bassett † ,Danielle S. Bassett , , , † Department of Physics & Astronomy, College of Arts & Sciences, University ofPennsylvania, Philadelphia, PA 19104, USA Department of Electrical & Systems Engineering, School of Engineering & AppliedScience, University of Pennsylvania, Philadelphia, PA 19104, USA Department of Bioengineering, School of Engineering & Applied Science, Universityof Pennsylvania, Philadelphia, PA 19104, USA Santa Fe Institute, Santa Fe, NM 87501, USAE-mail: [email protected] and [email protected]
January 2021
Abstract.
Across all scales of the physical world, dynamical systems can often beusefully represented as abstract networks that encode the system’s units and inter-unit interactions. Understanding how physical rules shape the topological structureof those networks can clarify a system’s function and enhance our ability to design,guide, or control its behavior. In the emerging area of quantum network science, a keychallenge lies in distinguishing between the topological properties that reflect a system’sunderlying physics and those that reflect the assumptions of the employed conceptualmodel. To elucidate and address this challenge, we study networks that representnon-equilibrium quantum-electronic transport through quantum antidot devices —an example of an open, mesoscopic quantum system. The network representationscorrespond to two different models of internal antidot states: a single-particle, non-interacting model and an effective model for collective excitations including Coulombinteractions. In these networks, nodes represent accessible energy states and edgesrepresent allowed transitions. We find that both models reflect spin conservation rulesin the network topology through bipartiteness and the presence of only even-lengthcycles. The models diverge, however, in the minimum length of cycle basis elements,in a manner that depends on whether electrons are considered to be distinguishable.Furthermore, the two models reflect spin-conserving relaxation effects differently,as evident in both the degree distribution and the cycle-basis length distribution.Collectively, these observations serve to elucidate the relationship between networkstructure and physical constraints in quantum-mechanical models. More generally,our approach underscores the utility of network science in understanding the dynamicsand control of quantum systems. ‡ Current Address: Committee on Computational and Applied Mathematics, Physical SciencesDivision, University of Chicago, Chicago, IL, 60637, USA † These authors contributed equally. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n etwork structure of effective models of non-equilibrium quantum transport
1. Introduction
The intersection of network science and quantum physics is an emerging areaof interdisciplinary research [1]. Methods from network science have been usedto characterize features of quantum networks that are relevant to the design ofquantum information and communication systems, such as quantum synchronization[2], transport efficiency [3], and robustness to noise [4]. Conversely, quantum effectsand dynamics have been applied to complex networks, such as by modeling quantumwalks [5] and representing nodes as entangled states [6]. In both directions of inquiry,this interdisciplinary work has focused on manipulating network structure in order tooptimize networks for quantum information processing, storage, and communicationtechnologies. Yet, this focus has necessarily neglected the important space of questionssurrounding how network structure emerges naturally and directly from quantumsystems themselves. In previous work, we sought to address this gap by considering thestructure of mesoscopic quantum networks, and by demonstrating the utility of networkcharacterizations in explaining transport properties [7]. Here we take a complementaryapproach and ask: What network topology emerges from different physical modelsof mesoscopic quantum systems? And what do those differences imply about systemdynamics and control?Mesoscopic quantum systems, such as quantum dots and quantum antidots,are of particular interest to those designing quantum information processing devices[8–10]. They are widely tunable and can be efficiently controlled electronically bycapacitive coupling to electrostatic gates that can alter their equilibrium charge [11–14].Mesoscopic quantum systems can be probed with transport experiments: electronstunneling between reservoirs weakly coupled to a mesoscopic system induce transitionsbetween quantum mechanical configurations, whose properties can be deduced frommeasurements of current and conductance. Features of mesoscopic systems, however,are difficult to characterize and predict since simulating a many-body interacting systemis computationally intractable, due to exponential scaling of the system’s Hilbert spacewith particle number [15,16]. Without true quantum simulators, the best tools availableto model mesoscopic systems are numerical, semi-classical models.In the recent literature, network science has emerged as a promising tool to offerintuition for the architectures of physical systems that produce mesoscopic dynamics[17–20]. For example, in a previous study employing a single-particle model ofquantum-electronic transport, we demonstrated that statistical characterizations fromnetwork science can capture physically-relevant emergent properties of non-equilibriumtransport [7]. Yet, the work left unanswered the question of how different physicalconstraints embedded in various models of quantum phenomena are reflected in thenetwork architecture, thereby informing appropriate control strategies [21, 22]. Thatsuch reflections might exist is intuitively plausible when one considers the nature ofthe physical models, which can represent quantum states and mechanisms for stateexcitation quite differently, for example by using different bases or at different levels of etwork structure of effective models of non-equilibrium quantum transport Figure 1. An overview of antidot physics. A.
Schematic cross section of anantidot. Tunneling occurs between edge states carrying a sea of electrons (green)and quantized antidot energy states. B. Equivalent capacitor network for the antidotelectrostatics. The quantized charge on the antidot is − N e , where N is the number ofelectrons (relative to a fixed reference configuration) and e is the electron charge. Theelectrostatic potential of the antidot can be determined by the capacitive couplings( C S , C D , C G ) to the source, drain, and gate voltages ( V S , V D , V G ). Any remainingcoupling to other elements of the device is modeled as a capacitive coupling to theground potential ( C R ), such that the total capacitance is C = C S + C D + C G + C R .Figure adapted from Ref. [25]. approximation [23, 24].To better understand the interdigitation between physics and topology, we studynetworks constructed from two models of non-equilibrium transport through a quantumantidot (see Figure 1A): a single-particle model and an effective model [25]. Both modelsproduce experimentally accurate time-averaged values of current and conductance fromtransport experiments, but describe the internal antidot configurations and mechanismsfor excitations in different ways [25]. The single-particle model treats quantumstates as composed of distinguishable, non-interacting elementary particles, whereasthe interacting model describes quantum states in terms of collective quasiparticleexcitations of a many-body liquid. That is, the two models are not merely differentbasis representations of the same physics. We performed a statistical investigation of themodels’ network topology, paying particular attention to the network’s cycle structureand degree distribution, which are high-order and lower-order, respectively, topologicalcharacteristics relevant to the propagation of information and control profiles of complexnetworks [26–28]. In comparing the networks built from these two models, we aim todistinguish the network characteristics that reflect the physically observable phenomenaof quantum transport (shared across both models) from those that reflect the underlyingphysical mechanisms of particular models (differing between models).
2. Methods
Here, we extend previous work that built a network model of the energy landscape ofnon-equilibrium transport through quantum antidots [7]. Expanding upon that study,we now consider an effective model in addition to a single-particle model of antidot etwork structure of effective models of non-equilibrium quantum transport
For an overview of transport through mesoscopic systems, we direct readers to seminalreviews such as Refs. [11–14]. Here, we focus on spin-resolved transport through asingle quantum antidot at filling factor ν AD = 2 in the integer quantum Hall regime atrelatively low magnetic fields ( B < I ,and the differential conductance, G = dIdV . Nonzero current indicates the presenceof accessible quantum states in the antidot within the energy window defined by therelative electrochemical potentials of the source, µ S , and drain, µ D (see Figure 1B).Differential conductance reflects changes in the transport configurations, generally dueto changes in the alignment of state transitions with µ S and µ D . Differential conductanceis typically positive but can become negative is certain configurations. Together,current and conductance are used as both qualitative and quantitative indicators ofquantum transport phenomena, and they are the key output of computational modelsfor comparison with experiments [7, 25].The number of antidot states involved with non-equilbrium quantum transportgrows rapidly with the applied bias (see Figure 1B). The additional states relevantfor non-equilibrium transport include excited states that represent different spin andorbital configurations [13]. The ways in which these spin and orbital configurationsare connected through tunneling and relaxation events are manifold, leading to arichly structured collective energy landscape. In fact, landscape complexity growsexponentially with particle number; it quickly becomes computationally intractable tocalculate transport characteristics analytically. As a result, the best tools availableto model non-equilibrium transport through mesoscopic systems are numerical, semi-classical models. Quantum antidot states can be modeled either in terms of the electronstate occupation number (see Section 2.2), or as edge-waves in the charge distribution(see Section 2.3) [30]. Here we provide a brief description of the single-particle model for transport throughquantum antidots (see Ref. [7] for further details). The single-particle energies are etwork structure of effective models of non-equilibrium quantum transport m = 0 , , , . . . ) and spin ( σ = ± ) quantum numbers, ε mσ = m ∆ E SP + σE Z , (1)yielding an energy spectrum of two ladders of equally-spaced energy levels. The spacingbetween orbital energy levels, ∆ E SP , is assumed to be constant, and the separationbetween energy ladders is the Zeeman energy E Z . Excitations are also governed bythese two energy scales, with possible values E ex = j ∆ E SP ± qE Z , (2)where q = 0 and q = 1 represent spin-conserving or spin-flip transitions, respectively,and j is any integer. Internal quantum states of the antidot are represented as a pair ofelectronic occupation vectors ( n ↑ , n ↓ ), with components n mσ = 0 or 1 for each orbital, m , and spin, σ . Since we can track whether electrons occupy specific orbitals, theelectrons in the single-particle model are distinguishable. Once the possible electronicoccupation vectors are enumerated, a Boolean set of selection rules can be calculated bydetermining which sets of electronic occupation vectors differ by exactly one electron.That is, if the XOR sum of two sets of electronic occupation vectors is exactly 1, thenthe transition is allowed. Otherwise, the transition is forbidden. We can also consider an effective model of antidot states, in which electrons areindistinguishable and excited states are described as collective excitations of a “quantumliquid” around the antidot edge [30, 31]. The effective model is based on the fullHamiltonian for a system of N interacting electrons, within the standard Born-Oppenheimer approximation in which the electronic degrees of freedom are decoupledfrom those of the lattice [32]. The Hamiltonian can be written in the formˆ H = N X i ˆ h i + e π(cid:15)(cid:15) N X i>j | x i − x j | , (3)where ˆ h i is the single-particle Hamiltonian acting on the i th electron, which is given byˆ h i = 12 m ∗ ( − i ~ ∇ i + e A ) − eϕ ( x i ) − gµ B B ˆ s zi . (4)This general Hamiltonian does not have any known analytic solutions for more than oneelectron.Using Hartree-Fock mean-field theory (see Supplement, Section 1.1), we assumethat each electron in the multi-electron system is described by its own single-particlewave function (Eq. 4). The multi-electron wave function Ψ can be written as a Slaterdeterminant of orthonomal single-particle spin orbitals, and we can obtain the totalenergy E for Ψ using the variational principle [25]. In this way, we obtain a ’fermionic’ etwork structure of effective models of non-equilibrium quantum transport M, S z ), using the rules for addition of angular momentum [30]. This process leads toa ‘bosonic’ basis, in which the neutral excitations are described by a spectrum of ‘edgewaves’ similar to the one-dimensional Tomonaga-Luttinger liquid model [33, 34].The antidot states and transition rules among them are defined as follows. Inthe effective model the antidot states are given by | N, S Z , n L , n S i , where N is theparticle number of the state, S Z is the total spin projection, n L ∈ Z + is the excitationof the density modes, and n S ∈ Z + is the excitation of the spin modes. Theconfiguration energy for a state in the interaction model is given by U ( | N, S Z , n L , n S i ) = U ( | N, S Z , , i ) + n L · E L + n S · E S where E L is the energy scale for the densitymode excitations and E S is the energy scale for the spin mode excitations. Atransition between two states of the effective model is allowed if ∆ N = ± n S ∈ { , − } − ∆ | S Z | . The binary selection rules are weighted by the ClebschGordan coefficient for the addition of the spins S Z and S Z ± corresponding to thetransition. This model of weighted selections rules qualitatively replicates asymmetriesin the conductance map over the voltage space (see Figure 2B) that are observed inexperiments [25]. The physics of the antidot enters the calculation in the form of a set of quantum states,its associated energy spectrum, and a set of matrix elements for transitions betweenstates. However, the method to construct and solve a rate-equation matrix for thesteady-state probabilities of the antidot is agnostic to the physical model used to obtainthe quantum states. We used the same master equation approach to obtain steady-stateprobability occupations for the antidot’s configurations as in Ref. [7].In Sections 2.2 & 2.3, we described two different models of internal antidotconfigurations, which yield different descriptions of the quantum states and transitionrules. In both cases, however, the total particle number, N , and the spin projection, S z , are good quantum numbers, and hence the selection rules between quantum statescan be written in block-matrix form, e.g., W + ↓ S z − · · · W −↓ S z − W −↑ S z − ... W + ↑ S z − W + ↓ S z W −↓ S z − W −↑ S z + ... W + ↑ S z W + ↓ S z +1 · · · W −↓ S z + . (5) etwork structure of effective models of non-equilibrium quantum transport {| N, S z i} , with S z increasing from top to bottom ( S z is the ground-state spin projection) and N alternating between two adjacent integer values. The sub-matrices W ± σS z contain theselection rules for adding (+) or removing ( − ) a particle of spin σ to a state with initialspin projection S z . The specific states included in the model (both the number of blocksand the number of states in each block) are determined through energy and dynamicalconsiderations for a given bias configuration.The transition rates γ s → s from antidot configuration | s i to | s i are calculatedaccording to a combination of antidot selection rules and Fermi’s golden rule (see Section1.2 of the Supplementary Information for a full derivation of the transition rates). Theresulting transition rate matrix, R , is defined by R ij = γ s i s j = γ s j → s i , where i and j represent different configurations. We seek the steady-state configuration where thetotal transition rate into and out of each state is equal, and the solution to the masterequation yields the steady-state occupation probabilities P (see Refs. [7, 25] for detailsabout the master equation approach). From P , we can compute the current flowingfrom each spin-polarized reservoir and the spin-resolved conductance [25].Using this computational model, we can simulate quantum transport as a functionof experimental parameter settings including gate voltages, drain-source bias, magneticfield, and temperature, and we can choose whether to use an energy spectrumbased on a single-particle model or an effective model. The settings chosen in thiswork are motivated by spin-resolved transport experiments in which the underlyingphysical parameters ( e.g. , ∆ E SP , E Z , and the spin-dependent tunneling rates) havebeen well characterized [25]. Unless indicated otherwise, the temperature is 50 mK,∆ E SP = 30 . µ eV, E Z = 45 . µ eV, E C = 85 µ eV, the effective spin-up tunneling rate is500 MHz, and the effective spin-down tunneling rate is 50 MHz. For details on how theexperimental parameter settings enter into quantum transport calculations, see Section1.4 of the Supplementary Materials. We can include spin-conserving relaxation effects within each set of antidotconfigurations with the same number of particles and total spin {| N, S z i} by addingblock matrices T describing these processes to the main diagonal of the matrix ineq. (5) [25]. For the single-particle model, we encoded the spin-conserving relaxations as T ij = 1 if state j results from moving one of the electrons in state i to the lowest availableorbital in state i , and T ij = 0 otherwise. For the effective model, we set T ij = 1 if state i (represented as | N i , S Zi , n Li , n Si i ) and state j (represented as | N j , S Zj , n Lj , n Sj i ) havethe same total spin ( N i = N j ) and spin mode excitation ( n Si = n Sj ), and state i is adensity mode excitation of state j ( n Lj < n Li ). For both models, the relaxation ratewas set at 500 MHz when relxation was included, which is on the same order as thetunneling rates into and out of the antidot. etwork structure of effective models of non-equilibrium quantum transport Figure 2. Single-particle model versus effective model of non-equilibriumtransport through quantum antidots.
Current and conductance calculationsbased on a single-particle model ( A ) and an effective model ( B ) of energy states inquantum antidots. Both models were run with the following parameters: T = 50 mK, B = 1 . ↑ = 500 MHz, and effective spin-downtunneling rate Γ ↓ = 50 MHz. All subsequent figures are based on models run withthese parameters unless noted otherwise. The number of nodes and the number ofedges for the networks corresponding to each set of voltage settings constructed basedon a single-particle model ( C ) and an effective model ( D ) of energy states. Note thatwe excluded networks corresponding to voltage settings that result in a current withmagnitude less than 1 pA from our analysis; the values in panels ( C ) and ( D ) that aredisplayed as NaN indicate that networks were excluded. Similar to constructing a master equation that determines the transition rate matrixand the occupation probabilities, the method to construct networks is agnostic tothe underlying physical model used to represent antidot configurations. In thenetworks representing transport through quantum antidots, the nodes represent antidotconfigurations and the edges represent possible transitions between configurations aftersingle electron tunneling events and relaxation events. We used the same method toconstruct networks based on the transition rate matrices R (see Figure 3 A & B andFigure 2A & B in the Supplement) and corresponding probability vectors P as reportedin Ref. [7]. The thresholding method for the probability vectors is presented in Section1.5 of the Supplementary Information.With these adjacency matrix representations of our network in hand, we can beginto perform rigorous statistical characterizations of network size, density, and topology. etwork structure of effective models of non-equilibrium quantum transport n in a network is given by the size of the matrix, and the number ofedges in the network is the number of non-zero elements in the matrix (see Figure 2C-D).To evaluate the network’s topology, we focus on two statistical measures relevant to anetwork’s capacity to propagate current, signals, nutrients, or other physical, biological,or informational items [26, 27]: the network’s degree distribution and cycle structure.The degree of a node i is the sum over i of A ij . A narrow degree distribution is indicativeof a particularly ordered systems; in a lattice, for example, the degree distributionis a delta function because every node has the same degree, given by the numberof its immediately adjacent neighbors [35]. A broad degree distribution is indicativeof more complexity, where some parts of the system are heavily connected (formingnetwork hubs), and other parts of the system are less connected [36]. In fact, degreeheterogeneity is a direct quantification of a network’s complexity as formalized in thenotion of entropy [37].The distribution of node degrees is a so-called lower-order topological characteristicthat considers only the edges in a node’s immediate neighborhood: those edges thatconnect the node directly to other nodes. Ongoing work in the field of networkscience, however, continues to demonstrate that higher-order topological characteristics– those that characterize the organization of edges which are more than 1 hop awayfrom a node – have important roles to play in system dynamics and control [37–41].It therefore seems prudent to consider both lower-order and higher-order topologicalstatistics in our evaluation. In choosing each, we considered the growing body ofevidence indicating that degree distribution (lower-order) and cycle structure (higher-order) are two specific network features that consistently shape the dynamics, capacityfor information storage, and control profile of a network [26, 36, 42–44]. We thereforecomplemented the examination of the degree distribution with an examination of thenetwork’s cycle basis, which can be algorithmically extracted using the python softwarepackage NetworkX [45, 46].A cycle is a closed walk that does not retrace any edges immediately after traversingthem. Cycles are particularly relevant for understanding transport because a densecycle structure has been shown to be optimal for transport in the face of spatially andtemporally varying loads [47]. A cycle basis is a basis for the vector space of all cycles,defined over Z , such that all cycles of a network can be expressed in terms of linearcombinations of elements in the cycle basis. Although a network can be decomposed intoa cycle basis in many different ways, the length distribution of the cycle basis elements isunique [48]. Since extracting the cycle basis of a network can be computed in polynomialtime, in contrast with an exhaustive enumeration of all possible cycles which requiresexponential time, our analysis of cycle structure is restricted to considering the length ofcycle basis elements in this paper [49]. Together, the degree distribution and cycle basisallow us to examine the interplay between network topology and mesoscopic physics. etwork structure of effective models of non-equilibrium quantum transport Figure 3. Example rate and adjacency matrices. ( A ) Example transition ratematrix R (left) and its corresponding adjacency matrix A (right) obtained using asingle-particle model of quantum transport. The network these matrices representcorresponds to voltage settings resulting in a current of | I | = 81 . B ) Example transition rate matrix R (left) and its corresponding adjacency matrix A (right) obtained using an effective model of quantum transport. The network thesematrices represent corresponds to voltage settings resulting in a current of | I | = 81 .
3. Results
By examining networks constructed from two models of non-equilibrium transportthrough quantum antidots, we can explore which aspects of energy-state transitionnetworks are common to the physical process of transport versus which aspects reflectparticular transport mechanisms. The former will manifest as characteristics commonto all transport networks, and the latter will manifest as characteristics that vary acrosstransport models. We begin by examining how spin conservation rules are reflected inthe network topology.
Using two models of non-equilibrium transport through quantum antidots — a single-particle model and an effective model — we constructed networks over a rangeof voltage configurations spanning two Coulomb diamonds (see Figure 2A & B).Transport calculations using both models agree with experimental values of currentand conductance (see Figure 2C & D), but the models assume quite different transportmechanisms. We first examined networks constructed from both models when excludingrelaxation effects. We found that all networks from both models are bipartite. Abipartite network has two classes of nodes and edges that connect only nodes of one classwith nodes of the other class [50]. Intuitively then, bipartiteness reflects the shared spinconservation rules that are a common underlying constraint upon both models. Edgesrepresent single tunneling events of electrons into or out of the antidot, so neighboringnodes differ by total spin (see Figure 4A). Understanding edges as single-electrontunneling events leads to a natural 2-color marking scheme of integer versus half-integertotal spins (Figure 4B). etwork structure of effective models of non-equilibrium quantum transport Figure 4. Spin constraints result in even-length cycles. ( A ) A schematic ofcycle trajectory through an adjacency matrix representation of a network. The blockdiagonal represents antidot energy state configurations that the system may occupy,and the grey off-diagonal blocks store transition rates between antidot states. Thecycle shown in the schematic corresponds to a cycle of length 6, where each node inthe cycle is a distinct antidot configuration in the diagonal block, and each edge isrepresented by two blue arrows that pass from one node through a transition rateto a new node. Spin-preserving relaxations occur within a diagonal block. Since theschematic shows an implementation of transport without relaxation, the system mustpass through a grey transition state in order to move from one node to another. Sincethe spin changes by a half-integer amount during each transition, all cycles have aneven length. ( B ) Sample networks with a 2-coloring scheme, where nodes having aninteger spin are shown in purple and nodes having a half-integer spin are shown in greenfor a single particle non-interacting model (left) and for an effective model (right). ( C )Distribution of cycle length in the cycle basis space for the single networks shown inpanel ( B ). ( D ) Distribution of cycle lengths for all networks over the voltage spacedisplayed in Figure 2 corresponding to a current greater than 1 pA for the networksconstructed from the single-particle model (top) and from the effective model (bottom).In each boxplot, the central mark represents the median, the top and bottom edgesindicate the third and first quartiles, the whiskers extend to ± . σ , and individualoutliers are displayed by ’-’. etwork structure of effective models of non-equilibrium quantum transport Figure 5. Enumeration of cycles. ( A ) All possible 4-cycles that involve twoelectrons tunneling into and two electrons tunneling out of an antidot are shown inschematics (i) - (v) . The color of the nodes indicates the number of electrons inthe antidot following the schematics in the clockwise direction; counter-clockwise cyclenode labeling is given by changing the sign of the added or subtracted electrons (e.g. N + 1 in the clockwise labeling becomes N − B )An example of a 6-cycle demonstrating a pattern of electrons tunneling into and outof the antidot that does not invalidate any of the energy or spin conservation rules forthe single-particle antidot model. ( C ) An example of a valid 4-cycle for the effectivemodel with the full quantum numbers labeled for each state. Notice that this cycle isthe same as the one shown in subpanel A(ii) . As a direct extension of their bipartite nature, we found that all networks containedonly even-length cycles in their cycle bases (see Figure 4C & D). Recall that a cycle isa closed walk that does not retrace any edges immediately after traversing them. Tocomplete a cycle and return to the starting antidot configuration (node), an even numberof tunneling events must occur since each tunneling event (edge) changes the total spinof a state (node) by . While networks constructed using an effective model have aminimum cycle length of 4 in their cycle bases elements — an expected minimum lengthsince in the network representations, a 2-cycle is simply an edge — networks constructedusing the single-particle model have a minimum cycle length of 6 (see Figure 4D).The difference in minimum cycle lengths stems from a fundamental differencebetween the single-particle model and the effective model. In the single-particle model, etwork structure of effective models of non-equilibrium quantum transport same electron since the internal antidot configurationdoes not track individual electrons, and therefore these 4-cycles are present in networksconstructed using the effective model. An example of such a 4-cycle for the effectivemodel is shown in Figure 5C. As described in Section 2.5, both the single-particle model and the effective model canincorporate spin-conserving relaxation effects by adding block matrices to the diagonal ofthe matrix describing selection rules between eigenstates. For the networks constructedfrom models that include spin-conserving relaxation effects, an edge represents either asequential tunneling event or a relaxation event. Including spin-conserving relaxationeffects results in a greater number of edges compared with networks excluding relaxationeffects corresponding to the same voltage settings (see Supplementary Figure 1), sincetwo nodes may be connected through a spin-conserving relaxation event. Relaxationevents do not suppress sequential tunneling events so long as the sequential tunnelingrate is on the same order as (or faster than) the relaxation rate; when the relaxationrate is much faster than the tunneling rate, the antidot will effectively remain in itsground state [13]. These relaxation pathways, however, violate the constraints neededto produce a bipartite structure; nodes connected by a relaxation event have the sametotal spin, so the 2-color marking scheme of integer versus half-integer spins discussedin Section 3.1 no longer holds.When we introduce spin-conserving relaxation effects to both models of quantumtransport, we observe odd-length cycles; yet, it is important to note that the models’distinct mechanisms of spin-conserving relaxation alter the network structures indifferent ways (Figure 6A). For the single-particle model, a relaxation between antidotconfigurations i and j is allowed if configuration j results from moving one of theelectrons in configuration i to the lowest available orbital in configuration i . Since thereare multiple possible excited configurations accessible for a given number of electrons, etwork structure of effective models of non-equilibrium quantum transport Figure 6. Spin-conserving relaxation effects introduce odd-length cycles. ( A ) Example networks with edges corresponding to spin-conserving relaxationshighlighted in blue for the single-particle model (top) and the effective model (bottom).( B ) Cycle length distribution for the single networks shown in panel ( A ). ( C )Distribution of cycle lengths for all networks over the voltage space displayed in Figure2 corresponding to a current greater than 1 pA for the networks constructed from thesingle-particle model (top) and the effective model (bottom). In each boxplot, thecentral mark represents the median, the top and bottom edges indicate the third andfirst quartiles, the whiskers extend to ± . σ , and individual outliers are displayed by’-’. a ‘ground configuration’ node has multiple edges corresponding to relaxations fromdifferent possible excited states (Figure 6A). As a result, cycles can contain multiplerelaxations; cycle basis elements containing an odd number of relaxations result in odd-length cycles in the cycle basis distribution, whereas cycle basis elements containing aneven number of relaxations result in even-length cycles (Figure 6B-C).For the effective model, a relaxation between states i and j is allowed if bothconfigurations have the same number of particles N and the same spin-excitationquantum number nS , and if the density spin excitation number of configuration j is less than the density spin excitation number of configuration i . Since the onlypossible density spin excitation numbers are 0 and 1, each node has at most one edgecorresponding to a spin-conserving relaxation (Figure 6A). This fact is reflected in theonly odd-length cycles in the cycle basis of the effective model as 3-length cycles, sincethere is at most one relaxation in a cycle basis element (Figure 6B-C). In sum, whileincluding spin-conserving relaxation effects results in odd-length elements in the cycle etwork structure of effective models of non-equilibrium quantum transport Figure 7. Physical constraints limit the tail of the degree distribution.
Thedegree distribution of networks constructed using the ( A ) single-particle model, ( B )single-particle model with spin-conserving relaxation effects, ( C ) effective model, and( D ) effective model with spin-conserving relaxation effects. Each network correspondsto voltage settings resulting in | I | > A ) and ( C ). Thenetworks corresponding to the same voltage settings in panel ( B ) as in panel ( A ) andin panel ( D ) as in panel ( C ) are highlighted in the corresponding color. basis for networks constructed using both the single-particle and the effective models,the models’ distinct mechanisms to describe excitations yield different cycle structures. While cycles are a higher-order topological characteristic that describes the organizationof edges which are more than 1 hop away from a given node, degree distribution is alower-order topological characteristic that concerns the organization of edges directlyconnected to a given node. While we observed that distinguishability of particles anddifferent mechanisms for relaxation impact the cycle structure of the networks, we seekto understand how the different physical representations of the antidot impact networkstructure locally. Degree distribution is one of the most fundamental properties ofa network used to discern well-known types of networks, such as scale-free networks,Erd˝os-R´enyi random graphs, and lattices, and it is particularly relevant for questionspertaining to controlling many-body quantum systems with applications in quantuminformation processing devices, since the distribution of edges in a network governshow information flows locally from one node to another [51]. More broadly, weare also interested in how the degree distribution of state-transition networks usingtwo different models of internal antidot configurations compares with other naturally-occurring networks. etwork structure of effective models of non-equilibrium quantum transport
4. Discussion
Many-body quantum systems are difficult to simulate on classical computing systems,because the number of parameters necessary to represent the system grows exponentiallywith particle number. As a result, physicists have devised effective models that can beimplemented numerically to describe many-body quantum systems. In using effectivemodels, physicists make explicit decisions about how to describe a system as well as etwork structure of effective models of non-equilibrium quantum transport
The ability to control many-body quantum systems has become increasingly importantin quantum information processing devices, since the ability to identify driver nodes anddevise control strategies for quantum systems is crucial in order to store, encode, and etwork structure of effective models of non-equilibrium quantum transport etwork structure of effective models of non-equilibrium quantum transport
Several methodological considerations are pertinent to this work. First, we comparedthe structure of quantum transport networks constructed using two underlying modelsof internal antidot configurations: a single-particle model and an effective model. Theimplementation of sequential transport used for both models does not include higher-order co-tunneling processes or spin-flip relaxation effects (due to hyperfine couplingor phonon-mediated spin-orbit interactions [13]), both of which may be present inexperiments. Yet, even without including these effects, we find close quantitativeagreement between the computational model and experiments [25]. Notably, ourapproach does allow spin-flip relaxation effects to be incorporated by adding blocktransition matrices between elements off of the main diagonal of Equation 5. Second,while the undirected networks allow us to probe the relationship between topology andquantum networks at the most fundamental level, certain physical effects that mayimpact dynamics are buried. For example, in the undirected networks, both tunneling etwork structure of effective models of non-equilibrium quantum transport
5. Conclusion
Using network science to study the energy-state transitions of non-equilibrium transportthrough a quantum antidot based on two different models of internal antidot states, wedemonstrated that structural properties of the network reflect model-specific spin andenergy constraints. These constraints result in different minimum-length elements in thecycle bases across models as well as different degree distributions. This understandingof how different physical models of mesoscopic quantum phenomena alters networkstructure may inform the design and control of quantum devices for quantum simulation,storage, or information processing.
6. Acknowledgements
ANP acknowledges support from the Benjamin Franklin Scholars Program andUniversity Scholars Program at the University of Pennsylvania. LCB and DSBacknowledge support from the NSF through the University of Pennsylvania MaterialsResearch Science and Engineering Center (MRSEC) DMR-1720530.
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Loops of any size and hamilton cycles in random scale-freenetworks. Journal of Statistical Mechanics: Theory and Experiment, 2005(06):P06005, 2005. upplementary Materials for “Network structureand dynamics paper of effective models ofnon-equlibrium quantum transport” Abigail N. Poteshman ‡ , Mathieu Ouellet , Lee C. Bassett † ,Danielle S. Bassett , , , † Department of Physics & Astronomy, College of Arts & Sciences, University ofPennsylvania, Philadelphia, PA 19104, USA Department of Electrical & Systems Engineering, School of Engineering & AppliedScience, University of Pennsylvania, Philadelphia, PA 19104, USA Department of Bioengineering, School of Engineering & Applied Science, Universityof Pennsylvania, Philadelphia, PA 19104, USA Santa Fe Institute, Santa Fe, NM 87501, USAE-mail: [email protected] and [email protected] ‡ Current Address: Committee on Computational and Applied Mathematics, Physical SciencesDivision, University of Chicago, Chicago, IL, 60637, USA † These authors contributed equally. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n upplementary Information Summary of the Contents of this Supplement
In this supplementary materials document, we provide additional methodologicaldetails and supplementary results that support the findings reported in the maintext. We divide the document into two main sections: Supplementary Methods andSupplementary Results. In the Supplementary Methods section, we begin by describingthe calculations that we performed to obtain a fermionic basis for the effective model.Next we describe calculations to estimate transition rates for the master equation. Thenwe describe the calculations that we performed to estimate current and conductancebased on the occupation probabilities obtained from the master equation. Finally, wedescribe the rational for our choice of threshold value for the transition probabilitymatrices. Next we turn to a Supplementary Results section where we report theresults for the current, conductance, and size of networks for models including spin-conserving relaxation effects. Then we give sample adjacency and rate matrices fornetworks including spin-conserving relaxation effects. Finally, we report results fromdegree-distribution preserving random matrices that demonstrate that the cycle basisis independent from the degree distribution in these quantum transport networks.
1. Supplementary Methods
Sections 1.1 - 1.4 were adapted from Ref. [1].
The Hartree-Fock (HF) method is one of several ‘mean field’ approaches to this problem,in which each electron in a system is influenced by an effective potential due to the chargedensity of all the other electrons. In particular, we assume that each electron in thesystem is described by its own single-particle wave function, such that the multielectronwave function may be written as a Slater determinant of orthonormal SP spin orbitals ψ i : Ψ = 1 √ N ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ( ξ ) ψ ( ξ ) · · · ψ ( ξ N ) ψ ( ξ ) ψ ( ξ ) · · · ψ ( ξ N )... ... . . . ... ψ N ( ξ ) ψ N ( ξ ) · · · ψ N ( ξ N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (1)where ξ i represents both the position and spin projection of the i th particle. ‡ Byconstruction, this wave function satisfies the antisymmetry requirement for fermions,that is Ψ( ξ , ξ , . . . , ξ i , . . . , ξ j , . . . , ξ N ) = − Ψ( ξ , ξ , . . . , ξ j , . . . , ξ i , . . . , ξ N ) , (2) ‡ It is assumed that the orbital and spin parts of the wave function are separable, i.e. that ψ i ( ξ ) = ψ n i ,m i ( x ) χ i ( s ), where χ = (cid:0) (cid:1) or (cid:0) (cid:1) in terms of the argument s = 1 , upplementary Information i and j corresponds to the interchange of two columnsof the determinant, and hence a change of sign. Thus the Pauli exclusion principle issatisfied: the wave function Ψ vanishes when ξ i = ξ j for any i = j . § General expressions for matrix elements between determinantal wave functions likeEq. (1) are well known, given for example in Ref. [2]. In the expectation value for theenergy of Ψ, many of the terms vanish due to the orthogonality of the ψ i , leaving h Ψ | ˆ H | Ψ i = X i h i | ˆ h | i i + X i 1) = − µ ( N ) defines an inflection point within each Coulomb blockaderegion. On one side of the inflection point we need only consider configurations withoccupation numbers ( N − , N ), while on the other we consider only ( N, N +1) states.In the plane of ( V G , V DS ), these become inflection lines that pass vertically through thecenter of each Coulomb diamond, and divide the calculation region by the occupationnumbers involved. upplementary Information N , N + 1) by their total spin projection S z . Suppose the ground-state spinfor the N -particle state is S z and for the N +1 particle state is S z − . Given thesevalues, we begin by constructing the vector of configurations: {| Ψ AD i} = {| N +1 , S z − i}{| N, S z − i}{| N +1 , S z − i}{| N, S z i}{| N +1 , S z + i}{| N, S z +1 i} , (23)where each {| N, S z i} corresponds to a vector of individual states | N, S z , q ↑ , q ↓ i , where q σ labels the configuration of the spin- σ particles. In the presence of interactions,these states are not true eigenstates of the Hamiltonian, but they provide a qualitativeapproximation to the excitation gaps in most cases.The number of excited states to include is determined through a consideration ofthe chemical potentials for transitions to or from the ground states with spin S z ± asdescribed above. With this arrangement for the configurations, the selection rules takethe block-matrix form W + ↓ S z − · · · W −↓ S z − W −↑ S z − ... W + ↑ S z − W + ↓ S z W −↓ S z − W −↑ S z + ... W + ↑ S z W + ↓ S z +1 · · · W −↓ S z + , (24)where, assuming the vectors of states {| N, S z i} are listed as subsequent groups of spin- ↑ states (labeled by q ↑ ) for each spin- ↓ state (labeled by q ↓ ), the sub-matrices W ± σS z aregiven by W ±↑ S z = ↓ ⊗ M ±↑ S z , (25a) W ±↓ S z = M ±↓ S z ⊗ ↑ . (25b)The matrices M ± σS z contain the selection rules for transitions in the spin- σ configurationindividually, and are easily worked out by comparing the occupation vectors n σ of theinitial and final states. For example, M + ↑ ij = 1 whenever the q ↑ = i state of the N +1configurations results from adding a single spin- ↑ particle to the q ↑ = j state of the N configurations, which we can write as M + ↑ ij = ( n ↑ ( i ) · [ − n ↑ ( j )] = 1 , . (26) upplementary Information R ± σij = X r =S , D Γ rσ ( µ ij ) W ± σij f ± r ( µ ij ) , (27)where f + r = f r is the Fermi function of lead r , and f − r = 1 − f r . As described in Sec.2.4 in the main text, we then add diagonal elements to the rate matrix to impose anet balance of rates in equilibrium, and an extra row of ones to enforce normalization,constructing the master equation in the form of Eq. (10) in Ref. [4]. The solution to thismaster equation gives the steady state occupation probability of each state | N, S z , q ↑ , q ↓ i ,which we then use to compute the current flowing through the system. The current ismost easily computed using Eq. (31), by isolating the transition rate involving only asingle lead, e.g. for the source, S ± σij = Γ S σ ( µ ij ) W ± σij f ± S ( µ ij ) . (28)Including signs to account for the direction of current flow, we can then write I = e X ij T ij P j , (29)where P j are the equilibrium occupation probabilities and T = S + ↓ S z − · · · − S −↓ S z − − S −↑ S z − ... S + ↑ S z − S + ↓ S z − S −↓ S z − − S −↑ S z + ... S + ↑ S z S + ↓ S z +1 · · · − S −↓ S z + . (30)The block-diagonal form of Eq. (30) also facilitates the straightforward calculation ofspin-resolved current components, simply by isolating the terms that correspond totunneling of each spin species. The spin-resolved conductance is calculated as in Eq. (37)from a finite difference of the currents computed at two different settings for V D .The procedure can be generalized to account for additional effects. For example,we can include spin-conserving relaxation of excited states within each set {| N, S z i} by adding block matrices describing these processes to the main diagonal of Eq. (24).Spin non-conserving relaxation due to spin-orbit coupling or the hyperfine interactioncould also be included by adding terms to the next off-diagonal blocks (connecting states {| N, S z i} with {| N, S z ± i} ). Note, however, that this model only obtains the steady-state ( t → ∞ ) configuration, and it assumes a Markovian ( i.e. , history-independent)interaction with the reservoirs; thus, we are not able to investigate coherent effects dueto quantum evolution with this procedure. upplementary Information S z -configurations and excited states until the occupation probability of the outermost statesfalls below a predetermined threshold. In the results reported here and in the main text,the total occupation probability threshold of the outermost states was constrained tobe less than 0.02. When performing simulations over a range of different parameters,the matrix-construction procedure is followed independently for each bias setting, andtherefore the set of states included in the calculation varies. When investigating therole of some network measures, it is important to maintain a constant network size( i.e. , the total number of states) and ideally the same state definitions. Therefore, afterperforming the calculation once over the full parameter space of interest, we determinethe union of all quantum states that appear at any point and subsequently repeat thecalculation at each bias point using the full set. A drawback of this approach is that,at every bias point, a large number of states have negligible occupation probability anddo not contribute to the dynamics. When appropriate, the non-participatory states canbe removed using a thresholding procedure as described in the next section. Once the master equation has been solved for the probabilities P ( s ), we can computethe current flowing out of each lead from the expression I r = e X ss (cid:2) γ + r,s → s P ( s ) − γ − r,s → s P ( s ) (cid:3) = e X ss (cid:2) γ + r,s → s − γ − r,s → s (cid:3) P ( s ) , (31)where we have used the single-particle Hamiltonian, given by Equation 1 in Ref [4] tosimplify the second term. Using the relation X r (cid:2) γ + r,s → s − γ − r,s → s (cid:3) = (cid:16) N ( s ) − N ( s ) (cid:17) γ s s , (32)it is straightforward to show that P r I r = 0, i.e. that the total current is conserved.Dropping the dependence on | ‘σ i , we can write Eq. (31) in the form I r = e X ss Γ r ( µ ss ) M ss h f r ( µ ss ) P ( s ) − (cid:0) − f r ( µ ss ) (cid:1) P ( s ) i , (33)where M ss = X ‘σ (cid:12)(cid:12) h s | a † ‘σ | s i (cid:12)(cid:12) (34)represents the selection rules for transitions between states s ↔ s and µ ss = E s − E s is the chemical potential for the antidot transition. Using current conservation we canderive the relation X ss M ss P ( s ) = X ss r Γ r Γ M ss (cid:2) P ( s ) + P ( s ) (cid:3) f r ( µ ss ) , (35) upplementary Information P r Γ r and we have suppressed the dependence of the Γ’s on µ ss . We use thisrelation to eliminate the term independent of f r in Eq. (33) to obtain a final expressionfor the current out of lead r , I r = e X ss r Γ r Γ r Γ M ss (cid:2) P ( s ) + P ( s ) (cid:3) × (cid:2) f r ( µ ss ) − f r ( µ ss ) (cid:3) . (36)We use this expression to calculate the current transmitted through the antidot, andcompute the conductance at finite bias by G ( V D ) = I ( V D + δV D ) − I ( V D − δV D )2 δV D , (37)which is typically a good approximation if eδV D . kT . The rate matrix inversion calculations in our model of sequential transport through anantidot produced numerical inaccuracies in probability values near zero. To excludethese numerical inaccuracies, we found a threshold value that excludes small positiveand negative values resulting from inaccuracies while maintaining the connectedness ofthe networks. Furthermore, we sought a low enough threshold that the sum of non-thresholded probability values remains close to 1.The threshold of 1 × − we used for all networks preserved both of theseconstraints; the probability vectors summed to greater than 0 . . . 2. Supplementary Results In this section we report the results of additional computations and analyses thatcomplement those that were reported in the main manuscript. We found that including spin-conserving relaxation effects in both models of non-equilibrium transport preserved the qualitative features of current and conductance(see Figure 1A-B), even though spin-conserving relaxation effects increase the numberof edges in a network by allowing transitions between nodes with the same total numberof particles (that is, transitions between two nodes with N particles is allowed through a upplementary Information Figure 1. Single-particle model versus effective model of non-equilibriumtransport through quantum antidots with spin-conserving relaxations. Current and conductance calculations based on a single-particle model A. and aneffective model B. of energy states in quantum antidots including spin-conservingrelaxation effects. The number of nodes and the number of edges for the networkscorresponding to each set of voltage settings constructed from a single-particle model C. and an effective model D. of energy states including spin-conserving relaxationeffects. Both models were run with the following parameters: T = 50 mK, B = 1 . ↑ = 0 . 005 Hz, effective spin-down tunneling rateΓ ↓ = 0 . . 005 Hz. All figures for simulations includingspin-conserving relaxation effects are drawn from models run with these parametersunless noted otherwise. Note that we excluded networks corresponding to voltagesettings that result in a current with magnitude less than 1 pA from our analysis, andvalues in panels C. and D. displayed as NaN indicate that networks were excluded. relaxation event). In the single-particle model, the spin-conserving relaxations increasedthe number of edges in a network by a factor of around 2 and increased the currentthrough the antidot for a given set of voltage settings, whereas the increase in thenumber of edges for the effective model is a much smaller effect (see Figure 1C & D).From a rudimentary analysis of network size (defined by the number of nodes and edges),we observe that the different mechanisms by which internal spin-conserving relaxationsare encoded in the two models result in varying network architecture and thus may havedifferent impacts on network functions such as transport and information storage. upplementary Information Figure 2. Example rate and adjacency matrices including spin-conservingrelaxation effects. A. Example transition rate matrix R (left) and its correspondingadjacency matrix A (right) obtained using a single-particle model of quantum transportincluding spin-conserving relaxation effects. The network these matrices representcorresponds to voltage settings resulting in a current of | I | = 81 . B. Example transition rate matrix R (left) and its corresponding adjacency matrix A (right) obtained using an effective model of quantum transport including spin-conserving relaxation effects. The network that these matrices represent correspondsto voltage settings resulting in a current of | I | = 81 . We included spin-conserving relaxation effects as described in Section 2.5 of the maintext. We observe the relaxations in the rate matrix as the non-zero elements in themain-diagonal block matrices (see the left panels of Figure 2A-B). The rate matrices arethen symmetrized and binarized, yielding the adjacency matrices on which we performedthe network analyses (see the right panels of Figure 2). From the rate matrices, we cansee the asymmetric nature of relaxations from an excited state to a ground state. Both degree distribution and cycle structure are network properties that emergefrom underlying quantum mechanical selection rules that govern mesoscopic quantumsystems. To assess whether the cycle structure is dependent upon the degreedistribution, we constructed degree-distribution preserving random networks basedon the quantum transport networks [5, 6]. We found that the cycle structure isnot dependent on the degree distribution, as the length distribution of cycle basiselements is different in the degree-distribution preserving random models (see Figure3) compared with the quantum transport networks. For instance, all of the cycle basesfrom the random models include cycles of even length, so the bipartite nature of thenetworks constructed from models excluding spin-conserving relaxations (see the leftpanels of Figure 3A & C) is a constraint that cannot be observed directly from thedegree distribution. The independence of cycle structure and degree distribution in upplementary Information Figure 3. Degree distribution and cycle structure are different aspectsof network structure. Distribution of cycle lengths in a cycle basis for allnetworks corresponding to a current greater than 1 pA for the networks (left) and thecorresponding degree-distribution preserving random models (right) constructed fromthe single-particle model excluding spin-conserving relaxation effects A. , the single-particle model including spin-conserving relaxation effects B. , the effective modelexcluding spin-conserving relaxation effects C. , and the effective model including spin-conserving relaxation effects D. . these quantum transport networks indicates that both may be important to take intoconsideration when designing network-based control strategies for quantum systems,given that both degree distribution and cycle structure play a role in the control profilesof complex networks [7]. 3. References [1] Lee C Bassett. Probing electron-electron interactions with a quantum antidot. arXiv preprintarXiv:1912.08006 , 2019.[2] H. A. Bethe and R. W. Jackiw. Intermediate Quantum Mechanics . 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