Transport and spectral properties of the XX+XXZ diode and stability to dephasing

Kang Hao Lee,1 Vinitha Balachandran,1, ∗ Chu Guo,2, 3 and Dario Poletti1, 4, †

1Science and Math Cluster, Singapore University of Technology and Design, 8 Somapah Road, 487372 Singapore2Henan Key Laboratory of Quantum Information and Cryptography, Zhengzhou,Henan 450000, China

3Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education,Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications,

Hunan Normal University, Changsha 410081, China4Engineering Product Development Pillar, Singapore University of Technology and Design, 8 Somapah Road, 487372 Singapore

(Dated: February 18, 2022)

We study the transport and spectral property of a segmented diode formed by an XX + XXZspin chain. This system has been shown to become an ideal rectifier for spin current for largeenough anisotropy. Here we show numerical evidence that the system in reverse bias has signaturespointing towards the existence of three different transport regimes depending on the value of theanisotropy: ballistic, diffusive and insulating. In forward bias we observe two regimes, ballistic anddiffusive. The system in forward and reverse bias shows significantly different spectral properties,with distribution of rapidities converging towards different functions. In the presence of dephasingthe system becomes diffusive, rectification is significantly reduced, the relaxation gap increasesand the spectral properties in forward and reverse bias tend to converge. For large dephasing therelaxation gap decreases again as a result of quantum Zeno physics.

I. INTRODUCTION

The study of transport at the nanoscale has raised in-terest both for fundamental and applied reasons [1–3].In particular, thanks to many-body interactions a dissi-patively boundary driven system can turn from ballisticto diffusive and even insulating, as shown for the XXZchain [1, 2, 4, 5]. We note that spin chains with tunableinteractions of lengths as large as 256 spins have been re-alized using quantum simulators [6] and that specificallythe XXZ chain was implemented in [7]. Furthermore, dif-ferent transport regimes of XXZ chain were also demon-strated using ultracold atoms of 7Li in [8]. Recently therehas been particular interest in studying the phenomenonof rectification in strongly interacting spin chain, both forspin and heat currents [9–14]. A setup of particular inter-est is the XX + XXZ chain, composed, for half, of an XX,and the other half, of an XXZ (semi-)chain [15, 16]. Withthis setup, it was shown that one can achieve perfect rec-tification in the thermodynamic limit [11]. This has beenexplained by the emergence of an insulating phase forlarge enough anisotropy/interaction in reverse bias, whilethe system still allows transport in forward bias. Hence,the emergence of significantly different transport proper-ties in this system can be linked to an out-of-equilibriumphase transition. Phase transitions in dissipative sys-tems have been studied with greater effort in the past tenyears [17–19] even with experimental setups [20]. In mod-els described by master equations in Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form [21, 22], one can rec-ognize the emergence of a dissipative phase transition bystudying the rapidities, i.e. the rate of relaxation of the

∗ vinitha balachandran@sutd.edu.sg† dario poletti@sutd.edu.sg

natural modes of the system. It has been shown that, atthe transition, the relaxation gap closes, i.e. more thanone mode has a vanishing relaxation rate [17, 18]. Thishowever does not apply to systems for which the dissipa-tion only acts at the boundaries, because in these systemsthe effects of the dissipation become a small perturbationin the thermodynamic limit [23]. In these cases, in thethermodynamic limit the relaxation gap always closes,but the scaling of the smallest rapidity with the systemsize varies, for example from algebraic to exponential,or between two different algebraic decays at the phasetransition [23].

ΔFIG. 1. The boundary driven segmented XXZ spin rectifierwith anisotropy ∆ only on the left half of the system. Thediagram above represents the reverse bias scenario where theleft bath tends to set the left most spin pointing down and theright bath tends to set the right most spin to a state which isan equal mixture of up and down spins.

In this work we achieve a deeper understanding of thetransport and spectral properties of XX + XXZ spindiode depicted in Fig.1 and studied in [11]. Our studiesare numerical and performed with exact diagonalizationand tensor-networks based algorithms [24, 25]. With theparameters considered, both the approaches are limitedin system size and this poses limitations to characterizethe system in the thermodynamic limit. Our transport

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studies suggest that in forward bias the system is bal-listic for lower interactions, and diffusive for larger ones.Furthermore, in reverse bias the system is ballistic, dif-fusive or insulating as one increases the interaction. Ourspectral analysis shows that the rapidities tend to dif-ferent distributions in forward and reverse bias. More-over, a qualitative change in the scaling of the relaxationgap of the rapidities is seen across the different transportregimes. Last, we will also add to the boundary dissi-pation that drives the system, dephasing at each site,something which affects significantly both the transportproperties of the system, and the distribution of the ra-pidities. Indeed, with dephasing we observe the increaseof relaxation gap as well as significant reduction of rectifi-cation along with the convergence of spectral propertiesof the forward and reverse bias. With very large de-phasing, the relaxation gap decreases again due to thequantum Zeno effect.

This work is organized in the following manner. InSec.II we describe the model. In Sec.III we consider thecurrents, in Sec.IV we describe the magnetization, and inSec.V the spectral properties. Finally in Sec.VI we dis-cuss the effect of dephasing and we give our conclusionsin Sec.VII.

II. MODEL

In the following we consider an even number L of spinsin a bipartite spin-1/2 chain described by the Hamilto-nian in [11] (pictorial representation in Fig.1)

H =

L/2−1∑n=1

[J(σxnσxn+1 + σynσ

yn+1 + ∆ σznσ

zn+1)]

+

L−1∑n=L/2+1

[J(σxnσxn+1 + σynσ

yn+1)]

+ Jm(σxL/2σxL/2+1 + σyL/2σ

yL/2+1). (1)

σxn, σyn and σzn are the Pauli matrices for the n-th spin, Jis the tunnelling amplitude, ∆ is the anisotropy in the lefthalf of the system, and Jm is the tunnelling amplitudebetween the two half chains. The left half chain thusfollows the XXZ model and the right half of the chainfollows the XX model.

The chain is coupled to two different spin baths at theboundaries and also subjected to bulk dephasing. Wemodel the effect of the baths via the following masterequation in GKSL form [21, 22]

dρ

dt= L(ρ)

= − i

~[H, ρ] +

4∑n=1

BnρB†n −

1

2

4∑n=1

{B†nBn, ρ},

+

L∑n=1

DnρD†n −

1

2

L∑n=1

{D†nDn, ρ} (2)

where the spin polarization processes at the boundariesare given by,

B1 =√

Γµlσ+1 , B2 =

√Γ(1− µl)σ−1 ,

B3 =√

Γµrσ+L , B4 =

√Γ(1− µr)σ−L ,

(3)

and the bulk dephasing of all spins,

Dn =√γσzn, n ∈ {1, ..., L}; (4)

γ controls the rate of dephasing, Γ is the magnitude ofthe boundary system-bath coupling and µl (µr) sets thespin magnetization imposed by the left (right) bath. Herewe point out that, in the absence of dephasing, and forµl = 0 and µr = 1, an XX chain is always ballistic, whilean XXZ chain would be ballistic for ∆ < ∆XXZ andinsulating for ∆ > ∆XXZ where ∆XXZ = 1.

Here, we set µl and µr to be equal to 0 or 0.5. Aspin current is induced when µl is different from µr.When µl = 0.5 > µr = 0, the left bath tends to setthe spins to be in an equal mixture of up and down spins(|↓〉n〈↓|+ |↑〉n〈↑|)/2 while the right bath tends to set thespins pointing down |↓〉n〈↓|. Spin current thus flows fromthe left to right and we refer to this as the forward bias.The reverse bias corresponds to the spin current flowingfrom right to left (µl = 0 < µr = 0.5).

The spin current J is calculated from the continuityequation for local observable σzn and is given by

J = Tr(jnρss), (5)

where ρss is the steady state such that dρss/dt = 0 and

jn =2J

~(σxnσ

yn+1 − σynσxn+1). (6)

We note that the spin current J is site independent inthe steady state, and for this reason J in Eq.(5) has noindex n. For the rest of the paper, we work in units suchthat J = ~ = 1.

The spin magnetization observable on site n is givenby

〈σzn〉 = Tr(σzn ρss), (7)

and their profiles are discussed in Sec.IV.To evaluate the performance of the system as a diode,

we consider the measure R [11]

R = −JfJr, (8)

where Jf and Jr are the currents in forward and reversebias respectively. When there is no rectification, Jf =−Jr and the measureR = 1. For a perfect diode, R =∞or 0 because either the forward or reverse current goes tozero. In [11] it was shown that the reverse bias system

becomes an insulator for ∆ > ∆R where ∆R = 1 +√

2.Here we would like to spend a few lines on how we

compute the steady state and the spectrum of the GKSL

3

master equation. To obtain the entirety of the spectrumwe write the right hand side of Eq.(2) as L(ρ) whereL(·) is a linear non-Hermitian superoperator which forour case, once written as a matrix, can be diagonalized.The eigenvalues are referred to as rapidities. These are,typically, complex numbers, whose real part describesthe decay of the eigenmode, while the imaginary partits oscillations. For the master equation we consider,there exist a single steady state, which corresponds toa rapidity equal to zero. It becomes apparent that thisprocedure to compute all the rapidities can only be per-formed for small systems because the size of the Hilbertspace grows approximately as 22L [26]. Another very im-portant issue regards the accuracy of these calculationswhen ∆ and L are larger. In fact, in this case there area number of rapidities which approach zero, thus mak-ing the diagonalization procedure less accurate. If wedo not require to obtain the full spectrum, we can usetwo other approaches. The first one is to use iterativediagonalization routines that only return the eigenvaluewith smallest modulus, which has modulus equal to zeroand its eigenmode corresponds to the steady state. Theother approach is to evolve the system in time, whichwe actually implement with a matrix product states al-gorithm [25]. However, even in these cases the problemof small rapidities still applies. For instance, one wouldneed to know when to stop the time evolution. For thiswe measure the current at each bond and compute thevariance. When the variance is small enough, we stopour calculations and study the steady state. However,when the steady state current is small, say of the orderof 10−5 it is very difficult to have a small enough variancebetween the different bonds, especially with such smallrapidities. In short this is to state that for the systemand bath parameters considered, it is still challenging tostudy large systems. For this reason, in this work we donot go beyond systems of size L = 20.

III. SPIN CURRENTS

To characterize the transport properties, we begin bystudying the finite size scaling of the spin current. Insome scenarios, significantly large system sizes are re-quired for the correct prediction of thermodynamic trans-port properties. As explained above, our studies are re-stricted to fairly small sizes. We will thus give our con-siderations from the sizes we observe and from our under-standing of the systems, laying the foundation for futureresearch on these systems.

Fig.2 shows the steady state spin current versus systemsizes for different anisotropies ∆. In general, the currentdecays algebraically with the system size, i.e. J ∝ 1/Lα.When α = 0, the current is size independent and thetransport is ballistic, while for α = 1 the system is diffu-sive. However, different values of α are also possible, andthe system is categorized as superdiffusive for 0 < α < 1and subdiffusive for α > 1. It is however important to

10 -2

10 0

6 8 10 12 14 16

10 -5

10 0

FIG. 2. Forward current Jf panel (a) and reverse current Jr

panel (b) versus the system size L for selected anisotropy ∆values. Other parameters: Jm = 1, Γ = 1, γ = 0.

note that the current may decay faster than algebraicallywith the system size, for example exponentially, in insu-lators. In Fig.2(a) we plot the currents in forward biasand we see two regimes of transport. For ∆ . ∆XXZ thesystem is ballistic, similar to [4, 27], while the system for∆ > ∆XXZ appears not to decay exponentially with thesystem size.

The reverse bias currents are plotted in Fig.2(b), andthey display a richer transport behavior with at leastthree clear regimes. The transport is ballistic for ∆ <∆XXZ , whereas for ∆XXZ < ∆ < ∆R, the current ap-pear to decay algebraically with the system size. Notethat the magnitude of the current in reverse bias is signif-icantly smaller compared to the forward bias case. Last,for ∆ > ∆R the indication from our numerical calcu-lations, and the insights developed in [11], indicate thatthe current decays exponentially with the system size. Tohelp the reader differentiate the three regimes, in Fig.2we have used different colors for each one, specifically redfor ∆ . ∆XXZ , green for ∆XXZ < ∆ < ∆R, and bluefor ∆ > ∆R. Furthermore, within each region, darkershades correspond to smaller values.

To gain deeper insight, we complement the analysis ofthe currents with that of the magnetization in the nextsection.

IV. SPIN MAGNETIZATION

To establish the existence of distinct regimes seen inthe transport in the previous section, we further investi-gate the spin magnetization in the two biases. First inFig.3, we analyze the system by studying its magnetiza-tion profile. We use a rescaled position (n − 1)/(L − 1)so that for each system size the plot is between 0 and1. In the left panels, i.e. Fig.3(a,c,e), we show theprofile for forward bias, while in the right panels, i.e.Fig.3(b,d,f), we depict results from the reverse bias. Inpanels Fig.3(a,b) we consider a small anisotropy ∆ = 0.5,Fig.3(c,d) an intermediate value ∆ = 2 and in Fig.3(e,f)

4

-1

-0.5

0

-1

-0.5

0

0 0.5 1-1

-0.5

0

0 0.5 1

FIG. 3. Spin magnetization profiles 〈σzn〉 against the spin

positions (scaled) (n − 1)/(L − 1) for ∆ = 0.5 (a,b), ∆ = 2(c,d), ∆ = 2.8 (e,f) in the forward (a,c,e) and reverse (b,d,f)directions. Other parameters: Γ = 1, Jm = 1, γ = 0.

a larger ∆ = 2.8. In all scenarios we observe that theright portion of the chain, for (n − 1)/(L − 1) > 0.5the profile is approximately constant, and this is due tothe fact that there the system is an XX chain which isnon-interacting and ballistic. The magnetization profilesare significantly different in the left portion of the chainwhere the anisotropy is non-zero. For both biases when∆ < ∆XXZ , we observe that the magnetization at theedges of the chain is significantly far from the one thebaths are trying to impose (either −1 or 0), and mostof the magnetization change occurs at the interface. Inthe left half of the chain we observe a gentle slope inthe magnetization profile indicating possible diffusive orsuperdiffusive transport.

The magnetization profile differs significantly from theforward and reverse bias once we consider larger valuesof ∆. In forward bias we observe that, for ∆ > ∆XXZ ,Fig.3(c,e), there is a distinct linear profile for the mag-netization which would be expected from the diffusivenature of the XXZ chain [2, 28, 29], although in thesestudies the bias from the baths was different. At the in-terface we also observe a significant jump in the magneti-zation, which is a typical signature of interface resistance.Similar results are seen for reverse bias with anisotropy∆XXZ < ∆ < ∆R as in Fig.3(d), for ∆ = 2, althoughthe magnetization slope is not a clear straight line as forthe forward bias, which could be a hint of subdiffusivebehavior. For large values of the anisotropy, ∆ > ∆R,shown in Fig.3(f) a steep change of the reverse bias mag-netization is seen which increases with the system size.Furthermore, the magnetization at the edge is extremelyclose to the values required by the baths. This is alignedwith what we would expect for an insulator.

We further analyze the magnetization profiles in Fig.4,for two different sizes L = 12 and L = 20 for different

-1

-0.5

0

0 0.5 1-1

-0.5

0

0 0.5 1

FIG. 4. Panels (a-d): Spin magnetization profiles 〈σzn〉 against

the spin positions (scaled) (n − 1)/(L − 1) for L = 12 (a,b)and L = 20 (c,d) for different anisotropies ∆ in the forward(a,c) and reverse (b,d) directions. Other parameters: Γ = 1,Jm = 1, γ = 0. The arrow indicates increasing magnitudes of∆, specifically ∆ = [0 to 3 in intervals of 0.2, and 4].

values of the anisotropy ∆. Specifically we use the values∆ = [0 to 3 in intervals of 0.2, and 4] increasing in thedirection of the arrows in Fig.4. To help differentiate thedifferent regimes, we use the same coloring schemes as inFig.2. The left panels of Fig.4 correspond to forward bias,while the right panels to the reverse bias. Focusing onthe forward bias, in Fig.4(a,c), consistently with Fig.2(a)and Fig.3(a,c,e), there appears to be two main regimes:one for ∆ < ∆XXZ for which the magnetization in theleft chain is approximately constant, and for ∆ > ∆XXZ

for which the magnetization changes linearly until theinterface. Furthermore, comparing the results for L =12, Fig.4(a), and for L = 20, Fig.4(c), the two differentregimes become more distinct, as the lines correspondingto different anisotropies becomes more clearly separated.In reverse bias, Fig.4(b,d), we also observe the three mainregimes which also become more distinct as one comparesthe magnetization profile for L = 12, Fig.4(b), and forL = 20, Fig.4(d).

We then focus on a single site n, and plot in Fig.5 themagnetization 〈σzn〉 versus the anisotropy ∆. In this fig-ure, each line corresponds to a different site (the colorbecomes darker as the site number increases, i.e. thelightest line corresponds to n = 1, and the darkest one ton = L as shown in the color bar). The forward bias case isconsidered in Fig.5(a,c) and reverse bias in Fig.5(b,d). Inthe reverse bias case we observe three different regions ap-pearing at the values of ∆ < ∆XXZ , ∆XXZ < ∆ < ∆R

and ∆ > ∆R. The different behaviors in the differentregions is much more marked for the larger system sizeL = 20. In contrast, for the forward bias no significantchange seems to occur at ∆ = ∆R, while a qualitativechange in the shape of the magnetization versus inter-action occurs at ∆ = ∆XXZ . This reinforces the obser-

5

-1

-0.5

0

1

3

6

9

12

0 2 4-1

-0.5

0

0 2 41

5

10

15

20

FIG. 5. Spin magnetization profiles 〈σzn〉 versus ∆ for different

sites n. The case for L = 12 is depicted in (a,b), and L = 20in (c,d), while the forward bias is represented in (a,c) andthe reverse bias in (b,d). Other parameters: Γ = 1, Jm = 1,γ = 0.

vation that there are two main transport regimes in theforward bias scenario.

Overall, our analysis of the current and of the mag-netization profiles is consistent in pointing that there isa qualitative change in the properties of the system, inforward bias at ∆ = ∆XXZ , while in reverse bias thereare two changes, one at ∆ = ∆XXZ and one at ∆ = ∆R.

V. SPECTRUM OF RAPIDITIES

To investigate further the different properties of for-ward and reverse bias systems, we now turn our attentionto the spectrum of the superoperator L from Eq.(2). Infact, the spectrum of the superoperator plays an impor-tant role in determining the relaxation dynamics as wellas the steady state properties of the open quantum sys-tem. One of the relevant quantities in this context is thespectral gap defined as |Re(λ1)| where λ1 is the first non-zero eigenvalue of L. The inverse of the spectral gap, alsocalled the asymptotic decay rate, determines the slowestrelaxation time scale in the dynamics of the system. In asystem with dissipation acting on an extensive number ofsites, the dissipative phase transitions is characterized bythe closing of the gap, which results in the divergence ofrelaxation times and emergence of multiple steady statesin the thermodynamic limit [17, 18, 31, 32]. However,when the dissipation is at the boundaries only, as thecase in our study, the gap always closes in the thermo-dynamic limit. Here, the phase transition is signalled bythe change in the scaling of the gap [23].

In Fig.6 we analyze the spectrum by plotting the ex-ponential of the rapidities eλi , where the λi are the ra-pidities in the complex plane. eλi is bound to be withinthe unit circle, with only one of them corresponding tothe steady state with λ = 0, on the unit circle [30]. Onthe left panels, Fig.6(a,c,e), we show the exponential of

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1 0 1-1

-0.5

0

0.5

1

-1 0 1

FIG. 6. Exponential of the rapidities λi in the unit circle(red continuous line) for anisotropies ∆ = 0.5 panels (a,b),∆ = 2 (c,d), ∆ = 5 (e,f). Left panels (a,c,e) are for forwardbias and right panels (b,d,f) are for reverse bias. The blackcontinuous inner circle shows the smallest possible radius ofthe exponential of rapidities [30]. Other parameters: L = 8,Γ = 1, Jm = 1, γ = 0.

the rapidities for the forward bias, while the right pan-els, Fig.6(b,d,f), depict the case of the reverse bias. Weobserve that for small ∆ the rapidities are distributedsimilarly in the forward and reverse bias, but as ∆ in-creases, in the reverse bias we see a number of rapiditiesapproaching the unit circle closing the spectral gap, in-dicating the emergence of slowly decaying modes.

To gain further insights we study the radial distribu-tion of the rapidities by plotting the number of rapiditieswithin concentric circles of radius dr = 0.05 denoted byp(r). These are depicted in Fig.7. In the left panelswe consider the forward bias case, while the right panelshows the reverse bias. In panels (a-h) we consider thesystem size L = 4 (black dotted), L = 6 (blue dashed)and L = 8 (red solid), and each panel for a different ∆.For L = 4 there are strong finite size effects, but alreadyfor L = 6 and L = 8 we see convergence in the distribu-tions. Interestingly, we find that the distribution differssignificantly in forward and in reverse bias. To see thismore clearly, in panels Fig.7(i,j) we plot the distributionfor L = 8 for different anisotropies. For large values ofanisotropy, forward and reverse bias distributions seems

6

0

0.1

0.2

0.3

0

0.1

0.2

0.3

0

0.1

0.2

0.3

0

0.1

0.2

0.3

0 0.5 10

0.1

0.2

0.3

0 0.5 1

FIG. 7. Distribution of rapidities p(r) against the radial dis-tance r in complex plane for different anisotropies ∆ = 0.5panels (a,b), ∆ = 2.0 (c,d) and ∆ = 2.8 (e,f). In panels (g,h)distribution p(r) is plotted against anisotropy ∆ for systemsize L = 8. Other parameters: Jm = 0.1, Γ = 1 and γ = 0.Left panels are for forward bias and right panels are for reversebias.

to tend towards a different, yet well defined, function.And the probability of rapidities close to r = 1 is signifi-cantly larger for the reverse bias compared to the forwardone, suggesting the presence of a significantly larger num-ber of slow decaying modes, which is expected due to themuch slower relaxation dynamics of the reverse bias sys-tem. The exact expression for these distributions can bea topic for future research.

Since the occurrence of an out-of-equilibrium phasetransition is related to the difference in the scaling ofrapidities which converge to zero in the thermodynamiclimit [23], we focus on the real part of the rapiditiesand plot it versus the anisotropy ∆, both for forward,Fig.8(b), and reverse bias, Fig.8(c). Furthermore, wesupplement these panels with Fig.8(a) in which we plotthe rectification R versus ∆. We observe that for ∆ >

10 -1

10 7

0

0.2

0.4

0 2 4 6 8 10

0

0.2

0.4

FIG. 8. Panel (a): Rectification measure R as a function ofanisotropy ∆. Panels (b) and (c) shows |Re(λi)| against ∆ forforward and reverse direction respectively. Other parameters:L = 8, Jm = 0.1, Γ = 1, γ = 0. Vertical dashed line indicates∆ = ∆R.

∆R (vertical dashed line), in reverse bias there is a sig-nificant number of rapidities which tends towards zero.Furthermore, in forward bias the rapidities tend towardszero, however their proximity to zero, and their density,is significantly smaller.

4 6 8 10 1210 -4

10 -2

10 0

10 -4

10 -2

10 0

FIG. 9. Log-log plot of |Re(λ1)| against L across ∆ =0.5, 1, 1.5, 2.8, 3 for forward bias (panel (a)) and reverse bias(panel (b)). Other parameters: Γ = 1, γ = 0, Jm = 0.1.

To have a more quantitative grasp of how the smallestrapidity approaches zero, in Fig.9 we plot the spectralgap versus the system size for different system lengths.In particular, in Fig.9(a) we show the forward bias case,while in Fig.9(b) the reverse bias one. For the smallervalue of the anisotropy shown, and the system sizes whichwe could explore, the decay of the relaxation gap withthe system size is consistent with an algebraic decay. For

7

large values of the anisotropy, e.g. ∆ = 2.4 and 3, forreverse bias the rapidity seems to decay faster than alge-braically, consistent with an expected exponential decayfor insulators, while in forward bias the relaxation is inagreement with an algebraic relaxation, although possi-bly with a different exponent. Though the system sizeswe used are not large, a fairly abrupt change of transportproperties are seen in Fig.5(b, d), and the fact that therelaxation gap possibly relaxes exponentially, hint to apossible first order phase transition in the reverse biasedsystem at ∆R.

VI. EFFECT OF DEPHASING

10 0

10 5

10 -7

10 0

10 -9 10 -5 0.01 10 5010 -4

10 -2

10 -1

FIG. 10. Rectification R (panel (a)), currents Jf and Jr

(panel (b)) and difference in magnetisation at the interfaceδSz (panel (c)) as a function of dephasing strength γ with∆ = 5 (solid) and ∆ = 2 (dashed). Forward bias (δSzf , Jf )are shown by red lines and reverse bias (δSzr, Jr) with bluelines. Other parameters: Jm = 0.1, Γ = 1 and L = 8.

Until now we have considered only the boundary driv-ing of XX+XXZ chain in which the baths are trying toimpose the system to be in the |↓〉〈↓| or (|↓〉〈↓|+ |↑〉〈↑|)/2.For the case of baths which are not exactly trying to im-pose these states one can refer to [11], where it is shownthat there is a significant decrease of the rectification.The effect of Hamiltonian perturbations inside the chainhave been studied in [14], where it was shown that gi-ant rectification can be obtained even in the presence ofHamiltonian perturbations, however there is no thresholdvalue of anisotropy beyond which the system becomes aperfect rectifier in the thermodynamic limit. For thermalbath it was also shown that significant rectification canemerge, although not to the extent of having a perfectrectifier in the thermodynamic limit [33].

The nature of the dissipation, whether it acts in thebulk or only at the boundaries, may significantly changethe transport as well as the spectral properties. It hasbeen shown in a number of articles [34–36], that dephas-ing (bulk dissipation) leads to diffusive transport, inde-pendent of the value of ∆. Furthermore, the scaling ofspectral gap can change from exponential to algebraic ina (not-segmented) XXZ chain in the presence of dephas-ing for ∆ > 1 [23]. Hence it is relevant to study howdephasing affects the properties that we have previouslyanalyzed. For this, we add a dephasing term on each siteas in Eq.(2) for γ 6= 0 in Eq.(4).

In Fig.10 we consider the rectificationR, panel (a), theforward (red lines) and reverse (blue lines) bias currents,panel (b), and the magnetization jump at the interfacebetween the XXZ and the XX parts of the chain, panel(c). It is clear from Fig.10(a) that even for values ofγ ≈ 10−3 there is a significant reduction of rectificationfor ∆ = 5 > ∆R whereas R does not show a strongdependence on γ for a small ∆ = 2 < ∆R, as there was nosignificant rectification for these values of the anisotropy.

These results can be understood from Fig.10(b) whichdepicts the forward and reverse bias current separately.We observe that when the current is significantly sup-pressed by the anisotropy, dephasing has first the effect ofincreasing the current (allowing transport which was im-peded by energy constraint due to the large anisotropy).Then, after reaching a maximum, both currents de-crease as dephasing increases because the dephasing de-stroys all coherences. In Fig.10(c) we study the depen-dence on γ of the magnetization jump at the interface,δSz = |σzL/2−1 − σzL/2|. Since this value is contained

between 0 and 1, in Fig.10(c) we plot 1 − δSz. When∆ > ∆R the system is insulating in the reverse bias lead-ing to a step magnetization profile with a maximal jumpat the interface. Hence, a minimum value of 1−δSz is ex-pected and is clearly seen in our studies. For small ∆ inthe reverse bias as well as for all ∆’s in the forward biasthe system is diffusive and hence a continuous change ofmagnetization at the interface is seen resulting in a largevalue of 1− δSz. However, we note that finite size effectslead to large jump at the interface for the system sizewe explore in these cases. Finally, as we increase γ, δSzbecomes identical in forward and reverse bias.

Next, we analyze how dephasing affects the relaxationspectrum. In Fig.11 we plot eλi , similarly to Fig.6,for forward (left panels) and reverse (right panels) bias.Fig.11(a,b) is for γ = 0 (absence of dephasing), and thelower panels have increasingly larger values of dephas-ing rate γ, i.e. γ = 0.1 in Fig.11(c,d) and γ = 1 inFig.11(e,f). In all panels we use a very large anisotropy∆ = 18 to further emphasize the significant impact of thedephasing rate γ. It is very apparent from Fig.11 thatdephasing forces the rapidities to become real except inthe surrounding of the origin, thus suppressing oscillatorydynamics. Dephasing also significantly reduces the num-ber of rapidities with real part close to zero. Importantly,already for γ ≈ 1 there is no apparent difference between

8

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1 -1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

FIG. 11. Exponential of rapidities in complex plane for dif-ferent dephasing rates γ = 0 (a,b), γ = 0.1 (c,d), γ = 1 (e,f).Left panels are for forward bias and right panels for reversebias. Here, L = 8, Jm = 0.1, Γ = 1 and ∆ = 18.

the forward and reverse biases, despite ∆ = 18� ∆R.

0

0.5

1

0 0.2 0.4 0.6 0.8 10

0.5

1

FIG. 12. Distribution of rapidities p(r) against the radialdirection r in complex plane for L = 8, Jm = 0.1, Γ = 1 fordifferent dephasing rates γ in the forward direction (a) andreverse direction (b).

To have a better grasp of the distribution of the ra-pidities, we plot them in Fig.12. Fig.12(a) corresponds

to the forward bias, while Fig.12(b) to the reverse bias.We observe that for small dephasing rate the distributionis different in two biases. However, as the γ increases thedistribution becomes similar and is completely localizedin the region around r = 0.

00.1

0.3

10 -1

10 7

00.1

0.3

1

20

0

0.2

0.4

0.8

3

0 2 4 6 8 100

0.5

1

0.9

1.2

FIG. 13. Rapidities |Re(λi)| vs ∆ for γ = 0 (a), γ = 0.01(b),γ = 0.1 (c),γ = 1 (d). Rectification is shown by reddashed lines on the right axis. Rlog and Rlin refers to Rplotted in logarithmic and linear scale respectively. Otherparameters are L = 8, Jm = 0.1 and Γ = 1.

We then analyze more in detail the real part of the ra-pidities |Re(λi)| versus ∆ for different magnitudes of thedephasing in Fig.13, where we also show the rectificationR with a red-dashed line. Each panel corresponds to adifferent value of the dephasing rate γ, specifically Fig.13(a-d) correspond to γ = 0, 0.01, 0.1 and 1 respec-tively. Rapidities are plotted only for the reverse biascase as the rectification is significantly determined by re-verse bias. From the figure we see that R is significantlyaffected by dephasing and reaches the value of R ≈ 20for γ = 0.01 and of R ≈ 3 for γ = 0.1. From studyingthe real part of the rapidities, we observe that the den-sity of rapidities close to zero significantly reduces as γincreases, indicating faster relaxation and it also corre-sponds to the disappearance of the insulating regime.

Last we focus only on the rapidity with the small-est real part different from zero, the relaxation gap. InFig.14(a,b) we plot the relaxation gap versus ∆ for dif-ferent values of γ, panel (a), and versus γ for differentvalues of ∆, panel (b). In Fig.14(a) we observe clearlythat the relaxation gap increases for larger (but not toolarge) γ, and it acquires similar values in the forward (redlines) and reverse biases (blue lines). This is an exampleof dephasing-assisted tunneling (see for instance [37]). InFig.14(b) we also observe how very large values of the de-phasing rate γ can lead to a smaller relaxation gap whichis identical in forward or reverse bias. In particular, therelaxation gap scales as the inverse of the dephasing rate,thus corresponding to quantum Zeno physics [23, 38–40].

9

0 5 10 15 2010-5

10-2

100

10-9 10-5 10-2 10110-3

10-2

10-1

100

FIG. 14. Panel (a): Relaxation Gap |Re(λ1)| against ∆ fordifferent dephasing rates γ = 0 (dashed), γ = 1 (solid). Panel(b): Gap |Re(λ1)| against γ for different anisotropies ∆ =1 (dashed) and ∆ = 5 (solid). Other parameters: L = 8,Jm = 0.1. Black dotted line indicates the inverse scaling ofrelaxation gap with dephasing rate.

VII. CONCLUSIONS

In this work we have studied the transport and spec-tral properties of the XX+XXZ diode. This diode con-sisting of a segmented chain coupled to magnetizationbaths trying to impose the spin down state on one sideand an infinite temperature state on the other side hasbeen shown to perform perfectly in the thermodynamiclimit. Large rectification stems from the insulating be-havior of the system in the reverse bias when anisotropy∆ > ∆R. Here we have shown that this setup is evenricher. Our analysis of the spin current and local mag-netization point towards an understanding that in theforward bias, transport goes from ballistic to diffusive at∆ = ∆XXZ . For larger anisotropies the spin current de-creases significantly, and this is mainly due to an increasein the interface resistance between the two parts. How-ever, because of the system sizes that can be reached,we are not able to conclude whether transport becomessub-diffusive. At the same time, the analysis of the mag-netization is still consistent with a diffusive behavior.

In reverse bias, the data indicates that there could bethree different regimes: ballistic for ∆ < ∆XXZ , diffusive

for ∆XXZ < ∆ < ∆R and insulating for ∆ > ∆R. Morestudies are required to better characterize the intermedi-ate region with ∆XXZ < ∆ < ∆R as the magnetizationprofile, for the system sizes we could reach, does not showa clear linear slope as it does in forward bias.

We also focused our attention on the spectral prop-erties of the system. We have seen that the rapiditiestend to different distributions in the forward and reversebias. In particular, in reverse bias the distributions havefar more rapidities with real part close to zero (i.e. theexponential of the rapidity is near the unit circle in thecomplex plane). This is expected as the system relaxesmuch more slowly in this regime. In the future it could beinteresting to derive an expression for such different dis-tributions and its relation to different transport regimes.

We then focused on the relaxation gap. Unlike forsystems with bulk dissipation, for systems in which dis-sipation only acts at the boundary, the spectral gap goesto zero with the system size. However, one would expecta change in the dependence of the spectral gap with thesystem size at a transition, and/or in different phases.We found that in reverse bias the spectral gap goes tozero much faster for ∆ > ∆R, in a manner consistentwith an exponential decay. Furthermore, the density ofrapidities close to zero is far more dense in reverse biascompared to forward bias.

We also investigate the effect of bulk dephasing onthe performance of the XX+XXZ diode and its spectralproperties. Our analysis shows that dephasing degradessignificantly and rapidly the rectifying property of thediode. As for the spectral properties, dephasing resultsin the exponential of the rapidities to be centered in theorigin of the unit circle, independent of whether the sys-tem is in forward or reverse bias. Furthermore, we areable to capture the emergence of a quantum Zeno regimein which the spectral gap becomes inversely proportionalto the dephasing rate.

We stress that our insights are captured from small tomedium-scale systems and could serve as a good startingpoint for more quantitative characterization of the in-triguing properties of the system. For instance, a futureresearch direction is how to obtain an accurate charac-terization of (almost) insulating boundary driven non-equilibrium steady states for which multiple rapiditiesare extremely close to zero. As mentioned in the intro-duction, XXZ chains can be realized in ultracold atomsexperiments. We however note that reaching a largeanisotropy may be challenging. For instance in [7] thelargest anisotropy reached is ∆ = 2. However, anothersetup in which this physics can be tested is in supercon-ducting circuits, as proposed in [9, 10].

ACKNOWLEDGMENTS

D.P. acknowledges support from the Ministry of Ed-ucation of Singapore AcRF MOE Tier-II (Project No.MOE2016-T2-1-065).

10

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