The Adiabatically Deformed Ensemble: Engineering Non-Thermal States of Matter
TThe Adiabatically Deformed Ensemble: Engineering Non-Thermal States of Matter
D. M. Kennes Department of Physics, Columbia University, New York, NY 10027, USA (Dated: September 3, 2018)We propose a route towards engineering non-thermal states of matter, which show largely unex-plored physics. The main idea relies on the adiabatic passage of a thermal ensemble under slowvariations of the system Hamiltonian. If the temperature of the initial thermal ensemble is either zeroor infinite the ensemble after the passage is a simple thermal one with the same vanishing or infinitetemperature. However, for any finite non-zero temperature intriguing non-thermal ensembles can beachieved. We exemplify this in: (a) a single oscillator (b) a dimerized interacting one dimensionalchain of spinless fermions, (c) a BCS-type superconductor and (d) the topological Kitaev chain.We solve these models with a combination of methods; either exactly, numerically using the densitymatrix renormalization group (DMRG) or within an approximate functional renormalization group(FRG) scheme. The designed states show strongly non-thermal behavior in each of the consideredmodels. For example, for the chain of spinless fermions we exemplify how long ranged non-thermalpower-law correlations can be stabilized and for the Kitaev chain we elucidate how the non-thermalensemble can largely alter the transition temperature separating topological and trivial phases.
Non-equilibrium states of matter have attracted agreat deal of interest lately. To achieve a steady non-equilibrium state one can, e.g., contact a system byleads to drive currents through it. Another routeto non-equilibrium is to isolate a system (sufficientlywell) from its environment and subsequently change itsHamiltonian. The second class includes (e.g.) pump-probe experiments or quenching ultra cold gases byabruptly tuning through a Feshbach resonance. Quan-tum quenches have been studied in great detail.
Here, we study the adiabatic deformation of a givensystem. In this context, the Kibble-Zurek mechanism has gained a lot of attention. At the critical point of asecond order phase transition the critical slowing downimplies that it is impossible to drive a system adiabati-cally through its transition. The finite speed of the pas-sage disturbs the ordering of the state after the transi-tion, leaving behind ordered domains (with size depend-ing on the rate of change and the critical exponents). The Kibble-Zurek mechanism has been tested extensivelynumerically as well as experimentally. We focus onadiabatic deformation of a given initial ensemble stayingaway from crossing a second order critical point. Studiesof such adiabatic deformations or in general finite timequenches (without crossing a phase boundary) mainly fo-cus on the initial state being the ground state.
Theseworks either cover the dynamics in the most general casein between sudden quenches or adiabatic deformations oraddress the important questions whether adiabatic evo-lution is possible at all and if so identify the leading cor-rections in the rate of change. This is a pressing matter,because in Ref. 16, it was shown that not all systemsexhibit adiabatic behavior. In this work the authors sep-arate systems into three generic classes with respect tothe behavior of the excess energy under slow variationsof the Hamiltonian: (a) analytic, where the correctionsto the adiabatic behavior vanish as the square of therate of change, (b) non-analytic, where the correctionsvanish following a non-quadratic behavior and (c) non- adiabatic, where the corrections depend on a power-lawin the system size and thus as the system size approachesinfinity the adiabatic limit ceases to exist. We note thatwe use the term ’adiabatic’ in contrast to this classifi-cation in a different convention, which is also commonin the literature: adiabatic variation here denotes suffi-ciently slow variations of a system parameter, such thatfurther decreasing the speed of the variation yields nochanges in physical observables (supported either on allenergy/inverse length scales or a subset thereof).A marked exception to the study of ground state prop-erties under adiabatic evolution is listed in Ref. 17, whereit was shown that adiabaticity in a Luttinger liquid (atleast for sufficiently smooth variations, such that the Lut-tinger liquid picture remains valid) falls into the ana-lytic class of Ref 16 when deforming the ground stateby changing the two-particle interaction. For finite T initial states this changes to a linear dependency of theexcess energy on the rate of change. However, within thisstudy the adiabatic deformation, even at finite tempera-ture, amounts simply to a change in temperature of theensemble after the deformation. This is a direct conse-quence of the linear dispersion of the assumed model andthe studied type of deformation, i.e. slowly changing theinteraction (see below).We find that in general and for a broad variety of quan-tum systems by adiabatically deforming an ensemble in-triguing non-thermal states can be prepared, which havethe potential to harbor interesting physics inaccessible bythermal pathways. For example we show that for Lut-tinger liquid non-thermal long ranged correlations canbe stabilized or that in the Kitaev chain the non-thermaldistribution function after the deformation can greatlyalter the critical temperature separating topological andtrivial phases.To illustrate the main idea, we concentrate on initialthermal states w.r.t. some Hamiltonian H and slowlyvary some system parameter(s) to a final Hamiltonian H (cid:48) . The system is assumed to be sufficiently isolated a r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l from its environment such that during this slow defor-mation of H → H (cid:48) it can be considered as closed. In thiscontext ”slow” refers to the regime in which the Gell-Mann and Low theorem holds for all excited states. Of course the applicability of the Gell-Mann and Low the-orem has to be checked on a case by case basis. The theo-rem states that the eigenstates of the initial Hamiltonianare mapped to the eigenstates of the final Hamiltonianunder an adiabatic passage. To simplify the general dis-cussion we assume a non-degenerate discrete spectrum ofthe family of Hamiltonians which describe the deforma-tion H → H (cid:48) . In our explicit examples studied below wewill relax this assumption and test the range of validityof the results presented explicitly.For the two Hamiltonians H and H (cid:48) with H | E n (cid:105) = E n | E n (cid:105) and H (cid:48) | E (cid:48) n (cid:105) = E (cid:48) n | E (cid:48) n (cid:105) the adiabatic deforma-tion of a thermal ensemble ρ = Z (cid:80) n e − E n /k B T | E n (cid:105) (cid:104) E n | leads to ρ adia → ρ deform = 1 Z (cid:88) n e − E n /k B T | E (cid:48) n (cid:105) (cid:104) E (cid:48) n | (1)with ρ being the density operator. The deformed ensem-ble appears similar to a thermal one, but with changedBoltzmann factors corresponding to the initial Hamilto-nian. A priori the consequences of this non-thermal dis-tribution, e.g. for observables, are unclear. It is how-ever obvious that the two limits of vanishing and infinitetemperature trivially deform into an ensemble with thesame zero or infinite temperature, respectively. For finitenon-zero temperature of the thermal ensemble before thepassage on the other hand, tuning the Boltzmann factors(by clever choice of the initial Hamiltonian) allows for en-gineering and tuning the properties of the state after thedeformation.We emphasize that the procedure we propose helpsconsiderably in this engineering process. To illustratethis let us consider the most simple case, where the fam-ily of Hamiltonians during the deformation H → H (cid:48) arenon-interacting. The eigenmodes of any non-interactingmodel are easy to determine. Now if we consider first asudden quench out of, e.g., the ground state of H engi-neering the properties of the final state after the quenchis still difficult as it involves the knowledge of the overlapof the eigenstates before and after the quench. Of courseone could simply solve this problem numerically varyingthe initial Hamiltonian H , but a straightforward, intu-itive picture remains illusive. On the contrary, for an adi-abatically deformed ensemble predicting the final state isvery simple. The mode occupancies remain constant andonly the energies of the modes follow the adiabatic pas-sage. This allows for example to engineer the followingsimple, but interesting example. Let us assume we wantto engineer a state which, e.g., displays a jump of arbi-trary height of the mode occupancy around the chemicalpotential in a non-interacting non-gapped system. Wecan easily solve this problem using nothing but generalarguments by considering a non-interacting band insula-tor in thermal equilibrium with the chemical potential in the middle of the band gap. If for simplicity the disper-sion is assumed to be symmetric, the occupancy of themodes of this band insulator will take continuous valuesfrom C to 1 / Z/ / − Z/ C (forthe part of the spectrum above the chemical potential),with some values C , C , 0 ≤ Z ≤ T → Z → T → ∞ , Z → C to 1 / Z/ / − Z/ C (below and above the chemicalpotential, respectively), displaying a ”quasi-particle”-likejump of height Z (which can be tuned by the tempera-ture before the adiabatic passage). This explicitly allowsto design all possible values of 0 ≤ Z ≤
1. No ther-mal ensemble can be used to obtain such a state in anon-interacting problem. Performing a similar Gedankenexperiment for the sudden quench to engineer such a non-thermal state is obscured by the difficulty of calculatingthe overlaps of final and initial states. Thus this con-stitutes one example where we have utilized adiabaticdeformation to engineer a non-thermal state with the de-sired property (jump at the chemical potential of height0 < Z <
1) we asked for. This example will be analyzedmore formally as the second example discussed below.Following this logic it is also clear why (effectively)non-interacting systems that follow a linear dispersionrelation cannot be adiabatically deformed away from athermal state, if the adiabatic deformation only changesthe prefactor of the linear dispersion from v → v (cid:48) ,as in the example of a Luttinger liquid with time de-pendent interactions mentioned above. The Boltzmannweights after the deformation ∼ e − β (cid:80) k vk can triviallybe rewritten as e − β (cid:48) (cid:80) k v (cid:48) k with changed inverse temper-ature β (cid:48) = βv/v (cid:48) .Identifying and engineering intriguing non-thermalphysics via the adiabatically deformed ensemble pro-posed above calls for case studies. One can argue that theassumption of adiabatic deformation of all eigenstates,severely limits the speed with which the passage has tobe performed in the absence of energy gaps. We will ex-plicitly test this in the examples studied below and findthat even in the absence of gaps the physics of the de-formed system behave adiabatic on energy (real-space)scales much larger (much smaller) than a scale set by theinverse speed (we will concentrate on linear ramps withspeed v in the following) of the deformation. . Since thegeneral idea of engineering non-thermal states by adia-batic passage applies to arbitrary physical systems, wewill put it to the test in a variety of different impor-tant models in the following, because we believe that thisstrategy is necessary to access the generality of the pro-posed concept. Every model study is self-contained andcan be accessed independently of each other. We thusstructure the rest of the paper in sections discussing theseparate examples: section I deals with a simple single E n − − − − p ( E n ) / p ( E ) g ini = { . , . , . , . , . , . } T = 2 . . . . . . g ini . . . . . . < ∆ n > / < n > ”Thermal””Deform.” T = 5 .FIG. 1. Main panel: adiabatic deformation of a single oscilla-tor. We show the relative weights p ( E (cid:48) n ) /p ( E (cid:48) ) of the eigen-states | E (cid:48) n (cid:105) to the ensemble after the adiabatic deformationfrom a thermal state with T = 2. For g ini > | E (cid:48) n (cid:105) fall off quicker than exponentially in their respective energies E (cid:48) n . The deformed ensemble thus suppresses fluctuations intohigher energy states. Inset: fluctuations in the mean bosonoccupancy ∆ n = (cid:113) (cid:104) n (cid:105) − (cid:104) n (cid:105) relative to the mean bosonoccupancy (cid:104) n (cid:105) = (cid:10) b † b (cid:11) itself after the adiabatic deformationfrom a thermal ensemble with T = 5 (solid black line) as wellas for a thermal state with temperature fixed by the meanoccupancy after the deformation (dashed blue line). oscillator, section II with a dimerized chain of interactingspinless fermions, section III with a BCS superconductorand section IV with the topological Kitaev chain. Finallyin section V we give a concluding summary. I. OSCILLATOR
First, we consider a single oscillator as a simple exam-ple. The oscillator is described by a harmonic as well asan anharmonic ( ∼ x ) contribution H = ω (cid:18) b † b + 12 (cid:19) + g ( t )4 ω (cid:0) b † + b (cid:1) (2)The single oscillator is initially in a thermal state withtemperature T and we set ω = 1. The anharmonicity g ( t ) ≥ g ini to 0. For thissimple (toy-)model it is easy to show that the dynamicsremains adiabatic as long as the speed of the ramp v (cid:28) E ≥ p ( E (cid:48) n ) of the eigenstates | E (cid:48) n (cid:105) to theensemble ρ deform = 1 Z (cid:88) n p ( E (cid:48) n ) | E (cid:48) n (cid:105) (cid:104) E (cid:48) n | (3) reached after the adiabatic deformation does not fol-low an exponential form ∼ e − E (cid:48) n /k B T in their respectiveEigenenergy E (cid:48) n as would be the case in a thermal en-semble. In fact they fall off quicker then exponentially,indicating a suppression of fluctuations into higher occu-pation number states. In the inset we show the fluctua-tions in the mean boson occupancy ∆ n = (cid:113) (cid:104) n (cid:105) − (cid:104) n (cid:105) relative to the mean boson occupancy (cid:104) n (cid:105) = (cid:10) b † b (cid:11) itselfafter the adiabatic deformation as a solid black line. Thissignal to noise measure can be enhanced compared to areference thermal ensemble (dashed blue line in the in-set), where the effective temperature T eff is determinedby the mean boson occupancy (cid:104) n (cid:105) ρ deform ! = (cid:104) n (cid:105) T eff =1 / ( e /T eff − n highlights the highly non-thermal nature of the deformed ensemble. We thus notethat adiabatic passage can be used to suppress fluctua-tions beyond the cooling paradigm. II. DIMERIZED CHAIN
Next we consider the 1D Hamiltonian H = N − (cid:88) j =1 (cid:104) J (cid:16) c † j c j +1 + H . c . (cid:17) + U n j n j +1 (cid:105) + N (cid:88) j =1 ( − j δ ( t )2 n j (4)describing a dimerized nearest-neighbor tight-bindingchain with open boundary conditions and density-densityinteraction U between adjacent fermions (density n j = c † j c j ) and set J = (cid:126) = k B = 1. We concentrate on half-filling. This model falls into the Luttinger liquid univer-sality class at non-zero | U | < δ = 0. Adiabatically Deformed Ensemble: U = 0 — We dis-cuss the non-interacting U = 0 case first. This limit al-lows for a simple and exact treatment and reveals essen-tial insights into the underlying physics of the problem.We calculate the adiabatic deformation of a thermal en-semble with T = 0 . δ = 1 to δ = 0 for dif-ferent deformation speeds v . The results are summarizedin Fig. 2. The main panel shows the distribution function n in dependency of the energy (cid:15) (we suppress the mo-mentum index k in the energies (cid:15) k in the following) bothinitially (thermal Fermi-Dirac distribution (thick dashedblue line in Fig. 2), note the energy gap) as well as afterthe adiabatic deformation for different speeds v and for N = 100. For slow enough deformation a jump in themode occupancy of finite height Z = lim (cid:15) → n ( − (cid:15) ) − n ( (cid:15) )occurs around the Fermi edge (cid:15) = 0. This is in starkcontrast to the thermal case at T >
0. We derive thedistribution function analytically (thin dashed black linein Fig. 2) n ( (cid:15) ) = 1 e sign( (cid:15) ) √ (cid:15) + δ (0) / + 1 . (5) − − (cid:15) . . . n ( (cid:15) ) v = 0 . v = 0 . v = 0 . ini − − − v . . . . . Z N = 50 N = 100 N = 200 10 N v . . . . . Z N = 50 N = 100 N = 200 FIG. 2. Main panel: We deform the dimerized chain Eq. (4)from δ = 1 to δ = 0 and U = 0. Shown is the distributionfunction before (black symbols) and after (colored symbols)the adiabatic passage for N = 100. We initially prepare athermal distribution with T = 0 . N isincreased slower time variations are needed to stay adiabatic,(right) all curves collapse when rescaling the x-axis by theminimum in single-particle energy gaps ∼ N − . It is clearly not a thermal one and one can engineer statesexhibiting a very sharp jump at the Fermi edge of height c = tanh (cid:18) δ (0)4 T (cid:19) . (6)resembling T = 0 Fermi liquids.Both insets demonstrate the dependency on systemsize. In the left inset we show that as the system size in-creases the energy gap between the states diminishes andtherefore the speed v for which the maximum Z (dashedred line) is reached must be lowered. The right insetshows, that if the results are scaled w.r.t. the minimumin the single particle energy gaps ∆ E ∼ N − , all curvescollapse. Thus, we identify v/ ∆ E (cid:28) Adiabatically Deformed Ensemble: finite U — For theinteracting model U (cid:54) = 0 we employ the DMRG, set upin the language of matrix product states, to describe its(thermo)dynamics. We concentrate on the same adi-abatic ramp in δ as described above and formulate theDMRG directly in the thermodynamic limit. An adi-abatic evolution is strictly speaking not possible in theinfinite system (see above). However, we find that if thesystem is deformed with given speed, local observablessupported on a spatial region of width ∼ /v still deformadiabatically. In this looser sense locally the system ap-pears adiabatic, where the degree of this locality can betuned by the speed of the deformation.As adiabatic deformation of initial states is not rou-tinely studied in DMRG (for an exception see Ref. 33)we first benchmark our DMRG result to those obtained − − (cid:15) . . . . n ( (cid:15) ) exactthermalDMRG . . . vt − . − . − . E ( t ) thermaladia. deform v=0.01v=0.005 FIG. 3. DMRG benchmark at U = 0 for the same protocolas studied in Fig. 2. Upper panel: Energy per lattice siteafter an adiabatic deformation of δ = 1 to δ = 0 of Eq. (4) fortwo deformation speeds and T = 1. We find good agreementwith the expectation from an adiabatically deformed ensemblewhile there is a significant deviation from the thermal valueat T = 1. Lower panel: Distribution function n ( (cid:15) ) after thesame adiabatic deformation as in the upper panel, with speed v = 0 .
01 at T = 1 in the non-interacting limit U = 0 (dashedorange line). We also show the prediction from Eq. (5) (solidblack line) as well as the thermal distribution (dashed redline). analytically in the model of Eq. (4) at U = 0. We con-centrate on an adiabatic deformation of δ = 1 to δ = 0as above. In all of our simulations we choose the param-eters such that the results are converged on the scale ofthe plots. In practice this means that we employ a secondorder Trotter decomposition for the imaginary time evo-lution to prepare the initial thermal state of the systemwith a step size of ∆ β = 0 .
005 at fixed bond dimension χ = 300. Subsequently, we subject this initial state to areal time evolution using a fourth order trotter schemewith step size ∆ t = 0 .
01 and a dynamically increasingbond dimension, keeping the truncation error below athreshold of 10 − at each time step.Fig. 3 shows the benchmark comparing the DMRG re-sults to the exact solution. The upper panel shows theenergy per lattice site after an adiabatic deformation of δ = 1 to δ = 0 of Eq. (4) for two deformation speedsand T = 1. This local observable behaves perfectly adia-batic at the speeds chosen although the system is infinite.We can predict the energy from Eq. (5) and find perfectagreement, while it deviates strongly from the thermalvalue at T = 1. The lower panels shows the distribu-tion function n ( (cid:15) ) of the deformed ensemble ( v = 0 . T = 1 in the non-interacting limit U = 0 extractedfrom the Fourier transform of the correlator c † L/ c L/ n .We restrict our calculation to | n | < (cid:15) which can be extracted reliably with | n | <
100 theagreement to the analytical result Eq. (5) is perfect.Our DMRG results beyond the consistency check j − − − − − − | S zz ( j ) | T=1.0 U=0.0 v → → → FIG. 4. We study the same adiabatic deformation of δ ofthe dimerized chain Eq. (4) as in Fig. 2, but also showingnon-zero U . We compare v = 0 .
01 (smaller filled symbols) to v → v = 0 .
005 inthe interacting case (larger open symbols). For comparisonwe show the thermal expectation w.r.t. a conservatively lowchosen temperature T = 1 / < T eff , where T eff is the effec-tive temperature fixing the correct energy expectation valueat the end of the deformation (blue crosses). In the adiabati-cally deformed ensemble the cut-off temperature is removed. of Fig. 3 are summarized in Fig. 4. We con-centrate on the density-density correlator S zz ( j ) = (cid:10) ( n L/ − / n L/ j − / (cid:11) . In thermal equilibriumand for δ = 0 this function decays exponentially fordistances j larger than a characteristic scale of the or-der of the inverse temperature 1 /T . The correlator S zz ( j ) calculated for a thermal ensemble with temper-ature T = 1 /
16 is shown as the x’es in Fig. 4 for U = 0and U = 1. We choose this value of the temperature as itis smaller than T eff , determined by (cid:104) H (cid:105) ρ deform ! = (cid:104) H (cid:105) T eff ,for all shown parameter sets. The small filled symbols inFig. 4 show S zz ( j ) obtained after a δ ramp from δ = 1to δ = 0 with velocity v = 0 .
01. We want to study thedegree to which these results are in the adiabatic limitand thus compare these results to a slower deformation.In the non-interacting case we can compare to results ob-tained in the limit v → v = 0 .
005 for comparison.The results obtained at this smaller velocity are given asopen larger symbols in Fig. 4. The results indicate thatthe more adiabatic the deformation becomes the morethe region extends in space over which the results for thetwo different deformation speeds agree. For large j thetwo curves start to deviate indicating that the adiabaticregime is left and to obtain converged results at thesevalues of j lower speeds need to be considered. Herewe want to concentrate on the spatial regime, where theresults are converged with respect to the speed of defor-mation v .In equilibrium at T = 0, S zz ( j ) displays long rangedcorrelations following a sum of two power-laws in real j − − − − | n j − / | T=0.0T=0.5T=1.0 T=2.0T=5.0T=10.0 . . . T − . − . − . − . α . . . v L FIG. 5. Quasi-steady occupancy deviations from one halffound at one of the boundaries of the system (dots) after anadiabatic deformation from δ = 1 to δ = 0 of Eq. (4) at U = 0 and a subsequent interaction quench to U = 1. Thequasi steady state found shows a power law dependent be-havior of the occupancy deviations, with an exponent Eq. (8)(lines). To show that the exponent has changed sufficiently todistinguish different temperatures we add also the power-lawof the largest temperature ( T = 10) to the dots of the smallestone ( T = 0) as a dashed line. The exponent α is temperaturedependent and can be tuned from its maximal value at T = 0to the non-interacting limit at T = ∞ . Blue crosses indicatethe corresponding result found in thermal equilibrium using T = 0 .
25. Clearly temperature in thermal equilibrium cutsof the power-law at site j ∼ /T . This cutoff is removedby the adiabatic deformation, because the jump at the Fermiedge is not softened in the adiabatically deformed ensemble.The inset shows the analytic prediction of the exponent of thepower-law (line) as well as the renormalization of the velocity v L . The velocity is tuned form its maximal value at T = 0 tothe non-interacting value at T = ∞ in resemblance to α . space. This type of power-law decay composed of a mo-mentum q = 0 and a q = 2 k F component (with k F be-ing the Fermi momentum) is characteristic for Luttingerliquids. For the adiabatically deformed ensemble wefind that as we lower the speed v , the regime over whichlong-ranged power-law correlations can be observed, sim-ilar to the T = 0 case, extends. Thus by lowering v , asin the equilibrium case by lowering T , one can stabi-lize long ranged correlations over an increasing spatialregime, where in the adiabatically deformed ensemblethe speed v has replaced the temperature T as a cut-off in real space. By comparing to a temperature chosensmaller than the effective temperature fixed by the energyexpectation value after the adiabatic deformation (illus-trated by x’es in Fig. 4), we see that we go beyond thecooling paradigm and extend the long ranged correlationsbeyond the thermal cutoff. Slow ramping thus opens upa route towards an experimental realization of Luttingerliquid behavior in systems which cannot be cooled downsufficiently. Probing an Adiabatically Deformed Ensemble at U = 0 with a Subsequent U Quench — To analyze the non-thermal behavior obtained by deforming the dimerizedchain Eq. (4) further, we additionally perform a studyof an interaction quench, by abruptly turning on thedensity-density type of interaction H U = N − (cid:88) j =1 U n j n j +1 after the adiabatic passage from δ = 1 to δ = 0 at U = 0has been completed ( δ = 0 was reached). This is the sameprotocol as studied above, with the crucial difference be-ing that the adiabatic deformation is done without in-teraction first and only subsequently the interaction isturned on abruptly. We use a functional renormalizationgroup (FRG) approach as described in Ref. 35 and 36to address the long-time asymptotic and analyze power-law correlations unambiguously. Being able to addresslarge system size and times comes at the cost of themethod being approximate in the interaction strength.We concentrate on the occupancy deviations from half-filling. We find that after the interaction quench onewave propagates from each end of the chain with velocity v L , leaving behind a distortion pattern in the occupan-cies. For times for which the wave of the one end of thechain has passed a certain site, but before the wave ofthe other end reaches this particular site, a quasi-steadyvalue of the occupancy is obtained. We will concentrateon these quasi-steady values of observables from nowon and briefly review the results found for the quenchin the thermal case first. The FRG approach em-ployed only captures the leading behavior in U of the exponent of boundary or impurity physics. To this orderthe equilibrium and the quenched non-equilibrium expo-nent of boundary and impurity physics agree. Here, weconcentrate on how the occupancies, exhibiting Friedeloscillations after the quench, fall off from one bound-ary of the chain. In the quenched case starting froma thermal T = 0 ensemble (ground state) those follow | n j − / | ∼ j − α . Expanding the expression for the ex-ponent α = − ( K + 1) / K = π/ [2 arccos( − U/ α ≈ − Uπ . (7)The interaction dependent (critical) power-law decay ofthe Friedel oscillations, being a hallmark of Luttingerliquid physics, is a consequence of the jump of the distri-bution function at the Fermi edge.
This can mosteasily be identified by considering the flow equations inequilibrium which show that the distribution n ( (cid:15) ) and theprefactor U of the flow equation show up only as a prod-uct U [1 − n ( (cid:15) )]. As a consequence reducing the heightof the jump of the Fermi function by a factor of c < U by the same factor when fo-cusing on low energy scales. Thus, in the case where the height of the jump at the Fermi edge is reduced below 1to c , we instead of Eq. (7) expect α ≈ − c Uπ . (8)The numerical results of our FRG study (colored dots)are summarized in Fig. 5 for different initial temperatures(of the ensemble before adiabatic deformation). We firstperform an adiabatic deformation from δ = 1 to δ = 0of Eq. (4) at speed v = 0 . N = 200 and thenabruptly turn on U = 1. We depict the quasi-steady oc-cupancies n j . For reference the same quench is performedout of a thermal equilibrium ensemble w.r.t. T = 0 . T = 10. The predic-tion Eq. (8) with c given by Eq. (6) (lines) is in goodagreement with the numerical results. The adiabaticallydeformed ensemble can be used to tune the boundary ex-ponent of the power-law decay of the occupancies fromits maximal value α ≈ − . T = 0 (same asfor quench starting from thermal T = 0 ensemble) allthe way to the non-interacting limit of α = − T = ∞ (where the adiabatically deformed ensembleremains a trivial thermal ensemble with T → ∞ ). Thisdependency of the boundary exponent is shown in theinset, which for completeness also shows the numericallyextracted velocity v L , the second parameter characteriz-ing a Luttinger liquid completely. v L is tuned w.r.t. thetemperature within the same limits (at T = 0, v L ≈ . T = 0 ensem-ble) and at T → ∞ , v L = 2, which is the non-interactingvalue).To sum up, also in the case of first deforming the en-semble with respect to δ at U = 0 and subsequently tun-ing on the interaction, we can stabilize a (critical) power-law behavior in observables (in this case how Friedel os-cillations fall off away from a boundary), over a spatialregion much larger than in the corresponding thermalcase. Temperature is effectively removed as a cutoff andreplaced by the inverse of the system size or the speed ofdeformation, whichever is larger. III. BCS SUPERCONDUCTOR
As another example we study a BCS superconductor H = (cid:88) k,σ = ↑↓ (cid:15) k c † k,σ c k,σ + ∆( t ) (cid:88) k c † k, ↑ c †− k, ↓ + H . c ., (9)with a time dependent gap ∆( t ). Here c ( † ) k,σ annihilates(creates) a fermion in the single particle state character-ized by momentum k and spin σ . We employ the lan-guage of Nambu-vectors Ψ † k = ( c † k, ↑ , c − k, ↓ ) to rewrite the Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0−0.50.00.51.0 ǫ =1.0 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0−0.50.00.51.0 ǫ =0.5 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0−0.50.00.51.0 ǫ =0.25 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0−0.50.00.51.0 ∆y∆x ǫ =0.1 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0−0.50.00.51.0 ǫ =1.0 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0−0.50.00.51.0 ǫ =0.5 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0−0.50.00.51.0 ǫ =0.25 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0 −0.5 0.0 0.5 1.0 Ω −1.0−0.50.00.51.0 ǫ =0.1 (cid:15) , ∆ x − − ∆ y − − f k . . . . . . (cid:15) , ∆ x − − ∆ y − − f k . . . . . . FIG. 6. Top two rows: Time evolution of (cid:126)
Ω (red line) for different (cid:15) and two T = 5 (upper row) and T = 10 (lower row) onthe Bloch sphere and T = 1. Blue dashed lines indicate the adiabatically deformed prediction. Large (cid:15) or T clearly lead to anadiabatic deformation in accordance with Eq. (16). The far right plot in the upper row includes the definition of ∆ x and ∆ y (black lines) used in the bottom row plots. Bottom row: Adiabatically deformed ensemble prediction (blue line) compared tofull numerics (red line) for T = 1 (left) and T = 5 (right) for T = 1. Red circles indicate the magnitude of deviation (definedas in the upper right plot) from the adiabatically deformed ensemble which lie in the plane perpendicular to the adiabaticallydeformed (cid:126) Ω. Clearly deviations vanish for 1 /(cid:15)/T (cid:28) Hamiltonian as H = (cid:88) k Ψ † k (cid:18) (cid:15) k ∆( t )∆( t ) ∗ − (cid:15) k (cid:19) Ψ k . (10)First, we concentrate on the case of a given gap func-tion ∆( t ) = ∆ (cid:20) tanh (cid:18) t − δT (cid:19) + 1 (cid:21) , (11)where we slowly tune from a normal conducting to a su-perconducting state. The unit of energy is set by ∆ = 1.We exploit the representation of Ref. 40, which re-expresses the dynamics of a BCS superconductor in terms of precessing spins. Using the language of Ref. 40, thenon-equilibrium two time contour Green’s functions G R (cid:15) k ( t, t (cid:48) ) = − i Θ( t − t (cid:48) ) (cid:68)(cid:2) Ψ k [ t ] , Ψ † k [ t (cid:48) ] (cid:3) + (cid:69) , (12) G K (cid:15) k ( t, t (cid:48) ) = − i (cid:68)(cid:2) Ψ k [ t ] , Ψ † k [ t (cid:48) ] (cid:3) − (cid:69) , (13) G A (cid:15) k ( t, t (cid:48) ) = i Θ( t (cid:48) − t ) (cid:68)(cid:2) Ψ k [ t ] , Ψ † k [ t (cid:48) ] (cid:3) + (cid:69) , (14)can be rephrased in terms of a vector (cid:126) Ω, where theKeldysh Green’s function in Nambu space at equaltimes takes the particularly simple form G K (cid:15) ( t, t ) =1 + f (cid:15) R † (cid:15) ( t ) τ R (cid:15) ( t ) , and (cid:126) Ω · (cid:126)τ = R † (cid:15) ( t ) τ R (cid:15) ( t ) , where ω . . . . σ ( ω ) / σ T = { . , . , . , . , . , . , . , . , . , . , . , . , . , . } ( a ) T . . . A ω . . . . σ ( ω ) / σ ( b ) T = { . , . , . , . , . , . , . , . , . , . , . , . , . , . } FIG. 7. Optical conductivity (black to yellow solid lines) for the thermal case (a) as well as the adiabatically deformed ensemble(b). For comparison the optical conductivity of the adiabatically deformed ensemble at initial temperature T = 0 . A for the thermal (multi-colored line) as well as theadiabatically deformed ensemble (blue line). The latter is zero. τ = ( τ , τ , τ ) T and τ i the Pauli matrices. The dynam-ics of (cid:126) Ω are determined by˙ (cid:126)
Ω = 2 (cid:126)b eff ( t ) × (cid:126) Ω , (15)with (cid:126)b eff = (∆( t ) , , (cid:15) ) T . Thus if (cid:126) Ω follows the externalramp of ∆( t ) (determining the effective field (cid:126)b eff ( t )) adia-batically so do the Green’s functions, which fully charac-terize the non-equilibrium dynamics. It is easy to check,that adiabatic evolution is guaranteed if1 (cid:15)T (cid:28) . (16)For a thermodynamic system (cid:15) can be arbitrarily small,and thus given any T one can always find a part ofthe energy spectrum, which does not follow the de-formation adiabatically. However, if T is sufficientlylarge, the extent of this part of the energy spectrum be-comes negligible. In this sense the evolution in the BCS-superconductor can be considered to be adiabatic if T issuch that the regime where (cid:15)T (cid:28) (cid:126) Ω( t )(red lines) on the Bloch sphere as well as the adiabaticprediction (straight blue line) for different (cid:15) and T (up-per row T = 5, lower row T = 10). As long as Eq. (16)is fulfilled the adiabatic prediction agrees with the timeevolved (cid:126) Ω( t ) at large times. In the regime where Eq. (16)is violated the vector (cid:126) Ω( t ) processes around the adiabaticprediction with constant radius r = ∆ x = ∆ y , where ∆ x and ∆ y are the deviations from the adiabatic predictionin the two directions orthogonal to it (see far upper rightpanel in Fig. 6). For the distribution of the quasi-modes(in Nambu space) f ( (cid:15) ) which fulfills f ( (cid:15) ) = (cid:126) Ω we can thus use the adiabatically deformed ensemble for all en-ergies, which fulfill Eq. (16). This is shown in the bottomrow of Fig. 6. The three dimensional plot shows the dis-tribution function in the perfectly adiabatically deformedensemble ( T → ∞ ) as a blue line in dependency of (cid:15) compared to the result of the adiabatic deformation atfinite T ( T = 1 left and T = 5 right) as red lines. Thedeviations from the adiabatic description ∆ x and ∆ y (asdefined in the rightmost plot of the first row of Fig. 6)are shown as circles around their mean value.Next, we consider the limit T → ∞ , which means thatthe adiabatically deformed ensemble description is validfor any energy scale (at finite T valid at energy scales (cid:15) (cid:29) /T ). We then calculate the frequency ω dependentdirty limit optical conductivity σ ( ω ) by replacing the en-ergy argument in the Fermi functions of Eqs. (3.9) and(3.10) in Ref. 41 by the corresponding adiabatic deformedone ( (cid:15) → ± (cid:112) (cid:15) − ∆ ). In Fig. 7(a) we show the real partof the optical conductivity calculated this way for initialtemperature T = 0 . T (black toyellow lines). None of the thermal curves can be usedto reproduce the adiabatically deformed one. The non-thermal additional in-gap content arises due to the factthat more energy is placed in the low energy modes com-pared to a thermal ensemble after deformation. The insetdemonstrates that the superfluid stiffness A is zero (blueline). The multi-colored line shows the equilibrium re-sult. This means that the deformed BCS theory predictsan optical conductivity similar to the thermal one (withquantitatively changed line shape), but without the char-acteristic delta distribution (superfluid stiffness) contri-bution at ω = 0, which supports the super-current re-sponse characteristic to a superconductor. In Fig. 7(b)we show the optical conductivity for the adiabaticallydeformed ensemble for additional initial temperatures.At very low temperature the strong divergence found inthe conductivity for the adiabatically deformed ensem-ble mimics the behavior of the delta-distribution in thethermal case. Adiabatic passage thus leads to a quanti-tatively non-thermal behavior of the optical conductivityin a BCS superconductor strictly lacking superfluid stiff-ness.As a final example we also consider the case wherea time dependent interaction U ( t ) is given and the gap∆( t ) is determined by the self-consistent equation ∆( t ) = U ( t ) (cid:88) k f ( k ) T r (cid:2) τ − R † (cid:15) ( t ) τ R (cid:15) ( t ) (cid:3) . (17)We concentrate on a three dimensional model of a cubiclattice with nearest neighbor hopping. Thus we study afeatureless semi-elliptic density of states and set units bychoosing the bandwidth to be 4. We choose a tempera-ture of T = 0 . U ( t ) following U + A (cid:20) erf (cid:18) t − δσ (cid:19) + erf (cid:18) δσ (cid:19)(cid:21) (18)Therefore σ controls the speed v ∼ /σ of the interactionramp, where σ → ∞ is the adiabatic limit. Representa-tive results are summarized in Fig. 8. The upper panelshows the interaction ramp for U = − . A = − . δ = 5 σ . The lower panel shows the self-consistentlydetermined gap for different values of σ . The oscillatorybehavior reminiscent of the abrupt quench, decrease inheight as σ is increased. The oscillations are a conse-quence of the residual precession of the spin analyzed inFig. 6 at large times, if the process is not sufficiently adi-abatic. For increasing σ the oscillations vanish and thesteady value is well described by the prediction of theadiabatically deformed ensemble (red line), while a ther-mal description of the system does not yield convincingresults (blue dashed line). The adiabatically deformedensemble thus provides a significantly simpler route tothe steady state than performing the transient time evo-lution. IV. MAJORANA CHAIN
As a final example we study the topological Kitaevchain H = L (cid:88) j =1 (cid:18) − c ( t ) c † j c j +1 + M c j c j +1 − m ( t )2 c † j c j + H . c . (cid:19) . (19)We choose periodic boundary conditions and for conve-nience | M | = 1. Using Nambu vectors Ψ k = ( c k , c †− k ) T the Hamiltonian can be written as H = (cid:80) k Ψ † k H k Ψ k t/σ − . − . − . − . − . U ( t ) t/σ . . . . . . . ∆ ( t ) σ = σ = σ = FIG. 8. Time evolution of the gap ∆( t ) (bottom panel) forsolving the self-consistent equation Eq. (17) given the timedependent interaction U ( t ) (top panel). σ controls the adia-baticity of the U ( t ) ramp. As σ → ∞ the asymptotic behaviorof ∆( t ) is correctly described by the adiabatically deformedensemble (red solid line), while a thermal description fails(blue dashed line). with H k = ∆ k n k · σ and n k = 2∆ k (0 , − sin k, − m ( t ) + c ( t ) cos k ) (20)∆ k = 2 (cid:113) ( − m + c cos k ) + sin k, (21)where σ = ( σ x , σ y , σ z ) T and σ i the Pauli matrices. Weconcentrate on adiabatic deformations in the regime c >m , which at T = 0 displays topological order and theBerry phase arg [cos( πω )] is π , with ω = 12 π (cid:73) (cid:32) ∂ k n ik n jk (cid:33) dk i (cid:54) = j. (22)To probe the topological properties for mixed stateswe use the generalization of the Berry phase, Φ U = arg (cid:40) cos( πω ) cos (cid:34)(cid:73) (cid:32) ∂ k n ik n jk (cid:33) sech (cid:18) ∆ k T (cid:19) dk (cid:35)(cid:41) . (23)Like the Berry phase at T = 0, Φ U = 0 (Φ U = π ) in thetrivial (topological) phase with a sharp transition sep-arating these two phase at T = T c . As we exclusivelyconcentrate on cases that show a finite gap ∆ k , the evo-lution must be slow on the scale of the gap v (cid:28) min(∆ k )to ensure adiabatic deformation. Adiabatic deformation0 FIG. 9. Critical temperature obtained from adiabaticallyvarying m ( t ) from m ini to m fin at fixed c ( t ) = 1. The thickdiagonal line indicates the cut c ini = c fin , where we recoverthe equilibrium result. The lower contour plot shows T c − T eq c ,where T eq c is the equilibrium critical temperature w.r.t. c fin on a diverging color map (white, blue, red colors indicate achange of 0, negative, positive magnitude, respectively). from a set ( c ini , m ini ) → ( c fin , m fin ) modifies Φ U toΦ U = arg (cid:26) cos( πω ( c fin , m fin )) × cos (cid:34)(cid:73) (cid:32) ∂ k n ik ( c fin , m fin )2 n jk ( c fin , m fin ) (cid:33) sech (cid:18) ∆ k ( c ini , m ini )2 T (cid:19) dk (cid:35) (cid:27) . (24)Therefore, for the adiabatically deformed ensemble thepreviously thermal distribution condensed in the argu-ment ∆ k T of the sech in Eq. (23) has to replaced by a sechwith a different (non-thermal) argument ∆ k ( c ini ,m ini )2 T .This non-thermal distribution in turn modifies T c . Re-sults for the critical temperature T c when adiabaticallyvarying m ( t ) are summarized in Fig. 9. Adiabatic defor-mation thus opens a route to alter the critical temper-ature, not due to cooling, but due to the non-thermaldistribution function (condensed in the argument of the sech) in Eq. (24). V. SUMMARY
We propose adiabatic deformation of (thermal) ensem-bles to engineer non-thermal states of matter exhibitinglargely unexplored physics. The speed with which thisadiabatic passage has to be undertaken depends on thesystem under scrutiny. A case by case study should beconducted to study, whether an adiabatically deformedensemble can be achieved before residual couplings to theenvironment spoil the closed-ness of the quantum systemand drive it back into a thermal state. We performedfour case studies for simple but important systems: (a)a single oscillator, (b) a dimerized finite one-dimensionalchain of spinless fermions, (c) a BCS superconductorwith time dependent gap-function as well as (d) a Ma-jorana chain. All show that the adiabatically deformedensemble harbors interesting physics inaccessible by ther-mal pathways. For the dimerized chain we show that longranged correlations can be achieved. This could opena route to study Luttinger liquid behavior in systemsfor which accessing low temperatures is difficult. Forthe BCS-type superconductor we investigated the opti-cal conductivity, one of the experimentally most relevantobservables. We find a clearly non-thermal behavior mostprominently reflected in a line shape qualitatively similarto the thermal one but lacking a superfluid stiffness. Fora topological system we find that due to the deformationthe critical temperature at which the topological phaseis lost can be increased by the non-thermal nature of thedistribution function after deformation. Future researchshould address whether the designed non-thermal statespresented here can harbor unknown (hidden) phases in-accessible by thermal equilibrium.
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