Time-dependent Real-space Renormalization-Group Approach: application to an adiabatic random quantum Ising model
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov Time-dependent Real-space Renormalization-GroupApproach: application to an adiabatic randomquantum Ising model
Peter Mason, Alexandre M. Zagoskin and Joseph J. Betouras
Department of Physics, Loughborough University, Loughborough, LE11 3TU, UnitedKingdom
Abstract.
We develop a time-dependent real-space renormalization-group approachwhich can be applied to Hamiltonians with time-dependent random terms. To illustratethe renormalization-group analysis, we focus on the quantum Ising Hamiltonian withrandom site- and time-dependent (adiabatic) transverse-field and nearest-neighbourexchange couplings. We demonstrate how the method works in detail, by calculatingthe off-critical flows and recovering the ground state properties of the Hamiltonian suchas magnetization and correlation functions. The adiabatic time allows us to traversethe parameter space, remaining near-to the ground state which is broadened if therate of change of the Hamiltonian is finite. The quantum critical point, or points,depend on time through the time-dependence of the parameters of the Hamiltonian.We, furthermore, make connections with Kibble-Zurek dynamics and provide a scalingargument for the density of defects as we adiabatically pass through the critical pointof the system.Corresponding authors; P. [email protected], J. [email protected]. ime-dependent RG on an adiabatic random quantum Ising model I. Introduction
An interacting quantum system evolving at zero temperature can demonstrate variousforms of approach to equilibrium even with no loss of phase coherence. In the past,detailed experimental [1] and theoretical studies [2, 3, 4, 5, 6, 7] for systems prepared bya quantum quench across a phase transition have studied this kind of out-of-equilibriumpath. A system is prepared in the ground state for certain values of parameters,which are then rapidly changed to values for which the ground state (GS) is knownto be in a different phase. Experiments on ultracold atoms are particularly usefulto probe this physics because they are essentially closed quantum systems on ratherlong time scales compared to the basic dynamical time scales of the system. As such,fundamental questions as to whether many physical properties equilibrate after thequench exponentially in time and the system thermalizes (so it can be described by aneffective temperature) can be addressed.Apart from instant quenches of an interacting quantum system, there is muchattention on those Hamiltonians which change smoothly across a quantum phasetransition [8, 9, 10, 11, 12]. In general a single tunable parameter, or perturbation, ischosen to address this transition, and the non-equilibrium dynamics such as the growthof density defects or entanglement entropy is then studied as a function of this parameter.For a second-order transition critical point, various averaged physical quantities suchas the excitation density and energy show power laws as the rate of change approacheszero, with exponents determined by the universal physics of the quantum critical point.The system evolves after such a sweep to a steady state for some quantities, but itsenergy distribution remains nonthermal [11].Quantum spin chains provide prime examples of the interacting quantum systemswhere the evolution of out-of-equilibrium many-body systems [13], that possess manydegrees of freedom, is theoretically illustrated. The addition of disorder in the systemprovides another dimension which needs to be addressed. The properties of quantumspin chains with quenched randomness at low temperatures have been the subject ofinterest for a number of years. In the case of a one-dimensional transverse-field quantumIsing model with randomness, a renormalization-group (RG) treatment shows that thedisorder grows without bound [14, 15, 16] - the flow is towards an infinite-randomnessfixed point. This strong-disorder renormalization-group approach has seen furthersuccessful progress and development to tackle issues related to many-body localisationin interacting models with disorder [17, 18], such as establishing the form of the growthof the entropy in an XXZ spin-chain [19, 20], as well as establishing the physics aroundthe phase transition between the many-body localised phase and the thermal phase fora general interacting model [21, 22, 23, 24, 25], and investigating Floquet dynamics inperiodically driven random quantum spin chains [26, 27].In the present work, we address the problem of a random quantum spin chain,requiring control of more than one parameter as we pass through a quantum criticalpoint. We will as such concentrate on an adiabatic quantum Hamiltonian. This is ime-dependent RG on an adiabatic random quantum Ising model H ( t ) = H initial ( t ) + H final ( t ) , (1)subject to initial conditions that H final (0) = 0 and final conditions that H initial ( T ) = 0,where T is the end-point of the computation. The ground, and easily reachable, stateof the initial Hamiltonian, H initial , is assumed known, whereas the ground state of thefinal Hamiltonian at t = T , H final , is unknown and encodes the desired solution to thecomputation. The adiabatic evolution of the Hamiltonian is such that the computationalways remains in, or close to, the local ground state at some given instant in time.Crucially, the computation will therefore terminate (ideally in) the ground state of H final .The inclusion of adiabatic parameters means that the energy gap between theground state and first excited state is now necessarily time-dependent, such that itcan vary depending on the value of time (i.e. because of the quantities H initial ( t ) and H final ( t )) in the computation. As such, remaining in the ground state, or removing anyexcited states, requires the adiabatic process to progress sufficiently slowly. However, ifthe quantum Hamiltonian exhibits a quantum critical point, the energy gap will, at thispoint, decrease to zero, and the computation will lose adiabaticity. For the quantumcomputation therefore, the transition through the quantum critical point will lead tothe development of excited states, or defects, the density of which depend on the rateof transition.We address the issue of a generic time dependence in the quantum Hamiltonian(Eq. (1)), focussing on developing a complete analysis of the state of the systemat any given time, as well as the dynamics across a given quantum critical point.The time-dependence (as we will see) provides a tool to shift in time the locationof the quantum critical point. It also allows us, through the choice of the adiabaticparameters, to increase (to greater than one) the number of critical points. This can beparticularly important in the development of a device to simulate the quantum criticalpoint: Optimisation problems commonly in the NP-complete and NP-hard classes canbe mapped onto an Ising model [28, 29] (indeed the architecture of, for example theD:Wave machines [30, 31], is designed to use a quantum adiabatic protocol, and runbased on an adiabatic quantum Ising Hamiltonian). As such we will concentrate ouranalysis on an adiabatic Ising Hamiltonian with random transverse-fields and exchangecouplings, but simplify the setup to consider an infinitely long one-dimensional chainof Ising spins. Such 1D spin chains governed by a quantum Ising Hamiltonian can beexperimentally realised as flux qubits; see for instance [32, 33].The strong-disorder RG approach that we develop is the time-dependent extensionof that formulated in detail for the time-independent transverse-field Ising Hamiltonianby D. S. Fisher in a series of papers in the 1990’s [14, 15, 16]. The work was basedon the perturbative technique developed by Dasgupta and Ma [34] that safely allowsfor the systematic removal of the high-energy degrees of freedom on the spin-1 / ime-dependent RG on an adiabatic random quantum Ising model II. The Real-Space Renormalization-Group Approach
In this section we present the analysis of the adiabatic Ising Hamiltonian in the strong-disorder limit. We concentrate on a renormalization-group (RG) approach and discussthe role that time plays as well as the underlying conditions of the analysis. Then,we proceed by writing down the time-dependent coupled integro-differential RG flowequations (in terms of the distributions functions of the exchange and site couplings)that govern the distribution of the random variables in our model. These flow equationsare then solved explicitly.
A. The Adiabatic Random Quantum Ising Hamiltonian
To illustrate our approach to the development of a time-dependent strong-disorder RGscheme, we study the nearest neighbour time-dependent random transverse-field IsingHamiltonian H ( t ) = − N X i =1 h i ( t ) σ xi − N X J ij ( t ) σ zi σ zj (2)on the one-dimensional infinite chain, where the parameters h i ( t ) and J ij ( t ) aretime-dependent random variables and denote the transverse fields and the nearest- ime-dependent RG on an adiabatic random quantum Ising model J ij (0) = 0 , h i ( T ) = 0 , (3)so that the evolution runs for times t ∈ [0 , T ], with beginning- and end-pointHamiltonians given by H initial ≡ H ( t = 0) = − N X i h i (0) σ xi , (4a) H final ≡ H ( t = T ) = − N X J ij ( T ) σ zi σ zj , (4b)respectively.The governing Hamiltonian, Equation (2), is termed the adiabatic random quantumIsing model (ARQIM). This is to be contrasted to the non-adiabatic random quantumIsing Hamiltonian, considered by, amongst others, Fisher [14], that can be thought of asa special case of the model studied here, and occurs when h i ( t ) and J ij ( t ) are no longerfunctions of time. A further special case of the ARQIM occurs when t = T , at whichpoint we obtain the classical Ising model (that is identical to the end-point Hamiltonian, H final , given above).A transformation can be performed on the ARQIM by taking s zi = Y j ≤ i σ xj , s xi = σ zi σ zi +1 , (5)such that, on exchange of the h i ( t ) and J ij ( t ), the ARQIM is recovered again: thisidentifies a duality transformation h i ( t ) ←→ J ij ( t ), a feature that will be used in theRG analysis later in this section. In analogy with the non-adiabatic Hamiltonian, subjectto certain conditions that we discuss below, the ARQIM contains a quantum criticalpoint which relates the magnitudes of the site and bond couplings, h i ( t ) or J ij ( t ). Fornow, there are no restrictions on these relative magnitudes, however at a later stage,the limit around, and the dynamics across, the critical point δ = 0, will be studied.An expression for δ has been obtained in the non-adiabatic case in [14]; we modify thisexpression to include the adiabatic parameters, to get δ ≈ ∆ h ( t ) − ∆ J ( t ) Var(log( J ( t ))) + Var(log( h ( t ))) , (6)where ∆ h ( t ) = log( h ( t )), ∆ J ( t ) = log( J ( t )), and where J ( t ) and h ( t ) are the sets thatcontain the J ij ( t ) and h i ( t ). The critical point is at δ = 0 (note that our analysis includesthe possibility of greater than one critical point in the time internal [0 , T ]), which occurswhen ∆ h ( t ) = ∆ J ( t ) . If δ > δ <
0, the ground state will beferromagnetic. Either side of the critical point rare regions effects are important; these ime-dependent RG on an adiabatic random quantum Ising model δ both paramagneticand ferromagnetic rare region effects are lost [38, 39, 40]. B. The Role of Time
At this stage it is appropriate to discuss the role of time in the renormalization procedure.We will allow for an explicit time dependence in the distributions of the J ( t ) ′ s and h ( t ) ′ s .In general, the renormalization leads to the low-energy fixed point ground state solution,that is either paramagnetic ( δ >
0) or ferromagnetic ( δ < h i ( t ) = B i ( t )˜ h i and J ij ( t ) = A ij ( t ) ˜ J ij , the time-dependence of h i ( t ) and J ij ( t ) can be separated out. Then ˜ h i and ˜ J ij are strictly time-independent independentrandom distributions, and A ij ( t ) and B i ( t ) are (adiabatic) bond/site-dependent non-random, hence controllable, functions. The form of the time-dependent functions A ij ( t )and B i ( t ) are, at this stage, left entirely general and independent, and for the majorityof our analysis we will work with the functions h i ( t ) and J ij ( t ), i.e. we will not assumethat the time dependence can be separated out from the random variables. However, inorder to clarify the RG procedure or the physics, we will at various stages separate outthe time dependence.In the case that we can separate out the time-dependence, Eq. (6) can be writtenas δ ≈ ∆ ˜ h − ∆ ˜ J − ∆ t Var(log( ˜ J )) + Var(log(˜ h )) , (7)where ∆ ˜ h = log(˜ h ), ∆ ˜ J = log( ˜ J ) and ∆ t = log( A ( t ) /B ( t )) (we note that we considera disorder average and that A ij ( t ) = A ( t ) and B i ( t ) = B ( t ) are non-random functionsof time as will be considered explicitly below), and ˜ J , ˜ h , are the sets that containthe ˜ J ij , ˜ h i respectively. In particular, ∆ t can take positive or negative values (note that∆ t ∈ [ −∞ , ∞ ]). The inclusion of a time dependence in the Hamiltonian gives rise to the∆ t parameter, the value of which (either positive or negative) can influence the groundstate properties of the system, and leads to the possibility of multiple, independent,critical points (dependent on the values of A ij ( t ) and B i ( t )). Schematically, the criticalline as a function of ∆ ˜ h − ∆ ˜ J and ∆ t is presented in Fig. 1. ime-dependent RG on an adiabatic random quantum Ising model ParamagneticGriffiths Phase δ = 0 Strongly FerromagneticFerromagneticGriffiths PhaseStrongly Paramagnetic ∆ t ∆ ˜ h − ∆ ˜ J Figure 1.
The critical line, δ = 0 separating the paramagnetic ( δ >
0) from theferromagnetic ( δ <
0) region. f T00.20.40.60.81 QCP B ( t ) A ( t ) t t T01 A ( t ) B ( t ) t t Figure 2.
Schematic time-flow under the supposition that ∆ ˜ h = ∆ ˜ J throughout. (a) A ( t ) = A ij ( t ) and B ( t ) = B i ( t ) ∀ i, j . There is a quantum critical point (QCP) whichoccurs when t = t f and δ = 0, here indicated at the point when A ( t f ) = B ( t f ). Theground state is paramagnetic for t < t f and ferromagnetic for t > t f . A duality intime exists across the critical point, as indicated by the double sided arrows. (b) The A ij ( t ) and B i ( t ) are taken to be in general distinct. There exist multiple (here three)quantum critical points when ∆ t = 0, at t = t , t and t . In Fig. 2 we give two schematic viewpoints of the time-flow under the suppositionthat ∆ ˜ h = ∆ ˜ J throughout. Figure 2(a) further supposes that A ( t ) = A ij ( t ) and B ( t ) = B i ( t ) ∀ i, j (here we choose A ( t ) = 1 − B ( t ) = ( t/T ) , but in general theycan take any form). There then exists a single quantum critical point at t = t f , where A ( t ) = B ( t ), such that for δ > t < t f ) the ground state is paramagnetic,whereas for δ < t > t f ) the ground state is ferromagnetic. In contrastFig. 2(b) takes the A ij ( t ) and B i ( t ) to be in general distinct, and shows the existenceof multiple independent quantum critical points, all located at ∆ t = 0. We note thatif ∆ ˜ h = ∆ ˜ J , then a quantum critical point can still exist: for instance if ∆ ˜ h = ˜ α ∆ ˜ J ,for some constant ˜ α , then the quantum critical point exists when ∆ t = ( ˜ α − ˜ J , orequivalently A ( t ) = B ( t ) exp(( ˜ α − ˜ J ). C. Renormalization-Group Formulation
The analysis that we formulate in this paper for the ARQIM will concentrate on astrong-disorder RG approach on the joint distribution functions p ( t, J ( t ) , l S ( t ); Ω( t ))and r ( t, h ( t ) , l B ( t ); Ω( t )), where l B ( t ) is the bond length scale and l S ( t ) is the site ime-dependent RG on an adiabatic random quantum Ising model t ) is introduced[14, 17], defined as the maximum of the bond and coupling strengths, i.e. Ω( t ) =max ij ( J ij ( t ) , h i ( t )). The RG analysis proceeds by perturbatively removing the high-energy degrees of freedom, thus reducing the energy scale. The RG flows will thusprovide the appropriate low-energy physical quantities of interest; after many RG stepsthe effective distribution functions read ˆ p ( t, ˆ J ( t ) , ˆ l S ( t ); Ω( t )) and ˆ r ( t, ˆ h ( t ) , ˆ l B ( t ); Ω( t )),with the ‘hat’ notation indicating that renormalization has taken place.Specifically, there are two cases to consider when analysing the governingHamiltonian: Ω( t ) takes either an element from J ( t ) or an element from h ( t ). Atthis stage a rescaling [14, 17] of the random variables is made, through the introductionof the following time-dependent functionsΓ( t ) = ln (cid:18) Ω I ( t )Ω( t ) (cid:19) , (8a) ζ ( t ) = ln Ω( t )ˆ J ( t ) ! , (8b) β ( t ) = ln Ω( t )ˆ h ( t ) ! , (8c)with Ω I ( t ) defined as the initial maximum bond or coupling strength. After eachrenormalization step we will, in general, find Ω( t ) < Ω I ( t ), so that Γ( t ) will becomea large parameter in the problem, representing the decrease in the energy scale. Notethat ζ ( t ) and β ( t ) are strictly non-negative, defined on the interval [0 , ∞ ).In the case of the problem considered in this work, the rescaled joint distributionfunctions ˆ p ( t, ˆ J ( t ) , ˆ l ( t ); Ω( t )) and ˆ r ( t, ˆ h ( t ) , ˆ l ( t ); Ω( t )) become ˆ P ( t, ˆ ζ ( t ) , ˆ l ( t ); Γ( t )) andˆ R ( t, ˆ β ( t ) , ˆ l ( t ); Γ( t )). We wish to find the form of these distribution functions for alltime. Given that the RG perturbation steps remain valid (see appendix), we can writedown coupled master equations that describe the renormalization flow. To begin, wenote that we are interested in the adiabatic limit of time evolution. As such we canconsider a perturbation in time to time-independent master equations. The solutionto these equations provides the form of the distribution functions for the off-criticalflow. The integro-differential coupled flow equations were originally written down byFisher [14, 17]. To form these equations, we must combine two effects. The first isthe change in either ζ or β as a result of the renormalisation that causes Γ to increase(equivalently Ω to decrease). The second effect comes from the decimation of either abond or site and the recombining of a new site or bond. Putting these together, alongwith a normalisation term gives usˆ P Γ − ˆ P ζ − ˆ R ˆ P ∗ ˆ P − (cid:16) ˆ P − ˆ R (cid:17) ˆ P = 0 , (9a)ˆ R Γ − ˆ R β − ˆ P ˆ R ∗ ˆ R − (cid:16) ˆ R − ˆ P (cid:17) ˆ R = 0 , (9b)where subscripts Γ, ζ and β indicate partial differentiation with respect to that variable,‘ ∗ ’ indicates a convolution, and ˆ P ≡ ˆ P | ζ ( t )=0 and ˆ R ≡ ˆ R | β ( t )=0 . ime-dependent RG on an adiabatic random quantum Ising model P ≡ ˆ P Γ − ˆ P ζ − ˆ R ˆ P ∗ ˆ P − (cid:16) ˆ P − ˆ R (cid:17) ˆ P , (10a)
R ≡ ˆ R Γ − ˆ R β − ˆ P ˆ R ∗ ˆ R − (cid:16) ˆ R − ˆ P (cid:17) ˆ R, (10b)so that, under the assumption that P → P + d ( P ) dt P , R → R + d ( R ) dt R , (11a)where d ( . ) /dt is the total derivative given by d ( P ) dt = ∂∂t + ˙ ζ ∂∂ζ , d ( R ) dt = ∂∂t + ˙ β ∂∂β , (12a)we obtain our set of coupled integro-differential flow equations for the time-dependentdistribution functions ˆ P and ˆ R as ∂∂t P + ˙ ζ ∂∂ζ P = 0 , ∂∂t R + ˙ β ∂∂β R = 0 . (13a)They govern the RG flows of the distribution functions for the two independent randomvariables. Their solutions provide the required information about the time-dependentcritical and off-critical flows.To solve these equations we suppose that the distribution functions ˆ P and ˆ R are split into two terms, a leading-order term that does not contain an explicit timedependence (rather the time comes parametrically through the ζ ( t ), β ( t ), l ( t ) and Γ( t )terms) and a small correction term that contains an explicit time dependence: i.e.ˆ P ( t, ζ ( t ) , l ( t ); Γ( t )) = ˆ P ( ζ ( t ) , l ( t ); Γ( t ))+ ˆ P ( t, ζ ( t ) , l ( t ); Γ( t )) and ˆ R ( t, β ( t ) , l ( t ); Γ( t )) =ˆ R ( β ( t ) , l ( t ); Γ( t )) + ˆ R ( t, β ( t ) , l ( t ); Γ( t )), with ˆ P and ˆ R small. Plugging these intoEquations (13) to leading order we obtainˆ P − ˆ P ζ − ˆ R ˆ P ∗ ˆ P − (cid:16) ˆ P − ˆ R (cid:17) ˆ P = 0 , (14a)ˆ R − ˆ R β − ˆ P ˆ R ∗ ˆ R − (cid:16) ˆ R − ˆ P (cid:17) ˆ R = 0 , (14b)where ˆ P ≡ ˆ P | ζ ( t )=0 and ˆ R ≡ ˆ R | β ( t )=0 . To first order we obtainˆ P − ˆ P ζ − R ˆ P ∗ ˆ P − ˆ R ˆ P ∗ ˆ P − (cid:16) ˆ P − ˆ R (cid:17) ˆ P − (cid:16) ˆ P − ˆ R (cid:17) ˆ P = ˆ g P , (15a)ˆ R − ˆ R β − P ˆ R ∗ ˆ R − ˆ P ˆ R ∗ ˆ R − (cid:16) ˆ R − ˆ P (cid:17) ˆ R − (cid:16) ˆ R − ˆ P (cid:17) ˆ R = ˆ g R , (15b)where ˆ P ≡ ˆ P | ζ ( t )=0 and ˆ R ≡ ˆ R | β ( t )=0 and where ˆ g P ( ζ ) and ˆ g R ( β ) are integrationconstants. ime-dependent RG on an adiabatic random quantum Ising model P and ˆ R , P ( t, ζ ( t ) , y ( t ); Γ( t )) = Z ∞ e − y ( t ) l ( t ) ˆ P ( t, ζ ( t ) , l ( t ); Γ( t )) dl ( t ) , (16a) R ( t, β ( t ) , y ( t ); Γ( t )) = Z ∞ e − y ( t ) l ( t ) ˆ R ( t, β ( t ) , l ( t ); Γ( t )) dl ( t ) , (16b)to get P − P ζ − R P ∗ P − (cid:0) P − R (cid:1) P = 0 , (17a) R − R β − P R ∗ R − (cid:0) R − P (cid:1) R = 0 , (17b)where P ≡ P (0 , y ( t )), R ≡ R (0 , y ( t )), P ≡ P (0 , R ≡ R (0 , P − P ζ − R P ∗ P − R P ∗ P − (cid:0) P − R (cid:1) P − (cid:0) P − R (cid:1) P = g P , (18a) R − R β − P R ∗ R − P R ∗ R − (cid:0) R − P (cid:1) R − (cid:0) R − P (cid:1) R = g R , (18b)where P ≡ P (0 , y ( t )), R ≡ R (0 , y ( t )), P ≡ P (0 , R ≡ R (0 , g P and g R are the Laplace transforms of ˆ g P and ˆ g R respectively.We can now work with the (small) inverse lengthscale y ( t ), instead of the (large)lengthscale l ( t ). At y ( t ) = 0 we have the normalisation conditions Z ∞ P | y =0 dζ = 1 , Z ∞ R | y =0 dβ = 1 . (19)We now suppose that P = P + P and R = R + R and make the ansatzes [14] that P = Y e − ζu , P = f P ˙ ζY e − ζu , (20a) R = S e − βv , R = f R ˙ βS e − βv , (20b)for unknown (to be determined) functions Y ( y ( t ); Γ( t )), Y ( y ( t ); Γ( t )), S ( y ( t ); Γ( t )), S ( y ( t ); Γ( t )), u ( y ( t ); Γ( t )), u ( y ( t ); Γ( t )), v ( y ( t ); Γ( t )), v ( y ( t ); Γ( t )), f P ( t ) and f R ( t ).Denoting Y ( y ( t ) = 0) ≡ Y and S ( y ( t ) = 0) ≡ S , to leading order we obtain u = − Y S , (21a) v = − Y S , (21b) Y = (cid:0) Y − S − u (cid:1) Y , (21c) S = (cid:0) S − Y − v (cid:1) S . (21d)The solution to these coupled equations follows directly (they closely follow the solutionspresented in [14], however the normalisation conditions lead to a slight change) as u ( y ( t ); Γ( t )) = − δ ( y ( t )) + ∆ ( y ( t )) coth (cid:2) (Γ( t ) + C ( y ( t ))) ∆ ( y ( t )) (cid:3) , (22a) v ( y ( t ); Γ( t )) = δ ( y ( t )) + ∆ ( y ( t )) coth (cid:2) (Γ( t ) + C ( y ( t ))) ∆ ( y ( t )) (cid:3) , (22b) Y ( y ( t ); Γ( t )) = ∆ ( y ( t ))sinh [(Γ( t ) + C ( y ( t ))) ∆ ( y ( t ))] e − D ( y ( t )) − [ S − Y − δ ( y ( t )) ] Γ( t ) , (22c) S ( y ( t ); Γ( t )) = ∆ ( y ( t ))sinh [(Γ( t ) + C ( y ( t ))) ∆ ( y ( t ))] e D ( y ( t ))+ [ S − Y − δ ( y ( t )) ] Γ( t ) , (22d) ime-dependent RG on an adiabatic random quantum Ising model δ ( y ( t )), ∆ ( y ( t )) = p y + δ ( y ( t )) , C ( y ( t )) and D ( y ( t )).For the first-order equations (18) we make the assumptions that u = u ≡ u and v = v ≡ v . This allows us to write down a further four coupled equations u Γ = f R f P S Y (cid:0) Y (cid:1) − Y S − g P Y , (23a) v Γ = f P f R Y S (cid:0) S (cid:1) − Y S − g R S , (23b) Y = (cid:0) Y − S − u (cid:1) Y + (cid:18) Y + f R f P S (cid:19) Y , (23c) S = (cid:0) S − Y − v (cid:1) S − (cid:18) S + f P f R Y (cid:19) S . (23d)Now, from the leading order solutions, we know that S = f e ¯ α and Y = f e − ¯ α , where f ( y ( t ); Γ( t )) = ∆ ( y ( t ))sinh [(Γ( t ) + C ( y ( t ))) ∆ ( y ( t ))] , (24a)¯ α ( y ( t ); Γ( t )) = D ( y ( t )) + (cid:2) S − Y − δ ( y ( t )) (cid:3) Γ( t ) . (24b)As such, we suppose that S = ge ¯ α and Y = ge − ¯ α , for function g ( y ( t ); Γ( t )) to be found.To make progress we examine the form of the solution at t = 0, and note that underour assumptions, the forms of Eq.’s (21a) and (23a) and Eq.’s (21b) and (23b) mustcorrespond. To ensure this correspondance we have that f P = f R and g P = g R = 0. Assuch Eq.’s (23) reduce to u Γ = S Y (cid:0) Y (cid:1) − Y S , (25a) v Γ = Y S (cid:0) S (cid:1) − Y S , (25b) Y = (cid:0) Y − S − u (cid:1) Y + (cid:0) Y + S (cid:1) Y , (25c) S = (cid:0) S − Y − v (cid:1) S − (cid:0) S + Y (cid:1) S . (25d)Substitution of the forms for S and Y and using the leading order solutions for theother functions, leads to a single second-order differential equation in g that can bereadily solved; g ΓΓ − g g = gf . (26)Using the form for f above we obtain g ( y ( t ); Γ( t )) = ∆ ( y ( t )) e − D ( y ( t ))Γ( t ) sinh [(Γ( t ) + C ( y ( t ))) ∆ ( y ( t ))] , (27)for constants of integration ∆ ( y ( t )) and D ( y ( t )).We are thus left with six constants of integration, namely δ ( y ( t )), ∆ ( y ( t )), C ( y ( t )), D ( y ( t )), ∆ ( y ( t )) and D ( y ( t )). The normalisation conditions (19) reduce to Y + f P ˙ ζ Y = u , (28a) S + f R ˙ βS = v , (28b) ime-dependent RG on an adiabatic random quantum Ising model u ( y ( t ) = 0) ≡ u and v ( y ( t ) = 0) ≡ v . These in turn lead to δ ( y ( t ) = 0) =∆ ( y ( t ) = 0), D ( y ( t ) = 0) = ∆ ( y ( t ) = 0) C ( y ( t ) = 0), S − Y = 2 δ ( y ( t ) = 0) and∆ ( y ( t ) = 0) = 0. It is this last condition that contains all the information on theadiabatic transfer. We recall that we began the analysis, at t = 0, by writing downa perturbation in time to an off-critical flow (about which we know the full physics[14]). We have then proceeded to find the form of the distribution functions at a given t >
0. These distribution functions contain leading-order terms ( P and R ) and explicittime-dependent correction terms ( P and R ), that are dependent on the rates ˙ ζ , ˙ β .To ensure that we remain in the vicinity of the ground state ( P , R smallperturbations), we impose that the rates ˙ ζ and ˙ β are small. These then become ouradiabatic conditions. If at any given time (say t = t ) we stop the time-flow and allowthe RG scheme to take over ( y → P and R →
0, as we haveshown above (∆ → P = P and R = R , both evaluated at t = t , i.e. they are the off-critical flows in the low-energyand long-length scales that have already been determined by Fisher [14]. This is notsurprising: time acts on our Hamiltonian (Eq. 2) as a simple rescaling of the site- andbond-coupling strengths. During our adiabatic transfer we would not expect to attainthese low-energy and long-length scales for all times. Those times where we do attainthese scales will correspond to the already found off-critical flows, whereas those timeswhere we do not attain these scales will correspond to a (small) error away from thelowest energy (ground) state. Therefore we expect that ∆ ( y ( t )) ∼ cy , in the limit y small, for some constant c .As the off-critical flows are fundamentally related to those of [14], we are ableto state that C ( y ( t )) = C and δ ( y ( t )) = δ (both constants), and D ( y ( t )) = 0 and D ( y ( t )) = 0. Therefore for large Γ, small y and small δ , we obtain u ( y ( t ); Γ( t )) = − δ + ∆ ( y ( t )) coth (cid:2) (Γ( t ) + C ) ∆ ( y ( t )) (cid:3) , (29a) v ( y ( t ); Γ( t )) = δ + ∆ ( y ( t )) coth (cid:2) (Γ( t ) + C ) ∆ ( y ( t )) (cid:3) , (29b) Y ( y ( t ); Γ( t )) = ∆ ( y ( t ))sinh [(Γ( t ) + C ) ∆ ( y ( t ))] e − δ Γ( t ) , (29c) S ( y ( t ); Γ( t )) = ∆ ( y ( t ))sinh [(Γ( t ) + C ) ∆ ( y ( t ))] e δ Γ( t ) , (29d) Y ( y ( t ); Γ( t )) = ∆ ( y ( t ))sinh [(Γ( t ) + C ) ∆ ( y ( t ))] e − δ Γ( t ) , (29e) S ( y ( t ); Γ( t )) = ∆ ( y ( t ))sinh [(Γ( t ) + C ) ∆ ( y ( t ))] e δ Γ( t ) . (29f)In terms of our joint distribution functions P and R , we therefore have P = e − ( ζu + δ Γ) sinh [(Γ + C ) ∆ ] (cid:16) ∆ + ˙ ζ ∆ (cid:17) , (30a) R = e − ( βv − δ Γ) sinh [(Γ + C ) ∆ ] (cid:16) ∆ + ˙ β ∆ (cid:17) . (30b)These solutions for the correction to the off-critical flows in the adiabatic time-limit ime-dependent RG on an adiabatic random quantum Ising model ζ and ˙ β are small) are the main results of our paper. The form for δ has beenwritten down in Eq. (6) and provides a measure of the ‘distance’ from the critical point. III. Transition Through the Quantum Critical Point
Up to now we have concentrated on the ground, or equilibrium, state properties ofthe time-dependent transverse-field Ising model, using the time as a weighting on therandom bond and site couplings. The inclusion of a time in the RG analysis provides theequilibrium critical exponents such as the location of the critical point, magnetisationand correlation length scalings, all parametrised in time. In this section we are interestedin the non-equilibrium dynamics that occur as we time-evolve the governing Hamiltonian(Eq. (2)) through a critical point. As noted earlier, our choice of an adiabatic timeevolution is deliberately made in order to remain in the local (in-time) ground statethroughout. We thus do not expect any defects - or excited states - to be generated,except in the vicinity of the critical point where the energy gap between the groundstate and first excited state will decrease and disappear exactly at the critical point.This is precisely where the computation loses adiabaticity.It turns out that the analytic scaling for the defect density (a non-equilibriumdynamics) is dependent on knowledge of the equilibrium critical exponents [9, 41]. Wewill assume for the transition dynamics that we can separate out the time functions fromthe random variables. Thus we can, by using the equilibrium scalings for the correlationlengths ξ ( t ), the distance from the critical point δ , and the form of the time functions A ij ( t ) and B i ( t ), find an estimate for the density of defects (or the number of domainwalls) that are formed as we transfer through this non-adiabatic region.To begin, we therefore consider a scaling analysis using the forms of the criticalparameter δ (Eq. (6)), critical correlation length ξ , given by ξ = 1 | δ | ν , (31)with ν = 2, and the dynamical z exponent, given by z ∼ | δ | + ∆ , (32)all in the region of the critical point. The energy gap is defined as ∆ gap ∼ | δ | zν ∼| δ | / [ | δ | +∆ ] . We are interested in the region close to the critical point, specifically theregion at which the computation loses adiabaticity. This occurs at a critical ˆ δ whenthe energy gap ∆ gap is equal to the rate of approach towards the critical point, definedas ∆ rate . To find ∆ rate we note that we must separately take account of the rate oftransition of both of the coupling strengths, J ij ( t ) and h i ( t ). From these, we expect tobe able to write down a characteristic rate 1 /τ as1 τ = 1 τ A ( t ) + 1 τ B ( t ) , (33) ime-dependent RG on an adiabatic random quantum Ising model /τ A ( t ) and 1 /τ B ( t ) represent the characteristic rates of the evolution of the bondand site couplings, respectively. These rates are found through the derivatives of thetime functions A ( t ) and B ( t ) evaluated at t f , the time when the system passes throughthe critical point. Then τ A ( t f ) = 1 / | ˙ A ( t f ) | and τ B ( t f ) = 1 / | ˙ B ( t f ) | . For example, if A ( t ) = ( t/T ) and B ( t ) = 1 − ( t/T ) (as in Fig. 2, where we assume ∆ ˜ h = ∆ ˜ J ), then t f = T / √ τ A ( t f ) = τ B ( t f ) = T / √ τ = T / (2 √ /τ introduces a natural energy scale on the adiabatic evolution of the RG scheme: we candefine a critical Γ, defined by Γ τ ∼ ln(Ω τ ). We must impose that Γ < Γ τ in order forthe RG scheme to be valid in the evolution of the system.Close to the critical point δ can be also linearised in time, such that δ ( t f ) ∼ t f /τ ,which gives ∆ rate = ( τ δ ) − . Thus, we can now find δ c as the solution to aτ δ c = δ / [ | δ c | +∆ ] c , (34)where a is a free parameter. Now, ∆ = p y + δ c , so we are left with two options:either we expand about small δ c , or we expand about small y . In the first case, usingthe result for δ c , together with the critical correlation length (Eq. (31)), we find that ξ c ≡ ξ | δ c approximates to ξ c ≈ log [log( τ /a )] − log( τ /a ) √ y log( τ /a ) − (cid:0) √ y + 2 (cid:1) log [log( τ /a )] ! ≈ log ( τ /a )log [log( τ /a )] , (35)since y is small, and therefore can be safely neglected. In the second case when weexpand about small y , we have ∆ ∼ δ c , in which case we revert back to the analysis of[9, 35], the result of which is identical to the right hand side of Eq. (35). It is importantto note that the choice of functions A ij ( t ) and B i ( t ) enters the above result throughthe characteristic rate 1 /τ ; in particular we can take A ij ( t ) and B i ( t ) to be nonlinearfunctions of t/T , as has been considered in the non-random transverse-field Ising model[42, 43, 44].For this adiabatic transition through the critical point, the logarithmic scaling re-flects the departure from the typical Kibble-Zurek ( ∝ τ / ) scaling that is evidenced in apure (non-random), or weakly disordered, Ising model. This is the same scaling as foundanalytically by [9] and numerically by [35], however our analysis, through the inclusionof the time-dependent parameters A ij ( t ) and B i ( t ) into the governing Hamiltonian, hasestablished a generic characteristic timescale that represents a transition through thecritical point that is influenced by both the bond strengths, J ij ( t ), and the site strengths, h i ( t ). IV. Discussion and Conclusion
In this work, we have developed and extended the well-established real-space RGapproach for quantum chains with randomness [14, 15, 16] to the case of time-dependent ime-dependent RG on an adiabatic random quantum Ising model N = 2. In particularthe site and bond coupling terms can be assumed to be independent and random, andcrucially individually modifiable. In the future we would expect that the length of thequbit chain could be increased to a few thousand qubits. This would put experiment inthe N -large limit where the conclusions of this paper could be tested.The RG approach that we introduce in this paper can be further extended toconsider non-integrable spin-chain models, such as the XXZ or next-nearest neighbourHamiltonians [19, 20, 22, 23] with random interactions. These out-of-equilibrium modelsadmit a many-body localization phase transition that is currently the focus of muchinterest (see for example [23, 24]). The inclusion of time-dependent functions into theseHamiltonians, along the lines of the adiabatic functions A ( t ) and B ( t ) considered in thispaper, will allow us to systematically probe (some of the) properties of this transitionin further detail. For example, we expect to be able to understand the identified fractalthermal Griffiths regions [24].The quantity that will enable the characterization of different phases in manydifferent settings is the entanglement entropy. In simpler cases [45], such as out-of-equilibrium steady states of the quantum Ising model, or when perturbations break theintegrability, critical boundaries were characterised by the value of the central charge.Following Ref.’s [46, 47], we expect to be able to calculate the distribution of theentanglement entropy for the ARQIM using the real-space RG technique developedin this paper, but in non-integrable spin models, how the time-dependence affects the ime-dependent RG on an adiabatic random quantum Ising model Acknowledgments
We would like to thank Alexander Balanov, Claudio Castelnovo, John Chalker, AndreyChubukov, Mike Gunn, Vedika Khemani, David Pekker, Anatoli Polkovnikov and ArturSowa for insightful discussions that took place during this work. The work was supportedby EPSRC through the grant EP/M006581/1.
Appendix
In this appendix we provide further details regarding the renormalization-group analysisof Sect. II. We start with Eq. (2), noting that in the first case when the largest couplingis a site, we have Ω( t ) = h i ( t ) for some i . The local Hamiltonian for this site and thetwo adjoining bonds is then H si − ,i +1 ( t ) = − h i ( t ) σ xi − J i − ,i ( t ) σ zi − σ zi − J i,i +1 ( t ) σ zi σ zi +1 , foradjoining bonds J i − ,i ( t ) and J i,i +1 ( t ). Under the RG procedure, these adjoining bondscan be treated perturbatively resulting in an effective Hamiltonian for H si,i +1 ( t ), writtenas ˆ H si − ,i +1 ( t ) ≈ − ˆ J i − ,i +1 ( t ) σ zi − σ zi +1 , (36a)where ˆ J i − ,i +1 ( t ) = J i − ,i ( t ) J i,i +1 ( t ) h i ( t ) . (36b)This is site decimation: a site, h i ( t ), and the two adjoining bonds, J i − ,i ( t ) and J i,i +1 ( t ),are removed and a new bond, ˆ J i − ,i +1 ( t ) created, joining sites h i − ( t ) and h i +1 ( t ). Thenew bond lengthscale under this decimation becomes ˆ l Bi − ,i +1 = l Bi − ,i + l Bi,i +1 + l Si . As in[14], we set each initial l Si and l Bi,i +1 equal to 1/2.In the second case when the largest coupling is a bond, we have Ω( t ) = J i,i +1 ( t )for some i . The local Hamiltonian for this bond and two adjoining sites is then H bi,i +1 ( t ) = − h i ( t ) σ xi − h i +1 ( t ) σ xi +1 − J i,i +1 ( t ) σ zi σ zi +1 , for adjoining sites h i ( t ) and h i +1 ( t ).Under the RG procedure, these adjoining sites can be treated similarly, resulting in aneffective Hamiltonian for H bi,i +1 ( t ), written asˆ H bi,i +1 ( t ) ≈ − ˆ h i ( t ) σ xi , (37a)where ˆ h i ( t ) = h i ( t ) h i +1 ( t ) J i,i +1 ( t ) . (37b)This is bond decimation: a bond, J i,i +1 ( t ), and the two adjoining sites, h i ( t ) and h i +1 ( t ), are removed and a new spin cluster, ˆ h i ( t ) created, with adjoining bonds J i − ,i ( t ) and J i +1 ,i +2 ( t ). The new spin cluster lengthscale under this decimation becomesˆ l Si = l Si + l Si +1 + l Bi,i +1 . ime-dependent RG on an adiabatic random quantum Ising model J i − ,i +1 ( t ) is small ( < ǫ )for a bond decimation and that ˆ h i ( t ) is small for a site decimation. However, thisstipulation can be somewhat relaxed: errors that result in the initial few steps of therenormalization from invalidity of the perturbation are gradually smoothed away as thenumber of RG steps increases such that the RG flows become asymptotically valid inthe low-energy, long-distance limit.In the case when we separate out the time functions the criteria for validity ofthe RG scheme becomes slightly more intricate. To see this, first note that the aboveexpressions for ˆ J i − ,i +1 ( t ) and ˆ h i ( t ) refer to the distribution functions after a singlerenormalization step of either bond or site decimation. In general, and after many (bondand site) renormalization steps, a site decimation will give the ˆ J i − ,i +1 ( t ) as (explicitlyseparating out the time dependence):ˆ J i − ,i +ˆ l − / ( t ) = Q k = i +ˆ l − / k = i − A k,k +1 ( t ) ˜ J k,k +1 Q k = i +ˆ l − / k = i B k ( t )˜ h k , (38a)while a bond decimation will give the ˆ h i ( t ) asˆ h i ( t ) = Q k = i +ˆ l − / k = i B k ( t )˜ h k Q k = i +ˆ l − / k = i A k,k +1 ( t ) ˜ J k,k +1 (38b)where ˆ l = n + 1 /
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