Quantum transitions driven by one-bond defects in quantum Ising rings

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by  Massimo Campostrini

Quantum transitions driven by one-bond defects in quantum Ising rings
Massimo Campostrini,1 Andrea Pelissetto,2 and Ettore Vicari1
1 Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Largo Pontecorvo 3, I-56127 Pisa, Italy and
2 Dipartimento di Fisica di “Sapienza” Universita` di Roma and INFN, Sezione di Roma I, I-00185 Roma, Italy
(Dated: January 15, 2015)
Weinvestigatequantumscalingphenomenadrivenbylower-dimensionaldefectsinquantumIsing-
likemodels. WeconsiderquantumIsingringsinthepresenceofabonddefect. Intheorderedphase,
5
the system undergoes a quantum transition driven by the bond defect between a magnet phase, in
1
which the gap decreases exponentially with increasing size, and a kink phase, in which the gap
0
decreases instead with a power of the size. Close to the transition, the system shows a universal
2
scaling behavior, which we characterize by computing, either analytically or numerically, scaling

n functions for the gap, the susceptibility, and the two-point correlation function. We discuss the
a implications of these results for the nonequilibrium dynamics in the presence of a slowly-varying
J
parallel magnetic field h, when going across the first-orderquantumtransition at h=0.

4
1 PACSnumbers: 05.30.Rt,05.70.Jk,64.70.qj,64.60.an

]
h
Quantumphasetransitions[1]arephenomenaofgreat A continuous transition occurs at g = 1, separating a
c
e interest in many different branches of physics. They disordered (g > 1) from an ordered (g < 1) phase [1].
m ariseinmany-bodysystemsinthepresenceofcompeting In the presence ofan additionalparallelmagnetic field h
- ground states. The driving parameters of the transition coupledtoσ(1),afirst-orderquantumtransition(FOQT)
at are usually bulk quantities, such as the chemical poten- occursath=i 0foranyg <1,henceweexpectthedefect
t tial in particle systems, or external magnetic fields in to be able to change bulk behavior for any g < 1. This
s
. spin systems. In the presence of first-order transitions, is the regime we shall consider below.
t
a bulk behavior is particularly sensitive to the boundary Analytic and accurate numerical results can be ob-
m
conditions or to localized defects, hence it is possible to tained by exploiting the equivalent quadratic fermionic
- induceaquantumcriticaltransitionbychangingonlythe Hamiltonian which is obtained by a Jordan-Wigner
d
parametersassociatedwiththedefectsortheboundaries. transformation[6,7]. Weanalyzethedependenceoflow-
n
o In this paper, we discuss an example of this type of energy properties on the defect parameter ζ [8]. In par-
c transitions, considering a quantum Ising ring in a trans- ticular, we consider the energy differences
[ verse magnetic field with a bond defect. In the ordered

1 phase,bulkbehaviorofthelow-energystatesdependson ∆L,n ≡En−E0, ∆L ≡∆L,1, (2)
v the defect coupling. One may have a magnet phase, in
where E is the energy of the ground state, and E
5 which the gap decreases exponentially, i.e., ∆ e−cL 0 n≥1
6 L ∼ are the (ordered) energies of the excited levels. The
with increasing the size L, or a kink phase, in which the
2 lowest states are one-kink states and ∆ 1/Lp. Here, magnetization σx(1) vanishes due to the global Z2 sym-
3 L ∼ metry. Thus, whe usie the two-point correlation function
0 we analyze the crossover region between these phases,
. showing the emergence of a universal scaling behavior G(x,y) σx(1)σy(1) to characterize the magnetic prop-
1 ≡ h i
controlled by the defect coupling. We also analyze the erties of the ground state.
0
5 slow nonequilibrium adiabatic dynamics [2, 3] across For g < 1, we should distinguish a magnet phase
1 this transition. We obtain general time-dependent scal- (ζ > 1)andakinkphase(ζ 1). Theloweststatesof
− ≤−
v: ing laws that generalize to the first-order transition case themagnetphasearesuperpositionsofstateswithoppo-
i those that characterize the Kibble-Zurek (KZ) mecha- site nonzero magnetization σx(1) = m0 (neglect-
X h±| |±i ±
nism at continuous transitions [3–5]. inglocaleffectsatthedefect),where[9]m0 =(1 g2)1/8.
ar We consider Ising rings of size L=2ℓ+1 in the pres- For a finite chain, tunneling effects between th−e states
enceofatransversemagneticfieldandofonebonddefect: + and lift the degeneracy, giving rise to an ex-
| i |−i
ponentially small gap ∆ [10, 11]. For example, [9]
L
ℓ−1 ℓ ∆ 2(1 g2)gL for ζ = 0 (OBC). An analytic cal-
L
Hr =−J X σi(1)σi(+1)1−g X σi(3)−ζ σ−(1ℓ)σℓ(1), (1) culat≈ion giv−es
i=−ℓ i=−ℓ
8g 1 g
∆ w2e−wL, w = − (1+ζ), (3)
L
where σ(a) are the Pauli matrices. We set J = 1, and ≈ 1 g g
i −
assume g ≥ 0. Note that periodic (PBC), open (OBC), for ζ 1+. The large-L two-point function is trivial,
and antiperiodic (ABC) boundary conditions are recov- →−
ered for ζ =1, 0, and 1, respectively. The bond defect G(x ,x )
1 2
− G (x ,x ) 1 (4)
generally breaks translation invariance, unless ζ = 1. r 1 2 ≡ m2 →
± 0
2
2
for x =x , keeping X x /ℓ fixed (but X = 1).
1 6 2 i ≡ i i 6 ± L=201 g=3/4
The low-energy behavior drastically changes for ζ L=801
1, in which the low-energy states are one-kink state≤s L=201 g=1/2
− L=801
(made of a nearest-neighbor pair of antiparallel spins), L=201 g=1/4
which behave as one-particle states with O(L−1) mo- ∆ (ζ) / ∆(ζ) L=801
L,2 L c
menta [1]. In particular, for ζ = 1 (ABC) we have
− 1
g π2 ∆(ζ) / ∆(ζ)
∆ = +O(L−4). (5) L L c
L 1 g L2
−
The first two excited states are degenerate, thus ∆ =
L,2
∆ ∆ . For ζ < 1, the ground state and the first
L,1 L
≡ −
excited state are superpositions with definite parity of 0−10 −5 0 5 10
ζ
the lowest kink and antikink states. The gap
| ↓↑i | ↑↓i s
scales as [11, 12] L−3; we obtain explicitly
FIG.1: (Coloronline)WeshowthescalingfunctionsD (ζ ),
8ζg2 π2 n s
∆ = +O(L−4). (6) cf. Eq.(10),andnumericaldatafortheratio∆L,n(ζ)/∆L(ζc),
L (1 ζ2)(1 g)2 L3 for n=1 (bottom) and n=2 (top) separated by the dotted
− −
line. Numericaldataclearlyapproachtheg-independentscal-
On the other hand, ∆L,n for n≥2 behaves as L−2, e.g., ing curvesDn(ζs) (differences are hardly visible).
3g π2 6(1 ζ)g2 π2
∆ = + − +O(L−4). (7)
L,2 (1 g) L2 (1+ζ)(1 g)2 L3 they confirm the scaling behavior (10). The asymptotic
− −
large-Lbehaviorisgenerallyapproachedwithcorrections
The two-point function G(x,y) can be perturbatively
of order L−1.
computed for small g, obtaining the asymptotic large-L
Other observables satisfy analogous scaling relations.
behaviors
The two-point function is expected to behave as
G(x ,x )=1 X X for ζ = 1, (8)
1 2 −| 1− 2| sin(πX )− sin(πX ) G(x1,x2;ζ)≈m20 G(X1,X2;ζs), Xi =x/ℓ, (12)
1 2
G(x ,x )=1 X X | − | (9)
1 2 −| 1− 2|− π where m = (1 g2)1/8. This scaling ansatz can be
0
−
checked by considering the zero-momentum quantities
for ζ < 1, where X x /ℓ. We conjecture (and verify
i i
− ≡
numerically) that the above formulas can be straightfor- 1
wardly generalized to the whole ordered phase g < 1 by χ=XG(0,x), ξ2 = Xx2G(0,x) (13)
2χ
simply introducing a multiplicative renormalization,i.e., x x
by replacing G with G G/m2.
r ≡ 0 (ξ is the second-moment length scale). Eq. (12) implies
These results suggest that ζ = 1 is a critical point,
c
−
separating the magnet and kink phases. We now show χ/L m2f (ζ ), ξ/L f (ζ ). (14)
that around ζ the system develops a universal scaling ≈ 0 χ s ≈ ξ s
c
behavior. We analytically compute (and verify numeri- Numerical data confirm them, see Fig. 2. In the lan-
cally) the asymptotic behavior of ∆ (ζ), obtaining the guage of renormalization-group (RG) theory, the defect
L,n
scaling behavior coupling ζ plays the role of a relevant parameter at the
magnet-kink transition, with RG dimension y =1.
ζ
∆ (ζ) ∆ (ζ )D (ζ ), (10)
L,n L c n s An interesting question is whether there is a quantity
≈
1 g playing the role of order parameter for the magnet-kink
ζ = − (ζ ζ )L, ζ = 1, (11)
s c c
g − − transition. This is provided by the center-defect correla-
tion b = lim G(0,ℓ). Indeed, b > 0 for ζ > ζ and
for L keeping the scaling variable ζ fixed. The L→∞ c
scaling→fu∞nctions D1 and D2 are shown ins Fig. 1 [13]. hba=vio0rf[o1r4]ζG≤(0ζ,cℓ.)MorLe−o1vfer(,ζw)e,owbistehrvfe(the s)ca=lin0gabned-
The cusp-like behavior at ζs = 0 is the consequence of f ( )= , see Fig∼. 3. b s b −∞
the crossing of the first two excited states at ζ = 1. b ∞ ∞
− An analogous magnet-to-kink quantum transition can
Generally, the scaling functions are universal apart from
be observedin the Ising chain by appropriately tuning a
normalizations of their arguments. In this case, the nor-
magneticfieldη, coupledto σ(1), localizedatthe bound-
malization of ζ is chosen so that the scaling curves for
s aries. Explicitly, we consider (we assume η 0)
different values of g are identical. Notice that, once the ≥
normalizationisfixedbyusingoneobservable,universal-
ℓ−1 ℓ
istuyltsshfoourldthheoelnderfogryadniffyeoretnhceerso∆bservaabreles.hNowunmienriFcaigl.re1-: Hc =− X σi(1)σi(+1)1−g X σi(3)−η (σ−(1ℓ)−σℓ(1)). (15)
L,n i=−ℓ i=−ℓ
3
0.20 LL==5210,1 g=3/4 ηc is the same, i.e., the two transitions belong to the
L=801 same universality class. This is confirmed by analytic
L=51, g=1/2
L=101 andnumericalcomputations,althoughthecomparisonof
L=201
L=401 the results is not straightforward,as the Ising chain (15)
ξ / L0.15 LLLL====852801001,11 g=1/4 0.2 bthreeaIkssintgherinZg2(s1y)m. Fmoertreyx,awmhpilceh, tihseingsatpeasdatpisrfieseesrsvceadlibnyg
relations analogous to Eq. (10). Explicitly,
L
ξ /
2√1 g
0.1 g=1/2 ∆L,n(η)=∆L(ηc)En(ηs), ηs = − (ηc η)L,
g −
−0.05 0.00 0.05
0.10 ζ−ζ (18)
c with [15] E (x) = D (x) for x 0 and E (x) =
−40 −20 0 20 40 n 2n−1 ≥ n
ζ D (x) for x 0. The reason of the peculiar mapping
2n
s is related to t≤he different behavior under Z of the two
2
models. Consider for instance the kink phase. While in
FIG.2: (Coloronline)Estimatesoftheratioξ/L,supporting
the scaling ansatz (14). Scaling corrections are only visible theIsingringtheloweststatesaresuperpositionsofkink
for L . 100. The dashed lines indicate the values of f (ζ ) andantikinkconfigurations,inthecaseofmodel(15)the
ξ s
for ζs and ζs = 0, obtained by matching the scaling parity symmetry is broken by the boundary fields, thus
→ ±∞
ansatz with the behaviors in the different phases: fξ( ) = onlykinkstatesareleft. Hence,model(15)hasonlyhalf
1/√24 from Eq. (4), f ( ) 0.098491 from Eq. (9)∞, and
f (0) = 1/√48 from Eξq.−(∞8). ≈The inset shows the crossing of the states of the Ising ring. Moreover, no degener-
ξ acy occurs at η = η so that levels must be smooth at
point of data for different Limplied by Eq. (14). c
thetransitionpoint,therebyexplainingwhythemapping
5 betweenthelevelsmustchangeatthetransitionpoint(in
10 the ring case, cusps occur at the transition).
Itisworthnotingthat, atthecriticalvalueg =1,cor-
4
) responding to the order-disorder continuous transition,
(0,)Gl 3 (0,Gl borulokfbtehheavpiroerseinscinedoefpdenefdeecntts,ohfetnhceebtohuendmaargynceotn-tdoi-tkioinnks
L transition only occurs for g strictly less than 1. For in-
2 0 −0.04 ζ0−.0ζ0 0.04 stance, the gap at g =1 behaves as ∆L ∼L−1 for any ζ
c or η. Of course, the prefactor depends on the boundary
conditions; see, e.g., the known results for PBC, OBC,
1 L=51
L=101 and ABC, summarized in Ref. [16].
g=1/2 L=201
LL==480011 It is interesting to reinterpretour results in the equiv-
0
−10 −8 −6 −4 −2 0 2 4 6 alent fermionic picture of models (1) and (15). In the
ζ
s magnet phase, i.e., for ζ > ζc and η < ηc, respectively,
the lowest eigenstates are superpositions of Majorana
FIG. 3: (Color online) Scaling behavior of the center-defect fermionic states localized at the boundaries or on the
correlation G(0,ℓ). The data of LG(0,ℓ) approach a scaling
defect [17, 18]. In finite systems, their overlap does not
functionofζs. TheinsetshowsG(0,ℓ)asafunctionofζ−ζc. vanish, giving rise to the splitting ∆ e−L/l0. The co-
∼
herence length l diverges at the kink-to-magnet transi-
0
tions as l−1 lns η η and l−1 ζ ζ in the
The analytic computation of the low-energy spectrum 0 ∼ | | ∼ c − 0 ∼ − c
twomodels,abehavioranalogoustothatobservedatthe
identifies a particular value of the boundary field, η =
c order-disordertransition g 1− where l−1 lng .
√1 g, separating the magnet and kink phases. In the → 0 ∼| |
− Inconclusion,wehaveshownthatquantumtransitions
magnet phase η <η we have
c
can be induced by tuning the boundary conditions or by
2gs√s2 1 1 η2 changing lower-dimensionaldefect parameters,when the
∆ = − s−L+O(s−2L), s= − . (16)
L s g g system is at a FOQT. We have explicitly discussed this
− behavior in the case of quantum Ising rings in a trans-
For η η we have instead
c verse field. If g < 1, this model shows a magnet and
≥
a kink phase, separated by a quantum transition point.
g π2
∆ =c(η) +O(L−3) (17) In its neighborhood, we can define general scaling laws,
L 1 g L2
− thatareanalogoustothosethatholdatcontinuoustran-
with c = 1 for η = η and c = 3 for any η > η . The sitions. The same scaling behavior is also observed in
c c
similarnatureofthecoexistingphasessuggeststhattheir the XYquantumringinwhichoneaddsadditionalbond
asymptotic large-L scaling behavior for ζ ζ and η couplings σ(2)σ(2) [19], and in the quantum Ising chain
≈ c ≈ i i+1
4
with opposite magnetic fields at the boundaries. The (t/τ)1+z/yh and L/ξ = L(t/t )1/yh = (t/τ)1/yhL/τ1/z,
h 0
universal scaling behavior is essentially the same and is with τ =tz/(z+yh) =t2/5 [23]. These considerations lead
0 0
uniquely determined by the structure of the low-energy us to conjecture the scaling behavior for the Ising chain
behavior in the two phases. We have characterized the
scaling variable and computed the scaling functions of hσx(1)i≈m0fm(cid:0)x/L, t/τ, τ/L2, ζs(cid:1), (20)
different observables. These scaling behaviors can be σ(1)σ(1) m2f x /L, t/τ, τ/L2, ζ , (21)
straightforwardly extended to allow for a nonzero tem- h x1 x2 i≈ 0 g(cid:0) i s(cid:1)
perature T, by considering a further dependence on the withζsgivenbyEq.(11). Analogousexpressionsapplyto
scaling variable TLz = TL2. Even though we have dis- other observables and to the model (15). Of course, the
cussed the issue in one dimension, we expect the same above nonequilibrium scaling theory should be further
type of behavior in quantum d-dimensional Ising models investigated,to geta thoroughunderstanding of KZ-like
defined inLd−1 M boxeswith L M,in the presence phenomena at FOQTs.
of a (d 1)-dim×ensional surface of≫defects or of opposite These issues may also be relevant in the context of
magnet−ic fields on the Ld−1 boundaries. quantum computing. Adiabatic algorithms rely on suf-
Thesizedependenceofthelow-energyspectrumisrele- ficiently large gaps during the variation of the model
vantfortheunderstandingofthenonequilibriumunitary parameters bringing to the ground state of the desired
dynamics, as it determines the conditions for a nearly Hamiltonian [24–26]. Thus, FOQTs, at which the gap
adiabatic quantum dynamics [2, 3]. Significantly differ- is exponentially small,representa hardproblem[27–29].
entbehaviorsareexpectedinthemagnetandkinkphases Asasimpleparadigmaticcasewemayconsiderthetime-
when we add a time-dependent parallel magnetic term, dependent Hamiltonian (19) for g < 1 and ζ = 1. Let
us assume that we want to adiabatically move from the
Ht =H −h(t/t0)Xσi(1), h(u)=h0u, (19) ground state with h = −h0 at time t = −Ta to the
i ground state with h = h0 at t = Ta. This requires an
exponentially large time scale, i.e. T & e2L/l0. Our
where t is the time scale of the time dependence. Adi- a
0
results for the ζ-dependence of the low-energy proper-
abatic evolutions across the FOQT (h(0) = 0) require
ties suggest a way to overcome this hard problem. In-
very different time scales t . In the magnet phase we
0
deed, instead of changing directly h, one could proceed
must have t & e2L/l0, while in the kink phase t & L4
0 0
at ζ (η ) and t & L6 for ζ < ζ [t & L4 for η > η as follows. First, one adiabatically changes the system
c c 0 c 0 c
varying the bond defect from ζ = 1 to ζ . ζ , which
in model (15)]. The dynamics in the magnetphase is es- c
correspondstoaddingafurthersingle-bondHamiltonian
sentially equivalent to that of a two-level Landau-Zener
term H = ζ(t/T )σ(1)σ(1). In the presence of a finite
model with an exponentially small gap [20, 21]. At ζc ζ − ζ −ℓ ℓ
parallel magnetic field h = h , the gap is finite. Then,
for model (1) and η η for model (15), dynamics be- 0
c −
≥ oneadiabaticallychangesh,accordingtoEq.(19),which
comes similar to that at a continuous transition with a
now requires a time scale T &L4 or L6 to adiabatically
dynamic exponent z = 2, since there is a tower of ex- a
cited states with ∆ = O(L−2). Model (1) for ζ < ζ gofrom h0 toh0. Finally,ζ isincreasedagaintoζ =1,
L,n c −
obtaining the ground state of the original problem in a
shows again a low-energy dynamics dominated by the
two lowest states with a gap of order L−3. The above totaltime thatscaleswithapower,andnotwith the ex-
ponential, of the size. Therefore, by taking advantage of
results suggestthat nonequilibrium scaling laws, such as
those describing the KZ mechanism at continuous tran- particular sensitivity of FOQTs to defects or boundary
sitions[3–5](whensystemsarerampedacrossacontinu- perturbations,onemayovercometheproblemofanexpo-
oustransitionatafiniterate),alsoholdatFOQTs,when nentially slow dynamics, which occurs at FOQTs within
the time-dependent h crosses the value h =0. The slow the more standard approaches.
nonequilibrium dynamics of the Hamiltonian (19) across
theFOQTh=0,andinparticulartheinterplayamongt,
t ,ζ (orη)andL,thus∆ ,canbedescribedbyascaling
0 L
theory which extends the equilibrium scaling behaviors
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| | −