Table Of ContentEntanglement Hamiltonians for chiral fermions with zero modes
Israel Klich, Diana Vaman and Gabriel Wong
Department of Physics, University of Virginia, Box 400714, Charlottesville, Virginia 22904, USA and
Department of Physics, University of Virginia, Charlottesville, VA, USA
In this Letter, we study the effect of topological zero modes on Entanglement Hamiltonians and
entropy of free chiral fermions in (1+1)d. We show how Riemann-Hilbert solutions combined with
finiterankperturbationtheoryallowustoobtainexactexpressionsforEntanglementHamiltonians.
In the absence of the zero mode, the resulting Entanglement Hamiltonians consists of local and bi-
6 local terms. In the periodic sector, the presence of a zero mode leads to an additional non-local
1 contributiontotheentanglementHamiltonian. Wederiveanexactexpressionforthistermandfor
0 the resulting change in the entanglement entropy.
2
c
Entanglement Hamiltonians (EH) are the next object the edge theory of the Quantum Hall state. Wherever
e
D to explore in a series of advances in our understanding available we make contact with previous results in the
the quantum structure of many-body states in field the- literature.
5 ory and condensed matter [1–8]. Indeed, much work has Our approach, as we will soon review, starts with
1
been devoted to understanding entanglement entropy, in the relation between the EH and the resolvent associ-
] avarietyofsystems,and,subsequently,moredetailedin- ated with an appropriately projected (onto the entan-
h formation about the spectrum of reduced density matri- gling region) free fermion Green’s function, and adding
c
ces is being investigated as well. Entanglement Hamilto- the contribution of zero modes with an essentially exact
e
m niansgoonestepfurtherinthattheycontaininformation re-summation. Forconcretenessweconcentrateonchiral
about the entanglement spectrum, as well as about the MajoranaandDiracfermionsin(1+1)d,withthespatial
-
t associatedeigenvectors,andthepossibilitytounderstand directiontakentobeafiniteradiuscircle. Aswefindbe-
a
t the reduced state as a thermal state. Perhaps the most low,inthiscase,thecomputationoftheresolventcanbe
s
strikingoftherecentresultsaboutEntanglementHamil- recastintoaRiemann-Hilbertproblem(RHP,seebelow),
.
at tonians is the realization that for a spherical entangling generalizing the approach of Casini and Huerta [1] who
m region in a conformal field theory (CFT), Entanglement computed the EH for free chiral fermions, albeit without
- Hamiltonians have a local form, that may be interpreted zero modes. We combine the RHP approach with ex-
d as the original hamiltonian energy density with a spa- actlysummableperturbationtheory, tocomputetheEH
n
tially dependent temperature [9–11]. This result is un- for chiral Majorana and Dirac fermions on a finite circle.
o
usualinitssimplicity,as,unfortunately,thecomputation Asourmainresultwefindanexactanalyticformforthe
c
[ of EHs is, in general, much more involved than entropy entanglement Hamiltonian Eq. (16), with a contribution
and spectrum and only a few results are available. Thus from the topological zero given separately in Eq. (22).
3
v it is a grand challenge to find additional solvable cases. ThereduceddensitymatrixρV onaspatialregionV is
2 defined to reproduce expectation values of all operators
8 In this Letter, we present a new method for comput- O inside V via the relation Tr (ρ O )=(cid:104)O (cid:105). The
V V V V V
4 ing Entanglement Hamiltonians of free fermions in the EH H is an effective Hamiltonian inside V given by
0 V
presence of zero modes. Fermionic modes localized on
0
. topological defects often reflect the topological nature e−HV
1 of the defect through unusual properties such as charge ρV = Z . (1)
0 V
fractionalization [12] and non-abelian braiding [13] and
5
appear in a variety of systems from polyacetylene [14] For free fermions and in the absence of a ground state
1
: to defects in topological insulators and superconductors degeneracy (i.e. in the absence of zero modes), Wick’s
v
[15–17]. The modes are usually intimately tied to the Theorem implies that the EH is a quadratic operator
i
X appearance of ground state degeneracies of a topological whose kernel (the single particle EH) H is uniquely de-
V
r nature. Here, we are able to present new results for the termined by the equal-time Green’s function
a
EHsforchiralfermions, andstudyindetailtheeffectsof
theboundaryconditions(BC),periodic/anti-periodicfor G(x,y)=(cid:104)Ψ(x)Ψ†(y)(cid:105) (2)
MajoranaandgenericforDiracfermions,andofthezero
modes (present in the case of periodic boundary condi- via the relation H =−log((P GP )−1−1) [19]. Here
V V V
tions) on entanglement. The edge theory of the p+ip P GP istheGreen’sfunctionrestrictedtotheregionV
V V
superconductor provides an explicit realization of such a by the projectors P . As we demonstrate in the supple-
V
scenario for Majorana fermions and serves as a concrete mentary materials, a similar relation holds for Majorana
physicalmodelforourcalculation(see, e.g. [18]). Mean- fermions ψ, given (cid:104)ψ(x)ψ(y)(cid:105), allowing us to treat the
while, our calculation in the Dirac sector is relevant to problem on a parallel footing.
2
The relation between H and G may be expressed in
V
integral form as:
(cid:90) ∞
H =− (L(β)+L(−β))dβ, (3)
V
1
2
where L(β) is the resolvent [1]
L(β)=(P GP −1/2+β)−1. (4)
V V
Thus the derivation of the EH reduces to a computation
of a projected Green’s function resolvent. It turns out
that finding the resolvent L(β) can be recast as an RHP.
This connection between the resolvent and the RHP
arises in the following way: for free fermions (and in
the absence of zero modes), the Green’s function G(x,y)
computedonthegroundstate, un-restrictedtosomeen- FIG.1: TheentanglementregionV =∪(aj,bj)forfermionson
tangling region V, acts as a single particle operator that aringofradiusRconsideredinthepaper,andanassociated
Riemann-Hilbert problem.
projects onto positive energy modes. For chiral fermions
in (1+1)d, the projection onto positive modes can be
interpreted as a projection onto analytic functions on can be manifested in a sub-leading contribution to the
the upper/lower half of the complex plane. For such a EE of a finite interval. While these calculations are usu-
G(x,y), the computation of the resolvent (4) (associated ally carried out by looking at the entire system, or at
with the projection of G onto V) can be mapped onto a EE, our EH calculation allows us to study the effect of
RHP (see, e.g. [20]), which, in the simple case treated degeneracy from the point of view of the local density
here is a scalar RHP and so exactly solvable. Succinctly matrix. In Eq. (24), we find the exact change in the en-
put, a RHP is a jump problem along some smooth con- tanglement entropy (EE) of free chiral fermions between
tourC inthecomplexplane,forapiecewiseanalyticma- periodic and anti-periodic BC, for an arbitrary subset of
trix function. Solving it amounts to finding the matrix the circle, and the term in the EH associated with it.
function X(z) which is analytic on C(cid:31)C, and which has In the limiting case when the entangling region spans
boundary values on both sides of C, X±(z) subject to a the circle, this ”local boundary entropy” reproduces the
jump condition X−−1(s)X+(s)=v(s) for s∈C, and with boundary entropy.
a given matrix v(s). The jump function for our case is a Concretely, we consider the EH on a region V =
scalar determined by the projector onto V, PV. ∪(aj,bj) as depicted in Fig. 1. Consider a chiral Dirac
Our next idea is that once we have access to the resol- fermion in on a ring with periodicity x∼x+2πR. As-
vent (4) (either through the RHP or any other method), suming generic BC, Ψ(x+2πR)=Ψ(x)e2παi, the mode
wteemcathnautsyeieitldtosiemxpplleorpeevrtaurriobuastidonefsoromf tahteioGnsreoefnt’hsefusnycs-- expansion is Ψ(x,t)=√21πR(cid:80)k∈Z+αbke−iRk(t−x), where
tion (2). In particular, topologically nontrivial back- α ∈ [0,1). Here α = 1 for the NS sector and α = 0
2
ground for the fermion will result in the appearance of for the R sector. For α (cid:54)= 0, evaluating the equal time
zero modes [21, 22]. As we show below, these manifest Green’s function on the ground state |Ω(cid:105), defined by
themselves as finite rank perturbations of the correla- bk>0|Ω(cid:105)=b†k<0|Ω(cid:105)=0, gives
tion matrices. Given the resolvent for the unperturbed
Green’s function, we solve for the perturbed resolvent Gα(x,y)≡(cid:104)Ω|Ψ(x)Ψ†(y)|Ω(cid:105)=eiα(xR−y)n(x,y). (5)
by summing up exactly the resulting perturbative series.
wheren(x,y)≡(cid:104)x|n|y(cid:105)isthekernelofthesingleparticle
Here we carry out this recipe for the simplest scenario of projector n onto momentum modes (cid:104)x|k(cid:105) = √1 eikRx
a chiral Dirac/Majorana fermion on a spatial circle with 2πR
with k a non-negative integer [29]. Explicitly:
general boundary conditions.
Starting with anti-periodic (Neveu-Schwarz, NS) 1 (cid:88)∞ ik(x−y+i0+)
fermionsonacirclewithnogaugefield,insertingaπflux n(x,y)≡ e R . (6)
2πR
then takes the anti-periodic sector to the periodic (Ra- k=0
mond, R) sector, which has a zero mode whose effect on We note that the Gα for α(cid:54)=0 are related by spectral
theEHwederiveexplicitly. Notethattheappearanceof flow, implemented by a similarity transformation
ground state degeneracy for period boundary conditions
Gα = U nU−1, α(cid:54)=0, (7)
has been studied extensively. In particular, Affleck and α α
Ludwig [23], introduced the notion of boundary entropy, where U are unitary transformations which translate
α
to understand the change in degeneracy due to impuri- momenta by α: U |k(cid:105) = |k +α(cid:105). Moreover, U com-
α α
ties. It was also noted in [24] that the boundary entropy muteswithspatialprojectionP ,whichimpliesthatthe
V
3
projected resolvents (4) and EH kernel (3) for different where
α are also related by a similarity transform. Thus, we
can relate all the α(cid:54)=0 resolvents Lα =UαNUα−1 to the Z±(x)=ln(cid:89)(ei(xR±i(cid:15)) −eiRaj ). (14)
following one: j ei(xR±i(cid:15)) −eiRbj
(cid:18) (cid:19)−1
N(β)≡ P nP +β− 1 (8) This form is the finite radius generalization of the result
V V 2 obtained in [1] for fermions on an infinite line.
The EH in the α (cid:54)= 0 sector is obtained by applying
AsanoperatoractingonfunctionsdefinedonV,N(β)
(7) to compute the resolvent and then integrating over β
is the resolvent for P nP , which is an integrable opera-
V V as in (3). Adapting the procedure described in [1] to our
torinthesenseof[20,25]. Ingeneral,suchoperatorsare
case, we find:
defined on a curve C on a manifold conformally related
to the Riemann Sphere and have a kernel of the form HVα(cid:54)=0 =4π2eiα(x−Ry)nPV(x,y)δ(Z+(x)−Z+(y)), (15)
(cid:88)
K(x,y)= fi(x)n(x,y)gi(y), (9) where nPV(x,y) = n(x,y)− 21δ(x−y) is the principal
i part of n. Using the distribution kernel (15) to compute
explicitly the EH, gives, owing to the δ function, a local
where n(x,y) is a projector onto analytic functions ”in-
contribution
side” Σ, and f ,g are a set of given functions of Σ. It
i i
is well known that the resolvent, (1+K)−1, is related to (cid:90) (cid:20) i d (1−2α) (cid:16) i (cid:17)(cid:21)
Hα(cid:54)=0 =−2π dxΨ† − − (cid:48) Ψ(,16)
the solution of a matrix RH problem [20]. In our case V,loc. |Z(cid:48)|dx 2R|Z(cid:48)| 2|Z(cid:48)|
V
C =S1 on the compactified complex plane Fig. 1. Since
weonlyhaveasingleterminK,thematrixRHPreduces as well as a bi-local contribution with kernel:
tothescalaroneandcanbestatedasfollows. Definethe
function X(x) on S1 in terms of the position space ele- Hα(cid:54)=0 =2π (cid:88) eiα(x−Ry)|Z(cid:48)(x)|−1δ(x−y (x)), (17)
msceanlatrs RfHaPndagmbouynXts(xto) =fin1di+ng1/fu(βnc−tio12n)sf(Xx)g/(Xx). tThhaet V,bi-loc. l;yl(x)(cid:54)=xR(cid:0)1−ei(xR−y)(cid:1) l
+ −
are analytic inside/outside the unit disc satisfying where y (x) are solutions of Z(x)=Z(y).
l
Let us pause to inspect the single interval result for
X(x)=X−−1(x)X+(x), x∈S1. (10) α = 1. Since the free fermion is a conformal field the-
2
ory, the CFT results of [10, 11] suggest that the single
Equivalently, we can characterize X /X as containing
+ − interval EH should be an integral of the energy density
onlyk positive/negativemomentummodesrespectively.
Q ∝ iΨ†∂Ψ. At first glance, (16) seems at odds with
This leads to the crucial identities nX−1n = X−1n and
+ + this form, as it contains an additional piece proportional
nX−−1n=nX−−1, and to the resolvent formula [30]: to the number density ρ(x)=Ψ†(x)Ψ(x). On closer in-
spection, we see that the CFT expectation and the di-
(cid:18) (cid:19)
1 1
N(β)= 1− fX−1nX g (11) rect calculation are, in fact, consistent. Indeed, the in-
β− 1 β− 1 + −
2 2 terpretation of the CFT local energy tensor in [10, 11]
as an operator, has to be done with care: in particu-
For our application f = g = ΘV where ΘV is the char- lar, the form −i(cid:82) dxc(x)Ψ†∂Ψ, is not Hermitian by it-
acteristic function of V (Θ (x) = 1 if x ∈ V and is 0
V self. However, integrationbypartsshowsthatitismade
otherwise). Eq. (10) written as lnX(x) = ln(X (x))−
+ Hermitian with the addition of a term proportional to
ln(X−(x))=ln(ββ−+121)ΘV(x), has the standard solution an integral over the density (i/2)(cid:82) dxc(cid:48)(x)ρ(x), which
2
is exactly the extra term in Eq. (16) for α = 1/2. In
(cid:88) eRi(x∓i(cid:15))−eiaj the path integral formalism of [11], this term can be de-
lnX (x)=ih(β) ln , (12)
± j eRi(x∓i(cid:15))−eibj rmivaeldrobtyatiidoennstififyxiinnggtthheeEeHndaspothinetsgeonferVa,towrhoifchcomnfuosrt-
also generate a phase rotation of the Dirac fermion due
where we introduced the shorthand notation
to its non-trivial conformal spin. This symmetry gener-
1 β+ 1 ator is the hermitian form of the local energy operator
h(β)= 2π lnβ− 212. (13) givenbyT00 = −iΨ†(∂xΨ)2+i(∂xΨ†)Ψ. G(cid:82)iventhisdefinition,
our result (16) takes the form H = dxβ(x)T , with
V V 00
Substituting X± from (12) into (11), we find for the re- β(x)=2π|Z(cid:48)(x)|−1 =4πRcsca−bsina−xsinb−x as a lo-
2R 2R 2R
solvent: cal entanglement temperature. Turning on a flux then
introduces a shift of T by a conserved charge density
(cid:104)x|N(β)|y(cid:105)= β−11/2δ(x−y)− e−ih(Zβ+2(−x)−14Z+(y))n(x,y) −µρ(x)withchemicalp0o0tentialµ= (1−2R2α). Asshownin
4
[11], this leads to a generalized first law of EE in which to the NS. The EE S = −Trρ lnρ can be expressed
V V V
small excitations of the vacuum causes a change in EE as in integral form:
(cid:82)
given by δS = dxβ(x)δ(cid:104)(T −µρ(x))(cid:105) .
The RamVondVsector in wh0i0ch α = 0 requires sepa- SV ≡(cid:82)1∞dβTr[(β−1/2)(L(β)−L(−β))−β+21β/2] (23)
2
rate consideration, since the zero mode k=0 acts on
a doubly degenerate ground state subspace of occu- Itfollows,using(8),thatallDiracfermionswithα(cid:54)=0
pied/unoccupiedmodes. Wechoosethestatedictatedby havethesameEE(disregardingpossibleanomalycontri-
the zero-temperature limit of the Fermi-Dirac distribu- butions coming from the UV cutoff).
tion. For a topological zero mode, Fermi-Dirac gives the Using(23)wefindthecontributiontotheEEfromthe
mixed state 1(|occupied(cid:105)(cid:104)occupied| + |empty(cid:105)(cid:104)empty|). change in BC, δS ≡SR−SNS:
2
In this state [31], the Green’s function is δS = lv (cid:82)∞dhtanh(lvh)(coth(hπ)−1). (24)
2R 0 2R
1
Gα=0 = n− 2|0(cid:105)(cid:104)0| (18) Expanding in the ratio lv gives:
R
and our Resolvent is δS∼(cid:88)∞ (cid:0)22n−1(cid:1)B2nζ(2n)lv2n∼ lv2 − lv4 , (25)
1 1 2n(2πR)2n 48R2 5760R4
Lα=0(β) = (PVnPV − 2 +β− 2PV|0(cid:105)(cid:104)0|PV)−1(19) n=1
whereB areBernoullinumbers. Remarkably,usingthe
where |0(cid:105) is the k =0 momentum mode (cid:104)x|0(cid:105)= √1 . 2n
2πR replica trick, the authors of [26] have given the change
The difference relative to the case we have already
in the EE induced by BC change for a massless Dirac
solved is that the zero modes introduce a shift of the
fermionintheformoftheseries(25),andnotedthatthe
projectornbyarankoneperturbation−1|0(cid:105)(cid:104)0|. Wepro-
2 seriesisnotconvergent. Herewehaveobtainedtheexact
ceed by treating the zero mode contribution as a Dyson
expression Eq. (24) for the EE and can now identify the
perturbative expansion, which we subsequently re-sum
replica trick result of [26] as an asymptotic expansion
(see supp. material), leading to
of the convergent integral (24). Finally, we consider the
limiting case where V is the entire circle. Plugging l →
N(β)P |0(cid:105)(cid:104)0|P N(β) v
Lα=0(β)=N(β)+ V V . (20) 2πR into (24), we find δS = log2, as expected from the
2−(cid:104)0|P N(β)P |0(cid:105)
V V degeneracy [32].
We note in passing that Eq. (20) may be applied for Finally,wefindthatasimilarstoryholdsfortheMajo-
computing the EH for single particle excitations of the rana fermion. We derived a similar relation between the
vacuumandthatthere-summationalsoworksforhigher Green’s function GM(x,y)=(cid:104)ψ(x)ψ(y)(cid:105) and EH [33]:
rank perturbations.
1
While the first term on r.h.s of Eq. (20) will yield HM = ln((P G P )−1−1) (26)
V 2 V M V
a similar Hamiltonian as we considered up to now, the
second term on the r.h.s. is different: it is due to the R ThisrelationdiffersfromtheDiraconebytheimportant
sector zero mode and produces a non-local contribution factor of −1/2, while G (x,y) agrees with the Dirac
M
to the EH. Explicitly caseintheNSandRsector. Inaddition,sinceψ(x)2 isa
constant for Majoranas, we can ignore the (x−y) terms
(cid:104)x|LR (β)|y(cid:105)=2sinh2(πh)eih(Z(y)−Z(x)), (21) in the EH kernel. Thus one finds that in th.e NS sector,
zero-mode lvh
πR(1+e R )
the Majorana EH has the local term
(cid:80)
where l = (b −a ) is the total length of V. Using
v i i i (cid:90) i
(3), the contribution from LR to the EH is: HNS =π dx ψ(x)∂ ψ(x) (27)
zero-mode V,loc. Z(cid:48)(x) x
V
∞
HVRzero-mode=−R1 (cid:82) dh 1lvheih(Z(y)−Z(x)) (22) while the bilocal kernel (17) remains the same. In the
−∞ 1+e R R sector the situation is more subtle, as the Majorana
=(cid:80)l |Z(cid:48)−(xπ)|Rδ(x−yl(x))+p.v.2πliv sinh(cid:0)πR(Z1(y)−Z(x))(cid:1) zero mode b0 has no natural partner to create a com-
lv plex fermion. To get the minimal, non trivial Hilbert
Thus, the zero mode induced part of the EH has a non- space representation of the Clifford algebra containing
local contribution, even for a single interval. It is re- b , we have to introduce an additional anti-commuting
0
markable that for one interval the δ(x−y) term exactly operator. A physical realization of such a situation is
cancelsthe1/2shiftinα(the”chemicalpotential”term) the edge of a p + ip superconductor, in the presence
in (16), and we get that, as far as the strictly local terms of a (π flux) vortex in the bulk (see, e.g. [18]). Such
are concerned, H (α=0)=H (α=1/2). a vortex acts to change the BC on the edge into R
V,loc. V,loc.
Thenon-localcontributionstotheresolventduetothe type and binds a Majorana zero energy mode to its
zero mode also changes the EE in the R sector relative core. We can combine the Majorana operator b(cid:48) for
0
5
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0
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particle momentum states |k(cid:105) are states of energy E =
Acknowledgement: The work of IK was supported by k
k.Thereforeaprojectorontok≥0momentummodesis
the NSF CAREER grant DMR-0956053. DV and GW
also a projector onto states with E ≥0.
k
were supported in part by the U.S. Department of En- [30] See supplementary material for more details.
ergy under Grant No. de-sc0007984. GW would like to [31] The following also applies to the pure state
thankLeoPandoZayasandtheMichiganCenterforThe- 1(|occupied(cid:105)+|empty(cid:105).
2
oretical Physics for hospitality during the initial stages [32] The requirement of Wick’s theorem to hold in the pres-
ence of zero modes, forces the state on the full circle to
ofthisproject,JeanMarieStephan,JihnhoBaik,Jeffrey
be a mixed state, hence the nonzero entropy for the full
Teo for helpful discussions. GW was supported in part
circle in the R sector.
by the UVA presidential fellowship.
[33] Supplementary Online Material.
[34] G. Wong, I. Klich and D. Vaman, in preparation.
[1] H.CasiniandM.Huerta,Class.Quant.Grav.26,185005
(2009). SUPPLEMENTARY MATERIAL
[2] I. Peschel and M.-C. Chung, EPL (Europhysics Letters)
96, 50006 (2011).
[3] A.M.La¨uchliandJ.Schliemann,PhysicalReviewB85, Entanglement Hamiltonian for Majorana Fermions
054403 (2012).
[4] X.-L.Qi,H.Katsura,andA.W.Ludwig,Physicalreview
Consider a chiral Majorana fermion in (1+1)d with
letters 108, 196402 (2012).
Lagrangian iψ(∂ +∂ )ψ, where the spatial dimension
[5] M.Hermanns,Y.Salimi,M.Haque,andL.Fritz,Journal 2 t x
of Statistical Mechanics: Theory and Experiment 2014, is periodic x∼x+2πR. The canonical equal-time anti-
P10030 (2014). commutation relation is {ψ(x),ψ(y)}=δ(x−y), and we
6
require ψ†(x)=ψ(x). Given the mode expansion ground states, or their mixed state. However, in deriv-
ingtheEH,weassumestatesthatareuniquelydescribed
1 (cid:88) k
ψ(t,x)= √ bke−iR(t−x), (28) by the Green’s function through Wick’s theorem. Such
2πR statesmusthaveavanishingexpectationvaluesforprod-
k
ucts of an odd number of Majorana operators, and re-
with k ∈ Z for the R sector and k ∈ Z + 1 for the strict our choice.
2
NS sector, the ground state satisfies bk|Ω(cid:105) = 0 for k > This is ensured by taking states that preserve fermion
0. As a single particle operator, the equal-time Green’s parity. Fermion Parity (−1)F acts by (−1)Fψ(−1)F =
functionintheNSsectorisessentiallythesameasinthe −ψ. For states ρ that commute with (−1)F, we have:
Dirac case, i.e.:
−Tr(ρψ)=Tr(ρ(−1)Fψ(−1)F)=Tr(ρψ) (35)
GNS(x,y)≡(cid:104)Ω|ψ(x+i0+)ψ(y)|Ω(cid:105)≡ei(x2−Ry)n(x,y),(29) ⇒Tr(ρψ)=0.
where n(x,y)≡(cid:104)x|n|y(cid:105) denotes the same projector as in In the R sector, taking into account b0, we find
the Dirac case. ∞
Given the reduced density matrix ρV on a region V, GR(x,y)=2π1R(−12 +(cid:88)eik(x−Ry+i0+))=Gα=0(x,y).(36)
the single particle EH for Majorana fermions, HM, is k=0
V
defined by:
ρV =ZV−1e−HMV ; (30) The relation bHetiwlbeeernptrhoeblreemsolsvoelnuttiaonnd. the Riemann
HM =(cid:82) dxdyHM(x,y)ψ(x)ψ(y). (31)
V V V
Here we check that the expression (11) is indeed the
We now show that that HM is obtained from the Majo-
V inverse of K. For this, one checks that
rana Green’s function by:
1
1 fng−fX−1nX g− fngfX−1nX g =0,(37)
HM = ln((P GP )−1−1), (32) + − β−1/2 + −
V 2 V V
where for simplicity we restricted to the case of i = 1.
where P is a projector onto V, with (cid:104)x|P |y(cid:105) =
V V Here f,g,X are understood to be operators which
±
Θ (x)δ(x−y). Here Θ is the characteristic function
V V act multiplicatively in position space, with fg = (β −
for V ∈ S1, with ΘV(x) = 1 if x ∈ V and ΘV(x) = 0 1/2)(X−1X −1). Substituting into the last term in the
otherwise. WenotethatEEhasbeenstudiedextensively − +
previousexpressionleadstof(n−X−1nX −n(X−1X −
also in terms of Majorana fermions (see e.g. [27]), but + − − +
1)X−1nX )g = 0. This equality can be checked using
are not aware of a simple treatment of the EH as we + −
the identities
present below. To derive Eq. (32), we take the Majo-
rana operators to be defined on a discrete lattice, with X−1n=nX−1n, (38)
+ +
{ψ ,ψ } = δ , and compute the correlation functions
i j ij nX−1 =nX−1n, (39)
from the trace formula for paired states in [28] adapted − −
to the present notation: which follow from the fact, that by their definitions, X
+
contains only positive ks and therefore multiplication by
(cid:113)
TreAijψiψjeBklψkψl = det(1+e2Ae2B) (33) X+ can only increases momentum while, similarly, mul-
tiplication by X can only decrease momentum.
−
where A,B are antisymmetric matrices. We have:
Gij ≡(cid:104)ψiψj(cid:105)= 21(cid:0)δij +∂αlogTre−HMV eα(ψiψj−ψjψi)|α=0(cid:1) Rank one perturbation and resummation
(cid:113)
= 12δij + 21∂αlog det(1+e−2HVMe2α(|i(cid:105)(cid:104)j|−|j(cid:105)(cid:104)i|))|α=0 Consider a matrix A and a rank one perturbation B=
(cid:16) (cid:17) |s(cid:105)(cid:104)s|wecanformallyexpandinA−1B andthenre-sum:
= 1 , (34)
1+e−2HVM ij (A+B)−1 =(1+A−1B)−1A−1 (40)
wheretheantisymmetryofHM wasusedingoingtothe =A−1−A−1|s(cid:105)((cid:80) (−1)n(cid:104)s|A−1|s(cid:105)n)(cid:104)s|A−1
V n=0
last line. Inverting these relations gives (32).
=A−1− A−1|s(cid:105)(cid:104)s|A−1
1+(cid:104)s|A−1|s(cid:105)
Green’s function in the Ramond sector Substituting A = PVnPV −1/2+β and |s(cid:105) = √i2PV|0(cid:105)
then gives the desired resolvent
In evaluating the Green’s function in the R sector, N(β)P |0(cid:105)(cid:104)0|P N(β)
Lα=0(β)=N(β)+ V V . (41)
we are allowed to choose any linear combination of the 2−(cid:104)0|P N(β)P |0(cid:105)
V V
7
Written explicitly using the kernels of the operators
above the resolvent kernel is given by:
1 (cid:82) dzdz(cid:48)(cid:104)x|N(β)|z(cid:105)(cid:104)z(cid:48)|N(β)|y(cid:105)
(cid:104)x|LR (β)|y(cid:105)= V .
zero-mode 2πR 2− 1 (cid:82) dzdz(cid:48)(cid:104)z|N(β)|z(cid:48)(cid:105)
2πR V
