Majorana-time-reversal symmetries: a fundamental principle for sign-problem-free quantum Monte Carlo simulations

0.83 MB·

by  Zi-Xiang Li

Majorana-Time-Reversal Symmetries:
A Fundamental Principle for Sign-Problem-Free Quantum Monte Carlo Simulations
Zi-Xiang Li,1 Yi-Fan Jiang,1,2 and Hong Yao1,∗
1Institute for Advanced Study, Tsinghua University, Beijing 100084, China
2Department of Physics, Stanford University, Stanford, California 94305, USA
(Dated: December 6, 2016)
A fundamental open issue in physics is whether and how the fermion sign problem in quantum
Monte Carlo (QMC) simulations can be solved generically. Here, we show that Majorana-time-
reversal (MTR) symmetries can provide a unifying principle to solve the fermion sign problem in
interacting fermionic models. By systematically classifying Majorana-bilinear operators according
totheanti-commutingMTRsymmetriestheyrespect, werigorouslyprovedthattherearetwoand
6
only two fundamental symmetry classes which are sign-problem-free and which we call the “Ma-
1
jorana class” and “Kramers class”, respectively. Novel sign-problem-free models in the Majorana
0
classincludeinteractingtopologicalsuperconductorsandinteractingmodelsofcharge-4esupercon-
2
ductors. We believe that our MTR unifying principle could shed new light on sign-problem-free
c QMC simulation on strongly correlated systems and interacting topological matters.
e
D
4 Interactions between particles are ubiquitous, and Number of anti-commuting Sign-problem-free Sign-problematic
studying interacting models of many-body systems is MTR symmetries

] of central importance in modern condensed matter 0 none 𝐼 = no symmetry
l
-e pothhyesricfis[e1l–d3s]., Hqouwaenvtuerm, alcmhroosmt oadllyinnatmeriaccstin(QgCmDo)d,elsanidn 1 none {{𝑇𝑇11+−}}
str two and three dimensions, especially those with strong 2 {{𝑇𝑇11+−,,𝑇𝑇22−−}} == MKraamjorearns-ac-lcalsass s {𝑇1+,𝑇2+}
t. correlations, are beyond the solvability of any known {𝑇1+,𝑇2+,𝑇3−}
a analytical methods. Consequently, developing efficient 3 {𝑇1+,𝑇2−,𝑇3−} {𝑇1+,𝑇2+,𝑇3+}
m and unbiased numerical methods plays a key role in {𝑇1−,𝑇2−,𝑇3−}
- understanding many-body physics in solid state mate- ≥4 all none
d
rials like high-temperature superconductors [4, 5] and
n
TABLE I. The “periodic table” of sign-problem-free symme-
o other systems such as quark matter. Quantum Monte
try classes defined by the set of anticommuting Majorana-
c Carlo (QMC) is among the most important approaches
[ to study interacting many-body systems[6–21], as it is twimheer-erepver=sal±syamndm(eTtr±ie)s2 {=T1±p11,.T2pW2,e··r·ig,oTrnopnu}slythperyovreedsptehcatt,
i i
3 numerically-exact and intrinsically-unbiased. Nonethe- there are two and only two fundamental symmetry classes
v less, QMC often encounters the notorious fermion-sign- whicharesign-problem-free: theMajoranaclassandKramers
0 problem, making it practically infeasible to study those class, respectively. For the former, Majorana-bilinear opera-
78 models with large sizes and at a low temperature[22]. t−oTrs−pTo+ss,easnsdtwitoiMs aTRgensyuminmelyetnrieewsTsi1+gna-npdroTb2l−emw-iftrheeTs1+yTm2−me=-
5 It has been highly desired to find solutions to the 2 1
try class introduced in Ref. [26], qualitatively different from
0 fermion-sign-problem in interesting models that are rel-
the latter, which is based on the conventional Kramers-time-
. evant to intriguing systems such as high-temperature
1 reversal symmetry studied in Ref. [25].
0 superconductors[23].
6 Even though a general solution of the fermion-sign-
1 problem is nondeterministic polynomial (NP) hard[24], erators hˆ , which is defined as having both time-reversal
: i
v many specific interacting models have been successfully symmetry Θˆ with Θˆ2 = −1 and charge conservation Qˆ.
i identified to be sign-problem-free. One prototype sign- With the Kramers symmetry, eigenvalues always appear
X
problem-free example is the repulsive Hubbard model at in Kramers pairs such that the Boltzmann weight can
r
a half filling[8]. In the language of auxiliary-field QMC[6– be shown to be positive definite[25]. Sign-problem-free
(cid:80)
8],thepartitionfunctionZ = ρ,wheretheBoltzmann models with Kramers-symmetry have been studied ex-
weightρ=Tr(cid:81)Nτ exp[hˆ ]withhˆ beingfermion-bilinear tensively during the past three decades. One naturally
i=1 i i
operatorsdependingonauxiliaryfieldsatimaginarytime asksifamorefundamentalsymmetryprincipleexistsfor
τ . If all Boltzmann weight ρ > 0, the simulation is free solving the fermion sign problem in models whose sign-
i
from the fermion sign problem and the needed computa- problem solutions remain unknown so far.
tion time grows only polynomially with the system size. Recently, Majorana representation was first intro-
Tremendous effort has been devoted to construct a fun- duced by three of us in Ref. [26] to solve the fermion
damental principle for solving the fermion sign problem. sign problem in models (including spinless and spinful
One successful strategy of solving the sign problem is fermion models) which are beyond the Kramers method.
toemploytheKramerssymmetryoffermion-bilinearop- Here we employ time-reversal symmetry in Majorana
2
representation[26] as a fundamental and unifying prin- Time-reversal symmetry plays an important role in
ciple to solve the fermion sign problem. We first classify classifying random matrices as well as topological
Majorana-bilinearoperatorshˆ =γThγ accordingtotheir insulators/superconducutors[42–46], and in avoiding the
Majorana-time-reversal (MTR) symmetries, where hT = fermion sign problem in QMC[25]. The Kramers-time-
−h is an antisymmetric matrix and γT = (γ ,··· ,γ ) reversal symmetry[25] has been a successful guiding
1 2N
are Majorana operators with {γ ,γ } = 2δ [27], and principle for sign-problem-free QMC simulations.
i j ij
then identify all symmetry classes which must be sign- Nonetheless, it requires the particle-number conser-
problem-free. Note that “time-reversal” here generally vation and is then not the most general time-reversal
represents “antiunitary”. Because Majorana operators symmetry one can utilize to prevent fermion-sign-
are real, Majorana-time-reversal transformation can be problem[26]. Thus, constructing a more fundamental
represented by T = UK, where U are real orthogonal andgenericsymmetryprincipletoavoidthefermion-sign
matrices and K is complex conjugation with T2 = ±1 problem is desired.
for UT = ±U. By systematically classifying Majorana- In Ref. [26] we proposed that time-reversal symme-
bilinear operators according to the maximal set of an- try in Majorana representation can be used to avoid
ticommuting MTR symmetries they respect, we prove the sign problem in interacting models. Namely,
thatthereareonlytwofundamental symmetryclassesof one can employ Majorana fermions to write fermion-
models which are sign-problem-free: the Majorana class bilinear operators: hˆ(τ ) ≡ hˆ = γTh γ, where γT =
i i i
andKramersclass,respectively. Othersign-problem-free (γ1,··· ,γ1 ,γ2,··· ,γ2 ) and h is a 2N×2N matrix. In
1 N 1 N i
symmetry classes have higher symmetries than the two the case that
fundamental ones, as shown in Table I. (cid:18) (cid:19)
B 0
For the Majorana class, the Majorana-bilinear oper- hi = 0i B∗ , (1)
ators possess two anticommuting symmetries T+ and i
1
T−, where (T±)2 = ±1. Majorana-bilinear operators in ρ = Tr(cid:81)Nτ exp[hˆ ] is positive definite because of the
2 i i=1 i
this class can always be transformed into two decoupled Majorana-time-reversal symmetry T+ = τxK, under
partswhicharetime-reversalpartnerstoeachothersuch whichγ1 →γ2,γ2 →γ1,andB →B∗ [26]. Hereτα are
i i i i i i
that it is sign-problem-free[26]. For the Kramers class, Pauli matrices acting in the Majorana space (1,2). Be-
from anticommuting T− and T−, the usual Kramers- causenocouplingbetweenγ1 andγ2 existsinhˆ ,tracing
1 2 i
time-reversal symmetry can be identified so that they over the Hilbert space of γ1 and γ2 can be done sepa-
are sign-problem-free. Recently, various correlated mod- rately and ρ = ρ∗ due to the Majorana-time-reversal
2 1
els in Kramers class were studied by QMC to investigate symmetry such that ρ=ρ ρ >0.
1 2
high-temperature superconductivity near quantum criti- Notethath inEq.(1)alsorespectsanotherMajorana-
i
cal points (QCPs) [28–34]. time-reversal symmetry T− =iτyK, besides T+ =τxK.
It is worth pointing out that sign-problem-free mod- Moreover, T−T+ = −T+T−. One naturally asks the
els in the genuinely new Majorana class include inter- following question: Can any Majorana-bilinear opera-
acting topological superconductors with helical Majo- tor respecting anti-commuting T+ and T− symmetries
rana edge states [35–37] and the minimal model for be transformed into the form in Eq. (1) such that it is
charge-4e superconductors[38]. Note that the sign prob- sign-problem-free? The answer is positive, as shown be-
lems of these models are beyond applicability of other low. This further motivates us to ask another question:
known approaches, especially those requiring particle- Can anticommuting Majorana-time-reversal symmetries
number conservation[39, 40]. In contrast, the Majo- provide a fundamental principle to classify Majorana-
rana approach here is general and can be applied to bilinear operators such that general sufficient conditions
generic models whether the particle number is conserved for sign-problem-free models can be constructed? Our
or not. As an application of our Majorana approach, we answer is also positive, as we prove below.
have performed large-scale sign-problem-free Majorana As Majorana fermion operators are real, Majorana-
QMC simulations on interacting time-reversal-invariant time-reversal symmetry can be represented by T± =
topological p+ip superconductors of spin-1/2 electrons U±K, where (T±)2 = ±1 and U± is a real orthogo-
and found that with increasing interactions the system nal matrix satisfying (U±)T = ±U±. We propose to
encounters a quantum phase transition from a topo- systematically classify generic Majorana-bilinear oper-
logical nontrivial superconducting phase to a topologi- ators hˆ according to the maximal set of anticommut-
i
cally trivial one by spontaneously breaking time-reversal ing MTR symmetries C = {Tp1,··· ,Tpn} they respect,
1 n
symmetry[41]. To the best of our knowledge, it is the namely [Tpj,h ] = 0 and TpiTpj + TpjTpi = p 2δ ,
j i i j j i i ij
first time that a topological quantum phase transition of where p = ±. Because of the sign choices of p = ±,
i i
spontaneous time-reversal symmetry in superconductors there are totally n + 1 distinct symmetry classes for
can be studied by numerically-exact and intrinsically- each n. For n = 0, there is only one symmetry class
unbiased simulations. {I}, which means that no Majorana-time-reversal sym-
Majorana-time-reversal symmetry classes: metry can be found for those Majorana-bilinear opera-
3
tors; while for n = 1 there are two symmetry classes: metry P = T+T− = U+U−. It is straightforward to
1 2 1 2
{T+} and {T−}. For n = 2 we have three symmetry see that P is a real symmetric matrix satisfying P2 =1.
1 1
classes: {T+,T+}, {T+,T−}, and {T−,T−}. Here, we Consequently,theeigenvaluesofP are±1. As[P,h ]=0,
1 2 1 2 1 2 i
are concerned with only the symmetries of h ; namely we can use P to block-diagonalize h .
i i
we assume that h are random matrices except respect- WedenotetheeigenvectorsofP witheigenvalue+1as
i
ingthespecifiedsetofanticommutingMTRsymmetries. χ , namely Pχ = χ , where a = 1,··· ,N. Because P
a a a
This classification scheme using anti-commuting symme- is a real-symmetric matrix, χ can be chosen to be real,
a
tries is, in spirit, similar to the one employed by Kitaev i.e. χ∗ = χ . Since T+ satisfies {T+,P} = 0, T+χ are
a a 1 1 1 a
using the Clifford algebra to classify random matrices eigenvectorsofP witheigenvalue−1. Now,weareready
andconstructtheperiodictableoftopologicalinsulators to use the basis χ˜ = (χ ,··· ,χ ,T+χ ,··· ,T+χ ) to
1 N 1 1 1 N
and superconductors[44]. block-diagonalize the 2N ×2N matrix h , as follows:
i
Obviously, if a symmetry class C is sign-problem-free, (cid:18) (cid:19)
B 0
anyhighersymmetryclassC(cid:48) whosesymmetriescangen- χ˜Th χ˜= i . (2)
i 0 B∗
erate all the symmetries of C must be sign-problem-free. i
For instance, the symmetry class {T1+,T2+,T3+,T4+} is Consequently,theBoltzmannweightsTr(cid:81)Nτ exp[hˆ ]are
higher than the symmetry class {T+,T−}, because T− i=1 i
1 2 2 positive definite, as required by the time-reversal sym-
in the latter can be generated from the former by identi-
metrybetweenthetwodecoupledblocksB andB∗. Be-
fyingT− =T+T+T+. Iftheformerissign-problem-free, i i
2 2 3 4 cause charge conservation is not required for this sym-
the latter must be sign-problem-free. Consequently, it metry class of {T+,T−}, it is a new sign-problem-free
wouldbesufficienttoderiveallthefundamental symme- 1 2
symmetry class, which was first studied in Ref. [26]. We
try classes which are sign-problem-free.
call it the Majorana class.
“Periodic Table” of fermion-sign-problem: It Kramers class: From T− and T− symmetries, a uni-
1 2
was known that fermion-sign problem can appear in tary symmetry Q = T−T− can be derived. Because Q
1 2
the following three symmetry classes: {I}, {T+}, and isantisymmetric, namelyQT =−Q, onecanconstructa
1
{T−}, as sign-problematic examples in these three sym- charge operator Qˆ =γT(iQ)γ such that [Qˆ,hˆ ]=0. It is
1 i
metry classes are known. For instance, Trexp[xγ1γ2] = clearthatthecombinationofT− andQchargeconserva-
1
2cosx, which is negative for x∈(π,π), even though the tioninthissymmetryclassisequivalenttotheKramers-
2
Majorana-bilinear operator xγ1γ2 respects the T− sym- symmetry since [T−,iQ] = 0 and (T−)2 = −1. The
1 1 1
metry (γ1→γ2, γ2→−γ1, plus complex conjugation). absence of the fermion-sign problem has been shown in
ThisillustratesthatthesymmetryofT− cannotguaran- Ref. [25] according to the observation of the Kramers
1
tee sign-problem-free. Consequently, symmetry itself for pairs in eigenvalues. Because of the Kramers symmetry,
these classes {I}, {T+}, {T−} is not sufficient to guar- we denote this symmetry class as the “Kramers class”.
1 1
antee the absence of the sign problem. We then move to For symmetry classes with n ≥ 3, it turns out that
symmetryclasseswithn=2anticommutingsymmetries: all of them, except only one class {T+,T+,T+}, can
1 2 3
{T+,T+}, {T+,T−}, and {T−,T−}. The symmetries generate the symmetries in either the Majorana-class
1 2 1 2 1 2
in the class {T+,T+} cannot guarantee sign-problem- or the Kramers-class. For instance, the symmetry class
1 2
free,becausethereareknownexampleswithfermion-sign {T+,T+,T−} has higher symmetry than {T+,T−} and
1 2 3 1 2
problem in this class, as shown explicitly below. How {T+,T−,T−}higherthanboth{T+,T−}and{T−,T−}.
1 2 3 1 2 1 2
about the other two classes {T+,T−} and {T−,T−}? It Theonlyremainingunclearclassis{T+,T+,T+},which
1 2 1 2 1 2 3
turns out these two are fundamental symmetry classes cannot generate {T+,T−} or {T−,T−}. Even though
1 2 1 2
which are sign-problem-free, as we shall prove below. the symmetry class {T+,T+,T+} has relatively high
1 2 3
If Majorana-bilinear operators hˆ respect MTR sym- symmetries, it can still suffer from the fermion-sign-
i
metries in one of the two symmetry classes {T+,T−} problemasweshowbelowthatthereexistexplicitexam-
1 2
and {T−,T−}, ρ = Tr(cid:81)Nτ exp[hˆ ] > 0. These two ples in this symmetry-class which are sign-problematic.
1 2 i=1 i
Now we explicitly demonstrate that the two symme-
are only fundamental symmetry classes which are sign-
try classes {T+,T+} and {T+,T+,T+} can be sign-
problem-free. Weshallprovethisbelowforthetwosym- 1 2 1 2 3
problematicbyconsideringthespin-1 repulsiveHubbard
metry classes separately. We call the former symmetry 2
model away from half-filling as an example. For the
classasthe“Majoranaclass”, whilethelatteroneasthe
repulsive-U Hubbard model
“Kramers class” for reasons which will be clear later.
Majorana class: In the Majorana class, the random H =−t (cid:88) (cid:2)c† c +h.c.(cid:3)+U(cid:88)n n −µ(cid:88)n ,(3)
matrix h respects two Majorana-time-reversal symme- iσ jσ i↑ i↓ iσ
i (cid:104)ij(cid:105)σ i iσ
tries T+ = U+K and T− = U−K, where U+ is a
1 1 2 2 1
real-symmetric orthogonal matrix but U− a real anti- where U>0, µ(cid:54)=0, and σ=↑,↓, we obtain the following
2
symmetric orthogonal matrix. From these two time- decoupled Majorana-bilinear Hamiltonian hˆ after uti-
n
reversal symmetries, one can construct a unitary sym- lizing the Majorana representations c = (γ1 +iγ2)/2
σ σ σ
4
and performing a Hubbard-Stratonovich (HS) transfor-
mation: hˆ = −t˜(cid:80) γTσ0τyγ +(cid:80) (cid:2)µ˜γTσ0τyγ −
n 2 (cid:104)ij(cid:105) i j i 4 i i
λφnγTiσyτzγ (cid:3), where φn are auxiliary fields on site i at
i i i i
imaginary time τ , γT = (γ1,γ1,γ2,γ2), σa is Pauli
n i i↑ i↓ i↑ i↓ FIG. 1. The quantum phase diagram of interacting topolog-
matrix in spin space and τa in Majorana space [41].
ical superconductors with a topological quantum phase tran-
It is straightforward to show that hˆn possesses three sitionaswellasthenatureofthequantumcriticalpointhave
MTR symmetries T+ = σxτxK, T+ = σzτxK, and beenstudiedbyoursign-problem-freeQMCsimulations[41].
1 2
T+ =σ0τzK. Eventhoughrespectingthesesymmetries,
3
theappearanceofsign-probleminthisdecoupledchannel
Majorana-space. Besides T−, it also possesses a unitary
forthedopedrepulsiveHubbardmodeliswell-known. In
symmetry P = σz, such that we can construct another
order to further confirm it, we have also computed the
Majorana-time-reversalsymmetryT+ =PT− =σxτzK.
Boltzmannweightsfordifferentauxiliary-fieldconfigura-
Inthecomplexfermionsbasis,thesetwoanti-commuting
tions and find that negative weights can indeed appear.
Namely,modelsinthesymmetry-class{T+,T+,T+}can TR symmetries are: T+ = σxK and T− = iσyK where
be sign-problematic. Since the class {T1+,T2+,T3+} has σa isPaulimatrixactinginspinspace(c↑,c↓). Notethat
highersymmetriesthan{T+,T+},thelat1ter2isals3osign- the unitary symmetry P = σz is not a U(1) symmetry
1 2 conservingthetotalSz becausethetripletpairinginthe
problematic.
Hamiltonian breaks the U(1) symmetry. Instead, it only
Combiningtheresultsabove,wehaveshownthatthere
aretwoandonlytwofundamental sign-problem-freesym- conserves the spin parity (−1)N↓, where N↓ is the num-
berofspin-downelectrons. Thetopologicalclassification
metry classes: the Majorana-class and the Kramers-
of superconductors respecting the time-reversal symme-
class. The“periodictable”ofsymmetryclasseswhichare
try and the spin-parity symmetry in the non-interacting
sign-problem-free or sign-problematic is shown in Table
limit is Z. In the presence of interactions, its topological
I. It provides a fundamental principle to identify sign-
classification was shown to be Z [35–37].
problem-free interacting fermion models. 8
FortheHubbardinteractions,weperformthefollowing
Interacting topological superconductors in the
HS transformation:
Majorana-class: So far novel interaction effects in
topological superconductors have not been investigated eU4∆τiγi1↑γi2↑iγi1↓γi2↓ = (cid:88) Aeλφi(iγi1↑γi2↑+iγi1↓γi2↓), (5)
by large-scale QMC mainly due to the lack of such sign-
φi=±1
problem-freemodels,althoughinteractingtopologicalin-
where φ represent auxiliary fields living on site i, ∆τ
sulators have been much studied by sign-problem-free i
is the imaginary time slice in the Trotter decomposition,
QMC[47–54]. As mentioned above, there is no require-
λ = 1cosh−1(eU∆τ/2), and A = 1e−U∆τ/4. It is clear
mentofcharge-conversationformodelsintheMajorana- 2 2
that the decoupled Majorana-bilinear operators also re-
class. Consequently, interesting sign-problem-free mod-
spect both T+ = σxτzK and T− = σxτzK such that
elsdescribinginteractingsuperconductorsmaybeidenti-
this model is sign-problem-free in the Majorana-class.
fiedinthisclasssuchthatwecanstudystrongcorrelation
We performed projector QMC [10–12] simulations which
effect in superconductors using sign-problem-free QMC.
show that, with increasing U, the system undergoes a
Indeed, we have found interacting models of topological
topological quantum phase transition from a topological
superconductorswithhelicalMajoranaedgestateswhich
SC to a trivial SC which breaks time-reversal symme-
are sign-problem-free in the Majorana-class.
try spontaneously [41], as schematically shown in Fig. 1.
WefirstconsidertheHamiltoniandescribingatopolog-
Theuniversalityclassofthistopologicalphasetransition
ical superconductor of spin-1/2 electrons on the square
is also obtained by our QMC simulations [41].
lattice with time-reversal symmetry:
Concluding remarks: We have shown that anti-
H=(cid:88)(cid:2)−t c† c +∆ c† c† +h.c.(cid:3)−U(cid:88)n n ,(4) commuting MTR symmetries can provide a fundamen-
ij iσ jσ ij,σ iσ jσ i↑ i↓
tal principle to identify sign-problem-free interacting
ij,σ i
models. Note that this does not contradict the no-go
where c† creates spin-1/2 electrons with spin polariza- theorem[24] as the symmetry principle introduced here
iσ
tion σ =↑,↓, n = c† c , t = t is nearest-neighbor may not directly provide recipes of the sign-problem for
iσ iσ iσ ij
hopping, and t = µ is the chemical potential. When allmodels. Assumingnootherrequirementthanrespect-
ii
∆ = ∆ for j = i+xˆ and ∆ = iσ∆ for j = i+yˆ, ing a set of anti-commuting MTR symmetries, we have
ij,σ ij,σ
the Hamiltonian in Eq. (4) describes a helical topologi- provedthattherearetwoandonlytwofundamental sign-
cal superconductor[44–46] with (p+ip) triplet-pairing of problem-free symmetry classes.
spin-upelectronsand(p−ip)triplet-pairingofspin-down Here, we focus on the generic symmetry principle for
electrons, which hosts helical Majorana edge states pro- sign-problem-free models. In case that the matrices h
i
tected by the Majorana-time-reversal symmetry T− = arenotfully randombesidesrespectingtherequiredsym-
iσyτzK, where σi acts in spin-space and τi in the metries, it is possible that sign-problem-free models can
5
be found in the symmetry classes {I}, {T+}, {T−}, 83, 3116 (1999).
1 1
{T+,T+}, and {T+,T+,T+} [20, 55]. For instance, it [21] N. Hatano and M. Suzuki, Physics Letters A 163, 246
1 2 1 2 3
was shown in Refs. [39, 55] that certain models in the (1992).
symmetryclass{T+}withspecialconditionsbesidesthe [22] E.Y.Loh,J.E.Gubernatis,R.T.Scalettar,S.R.White,
1 D.J.Scalapino,andR.L.Sugar,Phys.Rev.B41,9301
symmetry requirement could be sign-problem-free. In
(1990).
other words, it provides hope to avoid the fermion-sign-
[23] J. Zaanen, Science 319, 1205 (2008).
problem in various interesting models, such as the repul- [24] M. Troyer and U.-J. Wiese, Phys. Rev. Lett 94, 170201
sive Hubbard model away from half-filling [12], by utiliz- (2005).
ingacombinationofspecialfeaturesandsymmetryprin- [25] C.WuandS.-C.Zhang,Phys.Rev.B,71,155115(2005).
ciples. We believe that the Majorana-time-reversal sym- [26] Z.-X. Li, Y.-F. Jiang and H. Yao, Phys. Rev. B 91,
241117(R) (2015).
metry principle has shed new light on avoiding fermion-
[27] C.W.J.Beenakker,Annu.Rev.Con.Mat.Phys.4,113
sign-problem in strongly-correlated models describing
(2013).
systems such as high-temperature superconductors and
[28] E. Berg, M. A. Metlitski, and S. Sachdev, Science 338,
featuring exotic quantum critical phenomena[56–59]. 1606 (2012).
Acknowledgement: We would like to thank Shao-Kai [29] Y. Schattner, S. Lederer, S. A. Kivelson, and E. Berg,
Phys. Rev. X 6, 031028 (2016).
Jian and A. Kitaev for helpful discussions. This work
[30] Z.-X. Li, F. Wang, H. Yao, and D.-H. Lee,
was supported in part by the NSFC under Grant No.
arXiv:1512.04541 (2015).
11474175 at Tsinghua University (ZXL,YFJ, and HY).
[31] Y. Schattner, M. H. Gerlach, S. Trebst, and E. Berg,
Phys. Rev. Lett. 117, 097002 (2016).
[32] Z.-X. Li, F. Wang, H. Yao, and D.-H. Lee, Science Bul-
letin 61, 925 (2016).
[33] S. A. Kivelson, Science Bulletin 61, 911 (2016).
∗ [email protected] [34] P. T. Dumitrescu, M. Serbyn, R. T. Scalettar, and A.
[1] X.-G. Wen, Quantum Field Theory of Many-body Sys- Vishwanath, Phys. Rev. B 94, 155127 (2016).
tems, (Oxford University Press, New York, 2004). [35] X.-L. Qi, New J. Phys. 15, 065002 (2013).
[2] S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cam- [36] H. Yao and S. Ryu, Phys. Rev. B 88, 064507 (2013).
bridge University Press, Cambridge, 2011). [37] S.RyuandS.-C.Zhang,Phys.Rev.B85,245132(2012).
[3] E.Fradkin,FieldTheoriesofCondensedMatterPhysics, [38] Y.-F. Jiang, Z.-X. Li, S. A. Kivelson, and H. Yao,
2nded.(CambridgeUniversityPress,Cambridge,2013). arXiv:1607.01770.
[4] S.A.Kivelson,I.P.Bindloss,E.Fradkin,V.Oganesyan, [39] E. F. Huffman and S. Chandrasekharan, Phys. Rev. B
J. M. Tranquada, A. Kapitulnik, and C. Howald, Rev. 89, 111101 (2014).
Mod. Phys. 75, 1201 (2003). [40] L. Wang, Y. H. Liu, M. Iazzi, M. Troyer and G. Harcos,
[5] P.A.Lee,N.Nagaosa,andX.-G.Wen,Rev.Mod.Phys. Phys. Rev. Lett. 115, 250601 (2015).
78, 17 (2006). [41] See the Supplemental Material for the details.
[6] R.Blankenbecler,D.J.Scalapino,andR.L.Sugar,Phys. [42] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057
Rev. D 24, 2278 (1981). (2011).
[7] J. E. Hirsch, D. J. Scalapino, R. L. Sugar, and R. [43] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045
Blankenbecler, Phys. Rev. Lett. 47, 1628 (1981). (2010).
[8] J. E. Hirsch, Phys. Rev. B 31, 4403 (1985). [44] A. Kitaev, in Periodic Table for Topological Insulators
[9] J.E.HirschandH.Q.Lin,Phys.Rev.B37,5070(1988). and Superconductors (AIP, New York, 2009) [AIP Conf.
[10] G. Sugiyama and S. Koogin, Annals of Phys. 168, 1 Proc. 1134, 22 (2009)].
(1986). [45] A.P.Schnyder,S.Ryu,A.Furusaki,andA.W.W.Lud-
[11] S.Sorella,S.Baroni,R.CarandM.Parrinello.Europhys. wig, Phys. Rev. B 78, 195125 (2008).
8, 663 (1989). [46] X.-L.Qi,T.L.Hughes,S.Raghu,andS.-C.Zhang,Phys.
[12] S. R. White, D. J. Scalapino, R. L. Sugar, E. Y. Loh, J. Rev. Lett. 102, 187001 (2009).
E.GubernatisandR.T.Scalettar.Phys.Rev.B,40,506 [47] M.Hohenadler,T.C.Lang,andF.F.Assaad,Phys.Rev.
(1989). Lett. 106, 100403 (2011).
[13] D. Ceperley and B Alder, Science 231, 555 (1986). [48] D. Zheng, G.-M. Zhang, and C. Wu, Phys. Rev. B 84,
[14] S. Zhang, J. Carlson, and J. E. Gubernatis, Phys. Rev. 205121 (2011).
Lett. 74, 3652 (1995). [49] M. Hohenadler, Z. Y. Meng, T. C. Lang, S. Wessel, A.
[15] N. V. Prokofev, B. V. Svistunov, and I. S. Tupitsyn, Muramatsu,andF.F.Assaad,Phys.Rev.B85,115132
Physics Letters A 238, 253 (1998). (2012).
[16] A. W. Sandvik, Phys. Rev. B 59, R14157 (1999). [50] F. F.Assaad, M.Bercx, andM. Hohenadler, Phys.Rev.
[17] W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Ra- X 3, 011015 (2013).
jagopal, Rev. Mod. Phys. 73, 33 (2001). [51] H.-H. Hung, L. Wang, Z.-C. Gu, and G. A. Fiete, Phys.
[18] F. F. Assaad and H. G. Evertz, Computational Many Rev. B 87, 121113 (2013).
Particle Physics, Lecture Notes in Physics 739, 277 [52] T.C.Lang,A.M.Essin,V.Gurarie,andS.Wessel,Phys.
(2008). Rev. B 87, 205101 (2013).
[19] E.Gull,A.J.Millis,A.I.Lichtenstein,A.N.Rubtsov,M. [53] K.Slagle,Y.-Z.You,andC.Xu,Phys.Rev.B91,115121
Troyer,andP.Werner,Rev.Mod.Phys.83,349(2011). (2015).
[20] S. Chandrasekharan and U. J. Wiese, Phys. Rev. Lett. [54] Y.-Y.He,H.-Q.Wu,Y.-Z.You,C.Xu,Z.Y.Meng,and
6
Z.-Y. Lu, Phys. Rev. B 93, 115150 (2016). 335, 193 (2012).
[55] Z. C. Wei, C. Wu, Y. Li, S. Zhang, and T. Xiang, Phys. [58] Z.-X. Li, Y.-F. Jiang, S.-K. Jian, and H. Yao,
Rev. Lett. 116, 250601 (2016). arXiv:1512.07908 (2015).
[56] S.V.Isakov,Y.B.Kim,andA.Paramekanti,Phys.Rev. [59] H. Shao, W. Guo, and A. W. Sandvik, Science 352, 213
Lett. 97, 207204 (2006). (2016).
[57] S. V. Isakov, M. B. Hastings, and R. G. Melko, Science
I. Sign-problem-free condition for Projector QMC in Majorana class
ProjectorQMCisanalgorithmtoexplorethegroundpropertiesofthesystems. InprojectorQMC,theexpectation
value of an observable in the ground state can be evaluated as:
(cid:104)ψ |O|ψ (cid:105) (cid:104)ψ |e−θHOe−θH|ψ (cid:105)
0 0 = lim T T , (S1)
(cid:104)ψ0 |ψ0(cid:105) θ→∞ (cid:104)ψT|e−2θH|ψT(cid:105)
where ψ is the true ground state wave function and ψ is a trial wave function which should have a finite overlap
0 T
withthetruegroundstatewavefunction. Weshouldemphasizethatthefinite-temperatureproofofsign-problem-free
in the Majorana class can be generalized to projector QMC straightforwardly. Similar to the Boltzmann weight in
finite-temperature algorithm, the weight for each auxiliary field configuration in the projector algorithm is given by
the following expectation value
W =(cid:104)ψT|[(cid:89)Nτ e41γThiγ]|ψT(cid:105), (S2)
i=1
where γT = (γ1,··· ,γ1 ,γ2,··· ,γ2 ) represents a 2N-dimensional vector of Majorana fermions and |ψ (cid:105) =
1 N 1 N T
(cid:81)2Nf(γP˜) |0(cid:105) is the trial wave function. Here P˜ is a 2N × 2N projector matrix. This projector matrix can
a=1 a f
be rewritten as P˜ = χ˜ξ, where ξ = F ⊕F∗. F is a N ×N matrix, which can be written F = (φ ,··· ,φ ) with
f 1 Nf
φ an N-dimensional vector. φ is often chosen to be single-particle eigenstate of the non-interacting Hamiltonian. In
i i
the new Majorana fermion basis (α,T+α)=γχ˜, the weight W can be written as
1
W =ww∗, (S3)
with
w =(cid:104)0|(αF)†[(cid:89)Nτ e14αTBiα](αF)|0(cid:105). (S4)
i=1
Since W =ww∗, it is clear that it is positive definite. As w =±[det(F†(cid:81)Nτ eBiF)]1/2, we obtain
i=1
(cid:12) (cid:12)
(cid:12) (cid:89)Nτ (cid:12)
W =(cid:12)det(F† eBiF)(cid:12). (S5)
(cid:12) (cid:12)
(cid:12) (cid:12)
i=1
.
II. Sign-problematic examples in the symmetry classes {T+,T+} and {T+,T+,T+}
1 2 1 2 3
Even though the symmetry class {T+,T+,T+} has relatively high symmetries, it can still suffer from the fermion-
1 2 3
sign-problem. We consider spin-1 repulsive Hubbard model away from half filling as an example. The Hamiltonian
2
is:
H =−t (cid:88) (cid:2)c† c +h.c.(cid:3)+U(cid:88)n n −µ(cid:88)n , (S6)
iσ jσ i↑ i↓ iσ
(cid:104)ij(cid:105)σ i iσ
where U>0, µ(cid:54)=0, and σ=↑,↓. We take basis transformation c = (γ1 + iγ2)/2 and rewrite the Hamiltonian in
σ σ σ
Majorana representation:
H=−t (cid:88)γ˜Tσ0τyγ˜ +(cid:88)(cid:2)µγ˜Tσ0τyγ˜ +Uiγ1γ2iγ1γ2(cid:3) (S7)
2 i j 4 i i 4 i↑ i↑ i↓ i↓
(cid:104)ij(cid:105) i
7
whereγ˜T =(γ1,γ1,γ2,γ2)andσa isPaulimatrixinspinspaceandτa inMajoranaspace. ThenwetakeHubbard-
i i↑ i↓ i↑ i↓
Stratonovich(HS) transformation to decouple the Hubbard interaction term:
e−U4∆τiγi1↑γi2↑iγi1↓γi2↓ = 1 (cid:88) e21λφi(γi1↑γi1↓−γi2↑γi2↓)−U4∆τ (S8)
2
φi=±1
where λ = cosh−1(eU∆τ/2), φi is auxiliary field living on site i and ∆τ is imaginary time slice under Trotter decom-
position. After HS transformation the partition function can be expressed:
Tr[e−βH]= (cid:88) Tr[(cid:89)ehˆn(φn)] (S9)
φn=±1 n
i
hˆ is bilinear Majorana fermions operator at n-th imaginary time slice:
n
hˆ =−t˜(cid:88)γTσ0τyγ +(cid:88)(cid:2)µ˜γTσ0τyγ −λφnγTiσyτzγ (cid:3) (S10)
n 2 i j 4 i i i i i
(cid:104)ij(cid:105) i
where t˜= t∆ and µ˜ = µ∆ . It is straightforward to show that hˆ possesses three MTR symmetries T+ = σxτxK,
τ τ n 1
T+ =σzτxK, andT+ =σ0τzK. TheappearanceofsignprobleminthisdecoupledchannelofdopedHubbardmodel
2 3
iswell-known. Inordertofurtherconfirmit,wehavealsocomputedtheBoltzmannweightsfordifferentauxiliary-field
configurations and find that negative weights can indeed appear. This sign-problematic example illustrates that the
symmetry class {T+,T+,T+} cannot guarantee sign-problem-free. Since {T+,T+,T+} has higher symmetries than
1 2 3 1 2 3
{T+,T+} and {T+}, both {T+,T+} and {T+} are also sign-problematic.
1 2 1 1 2 1
III. Detailed proof of sign-problem-free for interacting topological superconductors of spin-1/2 electrons
Thetopologicalsuperconductorofspin-1/2electronsonthesquarelatticewithattractiveHubbardinteractionscan
be described by the following Hamiltonian:
H = (cid:88) (cid:2)−tc† c +∆ c† c† +h.c.(cid:3)−µ(cid:88)(n +n )−U(cid:88)n n , (S11)
iσ jσ ij,σ iσ jσ i↑ i↓ i↑ i↓
(cid:104)ij(cid:105),σ i i
where the triplet pairing amplitudes are given by ∆ = ∆ for j = i+xˆ and ∆ = iσ∆ for j = i+yˆ. We
ij,σ ij,σ
can express complex fermions by two components of Majorana fermions c = 1(γ1 +iγ2 ), and then rewrite the
jσ 2 jσ jσ
Hamiltonian as:
H =H +H ,
0 I
t (cid:88) i∆ (cid:88) i∆ (cid:88) µ(cid:88)
H =− γ˜Tσ0τyγ˜ + γ˜Tσ0τxγ˜ + γ˜Tσzτzγ˜ + γ˜Tσ0τyγ˜, (S12)
0 2 i j 2 i j 2 i j 4 i i
(cid:104)ij(cid:105) (cid:104)ij(cid:105) (cid:104)ij(cid:105) i
x y
U (cid:88)
H =− iγ1γ2iγ1γ2, (S13)
I 4 i↑ i↑ i↓ i↓
i
where σα and τα are Pauli matrices acting in the spin and Majorana space, respectively. We can perform Hubbard-
Stratonovich transformation of attractive Hubbard term in density channel:
eU4∆τiγi1↑γi2↑iγi1↓γi2↓ = (cid:88) Aeλφi(iγi1↑γi2↑+iγi1↓γi2↓), (S14)
φi=±1
where φ represent auxiliary fields living on site i, ∆τ is the imaginary time slice in the Trotter decomposition,
i
λ= 1cosh−1(eU∆τ/2), and A= 1e−U∆τ/4. Consequently, the decoupled Hamiltonian after HS transformation is:
2 2
hˆ(φ)= (cid:88) −t˜γTσ0τyγ + (cid:88) i∆˜γTσ0τxγ + (cid:88) i∆˜γTσzτzγ + µ˜ (cid:88)γTσ0τyγ +λ(cid:88)φ γTσ0τyγ , (S15)
2 i j 2 i j 2 i j 4 i i i i i
(cid:104)ij(cid:105),σ (cid:104)ij(cid:105) (cid:104)ij(cid:105) i i
x y
where γT = (γ1,γ2,γ1,γ2) and t˜= ∆τt,∆˜ = ∆τ∆,µ˜ = ∆τµ . Because it respects these two anti-commuting
i i↑ i↑ i↓ i↓
MTR symmetries: T+ = σxτzK and T− = iσyτzK, it belongs to the Majorana-class and is then sign-problem-free
1 2
according to the theorem we have proved in the main text. In the complex fermions basis, these two anti-commuting
TR symmetries are: T+ =σxK and T− =iσyK where σa is Pauli matrix acting in spin space c˜=(c ,c ).
1 2 ↑ ↓
8
2 .0
L = 1 6
1 .8 L = 1 8
L = 2 0
1 .6
h1+
L21 .4
M
1 .2
(b)
1 .0
-1 0 -5 (U 0-U )5L 1/n 1 0 1 5 2 0
c
FIG. S1. (a) The Majorana QMC results of the interacting topological superconductors of spin-1/2 electrons featuring a
topologicalquantumphasetransition. ThecrossingpointofBinderratiofortime-reversalsymmetrybreakingorderparameter
shows that the quantum phase transition occurs at U ≈4.48 for ∆=0.3 and µ=−0.5. (b) The data collapse analysis of the
c
critical behavior around the time-reversal symmetry breaking transition reveals that ν≈0.63 and η≈0.03.
IV. Numerical results of QMC simulations of the interacting topological superconductors
We have performed large-scale sign-problem-free Majorana projector QMC simulations of the correlation effect in
the interacting topological superconductors described by Eq. (S11) with ∆ = 0.3t and µ = −0.5t (hereafter we set
t = 1 for simplicity). It is clear that the topological superconductor is stable against weak interaction U. When U
is strong enough, we expect that the system shall possess a finite singlet pairing, which spontaneously breaks the
symmetryP =T+T− =σz. FromcomputingtheBinderratioB(L)ofthesingletorderparameter∆ =(cid:104)c† c† (cid:105)with
s i↑ i↓
size L×L, we can determine the critical values U of the spontaneous symmetry-breaking, as shown in Fig. S1(a).
c
The quantum critical point U ≈ 4.48 is obtained from the crossing point of Binder ratio of different system sizes.
c
Moreover, in the ordered phase, we found that the singlet pairing amplitude ∆ is pure imaginary because the value
s
of ∆2, obtained through the finite-size scaling of the correlation function (cid:104)c† c† c† c† (cid:105) with (cid:126)r = (cid:126)r +(L/2,L/2),
s i↑ i↓ j↑ j↓ j i
is negative. For instance, we obtain ∆ ≈ ±0.15i for U = 4.6, indicating that the system spontaneously breaks
s
time-reversal symmetries T+ and T− in the ordered phase. In other words, the system undergoes a topological
quantum phase transition from a topological SC to a topologically-trivial SC which breaks time-reversal symmetry
spontaneously.
Our sign-problem-free QMC can also study the critical behaviors of the topological quantum phase transition with
spontaneous time-reversal breaking. From the data collapse analysis, as shown in Fig. S1(b) , we obtain the critical
exponents ν≈0.63 and η≈0.03, which is quite consistent with the Ising quantum critical point in 2+1 dimensions.