Series-Expansion Thermal Tensor Network Approach for Quantum Lattice Models

Bin-Bin Chen,1 Yun-Jing Liu,1 Ziyu Chen,1 and Wei Li1, 2, ∗

1Department of Physics, Key Laboratory of Micro-Nano Measurement-Manipulationand Physics (Ministry of Education), Beihang University, Beijing 100191, China

2International Research Institute of Multidisciplinary Science, Beihang University, Beijing 100191, China

In this work we propose a series-expansion thermal tensor network (SETTN) approach for efficient simu-lations of quantum lattice models. This continuous-time SETTN method is based on the numerically exactTaylor series expansion of equilibrium density operator e−βH (with H the total Hamiltonian and β the imag-inary time), and is thus Trotter-error free. We discover, through simulating XXZ spin chain and square-latticequantum Ising models, that not only the Hamiltonian H , but also its powers Hn, can be efficiently expressedas matrix product operators, which enables us to calculate with high precision the equilibrium and dynamicalproperties of quantum lattice models at finite temperatures. Our SETTN method provides an alternative to con-ventional Trotter-Suzuki renormalization group (RG) approaches, and achieves an unprecedented standard ofthermal RG simulations in terms of accuracy and flexibility.

PACS numbers: 05.10.Cc, 02.70.-c, 05.30.-d, 75.10.Jm

Introduction.— Developing accurate methods in quantummanybody computations intrigues people in contemporarycondensed matter physics and also other relevant fields, whichat the same time also poses a great challenge. In particular, ef-ficient simulations of thermodynamics [1–4] are indispensableto relate theoretical models to experimental measurements atfinite temperatures [5, 6], and are also of fundamental interest.However, exact diagonalization (ED) calculations of many-body systems encounter the exponentially large Hilbert space,and one needs the full energy spectrum to compute the parti-tion function and other thermodynamic quantities, constitut-ing a severe constraint for large-scale simulations.

Therefore, people resort to various efficient numericalmethods to tackle manybody lattice models like spin-1/2 XXZHeisenberg chain whose Hamiltonian is

H =∑〈i,j〉

hi,j =∑〈i,j〉

J(Sxi Sxj + Syi S

yj + ∆Szi S

zj ), (1)

where {Sx, Sy, Sz} are spin operators, J = 1 is the exchangecoupling constant, and ∆ labels the z-axis anisotropy. Follow-ing Feynman’s path integral representation, the partition func-tion (in canonical ensemble) Z = Tr(e−βH) can be expressedas a discrete-time thermal tensor network (TTN) by Trotter-Suzuki decomposition ρ = e−βH ' (Π〈i,j〉e

−τhi,j )M , whereτ is a small Trotter slice and Mτ = β [3, 7]. In thisframework, to calculate thermal quantities such as energyand entropy, people developed stochastic approaches, namely,world-line quantum Monte Carlo (QMC) samplings [3, 8], aswell as deterministic renormalization-group (RG) methods in-cluding transfer-matrix RG (TMRG) [1], linearized tensor RG(LTRG) [2], finite-temperature density matrix RG [9, 10], etc.The RG methods, due to their high accuracy (sampling-errorfree) and wide versatility (sign-problem free), constitute animportant class of methods in computing thermodynamics oflow-dimensional quantum lattice models, especially in 1D andquasi-1D.

Nevertheless, these conventional RG approaches, despitetheir nice accuracies, suffer from two kinds of errors, i.e.,

Trotter and truncation errors. As a result, compared to groundstate simulations, the accuracies of finite-T calculations weredropped behind. The former, either for a finite or infinite 1Dsystem, can reach accuracy of 8 ∼ 9 significant digits (evennot machine accuracy), with only moderate efforts [9]. How-ever, in discrete-time thermal RG calculations, where the Trot-ter slice τ is typically set as 0.05 or 0.1 [1, 2], the relativeerrors are in the order of 10−4∼−5 (in free energy). One canchoose smaller step τ to reduce the Totter error and improvethe accuracy, but then significantly more RG steps M are de-manded correspondingly, making it impractical this way [11].More severely, the two errors may have different signs andthe accuracies of thermodynamic properties at any given tem-perature are not guaranteed to improve by keeping more RGstates in the calculations [2]. Therefore, it is appealing if acontinuous-time TTN method, which completely gets rid ofthe Trotter errors, can be devised.

This is possible by noticing that series-expansion QMC re-alizes Trotter-error-free calculations in discrete forms. Theinitial idea of series-expansion QMC was proposed by Hand-scomb in 1960’s, who noticed that Taylor expansion can beincorporated in QMC to simulate quantum ferromagnetic spinchains [12]. Though the utility of Handscomb’s method wasquite limited, through some efforts [13, 14], it thrives and de-velops into one of mainstream QMC techniques, dubbed asstochastic series expansion (SSE) [4].

In this work, inspired by the Handscomb’s and also SSEmethods, we propose a series-expansion tensor networkmethod for quantum manybody systems. Compared to pre-vious Trotter-Suzuki RG methods, by exploiting an efficienttensor-network representation of the powers of total Hamil-tonian (Hn), we remove the Trotter errors completely andachieve an unprecedentedly high accuracy in computing ther-modynamic properties. Compared to SSE, our method em-ploys a deterministic RG approach to do the series-expansioncalculations, which is in principle sign-problem free for frus-trated spin models. Moreover, time-dependent correlationfunctions are also accessible in this method, again calculated

arX

iv:1

609.

0126

3v2

[co

nd-m

at.s

tr-e

l] 2

6 M

ar 2

017

2

(a) H = D

{ }ˆ ˆ ˆˆ ˆ, , ,x y zP I S S S∈

(b) Z = 0ω Tr

1ω Tr+

nω Tr+

I

12

n

FIG. 1. (Color online) (a) MPO representation of total Hamiltonianof Heisenberg spin chain model,D labels geometric bond dimension,and local tensor P comprises spin operators and an identity. (b) Thepartition function Z can be expressed as a sum of ωnTr(Hn) terms,each of which constitutes a tensor network.

in a Trotter-error-free manner.Series-expansion thermal tensor networks.— Utilizing a

Taylor (precisely Maclaurin) series expansion, the partitionfunction can be written as:

Z(β) 'N∑n=0

κn(β) =

N∑n=0

ωn(β)Tr(Hn), (2)

where ωn(β) = (−β)n/n!, n runs over the expansion indexspace and N is the cut-off order of n. Owing to the fact thatωn(β) obeys Poisson distribution and centers at 〈n〉 = β,when N is sufficiently large the expansion is essentially freeof cut-off error. It thus offers a continuous-time approach(with β regarded as imaginary time) through a discrete formal-ism. In Handscomb’s method (and also in SSE), the total His decomposed into a sum of local terms, and the QMC sam-plings are performed in the space of index sequences [4, 12].

We point out that the series expansion Eq. (2) can be actu-ally encoded in an efficient representation without splitting Hinto local terms, in the framework of tensor networks (TNs).Exploiting this fact, we propose a continuous-time algorithmnamed series-expansion TTN (SETTN). Specifically, for 1Dmodels like XXZ spin chain, one can construct a matrix prod-uct operator (MPO) with small bond dimensions to representthe total Hamiltonian Eq. (1). As shown in Fig. 1(a), the localP tensor stores the spin and identity operators {Sx, Sy, Sz, I},which are all d× d matrices (d the dimension of local Hilbertspace). P tensors are connected via geometric bonds to con-stitute a MPO, with bond dimension D = 5(4) for XXZ(XY)model. For instance, in XY-chain model under open bound-ary conditions (OBCs) the nonzero elements in P tensor areP1,1 = P4,4 = I , P1,2 = P2,4 = Sx, P1,3 = P3,4 = Sy , andall operators in P apply vertically as shown in Fig. 1(a).

Despite some studies on the MPO representation of H , inthe context of ground state and also time evolution problems

β0 10 20 30 40

n

50

100

150

200

β0 10 20 30 40

n

50

100

150

200

β0 10 20 30 40

n

50

100

150

200

β0 20 40

n

50

100

150

200

0

0.5

1

(a) (b)

(c) (d)

FIG. 2. (Color online) Normalized weights κn(β) (for each fixed β,the κn(β) series is normalized such that the maximal value equalsunity) in partition function Eq. (2) for various system sizes L =12, 24, 36, 48, from (a) to (d), respectively. The sharp contrast ofbrightness (intensities of weights) reveals that only a relatively smallnumber of n terms are significant, around the linear “light” coneswhose slops are 3.55, 7.69, 11.17, 14.61, from (a) to (d), respectively.

[15–17], there are still much to be explored regarding theMPO representation of its power Hn. In Fig. 1(b), we plottheHn series, each term of which can be represented as a TN.The partition function can be obtained by summing over thetrace of each TN [multiplied by ωn(β) in Eq.(2], given thatwe can perform efficient contractions of them. This consti-tutes our main idea of SETTN method.

Two concerns immediately arise: Firstly, in order to geta converged and accurate summation for Z, one may won-der how many terms (N ) are generically sufficient, and howdoes N scale with β and system size L; The second possibleworry is about the compressibility of Hn series. To answerthe first question, in Fig. 2 we show the properly normalizedweights κn(β) in Eq. (2) for various system sizes. It can beseen that, for every β, only a small number of weights (in thebright cone-like region) essentially contribute to Z. The av-erage 〈n〉H , where the cone center resides, increases linearlywith β. In addition to that, the slop of the cone increases lin-early with system size L as well, and we can estimate that〈n〉H ≈ βL/3 from Fig. 2. This is remarkable since it meansthat in order to obtain thermal properties of an L-site systemat inverse temperature β, one only needs to prepare O(βL)number of Hn terms. We also numerically calculated XXZmodel with ∆ other than 0, and find this relation always valid.

Actually, such an observation can be understood as fol-lowing: Tr(Hn) is proportional to |Eln|n, where the exten-sive quantity Eln = elnL is the energy eigenvalue of Hwith largest norm. Therefore, the dominant term in seriesκn(β) = (βL|eln|)n

n! should be at 〈n〉H = βL|eln|. Rememberthat the ground-state energy per site (in the thermodynamiclimit) of XY-chain is −1/π (which happens to be also theeigenvalue with largest norm), thus 〈n〉H ' βL/π, in perfectagreement with the numerical results in Fig. 2.

Canonical optimization and compressibility of MPOHn.—Now we address the second question on whether each Hn has

3

(a)

(b)

(c)

(d)

U Λ V†

U Λ V†

Ti

αi

γi

pi

qi

Tγ

γ'χDχ

χDχD

I

I

α

α'

FIG. 3. (Color online) (a) Left and (b) right sweep, bringing the en-larged MPO into canonical form and perform optimal truncations.Diagrammatic representation of (c) right- and (d) left-canonical con-dition, I is identity matrix.

0 200 400 600 80010-20

10-10

100

2

n=20n=50n=100

=1, n=100=2, n=100

0 200 400n

0

100

200

300

L=12L=16L=20L=24

=0 (b)(a)

L = 48,XXZ chain

= 0,

= 10-8

FIG. 4. (Color online) (a) Fast decaying “entanglement” spectrum ofMPO Hn (n = 20, 50, 100), cut in the middle of the L = 48 XXZchains with ∆ = 0 (XY), 1 (Heisenberg), and 2 (XXZ). (b) showsthe required MPO bond dimensions χ to maintain a fixed truncationerror ε = 10−8 in XY chain of L = 12, 18, 20, 24.

an efficient MPO representation (compressibility) and how thebond dimensions χ scale as order n, as well as system size L,increases.

As afore mentioned, H has a simple MPO representation,which serves as a good starting point to prepare the Hn se-ries [in Fig.1(b)]. Suppose we have obtained MPO of Hn−1

(with bond dimension χ), in order to get the next MPOHn wehave to project a singleH ontoHn−1, obtaining an MPO with“fat” bond dimension χD. In order to prevent bond dimen-sions from this exponential growth as n increases, a propertruncation here is crucial.

Note that the enlarged MPO Hn generically deviatesfrom the so-called canonical form, and thus we first per-form a right-to-left sweep process, gauging the tensors intotheir right-canonical form. In each single operation, we re-shape T iαi,pi,qi,γi into a matrix and take a singular value de-composition T iαi;(pi,qi,γi) =

∑χDαi=1 Uαi,αiΛαiV

†αi,(pi,qi,γi)

.

Then, we update T i = V †, and associate Uαi,αiΛαi tothe T i−1 on the left [as shown in Fig. 3(a)]. It is clearthat T i has been gauged into a right canonical form, i.e.,∑

(pi,qi,γi)T iα,(pi,qi,γi)T

i∗α′,(pi,qi,γi)

= δα,α′ [Fig. 3(c)]. Weiteratively perform this canonicalization process on each site,all the way from right to left, gauging all tensors to theirright canonical forms. Subsequently, we perform a trunca-tion sweep, analogous to the previous procedure, but from leftto right. As this right-moving sweep proceeds, we have allleft-canonical tensors [see Fig. 3(d)] to the left of Λαi andall tensors right-canonical on the right-hand side. Note that

under this condition SVD actually plays the role as a globalSchmidt decomposition of the supervectorHn, and the singu-lar values in Λ are the (nonnegative) Schimdt weights, provid-ing a quasi-optimal truncation criteria. We keep the importantbond bases corresponding to large Λα and discard those hav-ing small weights.

In Fig. 4(a) we plot the normalized Λ2α in obtained canoni-

cal MPOsHn, from which we see very fast decaying weights.Therefore, it is only necessary to keep a small number (χ) ofstates out of the exponentially large bond space, while obtain-ing a good approximation to full Hn. In Fig. 4(b) we showthat to maintain the fixed truncation error ε (sum of discardedΛ2α weights, ε = 10−8 here), the required bond dimension

χ firstly increases and then saturates as order n increases. Itis also observed that as the system size L enhances, both thesaturation χ and convergence point n increases, revealing thatthe convergence behavior of χ is actually due to finite-sizeeffects. Interestingly, for small n (say, < 100), required χ de-creases as L enlarges; while for very large n one needs largerχ to maintain the same truncation error for longer L. Overall,in the models examined, we find that the MPO indeed con-stitutes an efficient representation for Hn, for any n and L[18].

In conventional Trotter-Suzuki TTN methods (TMRG,LTRG, etc), time costs of a local operation scale as O(χ3d6);In the SETTN algorithm, this time cost scales in thesame order on χ, as O(χ3D3d2). The memory cost ofSETTN [O(χ2D2d2)] is also slightly larger than that ofTMRG/LTRG [O(χ2d4)]. The number of total projectionsteps in TMRG/LTRG calculations is M = β/τ . For in-stance, when largest β = 50 and τ = 0.05, then M = 1000.In SETTN calculations, take XY-chain as an example, totalnumber of projection steps (to obtain Hn) is ∼ βL/3 (Fig.2), which is less(larger) than 1000 given L < 60(> 60). Toconclude, SETTN bears marginally larger time and memorycosts than LTRG, while achieves the flexibility of continuous(imaginary) time.

Thermodynamics results.— In Fig. 5(a), compared to exactdiagonalization (ED) data FED, we include errors of free en-ergy F , |F − FED|/|FED|, calculated by SETTN and LTRGwith both the first- and fourth-order Trotter-Suzuki decompo-sitions on a XY chain (∆ = 0). We find that the accuracyof SETTN improves continuously as χ enhances, and the rel-ative error reaches 10−11 when χ = 800 even at ultra-lowtemperatures (β = 150, T/J ' 0.0067), which is 5 ∼ 6orders of magnitude smaller than that of LTRG using first-order decomposition [19]. The latter suffers from the Trot-ter error, which is ∼ 10−5 for τ = 0.025 (and 10−6 forτ = 0.005), and cannot be improved by increasing χ. Notethat τ = 0.005 is even not very practical in real-life Trotter-Suzuki RG calculations since it leads to too many projectionsteps (thus costs 8 times the CPU time SETTN consumes withχ = 200). To reach similar accuracy as that of SETTN withχ > 200, extremely small Trotter slice τ is required in thefirst-order decomposition, making it very inefficient and im-practical. We have also compared the fourth-order decompo-

4

0 50 100 150

10-15

10-10

10-5R

elat

ive

Err

ors

of fr

ee e

nerg

y

D=200, =0.025D=200, =0.005D=200, =0.025

D=200, =0.05D=300, =0.025

0 2 4T

0

10

20

30

40

Hea

t Cap

acity

=0, LTRG, =200SETTN, =200

=1, LTRGSETTN

=2, LTRGSETTN

=200

=400

=600

=800

SETTN, N=900

=200L=1004th-Order LTRG

(a) (b)

1st-Order LTRG

FIG. 5. (Color online) (a) Relative errors of free energy of L = 14XY chain, obtained by SETTN and LTRG, down to ultra-low tem-peratures. (b) Heat capacity of L = 100 XXZ chains of ∆ = 0, 1, 2,where the SETTN and LTRG results agree perfectly with each other.

sition with SETTN in the high accuracy regime. To achievethe accuracy of ∼ 10−9 (10−10), the fourth-order calculationtakes 4(5) times the CPU time SETTN costs. Therefore, theSETTN approach proposed in this work, we think, establishesan unprecedented standard of finite-temperature calculations,surpassing previous (world-line) TTN methods in both accu-racy and efficiency.

Thermodynamic properties other than free energy canalso be conveniently evaluated in the SETTN frame-work. For instance, the energy E(β) = Tr(He−βH)

Tr(e−βH)=∑N

n=0 ωn(β)Tr(Hn+1)∑N

n=0 ωn(β)Tr(Hn)

can be easily calculated with obtainedTr(Hn) series. In Fig. 5(b) we show results of heat capac-ity C = −β2(∂E∂β ) of large-scale (L = 100) XXZ chain with∆ = 0, 1, 2. Since ED is no longer possible for such largesystem, we compare SETTN and LTRG (adapted to finite-sizesystems) results, and see perfect agreements in all cases.

Not limited in 1D, SETTN method can also be generalizedto calculate lattice models of higher dimensions [25]. In Fig.6(a) we show the results of the quantum Ising model on a 4×4square lattice, whose Hamiltonian can be represented by MPOfollowing a snake-like line, as depicted in the inset. As in 1D,the relative errors of free energy results (compared to ED data)improve continuously as χ enhances, and are stimulatinglysmall for even moderate χ’s, revealing that SETTN can beemployed to calculate thermodynamics of 2D lattice modelswith high accuracy.

Time-dependent correlation functions.— With efficientMPO representations of Hn series at hand, we can re-cycle them to calculate dynamical correlation functions〈Szi (t)Szj (0)〉β = Tr(e−βHeiHtSzi e

−iHtSzj ). By expand-

ing eiHδ =∑Nn=0

(iδ)n

n! Hn at a relatively small time δ

(= 0.1, say, in practical simulations), we can find a Trotter-error-free MPO representation Q(δ) for eiHδ , by minimizethe distance of Q(δ) with the Hn series. Given an ac-curate Q(δ) obtained, the Heisenberg operator O(−t) =e−iHtOeiHt = [Q(δ)†]KO[Q(δ)]K can also be efficiently ex-pressed as an MPO [where K = t/δ, O = ρ, Szi , etc.], withwhich the dynamical correlation function can be evaluated asTr{ρ(β,−t)Szi Szj (−t)}.

0 5 10 15 20

10 -15

10 -10

10 -5

Re

lati

ve

Err

ors

of

Fre

e E

ne

rgy

0

0.05

0.1

0.15

0.2

0.25

Re

[ S

iz(t

) S

jz

]

E.D. Heisenberg model, L = 12, i = j = 4

SETTN, N=10, =200

=400

0 5 10 15 20

t J

10 -8

10 -5

10 -2

Err

ors

LTRG, =0.05, =200

(a) (b)

(c)

=100,200,

400,600,800

SETTN N=140

FIG. 6. (Color online) (a) The relative errors of free energy inSETTN calculations, with χ = 100, 200, 400, 600, 800, respec-tively, for a 4×4 square-lattice Ising model H =

∑〈i,j〉 S

zi S

zj −

h∑i S

xi (with h = 1). (b) The real part of time-dependent corre-

lation functions 〈Szi (t)Szj (0)〉β at β = 1. (c) The errors of SETTNcalculations with χ = 200, 400, respectively, compared to the LTRGones with time step δ = 0.05.

Figs. 6(b,c) show the real part of 〈Szi (t)Szj (0)〉β for aHeisenberg chain model at finite temperature (β = 1). In thecalculations, we expand the exponential operators Q(δ) andρ(β) up to N = 10 orders, and retain χ = 200, 400 states onthe bonds of the MPOs O(−t) (O = ρ, Szj ). Fig. 6(b) showsthat, for time up to tJ = 20, the results agree perfectly withthe ED data. In Fig. 6(c), comparing with Trotter-Suzuki cal-culations (LTRG plus real-time evolution with step δ = 0.05)[21, 22], we find the accuracies of SETTN results are betterthan LTRG in short-time regime and of comparable accura-cies for relatively long time.

Conclusion and outlook.— We propose a series-expansionthermal tensor network (SETTN) approach for accurate cal-culations of equilibrium and dynamical properties of quan-tum manybody systems. We show that (a) only a small num-ber of weights in the summation

∑Nn=0

(−β)nn! Tr(Hn) con-

tribute to the partition function, and the upper bound of or-der is N ∼ βL|eln|, where eln is the energy eigenvalue oflargest norm; and (b) eachHn (regardless of the length L) canbe efficiently expressed as a matrix product operator, judgedfrom its fast-decaying “entanglement” spectrum. With thesetwo nice properties, we are able to realize a Trotter-error-freecontinuous-time thermal tensor network method, whose ex-traordinarily high accuracy and flexibility are benchmarkedby calculations of XXZ spin chains and square-lattice quan-tum Ising model.

There are still some open questions deserving further in-vestigations. To name a few, application of SETTN method tomore challenging 2D systems is surely appealing; Implemen-tation of symmetries can improve the efficiency in SETTN;In the Trotter-Suzuki framework, evaluating thermodynamicproperties of models with long-range interactions is tricky,whereas since the MPO representations of such Hamiltoni-ans can be systematically constructed [16], applying SETTNapproach to such systems is inspiring; Besides Taylor expan-sion, exploiting instead Chebyshev expansion in SETTN isalso quite motivated [26–29].

5

Acknowledgments.— WL is indebted to Andreas Weichsel-baum, Jan von Delft, Yong-Liang Dong, Hong-Hao Tu, LeiWang, and Zi-Yang Meng for stimulating discussions. Thiswork was supported by the National Natural Science Founda-tion of China (Grant Nos. 11504014, 11274033, 11474015and 61227902), the Research Fund for the Doctoral Programof Higher Education of China (Grant No. 20131102130005),and the Beijing Key Discipline Foundation of CondensedMatter Physics.

∗ w.li@buaa.edu.cn[1] R.J. Bursilly, T. Xiang and G.A. Gehring, The density matrix

renormalization group for a quantum spin chain at non-zerotemperature, J. Phys.: Condes. Matter 8, L583 (1996); X. Wangand T. Xiang, Transfer-matrix density-matrix renormalization-group theory for thermodynamics of one-dimensional quantumsystems, Phys. Rev. B 56, 5061 (1997); T. Xiang, Thermody-namics of quantum Heisenberg spin chain, Phys. Rev. B 589142 (1998).

[2] W. Li, S.J. Ran, S.S. Gong, Y. Zhao, B. Xi, F. Ye, and G. Su,Linearized Tensor Renormalization Group Algorithm for theCalculation of Thermodynamic Properties of Quantum LatticeModels, Phys. Rev. Lett. 106, 127202 (2011); S.-J. Ran, W.Li, B. Xi, Z. Zhang, and G. Su, Optimized decimation of ten-sor networks with super-orthogonalization for two-dimensionalquantum lattice models, Phys. Rev. B 86, 134429 (2012).

[3] M. Suzuki, Relationship between d-Dimensional Quantal SpinSystems and (d+1)-Dimensional Ising Systems: Equivalence,Critical Exponents and Systematic Approximants of the Par-tition Function and Spin Correlations, Prog. Theor. Phys. 56,1454 (1976).

[4] A. W. Sandvik and J. Kurkijarvi, Quantum Monte Carlo simula-tion method for spin systems, Phys. Rev. B 43, 5950 (1991); A.W. Sandvik, A generalization of Handscomb’s quantum MonteCarlo scheme — Application to the 1-D Hubbard model, J.Phys. A 25, 3667 (1992); A.W. Sandvik, Stochastic series ex-pansion method with operator-loop update, Phys. Rev. B 59,R14157 (1999); O. F. Syljuasen and A. W. Sandvik, QuantumMonte Carlo with Directed Loops, Phys. Rev. E 66, 046701(2002).

[5] X. Wang and L. Yu, Magnetic-Field Effects on Two-Leg Heisen-berg Antiferromagnetic Ladders: Thermodynamic Properties,Phys. Rev. Lett. 84, 5399 (2000); J. Lou, T. Xiang, and Z.B. Su,Thermodynamics of the Bilinear-Biquadratic Spin-One Heisen-berg Chain, Phys. Rev. Lett. 85, 2380 (2000); B. Gu andG. Su, Comment on “Experimental Observation of the 1=3Magnetization Plateau in the Diamond-Chain Compound Cu3

(CO3)2OH2”, Phys. Rev. Lett. 97, 089701 (2006).[6] X. Yan, W. Li, Y. Zhao, S.-J. Ran, and G. Su, Phase diagrams,

distinct conformal anomalies, and thermodynamics of spin-1bond-alternating Heisenberg antiferromagnetic chain in mag-netic fields, Phys. Rev. B 85, 134425 (2012).

[7] H.F. Trotter, On the product of semi-groups of operators, Proc.Amer. Math. Soc. 10, 545-551 (1959).

[8] J.E. Hirsch, R.L. Sugar, D.J. Scalapino, and R. Blankenbecler,Monte Carlo simulations of one-dimensional fermion systems,Phys. Rev. B 26, 5033 (1982).

[9] S.R. White, Density matrix formulation for quantum renormal-ization groups, Phys. Rev. Lett. 69, 2863 (1992).

[10] A.E. Feiguin and S.R. White, Finite-temperature density matrixrenormalization using an enlarged Hilbert space, Phys. Rev. B72, 220401(R) (2005).

[11] Higher-order Trotter-Suzuki decomposition could also reducethe Trotter error, but cost more projection steps as well (scalesexponentially in fractal decomposition construction as order en-hances); extrapolation based on finite τ calculations [see M.C.Banuls, K. Cichy, J.I. Cirac, K. Jansen, and H. Saito, Thermalevolution of the Schwinger model with matrix product opera-tors, Phys. Rev. D 92 034519 (2015)] will also be limited byexotrapolation errors, or even becomes inaccurate due to inter-correlated truncation and Trotter errors.

[12] D.C. Handscomb, D. C. Handscomb, The Monte Carlo methodin quantum statistical mechanics, Math. Proc. Cambridge Phil.Soc. 58, 594 (1962); A Monte Carlo method applied to theHeisenberg ferromagnet, Math. Proc. Cambridge Phil. Soc. 60,115 (1964).

[13] J.W. Lyklema, Quantum-Statistical Monte Carlo Method forHeisenberg Spins, Phys. Rev. Lett. 49, 88 (1982).

[14] D.H. Lee, J.D. Joannopoulos, and J.W. Negele, Monte Carlo so-lution of antiferromagnetic quantum Heisenberg spin systems,Phys. Rev. B 30, 1599 (1984).

[15] F. Frowis, V. Nebendahl, and W. Dur, Tensor operators: Con-structions and applications for long-range interaction systems,Phys. Rev. A 81, 062337 (2010).

[16] B. Pirvu, V. Murg, J.I. Cirac, and F. Verstraete, Matrix productoperator representations, New J. Phys. 12 025012 (2010).

[17] M.P. Zaletel, R.S.K. Mong, C. Karrasch, J.E. Moore, and F.Pollmann, Time-evolving a matrix product state with long-ranged interactions, Phys. Rev. B 91, 165112 (2015).

[18] Note that the compression scheme is based on the Frobeniusnorm of MPO, which is convenient to deal with in the tensornetwork algorithms and are widely adopted to compress MPOs.Nevertheless, an optimization scheme using instead the opera-tor norm still remains to be explored.

[19] The calculations here are performed using bilayer LTRGadapted to finite-size system, which is essentially equivalentto finite-T DMRG in Ref. 10 but reformulated in tensor-network language. Bilayer LTRG algorithm can be found inY.-L. Dong, L. Chen, Y.-J. Liu, and W. Li, Bilayer LinearizedTensor Renormalization Group Approach for Thermal TensorNetworks, arXiv:1612.01896 (2016).

[20] J. Sirker and A. Klumper, Real-time dynamics at Finite tem-perature by the density-matrix renormalization group: A path-integral approach, Phys. Rev. B 71, 241101(R) (2005).

[21] C. Karrasch, J.H. Bardarson, and J.E. Moore, Finite-Temperature Dynamical Density Matrix RenormalizationGroup and the Drude Weight of Spin-1=2 Chains, Phys. Rev.Lett. 108, 227206 (2012).

[22] M.C. Banuls, M.B. Hastings, F. Verstraete, and J.I. Cirac, Ma-trix Product States for Dynamical Simulation of Infinite Chains,Phys. Rev. Lett. 102, 240603 (2009).

[23] Y.-K. Huang, P. Chen, Y.-J. Kao, and T. Xiang, Long-time dy-namics of quantum chains: Transfer-matrix renormalizationgroup and entanglement of the maximal eigenvector, Phys. Rev.B 89, 201102(R) (2014).

[24] To calculate the time-dependent correlation functions, we fol-low the line developed in Ref. 21, i.e., not only the physicalbut also the ancillary degrees of freedom in the matrix productoperators are evolving, forward and backward in time, respec-tively.

[25] For 2D system, we can make use of a MPO or tensor prod-uct operator representation for the Hamiltonian: in the former,the system is treated as 1D chain with “long-range” interac-

6

tions, and thus the bond dimension of MPO scales linearly asthe width of the lattice; while the latter has a fixed bond dimen-sion regardless of the system size. The details of 2D SETTNcalculations will be published elsewhere.

[26] A. Holzner, A. Weichselbaum, I. P. McCulloch, U. Schollwockand J. von Delft, Chebyshev matrix product state approach forspectral functions, Phys. Rev. B 83, 195115 (2011).

[27] A. Braun and P. Schmitteckert, Numerical evaluation ofGreen’s functions based on the Chebyshev expansion, Phys.

Rev. B 90, 165112 (2014).[28] A.C. Tiegel, S. R. Manmana, T. Pruschke and A. Honecker,

Matrix product state formulation of frequency-space dynamicsat finite temperatures, Phys. Rev. B 90, 060406 (2014)

[29] B. Bruognolo, A. Weichselbaum, J. von Delft and M. Garst, Dy-namic structure factor of the spin-12 XXZ chain in a transversefield, Phys.Rev. B 94, 085136 (2016).