Estudos sobre as equações de Bethe

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Ficha catalográfica elaborada pelo DePT da Biblioteca Comunitária/UFSCar

V658es

Vieira, Ricardo Soares.

Estudos sobre as equações de Bethe / Ricardo Soares Vieira. -- São Carlos : UFSCar, 2015.

106 f.

Tese (Doutorado) -- Universidade Federal de São Carlos, 2015.

1. Física estatística. 2. Bethe, Equações de. 3. Modelo de seis vértices. 4. Polinômios auto-inversiveis. Polinômios de Salem. 5. Polinômios de Salem. 6. Ansatz, Bethe. I. Título.

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Resumo

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Abstract

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Sumário

Sumário . . . 11

1 INTRODUÇÃO . . . 15

1.1 O Ansatz de Bethe 15

1.2 As equações de Bethe 17

I

O ANSATZ DE BETHE

19

2.1 O Modelo de Heisenberg em uma dimensão 21

2.2 O Ansatz de Bethe de coordenadas em ação: diagonalizando o hamil-toniano 23

2.2.1 Estados de spins 23

2.2.2 O Estado de referência 24

2.2.3 Estados de um mágnon 24

2.2.4 Estados de dois mágnons 26

2.2.5 Estados de três mágnons 29

2.2.6 Estados deN mágnons 30

3 O ANSATZ DE BETHE ALGÉBRICO . . . 33

3.1 O modelo de seis vértices 33

3.1.1 A mecânica estatística do modelo de seis vértices 34

3.1.2 Representação matricial para a função de monodromia e de transferência 36 3.1.3 A equação de Yang-Baxter 38

3.1.4 A integrabilidade do modelo de seis vértices 40

3.2 O Ansatz de Bethe algébrico em ação: diagonalizando da matriz de transferência 41

3.2.1 As relações de comutação 42

3.2.2 O estado de referência 42

3.2.3 Estados de um mágnon 43

3.2.4 Estados de dois mágnons 45

3.2.5 Estados deN mágnons 47

II

AS EQUAÇÕES DE BETHE

55

4 RELAÇÃO ENTRE OS ANSÄTZE DE BETHE ALGÉBRICO E DE

COORDENADAS . . . 57

5 COMPARAÇÃOENTREASDUASFORMASDASEQUAÇÕESDE BETHE. . . 59

6 SOLUÇÕES DAS EQUAÇÕES DE BETHE PARA OS ESTADOS DE UM MÁGNON . . . 63

7 SOLUÇÕES DAS EQUAÇÕES DE BETHE PARA OS ESTADOS DE DOIS MÁGNONS . . . 65

7.1 A forma polinomial das equações de Bethe 65

7.2 Polinômios auto-inversivos 67

7.3 Teoremas sobre o número de raízes de um polinômio auto-inversivo sobre o círculo complexo unitário 68

7.4 Localização e distribuição das raízes dePa(z) 71

7.5 Polinômios de Salem 75

7.5.1 Um novo algoritmo para procurar por números de Salem pequenos 77

7.6 Distribuição das raízes de Bethe−A hipótese de strings 80

7.7 A completeza do Ansatz de Bethe paraN = 2 83

7.7.1 A questão da completeza para valores críticos de∆ 85

8 SOLUÇÕES DAS EQUAÇÕES DE BETHE PARA OS ESTADOS DE TRÊS MÁGNONS . . . 87

9 CONCLUSÃO E PERSPECTIVAS . . . 91

A SOLUÇÕESEXPLÍCITASDASEQUAÇÕESDEBETHEPARAN = 2EN = 3 . . . 93

A.1 Soluções explícitas das equações de Bethe paraN = 2 93

A.1.1 Soluções explícitas paraL = 1 93

A.1.2 Soluções explícitas paraL = 2 94

A.1.3 Soluções explícitas paraL = 3 94

A.2 Soluções explícitas das equações de Bethe paraN = 3. 95

A.2.1 Soluções explícitas paraL = 1 95

A.2.2 Soluções explícitas paraL = 2 96

A.2.4 Soluções explícitas paraL = 4 98

CAPÍTULO

1

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1.1 O Ansatz de Bethe

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NFOUF SFEV[JNPT BT FRVB§µFT EF #FUIF OP DBTP DPOTJEFSBEP BT FRVB§µFT BTTPDJBEBT BP NPEFMP 99; EF TFJT W©SUJDFT DPN DPOEJ§µFT EF DPOUPSOP QFSJ³EJDBT B VN TJTUFNB EF FRVB§µFT QPMJOPNJBJT EF NPEP RVF BT FRVB§µFT EF #FUIF QBTTBN B TFS VN PCKFUP EF FTUVEP EB HFPNFUSJB BMH©CSJDB F EB ¡MHFCSB DPNVUBUJWB 1BSB PT DBTPT NBJT TJNQMFT PT TFUPSFT EF F N¡HOPOT NPTUSBNPT RVF FTUF TJTUFNB QPEF TFS EFTBDPQMBEP F RVF BT TPMV§µFT SFTQFDUJWBT QPEFN TFS FTDSJUBT FN UFSNPT EBT SB­[FT EF DFSUPT QPMJO´NJPT BVUPJOWFSTJWPT 1PS NFJP EF UFPSFNBT EFEV[JEPT BRVJ BDFSDB EB " BO¡MJTF EB EJTUSJCVJ§£P EBT SB­[FT EPT QPMJO´NJPT BVUPJOWFSTJWPT <> ஷ஫[FNPT VNB BO¡MJTF QSPGVOEB EBT TPMV §µFT EBT FRVB§µFT EF #FUIF P RVF OPT QFSNJUJV FODPOUSBS B EJTUSJCVJ§£P EBT SB­[FT EF #FUIF EJTDVUJS B RVFTU£P EB DPNQMFUF[B EP "OTBU[ EF #FUIF FODPOUSBS VNB DPOFY£P FOUSF BT FRVB§µFT EF #FUIF F PT QPMJO´NJPT EF 4BMFN VTVBMNFOUF FTUVEBEPT OB UFPSJB BMH©CSJDB EPT OºNFSPT FUD

" QSFTFOUF UFTF FTU¡ PSHBOJ[BEB EB TFHVJOUF GPSNB OB QSJNFJSB QBSUF BQSFTFOUBSFNPT P "O TBU[ EF #FUIF POEF EJTDVUJSFNPT BT EVBT GPSNVMB§µFT NBJT DPNVOT EP "OTBU[ EF #FUIF B TBCFS P "OTBU[ EF DPPSEFOBEBT F P BMH©CSJDP /B TFHVOEB QBSUF GPDBSFNPT OBT FRVB§µFT EF #FUIF UFNB DFOUSBM EFTUB UFTF POEF B U©DOJDB EFTFOWPMWJEB QPS O³T QBSB PCUFS TVBT TPMV§µFT TFS¡ EFTDSJUB %JTDVUJ SFNPT FN EFUBMIFT BT FRVB§µFT EF #FUIF QBSB P NPEFMPX X ZEF )FJTFOCFSH BTTPDJBEBT BPT TFUPSFT EF VN F EPJT N¡HOPOT 5FPSFNBT TPCSF QPMJO´NJPT BVUPJOWFSTJWPT EFEV[JEPT EVSBOUF B OPTTB QFT RVJTB TFS£P BQSFTFOUBEPT F B DPOFY£P FOUSF BT FRVB§µFT EF #FUIF DPN PT DIBNBEPT QPMJO´NJPT EF 4BMFN TFS£P UBNC©N BCPSEBEPT 5BNC©N BQSFTFOUBSFNPT VN BMHPSJUNP OPWP QBSB TF QSPDVSBS QPS OºNFSPT EF 4BMFN QFRVFOPT " RVFTU£P EB DPNQMFUF[B EP "OTBU[ EF #FUIF QBSB P TFUPS EF EPJT N¡HOPOT TFS¡ EJTDVUJEB FN TFHVJEB F OP PVUSP DBQ­UVMP BMHVOT SFTVMUBEPT QBSB P TFUPS EF USªT N¡H OPOT TFS£P BQSFTFOUBEPT 1PS ஷ஫N OPT BQªOEJDFT BQSFTFOUBSFNPT TPMV§µFT FYQMJDJUBT EBT FRVB§µFT EF #FUIF QBSB PT TFUPSFT EF EPJT F USªT N¡HOPOT QBSB QFRVFOPT WBMPSFT EFL

Parte I

CAPÍTULO

2

0 "/4"5; %& #&5)& %& $003%&/"%"4

2.1 O Modelo de Heisenberg em uma dimensão

"T FRVB§µFT EF #FUIF FNFSHJSBN FN ଍­TJDB OP FTUVEP QJPOFJSP EF )BOT #FUIF TPCSF P NP EFMP EF )FJTFOCFSH FN VNB EJNFOT£P <> 0 NPEFMP EF )FJTFOCFSH DPOTJTUF FN VNB SFEF EF QBSU­ DVMBT RVF JOUFSBHFN BUSBW©T EF TFVT TQJOT PT RVBJT QPEFN OP DBTP NBJT HFSBM BQPOUBS QBSB RVBMRVFS EJSF§£P EP FTQB§P <> /P DBTP VOJEJNFOTJPOBM RVF WBNPT DPOTJEFSBS BT QBSU­DVMBT FTUBS£P EJTQPT UBT TPCSF VNB SFUB JOஷ஫OJUB PV DBTP TFKB JNQPTUP DPOEJ§µFT EF DPOUPSOP QFSJ³EJDBT FN VN D­SDVMP $POTJEFSBSFNPT UBNC©N RVF BT QBSU­DVMBT JOUFSBHFN BQFOBT DPN PT TFVT WJ[JOIPT NBJT QS³YJNPT

0 IBNJMUPOJBOP EF )FJTFOCFSH RVF EFTDSFWF VN TJTUFNB EF L QBSU­DVMBT RVF JOUFSBHFN BUSBW©T EF TFVT TQJOT QPEF TFS FTDSJUP DPNP

H =H0− 1

2

L

!

k=1

Jxσ_kxσ_kx₊₁+Jyσ y

kσ

y

k+1+Jzσ

z kσ

z k+1,

POEF H0 © VNB DPOTUBOUF BSCJUS¡SJB RVF EFஷ஫OF P [FSP EF FOFSHJB Jx Jy F Jz T£P DPOTUBOUFT EF BDPQMBNFOUP RVF EFTDSFWFN BT JOUFSB§µFT OBT EJSF§µFTXYFZF

σx = ⎡⎢⎢⎢

⎢⎣

0 1 1 0

⎤⎥ ⎥⎥

⎥⎦, σy = ⎡⎢⎢⎢⎢⎣

0 ₋i i 0

⎤⎥ ⎥⎥

⎥⎦, σz = ⎡⎢⎢⎢⎢⎣

1 0 0 ₋1

⎤⎥ ⎥⎥ ⎥⎦,

T£P BT NBUSJ[FT EF 1BVMJ POEF

σ_ka= I1 ⊗· · ·⊗Ik−1⊗!σa"k ⊗ Ik+1⊗· · ·⊗IL, a ∈{x,y,z",

T£P PQFSBEPSFT EFஷ஫OJEPT_{OP &OE(V}

1⊗· · ·⊗Vk ⊗· · ·⊗VL)RVF BUVBN O£PUSJWJBMNFOUF BQFOBT

OP FTQB§P WFUPSJBMVk

/PUFNPT RVF B JOUFSB§£P EF FOUSF EVBT QBSU­DVMBT WJ[JOIBT EB SFEF © EFTDSJUB QFMP IBNJMUP OJBOP

Hk,k+1 =−

1 2

"

Jxσ_kxσ_kx₊₁+Jyσ y

kσ

y

k+1+Jzσ

z kσ z k+1 # ,

EF NPEP RVF QPEFNPT FTDSFWFS

H =H0+

L

!

k=1

Hk,k+1.

$BTPT FTQFDJBJT EP NPEFMPXY Z QPEFN TFS PCUJEPT JNQPOEP SFTUSJ§µFT  T DPOTUBOUFT EF BDPQMBNFOUP 1PS FYFNQMP DPMPDBOEPTF Jx = Jy PCUFNPT P DIBNBEP NPEFMP X X Z OP RVBM BQFOBT B JOUFSB§£P OB EJSF§£PZ© EJGFSFOUF EBT EFNBJT₋P TQJO EBT QBSU­DVMBT QPEF UFS BHPSB BQFOBT EPJT WBMPSFTupPV down 4F BM©N EJTTP DPMPDBSNPT Jx = Jy = Jz PCUFNPT P NPEFMPX X X DVKBT JOUFSB§µFT O£P EFQFOEFN EB EJSF§£P EF TQJO FTTF GPJ P NPEFMP DPOTJEFSBEP QSJNFJSBNFOUF QPS #FUIF

/B TFRVªODJB WBNPT DPOTJEFSBS BQFOBT P NPEFMP X X Z QPJT WFSFNPT NBJT BEJBOUF RVF FTUF NPEFMP VOJEJNFOTJPOBM FTU¡ SFMBDJPOBEP DPN P NPEFMP EF TFJT W©SUJDFT EFஷ஫OJEP FN VNB SFEF CJEJNFOTJPOBM 1BSB P NPEFMPX X Z P IBNJMUPOJBOP UPSOBTF

HX X Z = H0+

L

!

k=1

Hk,k+1= H0−

J

2

L

!

k=1

σx_kσ_kx₊₁+σy_kσy_k+1+∆σ_kzσz_k+1,

POEF DPMPDBNPTJy = Jx = JFJz = J∆ EF NPEP RVF

Hk,k+1=−

J

2

"

σx_kσx_k+1+σy

kσ

y

k+1+∆σ

z kσ z k+1 # .

4FS¡ DPOWFOJFOUF JOUSPEV[JS PT PQFSBEPSFT

Pk,k+1 =σx_kσx_k+1+σ_kyσ_ky₊₁, F Qk,k+1 =σ_kzσ_kz₊₁,

F EFஷ஫OJS

P =

L

!

k=1

Pk,k+1, Q=

L

!

k=1

Qk,k+1,

EF NPEP RVF P IBNJMUPOJBOP QPTTB TFS FTDSJUP TJNQMFTNFOUF DPNP

Hk,k+1 =−

J

2 (Pk,k+1+∆Qk,k+1) F H =H0−

J

2(P+∆Q).

2.2 OAnsatz de Bethe de coordenadas em ação:diagonalizando

o hamiltoniano

" TFHVJS WBNPT NPTUSBS DPNP P "OTBU[ EF #FUIF EF DPPSEFOBEBT QFSNJUF EJBHPOBMJ[BS P IBNJMUPOJBOP EF BDPSEP DPN P OºNFSP EF N¡HOPOT EPT FTUBEPT RV¢OUJDPT

2.2.1 Estados de spins

6NB WF[ RVF BT QBSU­DVMBT DPNQPOFOUFT EP TJTUFNB FN DPOTJEFSB§£P JOUFSBHFN BQFOBT QPS NFJP EF TFVT TQJOT PT FTUBEPT RV¢OUJDPT BTTPDJBEPT QPEFN TFS FTDSJUPT FN UFSNPT EPT TQJOT EBT QBSU­DVMBT RVF QPEFN BQPOUBS BQFOBT OBT EJSF§µFTupF down 3FQSFTFOUBOEP QPS |↑⟩F |↓⟩ PT

FTUBEPT BTTPDJBEPT B VNB QBSU­DVMB DPN TQJOupF down SFTQFDUJWBNFOUF QPEFNPT JOUSPEV[JS B

TFHVJOUF SFQSFTFOUB§£P NBUSJDJBM

|_↑⟩= ⎡⎢⎢⎢

⎢⎣

1 0

⎤⎥ ⎥⎥

⎥⎦, |↓⟩= ⎡⎢⎢⎢⎢⎣

0 1

⎤⎥ ⎥⎥ ⎥⎦,

DPN|_↑⟩F|_↓⟩EFஷ஫OJEPT OP FTQB§P EF )JMCFSUV EF EJNFOT£P %FTUF NPEP QPEFNPT FODPOUSBS B B§£P EBT NBUSJ[FT EF 1BVMJ TPCSF|_↑⟩F|_↓⟩

σx|_↑⟩ = |_↓⟩,

σy_{|↑⟩ = i|↓⟩,} σz|_↑⟩ = |_↑⟩,

σx|_↓⟩ = |_↑⟩,

σy_{|↓⟩ = −i|↑⟩,} σz|_↓⟩ = −|_↓⟩.

"USBW©T EFTUBT SFMB§µFT QPEFNPT FODPOUSBS GBDJMNFOUF B B§£P EPT PQFSBEPSFTPk,k+1FQk,k+1TPCSF

PT FTUBEPT BTTPDJBEPT B EVBT QBSU­DVMBT WJ[JOIBT EB SFEF MPDBMJ[BEBT EJHBNPT OB QPTJ§£P xk Fxk+1

EB SFEF POEF1≤ k _≤ L JTUP © PT FTUBEPT

|_{↑↑⟩k,k+1}=|_{↑⟩k ⊗} |_↑⟩k+1, |↑↓⟩_k,k+1 =|↑⟩k ⊗|↓⟩k+1, |_↓↑⟩k,k+1=|↓⟩k ⊗ |↑⟩k+1, |↓↓⟩k,k+1 =|↓⟩k ⊗|↓⟩k+1,

RVF FTU£P EFஷ஫OJEPT OP FTQB§P UFOTPSJBMVk,k+1 = Vk ⊗Vk+1 1FMBT FRVB§µFT © G¡DJM WFSJஷ஫DBS

RVF

Pk,k+1|↑↑⟩k,k+1 = 0|↑↑⟩k,k+1,

Pk,k+1|↑↓⟩k,k+1 = 2|↓↑⟩k,k+1,

Pk,k+1|↓↑⟩k,k+1 = 2|↑↓⟩k,k+1,

Pk,k+1|↓↓⟩k,k+1 = 0|↓↓⟩k,k+1,

Qk,k+1|↑↑⟩k,k+1 = +|↑↑⟩k,k+1,

Qk,k+1|↑↓⟩k,k+1 = −|↑↓⟩k,k+1,

Qk,k+1|↓↑⟩k,k+1 = −|↑↓⟩k,k+1,

Qk,k+1|↓↓⟩k,k+1 = +|↓↓⟩k,k+1,

_{" NFMIPS BQSFTFOUB§£P RVF QVEF FODPOUSBS EP "OTBU[ EF #FUIF EF DPPSEFOBEBT FTU¡ QSFTFOUF OP DBQ­UVMP EP MJWSP}

-FDUVSॻ PO 1IZTJDT 7PM *** EF 3 1 'FZONBO <> &NCPSB 'FZONBO O£P UFOIB BQSFTFOUBEP BT FRVB§µFT EF #FUIF OFTUB EJTDVTT£P FMF DPOTJEFSB VNB BQSPYJNB§£P TVஷ஫DJFOUF QBSB P DPOUFYUP EP MJWSP B TJNQMJDJEBEF EF TVBT JE©JBT F B EJTDVTT£P ଍­TJDB EBEB QFMP BVUPS T£P DPNP TFNQSF HFOJBJT 'FZONBO FTUFWF JOUFSFTTBEP OP "OTBU[ EF #FUIF OPT ºMUJNPT BOPT EF TVB WJEB " SFMB§£P EF 'FZONBO DPN P "OTBU[ EF #FUIF © NVJUP CFN DPOUBEB FN <>

&TUBT SFMB§µFT TFS£P NVJUP ºUFJT RVBOEP EJBHPOBMJ[BSNPT P IBNJMUPOJBOP BUSBW©T EP "OTBU[ EF #FUIF EF DPPSEFOBEBT 0 QSJNFJSP QBTTP QBSB TF EJBHPOBMJ[BS P IBNJMUPOJBOP © WFSJஷ஫DBS RVF FTUF IBNJMUPOJBOP DPNVUB DPN B DPNQPOFOUFZEP TQJO UPUBM EP TJTUFNB

Sz = 1 2

L

!

k=1

σz_k,

P RVF © VNB DPOTFRVªODJB EB SFMB§£P EF DPNVUB§£P[Sx,Sy] =iSz *TTP TJHOJஷ஫DB RVF B DPNQPOFOUF

Z EP TQJO UPUBM EP TJTUFNB © DPOTFSWBEB F QPSUBOUP B B§£P EP IBNJMUPOJBOP FN VN EBEP FTUBEP RV¢OUJDP O£P BMUFSB P OºNFSP EF TQJOTdownFupBTTPDJBEPT B FTUF FTUBEP &TUB QSPQSJFEBEF © NVJUP

CFN WJOEB QPJT QFSNJUF EJBHPOBMJ[BS P IBNJMUPOJBOP QPS TFUPSFT EF BDPSEP DPN OºNFSP EF TQJOT downFupEPT FTUBEPT RV¢OUJDPT 6N FTUBEP RV¢OUJDP DPNNTQJOTdown© DIBNBEP EF VN FTUBEP

EFNN¡HOPOT /PUF RVF B DPNQPOFOUFZEP TQJO UPUBM EB SFEF FTU¡ SFMBDJPOBEB DPN P OºNFSP EF N¡HOPOT QFMB G³SNVMBSz = L/2−N.

2.2.2 O Estado de referência

$PNFDFNPT QFMP FTUBEP NBJT TJNQMFT RVF © P FTUBEP EF [FSP N¡HOPOT &YJTUF BQFOBT VN FTUBEP EFTTF UJQP OP RVBM UPEPT PT TQJOT EBT QBSU­DVMBT T£P EP UJQPup &TUF FTUBEP © DIBNBEP EF FTUBEP EF SFGFSªODJB F © EBEP QPS

Ψ0 =|↑⟩1 ⊗· · ·⊗|↑⟩L = ⎡⎢ ⎢⎢ ⎢⎣ 1 0 ⎤⎥ ⎥⎥ ⎥⎦1 ⊗ ...⊗ ⎡⎢⎢⎢ ⎢⎣ 1 0 ⎤⎥ ⎥⎥ ⎥⎦L .

1PEFNPT GBDJMNFOUF FODPOUSBS B B§£P EP IBNJMUPOJBOP FNΨ0 %F GBUP BUSBW©T EBT

FRVB§µFT QPEFNPT WFS RVFPBOVMB_Ψ0 FORVBOUP RVFQDPOUSJCVJ DPN P WBMPSL 1PSUBOUP

QPS NFJP EF PCUFNPT EJSFUBNFOUF RVF

HΨ0 =

$

H0−

J

2∆L

%

Ψ0.

1PEFNPT BHPSB DPMPDBSH0 = J₂∆LB ஷ஫N EF RVF B FOFSHJB SFMBUJWB BP FTUBEP EF SFGFSªODJB TFKB OVMB

2.2.3 Estados de um mágnon

$POTJEFSF BHPSB BT DPOஷ஫HVSB§µFT OBT RVBJT BQFOBT VNB EBT QBSU­DVMBT EB SFEF UFN TQJO down 0T FTUBEPT RV¢OUJDPT BTTPDJBEPT B FTTBT DPOஷ஫HVSB§µFT T£P DIBNBEPT EF FTUBEPT EF VN N¡H

OPO 4FKBφm(xm) ≡ φm B GVO§£P EF POEB BTTPDJBEB B VNB EBEB DPOஷ஫HVSB§£P OB RVBM BQFOBT B

QBSU­DVMB OB QPTJ§£PxmUFN TQJOdown JTUP ©

φm =|↑⟩1 ⊗· · ·⊗|↑⟩m−1⊗|↓⟩m ⊗|↑⟩m+1⊗· · ·⊗ |↑⟩L.

" B§£P EP PQFSBEPSPTPCSFφmQPEF TFS GBDJMNFOUF FODPOUSBEB %F GBUP OPUFNPT QSJNFJSP

RVFφm TFS¡ BOVMBEP QPS UPEPT PT PQFSBEPSFTPk,k+11 ≤ k ≤ L DPN FYDFTT£P EPT PQFSBEPSFT

Pm−1,mFPm,m+1 PT RVBJT GPSOFDFN

Pm−1,mφm = 2φm−1, Pm,m+1φm = 2φm+1.

"TTJN UFNPT RVF

Pφm = 2φm−1+ 2φm+1.

+¡ QBSB B B§£P EFQTPCSFφm WFNPT B QBSUJS EBT FRVB§µFT RVF

Qm−1,mφm =−φm, Qm,m+1φm =−φm,

F

Qk,k+1φm =φm, TF k ! {m−1,m},

F QPSUBOUP PCUFNPT

Qφm = (L−2)φm−2φm = (L−4)φm.

$PN JTTP FODPOUSBNPT RVF

Hφm =H0φm−

J

2 (P+∆Q)φm =−J(φm−1+φm+1−2∆φm).

&ODPOUSBNPT BTTJN DPNP P IBNJMUPOJBOP BUVB FN VN EBEP FTUBEP EF TQJO BTTPDJBEP BP TFUPS EF VN N¡HOPO " GVO§£P EF POEB NBJT HFSBM QFSUFODFOUF BP TFUPS EF VN N¡HOPO DPSSFTQPOEF UPEBWJB B VNB TVQFSQPTJ§£P EBT GVO§µFT EF POEB EFTUF TFUPS PV TFKB EFWFNPT DPOTJEFSBS B DPNCJOB§£P MJOFBS

Ψ1(x1, ...,xL) = L

!

k=1

ak(xk)φk(xk),

POEF PT DPFஷ஫DJFOUFT am(xm) ≡ am DPSSFTQPOEFN   BNQMJUVEF EF QSPCBCJMJEBEF QBSB RVF VNB

QBSU­DVMB OB QPTJ§£PxmUFOIB TQJOdown &TUFT DPFஷ஫DJFOUFT EFWFN BJOEB TFS EFUFSNJOBEPT

1SPKFUBOEP BHPSB P FTUBEPΨ1(x1, ...,xL)OPT FTUBEPT EF TQJOφm(xm)F VTBOEP B FRVB§£P

EF 4DIS¶EJOHFS PCUFNPT B SFMB§£P

E1am =−J(am−1+am+1−2∆am).

6NB WF[ RVF FTUB FRVB§£P TFKB SFTPMWJEB QBSB PT DPFஷ஫DJFOUFTam(xm) P BVUPFTUBEPΨ1(x1, ...,xL)

ஷ஫DBS¡ EFUFSNJOBEP F PT DPSSFTQPOEFOUFT BVUPWBMPSFT QPEFS£P TFS FN TFHVJEB DBMDVMBEPT 1PEFNPT WFSJஷ஫DBS EJSFUBNFOUF RVF P "OTBU[

am(xm) = A1eik1xm,

TBUJTGB[ &TTF "OTBU[ © GBDJMNFOUF KVTUJஷ஫DBEP EP QPOUP EF WJTUB ଍­TJDP 0T FTUBEPT EF VN N¡H OPO QPEFN TFS QFOTBEPT DPNP FTUBEPT BTTPDJBEPT B VNB QTFVEPQBSU­DVMB MJWSF RVF TF NPWF DPN NPNFOUVNk1

4VCTUJUVJOEP FN QPEFNPT FODPOUSBS GBDJMNFOUF B FOFSHJB BTTPDJBEB B FTUFT FT UBEPT 0 SFTVMUBEP ©

E1 =−2J(DPTk1−∆)

F BP JNQPSNPT VNB DPOEJ§£P QFSJ³EJDB EF DPOUPSOP FODPOUSBNPT RVFeik1xm = _eik1xm+L = ei(k1+L) PV TFKB EFWFNPT UFS

eik1L _{= 1}

.

&TUB ºMUJNB SFMB§£P OPT NPTUSB RVFk1QPEF BTTVNJS BQFOBT PT WBMPSFTkn = 2_Lπn,1 ≤ n ≤ L "

FRVB§£P BDJNB © B FRVB§£P EF #FUIF QBSB P TFUPS EF VN N¡HOPO EP NPEFMPX X ZEF )FJTFOCFSH 1BSB DBEB WBMPS EFn JF QBSB DBEB TPMV§£P EB FRVB§£P EF #FUIF PCUFNPT VN WBMPS QBSB B FOFSHJB E1F VNB DPSSFTQPOEFOUF BVUPGVO§£PΨ1

2.2.4 Estados de dois mágnons

0 QS³YJNP DBTP B TFS DPOTJEFSBEP © P DBTP POEF EVBT QBSU­DVMBT EB SFEF UªN TQJOTdown

FORVBOUP RVF BT SFTUBOUFT UªN TQJOup 3FQSFTFOUFNPT QPSφm,n(xm,xn)≡ φm,nB GVO§£P EF POEB

EF VNB EBEB DPOஷ஫HVSB§£P OB RVBM EVBT QBSU­DVMBT EB SFEF OBT QPTJ§µFT xm F xn QPTTVFN TQJO

down 5FNPT BTTJN RVF

φm,n= |↑⟩1⊗· · ·⊗|↑⟩m−1⊗|↓⟩m⊗|↑⟩m+1⊗· · ·⊗|↑⟩n−1⊗|↓⟩n⊗|↑⟩n+1⊗· · ·⊗|↑⟩L.

%FTFKBNPT DBMDVMBS B B§£P EFH TPCSF PT FTUBEPTφm,n 1BSB JTTP DPOW©N DPOTJEFSBS EPJT

DBTPT EJTUJOUPT B TBCFS RVBOEPxn = xm + 1 JTUP © RVBOEP PT EPJT TQJOTdownT£P BEKBDFOUFT F

RVBOEPxn > xm+ 1

ߕVBOEP PT TQJOT O£P T£P BEKBDFOUFT B B§£P EFHTPCSFφm,nQPEF TFS GBDJMNFOUF DBMDVMBEB

VNB WF[ RVF FMB © TFNFMIBOUF BP DBTP BOUFSJPS SFGFSFOUF BPT FTUBEPT EF VN N¡HOPO %F GBUP OB B§£P EFPTPCSFφm,nT³ TPCSFWJWFN RVBUSP UFSNPT B TBCFS

Pm−1,mφm,n = 2φm−1,n, Pn−1,nφm,n = 2φm,n−1,

Pm,m+1φm,n= 2φm+1,n, Pn,n+1φm,n= 2φm,n+1,

P RVF OPT GPSOFDF

P|xm,xn⟩ = (Pm−1,m +Pm,m+1+Pn−1,n+Pn,n+1)φm,n

= 2φm−1,n+ 2φm+1,n+ 2φm,n−1+ 2φm,n+1,

(xn > xm+ 1). _{%FWJEP   DPOEJ§£P QFSJ³EJDB EF DPOUPSOP © TVஷ஫DJFOUF DPOTJEFSBSx}

n >xm

%F NPEP TFNFMIBOUF UFNPT RVF

Qm−1,mφm,n =−φm,n, Qn−1,nφm,n =−φm,n,

Qm,m+1φm,n =−φm,n, Qn,n+1φm,n=−φm,n,

FQk,k+1φm,n =φm,nOPT PVUSPT DBTPT 1PSUBOUP ஷ஫DBNPT DPN

Qφm,n = (L−4)φm,n−4φm,n = (L−8)φm,n, (xn> xm + 1).

"TTJN PCUFNPT RVF

H|xm,xn⟩ = −J(φm−1,n+φm+1,n+φm,n−1+φm,n+1−4∆φm,n)

(xn> xm + 1),

$POTJEFSF BHPSB P DBTP FN RVF PT TQJOTdownT£P BEKBDFOUFT DPNP FNφm,m+1 /FTUF DBTP

OB B§£P EFPFNφm,m+1TPCSFWJWFN BQFOBT PT UFSNPT

Pm−1,mφm,m+1 = 2φm−1,m+1, Pm+1,m+2φm,m+1 = 2φm,m+2,

F BTTJN

Pφm,m+1= 2φm−1,m+1+ 2φm,m+2.

%B NFTNB GPSNB QBSB P PQFSBEPSQ UFNPT RVFQk,k+1φm,m+1 = φm,m+1FYDFUP RVBOEPk =

m₋1PVk =m RVBOEP UFNPT RVF

Qm−1,mφm,m+1 =−φm,m+1, Qm,m+1φm,m+1 =−φm,m+1,

F QPSUBOUP UFNPT

Qφm,m+1= (L−2)φm,m+1−2φm,m+1= (L−4)φm,m+1.

1PS DPOTFHVJOUF

Hφm,m+1=−J(φm−1,m+1+φm,m+2−2∆φm,m+1),

F QBSB B B§£P EFHTPCSFφm−1,m WBNPT PCUFS VN SFTVMUBEP TFNFMIBOUF

Hφm−1,m =−J(φm−2,m+φm−1,m+1−2∆φm−1,m).

0CUFNPT BTTJN B B§£P EP IBNJMUPOJBOP OPT FTUBEPT EF TQJO BTTPDJBEPT BP TFUPS EF EPJT N¡H OPOT 0 FTUBEP NBJT HFSBM QPTT­WFM QBSB P TFUPS EF EPJT N¡HOPOT © DPNP BOUFT EBEP QFMB DPNCJOB §£P MJOFBS EPT FTUBEPT EF TQJO QFSUFODFOUFT B FTUF TFUPS JTUP ©

Ψ2(x1, ...,xL) =

!

1≤k<l≤L

ak,l(xk,xl)φk,l(xk,xl),

POEF EFWFNPT BHPSB EFUFSNJOBS BT BNQMJUVEFTam,n(xm,xn)≡ am,n 1BSB JTTP QSPKFUFNPTΨ2(x1, ...,xL)

TPCSFφm,n(xm,xn) DPN P RVF WBNPT PCUFS BT TFHVJOUFT SFMB§µFT

−J(am−1,n+am,n−1+am+1,n+am,n+1−4∆am,n) = am,nE2, (n> m+ 1)

F

−J(am−1,m+1+am,m+2−2∆am,m+1) = am,m+1E2, (n=m+ 1).

1BSB SFTPMWFS FTTBT FRVB§µFT GVODJPOBJT QBSB PT DPFஷ஫DJFOUFTam,nQPEFS­BNPT UFOUBS P "OTBU[

am,n(xm,xn) = A12eik1xm+ik2xn QPS©N QPEFNPT GBDJMNFOUF WFSJஷ஫DBS RVF FTUF "OTBU[ O£P TBUJTGB[

B FRVB§£P FNCPSB FMF TBUJTGB§B B FRVB§£P 1BSB SFTPMWFS FTUB JODPNQBUJCJMJEBEF #FUIF DPOTJEFSPV VNB DPNCJOB§£P MJOFBS EF GVO§µFT EF POEBT DPN NFTNB FOFSHJB " ºOJDB QPTTJCJMJEBEF © BDSFTDFOUBS VN UFSNP EB GPSNB A21(xm,xn)eik1xn+ik2xm OP RVBM USPDBNPT P NPNFOUVN EBT

QBSU­DVMBT 1PSUBOUP P "OTBU[ DPSSFUP BRVJ © P TFHVJOUF

am,n(xm,xn) = A12eik1xm+ik2xn +A21eik1xn+ik2xm,

P RVBM TBUJTGB[ BNCBT BT FRVB§µFT F %F GBUP B TVCTUJUVJ§£P EF OB FRVB§£P OPT GPSOFDF PT BVUPWBMPSFT EP IBNJMUPOJBOP

E2 =−2J(DPTk1+DPTk2−2∆),

F B TVCTUJUVJ§£P EFTUFT WBMPSFT FN OPT GPSOFDF B JEFOUJEBEF

A12

A21

=₋e

i(k1+k2)

−2∆eik1 _{+ 1} ei(k1+k2) ₋2∆_eik2 + 1.

/PUF BHPSB RVF TF MFWBSNPT QPS FYFNQMP B QBSU­DVMB OB QPTJ§£Pxm QBSB B QPTJ§£Pxm+L

FOU£P EFWJEP   DPOEJ§£P QFSJ³EJDB EF DPOUPSOP WBNPT PCUFS B NFTNB DPOஷ஫HVSB§£P EF BOUFT *TTP TJHOJஷ஫DB RVF BT BNQMJUVEFTam,nFan,m+L EFWFN TFS JHVBJT &TUB TJNFUSJB OPT GPSOFDF BT TFHVJOUFT

SFMB§µFT FOUSFA12FA21

A12 = A21eik1L, A21= A12eik2L F ei(k1+k2)L = 1.

EF NPEP RVF PCUFNPT BT FRVB§µFT

eik1L ₌

−e

i(k1+k2)_−2∆eik1_{+ 1} ei(k1+k2)₋2∆_eik2+ 1

, eik2L =−e

i(k1+k2) _−2∆eik2_{+ 1} ei(k1+k2) ₋2∆_eik1+ 1

.

&TUBT T£P BT FRVB§µFT EF #FUIF QBSB P TFUPS EF EPJT N¡HOPOT EP NPEFMPX X ZEF )FJTFOCFSH

2.2.5 Estados de três mágnons

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xm < xn < xo QPEF TFS SFQSFTFOUBEB QPSφm,n,o(xm,xn,xo)≡ φm,n,o EF NPEP RVF P FTUBEP NBJT

HFSBM BTTPDJBEP B FTUF TFUPS TFS¡ EBEP QFMB DPNCJOB§£P MJOFBS

Ψ3(x1, ...,xL) =

!

1≤i<j<k≤L

ai,j,k(xi,xj,xk)φi,j,k(xi,xj,xk),

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NPT FODPOUSBS B B§£P EFHTPCSFφm,n,o 4F BT USªT QBSU­DVMBT DPN TQJOdownO£P T£P BEKBDFOUFTJF

TFxn− xm >1Fxo−xn > 1 PCUFSFNPT BTTJN B SFMB§£P

− J(am−1,n,o+am+1,n,o+am,n−1,o+am,n+1,o+am,n,o−1+am,n,o+1)

+ 6J∆am,n,o=am,n,oE3.

+¡ RVBOEP EPJT TQJOT T£P BEKBDFOUFT UFNPT RVF xn = xm+1Fxo − xn > 1 PVxn − xm > 1F

xo =xn+1 P RVF OPT GPSOFDF BT SFMB§µFT

−J(am−1,m+1,o+am,m+2,o+am−1,n+1,o−1+am,m+1,o+1−4∆am,m+1,o) = am,m+1,oE3,

−J(am−1,n,n+1+am+1,n,n+1+am,n−1,n+1+am,n,n+2−4∆am,n,n+1) = am,n,n+1E3,

1PS ஷ஫N RVBOEP PT USªT TQJOT down T£P BEKBDFOUFT JTUP © RVBOEPxo = xm+2 F xn = xm+1

UFSFNPT

−J(am−1,m+1,m+2+am,m+1,m+3−2∆am,m+1,m+2) = am,m+1,m+2E3.

/PUF RVF BT SFMB§µFT F QPEFN TFS GBDJMNFOUF EFEV[JEBT EF

0 "OTBU[ EF #FUIF BEFRVBEP QBSB SFTPMWFS BT FRVB§µFT GVODJPOBJT F © P TFHVJOUF

am,n,o(xm,xn,xo) = A123eik1xm+ik2xn+ik3xo +A132eik1xm+ik3xn+ik2xo

+ A213eik2xm+ik1xn+ik3xo +A231eik2xm+ik3xn+ik1xo

+ A312eik3xm+ik1xn+ik2xo +A321eik3xm+ik2xn+ik1xo.

%F GBUP B TVCTUJUVJ§£P EF FN OPT GPSOFDF EJSFUBNFOUF PT BVUPWBMPSFT EF FOFSHJB

E3=−2J(DPTk1+DPTk2+DPTk3−3∆),

F B TVCTUJUVJ§£P EF F FN OPT GPSOFDF BT SFMB§µFT

A312

A321 =−

s12, A213

A231 =−

s13, A123

A132 =−

s23,

POEF JOUSPEV[JNPT BT RVBOUJEBEFT

si j =

ei(ki+kj)_−2∆eiki _{+ 1} ei(kj+ki)₋2_∆eikj + 1 ,

DIBNBEBT EFGBTॻ EF FTQBMIBNFOUP EF #FUIF

"M©N EJTTP EFWJEP  T DPOEJ§µFT QFSJ³EJDBT EF DPOUPSOP BT BNQMJUVEFTam,n,oFan,o,m+LFUD

EFWFN TFS JHVBJT P RVF OPT MFWB B PVUSBT SFMB§µFT FOUSF BT DPOTUBOUFTAi j k

A123

A231 =

A132

A321 =

eik1L

, A231

A312 =

A213

A132 =

eik2L

, A312

A123 =

A321

A213 =

eik3L

,

P RVF JNQMJDB UBNC©N OB SFMB§£P

ei(k1+k2+k3)L _{= 1}

.

"USBW©T EBT SFMB§µFT F QPEFNPT ஷ஫OBMNFOUF WFSJஷ஫DBS RVF PT OºNFSPT EF #FUIF k1k2Fk3EFWFN TBUJTGB[FS P TFHVJOUF TJTUFNB EF FRVB§µFT

eik1L ₌ '

(

ei(k1+k2)_−2∆eik1_{+ 1} ei(k1+k2)₋2_∆eik2+ 1

) *'(

ei(k1+k3)_−2∆eik1_{+ 1} ei(k1+k3)₋2_∆eik3+ 1

) *,

eik2L ₌ '

(

ei(k1+k2)_−2∆eik2_{+ 1} ei(k1+k2)₋2_∆eik1+ 1

) *'(

ei(k2+k3)_−2∆eik2_{+ 1} ei(k2+k3)₋2_∆eik3+ 1

) *,

eik3L ₌ '

(

ei(k1+k3)_−2∆eik3_{+ 1} ei(k1+k3)₋2_∆eik1+ 1

) *'(

ei(k2+k3)_−2∆eik3_{+ 1} ei(k2+k3)₋2_∆eik2+ 1

) *.

&TUBT T£P BT FRVB§µFT EF #FUIF QBSB P TFUPS EF USªT N¡HOPOT 1PEFNPT WFSJஷ஫DBS RVF TF BT FRVB§µFT EF #FUIF T£P TBUJTGFJUBT UPEBT BT FRVB§µFT GVODJPOBJT F T£P TBUJTGFJUBT

2.2.6 Estados de

N

mágnons

"OBMJTBOEP PT DBTPT BOUFSJPSFT QPEFNPT JOGFSJS P RVF BDPOUFDF DPN P TFUPS EFNN¡HOPOT /PUF RVF EP QPOUP EF WJTUB ଍­TJDP EFWFNPT DPOTJEFSBSL _≥ N VNB WF[ RVF P OºNFSP N¡YJNP EF TQJOTdownEB SFEF © MJNJUBEP QFMP OºNFSP EF QBSU­DVMBT RVF FMB DPOU©N %FWJEP B TJNFUSJB EF

SFWFST£P EPT TQJOT © TVஷ஫DJFOUF BJOEB DPOTJEFSBS RVFL _≥ 2N

$POஷ஫HVSB§µFT OBT RVBJTN QBSU­DVMBT EB SFEF EJHBNPT BRVFMBT OBT QPTJ§µFTxm1, ...,xmN QPTTVFN TQJO downEFஷ஫OFN PT FTUBEPT RV¢OUJDPT BTTPDJBEPT BP TFUPS EFN N¡HOPOT 1PEFNPT

SFQSFTFOUBS B GVO§£P EF POEB EF VN UBM FTUBEP QPSφm1,...,mN(xm1, ...,xmN)≡ φm1,...,mN 0 FTUBEP

NBJT HFSBM QPTT­WFM QFSUFODFOUF BP TFUPS EF NN¡HOPOT DPOTJTUF FN VNB TVQFSQPTJ§£P EPT FTUBEPT EF TQJO EFTUF TFUPS

ΨN(x1, ...,xL) =

!

1≤k1<k2<...<kN≤L

ak1,...,kN(xk1, ...,xkN)φk1,...,kN(xk1, ...,xkN),

F EFWFNPT EFUFSNJOBS BT BNQMJUVEFTak1,...,kN(xk1, ...,xkN)≡ ak1,...,kN

ߕVBOEP OFOIVN QBS EF TQJOTdownT£P BEKBDFOUFT B B§£P EFHTPCSF PT FTUBEPTφm1,...,mN OPT GPSOFDF B TFHVJOUF SFMB§£P GVODJPOBM

− J "am1−1,m2,...,mN +am1+1,m2,...,mN +...+am1,m2,...,mN+1+am1,m2,...,mN−1

#

+ 2N J∆am1,m2,...,mN =am1,m2,...,mNEN,

+¡ RVBOEP VN QBS EF TQJOT T£P BEKBDFOUFT EJHBNPT OB QPTJ§£Pxm1 Fxm1+1 = xm + 1 PCUFNPT SFMB§µFT EB GPSNB

− J "am1−1,m1+1,m3,...,mN +am1,m1+2,m3,...,mN +...+am1,m1+1,...,mN+1+am1,m1+1,...,mN−1

#

+ 2J(N ₋1)∆am1,m1+1,...,mN =am1,m1+1,...,mNEN,

F FRVB§µFT TFNFMIBOUFT QBSB RVBOEP PT TQJOT BEKBDFOUFT FTU£P OBT QPTJ§µFT xjFxj+11 ≤ j ≤ L

"M©N EJTTP RVBOEP UJWFSNPT USªT PV NBJT TQJOTdownBEKBDFOUFT FRVB§µFT TFNFMIBOUFT  T EF DJNB

EFWFN TFS DPOTJEFSBEBT

0 "OTBU[ DPSSFUP QBSB SFTPMWFS BT FRVB§µFT GVODJPOBJT BDJNB EFWF MFWBS FN DPOUB UPEBT BT QFSNVUB§µFT QPTT­WFJT EPT NPNFOUB EBT QBSU­DVMBT JTUP © EFWF UFS B GPSNB

am1···mN(xm1, ...,xmN) = !

pi∈Sn

Ap1...pNe

ikp1xm1+...+ikp Nxm N_.

" TVCTUJUVJ§£P EF FN OPT GPSOFDF PT BVUPWBMPSFT EP IBNJMUPOJBOP

EN =−2J N

!

j=1

"

DPTkj−∆

# ,

F DPN FTUF WBMPS EFEN B TVCTUJUVJ§£P EF FN OPT QFSNJUF ஷ஫YBS SB[µFT FOUSF BT BNQMJUVEFT

Ak1,...,kN %F GBUP PCUFNPT SFMB§µFT EB GPSNB Ap1...pjpj+1...pN Ap1...pj+1pj...pN

=−spj,pj+1, ∀pj ∈ S

n

,

POEF

si j =

ei(ki+kj)₋2_∆eiki + 1 ei(kj+ki)_−2∆eikj _{+ 1}. *NQPOEP DPOEJ§µFT QFSJ³EJDBT EF DPOUPSOP PCUFNPT UBNC©N RVF

Ap1p2...pN−1pN Ap2p3...pNp1

=eikp1L_, ∀p

j ∈Sn,

P RVF GPSOFDF UBNC©N B SFMB§£P

ei(k1+...+kN) _{= 1} .

"USBW©T EBT SFMB§µFT F UPEPT PT DPFஷ஫DJFOUFTAk1...kNQPEFN TFS EFUFSNJOBEPT F FTDSJUPT QPS FYFNQMP FN UFSNPT EFA12...N &N QBSUJDVMBS QPEFNPT BT JEFOUJEBEFT

(₋1)N−1

sm,1...sm,N =eikmL, 1≤ m≤ N,

BT RVBJT OPT GPSOFDFN P TFHVJOUF TJTUFNB EF FRVB§µFT O£PMJOFBSFT RVF ஷ஫YBN JNQMJDJUBNFOUF PT OºNFSPT EF #FUIFk1, ...,kN

eikmL _{= (−1)}N−1

N

&

j=1

n!m ' (

ei(km+kj)_−2∆eikm _{+ 1} ei(km+kj)₋2_∆eikj + 1

)

*, 1≤ m ≤ N,

&TUBT T£P BT FRVB§µFT EF #FUIF QBSB P TFUPS EFN N¡HOPOT EP NPEFMP EF TFJT W©SUJDFTX X Z DPN DPOEJ§µFT EF DPOUPSOP QFSJ³EJDBT /PUF RVF PT BVUPWBMPSFT F BVUPFTUBEPT EP IBNJMUPOJBOP T³ QP EFN TFS FYQMJDJUBNFOUF FTDSJUPT RVBOEP BT FRVB§µFT EF #FUIF GPSFN SFTPMWJEBT QBSB PT OºNFSPT EF #FUIFk1, ...,kN.

CAPÍTULO

3

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3.1 O modelo de seis vértices

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3

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3.1.1 A mecânica estatística do modelo de seis vértices

/FTUB TF§£P QSFUFOEFNPT BOBMJTBS P NPEFMP EF TFJT W©SUJDFT BUSBW©T EB NFD¢OJDB FTUBU­TUJDB $POTJEFSFNPT QBSB FTUF ஷ஫N VN SFUJDVMBEP DPNLDPMVOBT FNMJOIBT 7BNPT JNQPS DPOEJ§µFT QF SJ³EJDBT EF DPOUPSOP EF NPEP P W©SUJDF OB QPTJ§£P(i+L, j+N)TFS¡ JEFOUJஷ஫DBEP DPN P W©SUJDF _{&NCPSB B FTUSVUVSB EP HFMP SFBM TFKB USJEJNFOTJPOBM TPC NVJUPT BTQFDUPT © TBUJTGBU³SJP DPOTJEFSBS VN NPEFMP CJEJ} NFOTJPOBM 1PS FYFNQMP FOUSPQJB SFTJEVBM EP HFMP DBMDVMBEB BUSBW©T EF EF VN NPEFMP EP HFMP FN EVBT EJNFOTµFT UFN VN CPN BDPSEP DPN PT FYQFSJNFOUPT <>

OB QPTJ§£P(i,j) " DBEB W©SUJDF EP SFUJDVMBEP TFKBλVNB WBSJ¡WFM RVF EFTDSFWF B DPOஷ஫HVSB§£P EFTUF

W©SUJDF _{" FOFSHJB EP W©SUJDF EFQFOEF EF TVB DPOஷ஫HVSB§£P F QPEF TFS FTDSJUB DPNPE}

i j(λ) 0 QFTP

EF #PMU[NBOO SFGFSFOUF B FTTB DPOஷ஫HVSB§£P EP W©SUJDF © FOU£P Ri j(λ) = FYQ

"

−Ei j(λ)/kBT

#

POEFkB© B DPOTUBOUF EF #PMU[NBOO FTB UFNQFSBUVSB 1BSB B SFEF DPNP VN UPEP UFNPT RVF B TVB

FOFSHJB TFS¡ JHVBM B TPNB EBT FOFSHJBT EF DBEB W©SUJDF PV TFKB

E(λ) =

N ! i=1 L ! j=1

Ei j(λ),

0 QFTP EF #PMU[NBOO BTTPDJBEP B VNB EBEB DPOஷ஫HVSB§£P EP TJTUFNB DVKB FOFSHJB ©E(λ) © EBEB

FOU£P QPS

R(λ) = FYQ

$

−E(λ) kBT

%

=FYQ'+

(−

1

kBT N ! i=1 L ! j=1

Ei j(λ)), * = N & i=1 L & j=1 FYQ $

−Ei j(λ) kBT

% = N & i=1 L & j=1

Ri j(λ),

F QPS ஷ஫N B GVO§£P EF QBSUJ§£P DPSSFTQPOEF B TPNB EPT QFTPT EF #PMU[NBOO QBSB UPEBT BT QPTT­WFJT DPOஷ஫HVSB§µFT EP TJTUFNB JTUP ©

Z =!

λ

R(λ) =!

λi j

'+ ( N & i=1 L & j=1

Ri j(λ)), *

.

6TVBMNFOUF B GVO§£P EF QBSUJ§£P © DBMDVMBEB EB TFHVJOUF GPSNB DPOTJEFSBNPT VNB EBEB MJOIBi = aEB SFEF F DBMDVMBNPT B GVO§£P EF QBSUJ§£P EFWJEP BQFOBT  T JOUFSB§µFT EBT QBSU­DVMBT OFTUB MJOIB &TUB GVO§£P EF QBSUJ§£P NBJT SFTUSJUB © DIBNBEB EF USBOTGFSªODJB %FQPJT DPOTJEFSBNPT BT JOUFSB§µFT WFSUJDBJT FOUSF FTUBT MJOIBT PCUFOEPTF BTTJN B GVO§£P EF QBSUJ§£P

$POW©N QPSUBOUP EJTUJOHVJS BT JOUFSB§µFT EBT QBSU­DVMBT OBT EJSF§µFT IPSJ[POUBM F WFSUJDBM 1BSB JTTP JOUSPEV[JNPT OP MVHBS EF λBT WBSJ¡WFJT µFν RVF DPSSFTQPOEFN BP OºNFSP EF DPOஷ஫

HVSB§µFT QPTT­WFJT EBT BSFTUBT IPSJ[POUBJT F WFSUJDBJT EF VN EBEP W©SUJDF SFTQFDUJWBNFOUF "TTJN B FOFSHJB EF VNB EBEB MJOIBi =aEB SFEF QPEF TFS FTDSJUB DPNP

Ea(µ,ν) = Ea1(µ,ν) +...+EaL(µ,ν),

F P QFTP EF #PMU[NBOO BTTPDJBEP B FTUB MJOIB RVF © DIBNBEP EFNPOPESPNJB © EBEP QPS

Ma(µ,ν) = FYQ'+ (−

1

kBT L

!

j=1

Ea j(µ,ν)), *

=

L

&

j=1

FYQ$−Ea j(µ,ν) kBT

%

=

L

&

j=1

Ra j(µ,ν).

_{1BSB P NPEFMP EF W©SUJDFTλQPEF BTTVNJS PT WBMPSFT BQFOBT TFJT WBMPSFT}

i=a

_L

Ea1(λ) Ea2(λ) Ea3(λ) Ea4(λ) EaL(λ)

'JHVSB ؜ 6NB MJOIB EP SFUJDVMBEP & FOFSHJB EF DBEB QBSU­DVMB EFQFOEF EB DPOஷ஫HVSB§£PλEP W©S UJDF RVF MIF © BTTPDJBEP

"TTJN B USBOTGFSªODJB EB MJOIBa© PCUJEB QFMB TPNB TPC UPEBT BT DPOஷ஫HVSB§µFT IPSJ[POUBJT EB NP OPESPNJB BTTPDJBEBT B FTUB MJOIB

Ta(ν) =

!

µ

Ma(µ,ν) = !

µ

'+ (

L

&

j=1

Ra j(µ,ν)),

*

.

1BSB DBMDVMBS B GVO§£P EF QBSUJ§£P EFWFNPT BHPSB MFWBS FN DPOUB B JOUFSB§£P EF DBEB MJOIB EB SFEF DPN BT MJOIBT BEKBDFOUFT 3FQFUJOEP P D¡MDVMP PCUFNPT RVF B GVO§£P EF QBSUJ§£PZ TFS¡ EBEB QFMB G³SNVMB

Z =!

ν

' (

N

&

a=1

Ta(ν)) *.

/PUF RVF FTUF QSPDFEJNFOUP QPEF TFS GBDJMNFOUF FYUFOEJEP B EJNFOTµFT NBJT BMUBT

3.1.2 Representação matricial para a função de monodromia e de

transfe-rência

1PEFNPT GBDJMNFOUF DPOTUSVJS VNB SFQSFTFOUB§£P NBUSJDJBM QBSB FTUBT RVBOUJEBEFT 1BSB JTUP QPEFNPT BTTPDJBS B DBEB W©SUJDF EP SFUJDVMBEP VN FTQB§P EF )JMCFSU EF NPEP RVF BT HSBOEF[BT BTTPDJBEBT B FTUF W©SUJDF QPTTBN TFS SFQSFTFOUBEBT QPS PQFSBEPSFT NBUSJ[FT BUVBOEP OFTUF FTQB§P 6NB WF[ RVF DBEB W©SUJDF BENJUF B QSJPSJ DPOஷ஫HVSB§µFT FTUBT NBUSJ[FT EFWFN UFS PSEFN4×4

"TTJN B DBEB W©SUJDF EFWFNPT BTTPDJBS VN FTQB§P EF )JMCFSU EF EJNFOT£P4 $POW©N EJGFSFODJBS BT

EJSF§µFT WFSUJDBM EB IPSJ[POUBM F EFஷ஫OJS P FTQB§P EF )JMCFSU BTTPDJBEP B VN EBEP W©SUJDF EP SFUJDV MBEP BUSBW©T EP FTQB§P UFOTPSJBMH = H _⊗V POEFHFV T£P EPJT FTQB§PT EF )JMCFSU JTPNPSGPT B

C2

%FTTF NPEP P QFTP EF #PMU[NBOO BTTPDJBEP B VN EBEP W©SUJDF OB MJOIBaF DPMVOBbEP SFUJDVMBEP DPSSFTQPOEFS¡ B VNB NBUSJ[Rab(λ)EF PSEFN4×4 EFஷ஫OJEB OP &OE(Ha⊗Vb) "RVJ

λ © VNB WBSJ¡WFM SFMBDJPOBEB DPN BT JOUFSB§µFT EP TJTUFNB₋P DIBNBEPQBS¢NFUSP FTQFDUSBM 0T FMFNFOUPT EB NBUSJ[R(λ)QPEFN TFS FTDSJUPT DPNPRj1j2

i1i2(λ) POEF PT ­OEJDFT(i1, j1)SFGFSFNTF BP FTQB§PHaF(i2,j2)FTQB§PVb

1BSB P NPEFMP EF TFJT W©SUJDFT BQFOBT TFJT FMFNFOUPT EB NBUSJ[ REFWFN TFS EJGFSFOUFT EF [FSP−FMFT DPSSFTQPOEFN  T TFJT QPTTJCJMJEBEFT EF DPOஷ஫HVSB§µFT EF W©SUJDFT NPTUSBEBT OB ஷ஫HVSB "M©N EJTTP EFWJEP  T TJNFUSJBT EP NPEFMP BT DPOஷ஫HVSB§µFT RVF EJGFSFN QPS VNB JOWFST£P DPNQMFUB EB DPOஷ஫HVSB§£P EP W©SUJDF UBOUP OB IPSJ[POUBM RVBOUP OB WFSUJDBM EFWFN QPTTVJS NFTNB FOFSHJB F QPSUBOUP FMFNFOUPT EF NBUSJ[ EFR(λ)DPSSFTQPOEFOUFT B FTTBT DPOஷ஫HVSB§µFT EFWFN TFS JHVBJT "T

TJN B NBUSJ[R(λ)EP NPEFMP EF TFJT W©SUJDFT EFWFS¡ UFS BQFOBT USªT FMFNFOUPT EJTUJOUPT &TDPMIFOEP

VNB CBTF BQSPQSJBEB QPEFNPT FTDSFWFS FTUB NBUSJ[RDPNP

R(λ) =

⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎣

a(λ) 0 0 0 0 b(λ) c(λ) 0 0 c(λ) b(λ) 0

0 0 0 d(λ)

⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦

,

POEF BT GVO§µFTa(λ)b(λ)Fc(λ)EFWFN BJOEB TFS EFUFSNJOBEBT

$POTJEFSF BHPSB EBEB MJOIB EP SFUJDVMBEP EJHBNPT B MJOIBi =a 1BSB DPOTUSVJS VNB SFQSF TFOUB§£P NBUSJDJBM QBSB B NPOPESPNJB QPEFNPT BTTPDJBS VN FTQB§P EF )JMCFSU B FTUB MJOIB BUSBW©T EP QSPEVUP UFOTPSJBM EPT FTQB§PT EF )JMCFSU BTTPDJBEPT B DBEB W©SUJDF RVF FMB DPOU©N JTUP ©

V =Va1⊗Va2⊗· · ·⊗VaL.

$PN JTTP JOUSPEV[JNPT BNBUSJ[ EF NPOPESPNJBDPNP VNB NBUSJ[ RVF BUVB OP &OE(Ha⊗V)BUSB

W©T EF

Ma(λ) = Ra1(λ)Ra2(λ)· · ·RaL(λ),

POEF Ra j(λ)1 ≤ j ≤ L © VNB NBUSJ[ EFஷ஫OJEB FN &OE(Ha⊗Va1⊗· · ·⊗VaL)RVF BUVB DPNP

R(λ)OP FTQB§P UFOTPSJBMHa ⊗VjF DPNP JEFOUJEBEF OPT EFNBJT FTQB§PT WFUPSJBJT

"NBUSJ[ EF USBOTGFSªODJB QPS TVB WF[ ஷ஫DB EFஷ஫OJEB QFMP USB§P OP FTQB§PHaEB NBUSJ[ EF

NPOPESPNJB JTUP ©

Ta(λ) =USaMa(λ) = USa(Ra1(λ)Ra2(λ)· · ·RaL(λ)),

1PS ஷ஫N B GVO§£P EF QBSUJ§£P QPEF TFS DBMDVMBEB BUSBW©T EB G³SNVMB

Z =US(T1(λ)T2(λ)· · ·TN(λ)).

POEF P USB§P © UPNBEP BHPSB OPT FTQB§PT WFSUJDBJT

1PEFNPT WFS EFTUF NPEP RVF TF GPS QPTT­WFM EJBHPOBMJ[BS B NBUSJ[ EF USBOTGFSªODJB FOU£P B GVO§£P EF QBSUJ§£P QPEF TFS GBDJMNFOUF DBMDVMBEB %F GBUP EFTEF DBEB MJOIB EP SFUJDVMBEP © FRVJ WBMFOUF BT EFNBJT TFHVF RVFTi EFWF QPTTVJS PT NFTNPT BVUPWBMPSFT QBSB RVBMRVFSi F QPEFNPT

QPSUBOUP FTDSFWFS

Z =US(T(λ))N,

_{1PS FYFNQMP OB ஷ஫H UFNPT RVF BT DPOஷ஫HVSB§µFT{1,6}{2,4}F{3,5}EFWFN QPTTVJS FOFSHJBT JHVBJT}

POEF TVQPNPT BRVJ RVFT(λ)FTU¡ FTDSJUB OVNB CBTF POEF FMB © EJBHPOBM 7FNPT BTTJN RVF B NBUSJ[

EF USBOTGFSªODJB QPEF TFS WJTUB DPNP VN PQFSBEPS EF FWPMV§£P UFNQPSBM TF JOUFSQSFUBSNPT B EJSF§£P WFSUJDBM DPNP UFNQP EF NPEP RVF DBEB MJOIB EP SFUJDVMBEP DPSSFTQPOEB B VN TJTUFNB ଍­TJDP FN VNB EJNFOT£P FN VN EBEP JOTUBOUF EF UFNQP 1BSB P DBTP EP NPEFMP EF W©SUJDF WFSFNPT NBJT BEJBOUF RVF FTTF NPEFMP VOJEJNFOTJPOBM © KVTUBNFOUF P NPEFMP EF )FJTFOCFSH

3.1.3 A equação de Yang-Baxter

" FTUSVUVSB BMH©CSJDB EP "OTBU[ EF #FUIF RVF QFSNJUF EJBHPOBMJ[BS B NBUSJ[ EF USBOTGFSªODJB EF VN NPEFMP JOUFHS¡WFM © GPSOFDJEB QFMB FRVB§£P EF :BOH#BYUFS

R12(λ−µ)R13(λ)R23(µ) = R23(µ)R13(λ)R12(λ− µ).

&TUB FRVB§£P FTU¡ EFஷ஫OJEB FN &OE(V1⊗V2 ⊗V3)FRi j(λ)T£P PQFSBEPSFT RVF BUVBN DPNP

B NBUSJ[R(λ)OPT FTQB§P UFOTPSJBMVi j ≡Vi ⊗VjF DPNP B JEFOUJEBEF OP PVUSP FTQB§P WFUPSJBMVk

" FRVB§£P EF :BOH#BYUFS GPJ EFEV[JEB RVBTF FN EPJT DPOUFYUPT EJGFSFOUFT /B UFPSJB EP FTQBMIBNFOUP RV¢OUJDP FMB GPJ EFEV[JEB QPS .D(VJSF F :BOH F DPNP VNB DPOEJ§£P QBSB RVF P FT QBMIBNFOUP EF VN FOTBNCMF EF QBSU­DVMBT QPTTB TFS EFTDSJUP DPNP VNB DPNQPTJ§£P EF FTQBMIBNFO UPT FOUSF QBSFT EF QBSU­DVMBT <؜> &N NFD¢OJDB FTUBU­TUJDB FMB GPS EFEV[JEB QPS #BYUFS DPNP VNB DPOEJ§£P TVஷ஫DJFOUF QBSB RVF B NBUSJ[ EF USBOTGFSªODJB EF VN EBEP NPEFMP DPNVUF QBSB EJGFSFOUFT WBMPSFT EP QBS¢NFUSP FTQFDUSBM <> NVJUP FNCPSB B TVB PSJHFN SFNPOUF   SFMB§£PTUBSUSJBOHMF NFODJPOBEB OP USBCBMIP EF 0OTBHFS TPCSF P NPEFMP EF *TJOH <> " DPNVUBUJWJEBEF EB NBUSJ[ EF USBOTGFSªODJB © JNQPSUBOUF OP "OTBU[ EF #FUIF BMH©CSJDP QPSRVF FMB QFSNJUF JOUFSQSFUBS B NBUSJ[ EF USBOTGFSªODJB DPNP VNB HFSBEPSB EF JOºNFSBT JOஷ஫OJUBT OP MJNJUF UFSNPEJO¢NJDP RVBOUJEBEFT

R23

R₁₂ R13

R12

R₂₃

R13

'JHVSB ؜ 3FQSFTFOUB§£P EJBHSBN¡UJDB EB FRVB§£P EF :BOH#BYUFS " ஷ஫HVSB JMVTUSB P FTQBMIBNFOUP EF USªT QBSU­DVMBT WJTUP DPNP B DPNQPTJ§£P EF FTQBMIBNFOUP FOUSF QBSFT EF QBSU­DVMBT -FOEPTF P EJBHSBNB EF CBJYP QBSB DJNB TFUB EP UFNQP B FRVB§£P EF :BOH#BYUFS QPEF TFS JEFOUJஷ஫DBEB

DPOTFSWBEBT FN JOWPMV§£P QPJT OFTUFT DBTPT FMB DPNVUB UBNC©N DPN B IBNJMUPOJBOB BTTPDJBEB BP NPEFMP *TTP GPSOFDF P TUBUVT EF JOUFHS¡WFM BPT NPEFMPT TPMºWFJT QFMP "OTBU[ EF #FUIF

4F TF VN NPEFMP © JOUFHS¡WFM FOU£P B NBUSJ[R(λ)BTTPDJBEP B FTUF NPEFMP TFS¡ VNB TPMV§£P

EB FRVB§£P EF :BOH#BYUFS &N DPOUSBQBSUJEB UPEB TPMV§£P EB FRVB§£P EF :BOH#BYUFS QPEF TFS BTTPDJBEP B VN NPEFMP JOUFHS¡WFM $POTJEFSF FOU£P B NBUSJ[R(λ)EP NPEFMP EF TFJT W©SUJDFT

7FSFNPT B TFHVJS DPNP QPEFNPT ஷ஫YBS BT GVO§µFTx(λ)y(λ)Fz(λ)JNQPOEP RVFR(λ)TBUJTGB§B B

FRVB§£P EF :BOH#BYUFS

1BSB P NPEFMP EF TFJT W©SUJDFT B FRVB§£P EF :BOH#BYUFS DPOTJTUF FN VN TJTUFNB EF FRVB §µFT GVODJPOBJT OBT JOD³HOJUBTx(λ)y(λ)Fz(λ) %FWJEP  T TJNFUSJBT EP NPEFMP B NBJPSJB EFTTBT

FRVB§µFT T£P =JEFOUJDBNFOUF TBUJTGFJUBT PV T£P MJOFBSNFOUF EFQFOEFOUF EF PVUSBT $PN FGFJUP UF NPT RVF BQFOBT EBT FRVB§µFT GVODJPOBJT T£P JOEFQFOEFOUFT B TBCFS BT TFHVJOUFT

x(λ)z(µ)x(λ− µ) = y(λ)z(µ)y(λ−µ) +z(λ)x(µ)z(λ− µ),

x(λ)y(µ)z(λ− µ) = y(λ)x(µ)z(λ− µ) +z(λ)z(µ)y(λ− µ),

z(λ)y(µ)x(λ− µ) = z(λ)x(µ)y(λ− µ) + y(λ)z(µ)z(λ− µ).

&TUF TJTUFNB EF FRVB§µFT GVODJPOBJT GPJ SFTPMWJEP QSJNFJSBNFOUF QPS #BYUFS < > 1BSB JTTP FMF FMJNJOPV EBT FRVB§µFT BT BNQMJUVEFT DPN VNB EBEB EFQFOEªODJB 1PS FYFNQMP FMJNJOBOEP BT BNQMJUVEFT DPN B EFQFOEªODJB FNλ₋ µ PCUFNPT B TFHVJOUF FRVB§£P

x(λ)2+y(λ)2− z(λ)2

x(λ)y(λ) =

x(µ)2+y(µ)2−z(µ)2

x(µ)y(µ) .

*TTP GPSOFDF VNB TFQBSB§£P EF WBSJ¡WFJT K¡ RVF DBEB MBEP EFTUB FRVB§£P EFQFOEF EF VNB WBSJ¡WFM EJGFSFOUF *TTP JNQMJDB RVF BNCPT PT MBEPT EFTUB FRVB§µFT EFWFN JHVBMBS VNB EBEB DPOTUBOUF EF NPEP RVF QPEFNPT FTDSFWFS

x(λ)2+y(λ)2−z(λ)2

x(λ)y(λ) =∆.

"HPSB UVEP P RVF © QSFDJTP QBSB BDIBS B TPMV§£P © FODPOUSBS USªT GVO§µFTx(λ)y(λ)Fz(λ)

RVF TBUJTGB§BN B SFMB§£P GVODJPOBM BDJNB 7¡SJBT GVO§µFT TBUJTGB[FN FTUB JEFOUJEBEF F DBEB TPMV§£P QPTT­WFM EFTDSFWF VN NPEFMP JOUFHS¡WFM 1BSB P NPEFMP EF TFJT W©SUJDFT DPN TJNFUSJBX X Z B TPMV§£P QSPDVSBEB ©

x(λ) = TJOI(λ+η), y(λ) =TJOIλ, z(λ) = TJOIη,

EF NPEP RVF UFNPT

∆=DPTIη.

"RVJη © VN QBS¢NFUSP MJWSF P RVBM QPEF TFS VTBEP QBSB BEFRVBS PT SFTVMUBEPT UF³SJDPT DPN PT FYQFSJNFOUBJT

3.1.4 A integrabilidade do modelo de seis vértices

.PTUSBSFNPT BHPSB DPNP B FRVB§£P EF :BOH#BYUFS QBSB P NPEFMP EF W©SUJDFT JNQMJDB B DPNVUBUJWJEBEF EB NBUSJ[ EF USBOTGFSªODJB 1BSB JTTP TFS¡ DPOWFOJFOUF JOUSPEV[JS VNB OPWB SFQSF TFOUB§£P NBUSJDJBM−B DIBNBEB SFQSFTFOUB§£P EF -BY /FTUB SFQSFTFOUB§£P B NBUSJ[Rab(λ) BTTPDJ

BEB BP W©SUJDF EB MJOIBaF DPMVOBk © WJTUB DPNP VNB NBUSJ[ EF PSEFN2×2RVF BUVB OP FTQB§P

IPSJ[POUBMHaNBT DVKPT FMFNFOUPT T£P PQFSBEPSFT BUVBOEP OP FTQB§P WFSUJDBMVb *TUP ©

R(λ) = ⎡⎢⎢⎢

⎢⎣

L1₁(λ) L2₁(λ)

L1₂(λ) L2₂(λ)

⎤⎥ ⎥⎥ ⎥⎦,

POEF

L₁1(λ) = -x(₀λ) _y(0_λ)., L2₁(λ) = /_z(0_{λ) 0}00, L1₂(λ) = /₀0z(₀λ)0, L2₂(λ) =

-y(λ) 0

0 x(λ)

.

.

" NBUSJ[ EF NPOPESPNJB QPS TVB WF[ UBNC©N QPEF TFS SFQSFTFOUBEB DPNP VNB NBUSJ[

2× 2RVF BUVB FN HaNBT DVKPT FMFNFOUPT T£P PQFSBEPSFT RVF BUVBN OP FTQB§P UFOTPSJBMV ≡

V1⊗...⊗VL &TDSFWFNPT BTTJN

M(λ) = ⎡⎢⎢⎢

⎢⎣

M₁1(λ) M₁2(λ)

M₂1(λ) M₂2(λ)

⎤⎥ ⎥⎥ ⎥⎦ ≡ ⎡⎢⎢⎢⎢⎣

A(λ) B(λ)

C(λ) D(λ)

⎤⎥ ⎥⎥ ⎥⎦.

0T FMFNFOUPT EFM(λ)QPEFN FOU£P TFS DPNQVUBEPT QFMB G³SNVMB

M_ij(λ) =

2

!

{k1,...,kL−1}=1 Lk1

i (λ)⊗ L k2

k1(λ)⊗ L

k3

k2(λ)⊗...⊗ L

j

kL−1(λ).

RVF QPEF TFS GBDJMNFOUF DPNQSFFOEJEB PT OPUBSNPT RVF PT QSPEVUPT UFOTPSJBJT QSFTFOUFT JHVBMBN FN OºNFSP B RVBOUJEBEFLEF W©SUJDFT EP SFUJDVMBEP F RVF B TPNBU³SJB TF FYUFOEF B UPEBT BT DPN CJOB§µFT QPTT­WFJT EPT ­OEJDFT NVEPT PT RVBJT QPEFN BTTVNJS EPT WBMPSFT1F2TPNFOUF

'JOBMNFOUF UFNPT RVF B NBUSJ[ EF USBOTGFSªODJB QPEF TFS FTDSJUB OFTUB SFQSFTFOUB§£P TJN QMFTNFOUF DPNP

T(λ) = A(λ) +D(λ). _{1PS FYFNQMP QBSB P DBTPL= 3 UFNPT}

M11(λ) =L 1 1(λ)⊗L

1 1(λ)⊗L

1 1(λ)+L

1 1(λ)⊗L

2 1(λ)⊗L

1 2(λ)+L

2 1(λ)⊗L

1 2(λ)⊗L

1 1(λ)+L

2 1(λ)⊗L

2 2(λ)⊗L

1 2(λ),

M1 2(λ) =L

1 2(λ)⊗L

1 1(λ)⊗L

1 1(λ)+L

1 2(λ)⊗L

2 1(λ)⊗L

1 2(λ)+L

2 2(λ)⊗L

1 2(λ)⊗L

1 1(λ)+L

2 2(λ)⊗L

2 2(λ)⊗L

1 2(λ),

M12(λ) =L 1 1(λ)⊗L

1 1(λ)⊗L

2 1(λ)+L

1 1(λ)⊗L

2 1(λ)⊗L

2 2(λ)+L

2 1(λ)⊗L

1 2(λ)⊗L

2 1(λ)+L

2 1(λ)⊗L

2 2(λ)⊗L

2 2(λ),

M22(λ) =L 1 2(λ)⊗L

1 1(λ)⊗L

2 1(λ)+L

1 2(λ)⊗L

2 1(λ)⊗L

2 2(λ)+L

2 2(λ)⊗L

1 2(λ)⊗L

2 1(λ)+L

2 2(λ)⊗L

2 2(λ)⊗L

2 2(λ).

$PN JTTP EFEV[TF EB FRVB§£P EF :BOH#BYUFS B DIBNBEBFRVB§£P GVOEBNFOUBM EP "OTBU[ EF #FUIF B TBCFS B FRVB§£P

S(λ₋ µ)!M(λ)⊗ M(µ)"=!M(µ)⊗ M(λ)"S(λ− µ).

&TUB FRVB§£P NBUSJDJBM © © EFஷ஫OJEB FN &OE(Ha⊗V) DPNV =V1⊗· · ·⊗VLF POEF

S(λ) =

⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎣

x(λ) 0 0 0 0 z(λ) y(λ) 0

0 y(λ) z(λ) 0

0 0 0 x(λ)

⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦ ,

© DIBNBEB EF NBUSJ[ EF FTQBMIBNFOUP−FMB FTU¡ SFMBDJPOBEB   NBUSJ[R(λ)QFMB G³SNVMB S(λ) =

PR(λ) POEF P © P DIBNBEP QFSNVUBEPS EFஷ஫OJEP QFMB SFMB§£P P(A_⊗ B)P = B _⊗ A W¡MJEB QBSB RVBJTRVFS PQFSBEPSFT AF B EFஷ஫OJEPT FN &OE(V1⊗V2) " NBUSJ[M EFWF FTUBS FTDSJUB OB

SFQSFTFOUB§£P EF -BY QBSB RVF P QSPEVUP NBUSJDJBM GB§B TFOUJEP 5PNBOEP P USB§P OP FTQB§PHaEB FRVB§£P UFNPT RVF

USa!S(λ− µ)M(λ)⊗ M(µ)" = USa!M(µ)⊗ M(λ)S(λ− µ)",

USa!M(λ)⊗ M(µ)" = USa

#

S(λ₋ µ)−1M(µ)⊗ M(λ)S(λ− µ) $

,

USa!M(λ)⊗ M(µ)" = USa

#

S(λ₋ µ)S(λ− µ)−1M(µ)⊗ M(λ) $

,

USa!M(λ)⊗ M(µ)" = USa!M(µ)⊗ M(λ)",

USaM(λ)USaM(µ) = USaM(µ)USaM(λ),

PV TFKB EFTEF RVFT(λ) =USaM(λ) PCUFNPT

[T(λ),T(µ)] = 0.

" NBUSJ[ EF USBOTGFSªODJB EP NPEFMP EF W©SUJDFT DPNVUB QPSUBOUP QBSB RVBJTRVFS WBMPSFT EPT QBS¢NFUSPT FTQFDUSBJTλFµ P RVF NPTUSB RVFT © HFSBEPSB EF VNB JOஷ஫OJEBEF EF RVBOUJEBEFT

DPOTFSWBEBT FN FTQFDJBM B IBNJMUPOJBOB EP NPEFMP

3.2 O Ansatz de Bethe algébrico em ação: diagonalizando da

matriz de transferência

/FTUB TF§£P NPTUSBSFNPT DPNP P "OTBU[ EF #FUIF BMH©CSJDP QPEF TFS VTBEP QBSB EJBHPOB MJ[BS B NBUSJ[ EF USBOTGFSªODJB EP NPEFMP EF TFJT W©SUJDFT 7FSFNPT RVF NBJT VNB WF[ B FRVB§£P EF :BOH#BYUFS © EF JNQPSU¢ODJB GVOEBNFOUBM OFTUB DPOTUSV§£P

3.2.1 As relações de comutação

/P "OTBU[ EF #FUIF BMH©CSJDP PT FTUBEPT FYDJUBEPT T£P DPOTUSV­EPT BUVBOEPTF TVDFTTJWB NFOUF PT PQFSBEPSFT B(λ1) B(λ2)FUD FN VN EBEP BVUPFTUBEP EB NBUSJ[ EF USBOTGFSªODJB DP

OIFDJEP DPNP FTUBEP EF SFGFSªODJB 1BSB DPNQVUBSNPT B B§£P EB NBUSJ[ EF USBOTGFSªODJBT(λ) =

A(λ) +D(λ)TPCSF FTUFT FTUBEPT UFNPT EF QBTTBS PT PQFSBEPSFTA(λ)FD(λ)TPCSF PT PQFSBEPSFT

B(λ1)B(λ2)FUD PV TFKB UFNPT EF TBCFS BT SFMB§µFT EF DPNVUB§£P FOUSF FTTFT PQFSBEPSFT 0DPSSF

FTTBT SFMB§µFT EF DPNVUB§£P QPEFN TFS FODPOUSBEBT QFMB FRVB§£P GVOEBNFOUBM EBEB FN %F GBUP B FRVB§£P RVBOEP FTDSJUB OB SFQSFTFOUB§£P EF -BY SFQSFTFOUB VN TJTUFNB EF FRVB§µFT MJOFBSFT GVODJPOBJT QBSB WBSJ¡WFJT O£PDPNVUBUJWBTA,BCFD PT FMFNFOUPT EB NBUSJ[ EF NPOPESP

NJB 1PEFNPT NBOJQVMBS FTTBT FRVB§µFT SFTPMWFOEPTF P TJTUFNB EF NPEP B FODPOUSBS BT SFMB§µFT EF DPNVUB§£P EFTFKBEBT

"QSFTFOUBSFNPT B TFHVJS BQFOBT BT SFMB§µFT EF DPNVUB§£P RVF TFS£P VUJMJ[BEBT NBJT BEJBOUF 4£P FMBT

B(λ)B(µ) = B(µ)B(λ),

A(λ)B(µ) = a(µ−λ)B(µ)A(λ)−b(µ−λ)B(λ)A(µ),

D(λ)B(µ) = a(λ− µ)B(µ)D(λ)−b(λ−µ)B(λ)D(µ),

POEF QPS DPOWFOJªODJB JOUSPEV[JNPT BT RVBOUJEBEFT

a(λ) = x(λ)

y(λ) =

TJOI(λ+η)

TJOI(λ) , F b(λ) =

z(λ)

y(λ) =

TJOI(η)

TJOI(λ).

3.2.2 O estado de referência

0 QPOUP EF QBSUJEB EP "OTBU[ EF #FUIF BMH©CSJDP DPOTJTUF FN FODPOUSBS VN FTUBEP RVF TFKB VN BVUPFTUBEP EB NBUSJ[ EF USBOTGFSªODJB 1BSB P NPEFMP EF W©SUJDFT DPN DPOEJ§µFT EF DPOUPSOP QFSJ³EJDBT JTTP © CFN TJNQMFT EF TFS GFJUP F EF GBUP UFNPT B TFHVJOUF

1SPQPTJ§£P 0 FTUBEP EF׾OJEP QPS

|0⟩= ⎡⎢⎢⎢

⎢⎣ 1 0 ⎤⎥ ⎥⎥ ⎥⎦ ⊗...⊗ ⎡⎢⎢⎢⎢⎣ 1 0 ⎤⎥ ⎥⎥ ⎥⎦ = L ' k=1 ⎡⎢ ⎢⎢ ⎢⎣ 1 0 ⎤⎥ ⎥⎥ ⎥⎦,

© VN BVUPFTUBEP EB NBUSJ[ EF USBOTGFSªODJBT(λ) = A(λ) +D(λ)F

T(λ)|0_⟩= "x(λ)L ₊

y(λ)L #

|0_⟩.

"M©N EJTTP PT FMFNFOUPT A(λ)FD(λ)EB NBUSJ[ EF NPOPESPNJB BUVBN FN |0⟩DPNP

A(λ)|0⟩= x(λ)L|0⟩, B(λ)|0⟩ ̸∝ |0⟩, C(λ)|0⟩= 0|0⟩, D(λ)|0⟩= y(λ)L|0⟩.

%FNPOTUSB§£P " QSPWB EFTTBT Bஷ஫SNB§µFT © EJSFUB 1SJNFJSP OPUF RVF PT FMFNFOUPT EB NBUSJ[ R BUVBN FN|0_⟩ RVBOEP FTDSJUBT OB SFQSFTFOUB§£P EF -BY EB TFHVJOUF GPSNB

L1₁(λ)|0⟩=x(λ)|0⟩, L2₁(λ)|0⟩ ̸∝ |0⟩, L1₂(λ)|0⟩= 0|0⟩, L₂2(λ)|0⟩=y(λ)|0⟩,

-PHP QBSB QSPWBS B QSJNFJSB Bஷ஫SNB§£P CBTUB PMIBS QBSB B EFஷ஫OJ§BP EP PQFSBEPS A(λ) EBEB QFMB

FRVB§£P " QBSUJS EFTUB EFஷ஫OJ§£P QPEFNPT WFS RVF P ºOJDP UFSNP RVF O£P DPOU©N P PQFSB EPSL1₂(λ)© P QSJNFJSP POEF UPEPT PT ­OEJDFT NVEPT T£P JHVBJT B1 $POTFRVFOUFNFOUF BQFOBT P

QSJNFJSP UFSNPL1₁(λ)_⊗...⊗L1₁(λ)TPCSFWJWF F B TVB B§£P TPCSF|0⟩© KVTUBNFOUFx(λ)L|0⟩

%P NFTNP NPEP QBSB QSPWBS B ºMUJNB Bஷ஫SNB§£P © TVஷ஫DJFOUF QFSDFCFS RVF QFMB EFஷ஫OJ§£P EFD(λ) BQFOBT P ºMUJNP UFSNP O£P DPOU©N P PQFSBEPSL1₂(λ) POEF PT ­OEJDFT NVEPT T£P UPEPT

JHVBJT B2 "TTJN EFTEF RVF B B§£P EFL2₂(λ)_⊗...⊗L2₂(λ)FN|0⟩© FYBUBNFOUFy(λ)L|0⟩B Bஷ஫SNB§£P

© KVTUJஷ஫DBEB

/PWBNFOUF B QBSUJS EB EFஷ஫OJ§£P EFB(λ)UBNC©N QPEFNPT WFS RVF B TVB B§£P TPCSF|0_⟩

O£P QPEF TFS QSPQPSDJPOBM B|0_⟩ K¡ RVF FYJTUFN UFSNPT RVF DPOUªNL2₁(λ)DPNP VN GBUPS NBT RVF

O£P DPOUªNL1₂(λ) 1PS ஷ஫N QBSB QSPWBS RVFC(λ)|0⟩= 0|0⟩CBTUB OPUBS RVF OB EFஷ஫OJ§£P EFC(λ)

UPEPT PT UFSNPT DPOUªNL1₂(λ)DPNP VN GBUPS EF NPEP RVF P FTUBEP EF SFGFSªODJB TFS¡ BOJRVJMBEP

QPS UPEPT FTUFT UFSNPT " QSPQPTJ§£P ஷ஫DB BTTJN EFNPOTUSBEB !

&N SFTVNP P FTUBEP|0_⟩= [1

0]⊗...⊗[10]© VN BVUPFTUBEP EB NBUSJ[ EF USBOTGFSªODJB DVKP

BVUPWBMPS ©

τ0(λ) = x(λ)L+y(λ)L =TJOI(λ+η)L+TJOI(λ)L.

3.2.3 Estados de um mágnon

"USBW©T EB B§£P EB NBUSJ[ EF NPOPESPNJB TPCSF P FTUBEP EF SFGFSªODJB QPEFNPT OPUBS EVBT DPJTBT FN QSJNFJSP MVHBS WFNPT RVF P PQFSBEPSCTF DPNQPSUB DPNP VN PQFSBEPS EF BOJRVJMB§£P F FN TFHVOEP MVHBS RVF P PQFSBEPSBBUVB DPNP VN PQFSBEPS EF DSJB§£P 0T FTUBEPT FYDJUBEPT QPEFN QPS DPOTFHVJOUF TFS DPOTUSV­EPT QFMB B§£P EP PQFSBEPS EF DSJB§£PBTPCSF P FTUBEP EF SFGF SªODJB|0⟩ 7FSFNPT BHPSB RVF © FYBUBNFOUF JTTP RVF © GFJUP OP "OTBU[ EF #FUIF BMH©CSJDP %Fஷ஫OJNPT

BTTJN P FTUBEP EF VN N¡HOPO EB TFHVJOUF GPSNB

|λ1⟩=B(λ1)|0⟩.

/PUF RVF P PQFSBEPS BEFQFOEF EB DFMFSJEBEF λ1 F O£P EFλ *TTP © GFJUP QBSB RVF QPTTBNPT EF

UFSNJOBSλ1EF NPEP RVF|λ1⟩TFKB EF GBUP VN BVUPFTUBEP EB NBUSJ[ EF USBOTGFSªODJB WFSFNPT OB

TFRVªODJB RVFλ1TFS¡ ஷ஫YBEB JNQMJDJUBNFOUF QFMB FRVB§£P EF #FUIF BTTPDJBEB BP TFUPS EF VN N¡H

OPO

1BTTFNPT BHPSB BP D¡MDVMP EB B B§£P EB NBUSJ[ EF USBOTGFSªODJB TPCSF|λ1⟩ 5FNPT RVF

T(λ)|λ1⟩= A(λ)B(λ1)|0⟩+D(λ)B(λ1)|0⟩,

/P "OTBU[ EF #FUIF BMH©CSJDP O£P QSPDVSBNPT FODPOUSBS B B§£P EP PQFSBEPSBFN|0_⟩ BOUFT VTB

NPT BT SFMB§µFT EF DPNVUB§£P FOUSF PT PQFSBEPSFTA(λ)FD(λ)DPNB(µ) RVF T£P GPSOFDJEBT QFMB

FRVB§£P GVOEBNFOUBM %F BDPSEP DPN BT FRVB§µFT F UFSFNPT FOU£P

A(λ)B(λ1)|0⟩ = a(λ1−λ)B(λ1)A(λ)|0⟩ −b(λ1− λ)B(λ)A(λ1)|0⟩,

D(λ)B(λ1)|0⟩ = a(λ−λ1)B(λ1)D(λ)|0⟩ −b(λ−λ1)B(λ)D(λ1)|0⟩.

$PN JTTP QPEFNPT BHPSB BUVBS PT PQFSBEPSFTAFDOP FTUBEP EF SFGFSªODJB|0⟩ DPN P RVF WBNPT

PCUFS

A(λ)B(λ1)|0⟩ = x(λ)La(λ1−λ)B(λ1)|0⟩ −x(λ1)Lb(λ1−λ)B(λ)|0⟩,

D(λ)B(λ1)|0⟩ = y(λ)La(λ−λ1)B(λ1)|0⟩ −y(λ1)Lb(λ−λ1)B(λ)|0⟩.

1PSUBOUP WFNPT RVF B B§£P EB NBUSJ[ EF USBOTGFSªODJB FN|λ1⟩ஷ஫DB EBEB QPS

T(λ)|λ1⟩ =

"

x(λ)La(λ1−λ) +y(λ)La(λ−λ1)

# |λ1⟩

− "x(λ1)Lb(λ1−λ) + y(λ1)Lb(λ−λ1)

# |λ_⟩.

/PUF BHPSB RVF|λ1⟩TPNFOUF TFS¡ VN BVUPFTUBEP EFT(λ)TF P TFHVOEP UFSNP TF BOVMBS K¡

RVF FTUF UFSNP O£P QPEF TFS QSPQPSDJPOBM B|λ1⟩ VNB WF[ RVF P QBS¢NFUSP FTQFDUSBMλ© BSCJUS¡SJP

*TTP E¡ PSJHFN B FRVB§£P EF #FUIF

x(λ1)L

y(λ1)L

=−b(λ−λ1) b(λ1−λ) = 1

,

PV TVCTUJUVJOEP BT BNQMJUVEFT

$TJOI

(λ1+η)

TJOIλ1

%L

= 1.

"TTJN TFλ1TBUJTGB[ FTUB SFMB§£P FOU£P|λ1⟩TFS¡ VN BVUPFTUBEP EFT(λ)F P DPSSFTQPOEFOUF

BVUPWBMPS TFS¡ EBEP QPS

τ1(λ|λ1) = a(λ1−λ)x(λ)L+a(λ−λ1)y(λ)L

= TJOI(λ1−λ+η)

TJOI(λ1−λ) TJOI(

λ+η)L+ TJOI(λ− λ1+η)

TJOI(λ₋λ1) TJOI(

λ)L